durham e-theses spectral characterisation of variable
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Durham E-Theses
Spectral characterisation of variable reactance devices
Fifa, Ekaterini
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Fifa, Ekaterini (1987) Spectral characterisation of variable reactance devices, Durham theses, DurhamUniversity. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/6763/
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To
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SPECTRAL CHARACTERISATION OF VARIABLE REACTANCE DEVICES
By
Ekaterini Fifa
Graduate Society
A thesis submitted to the Faculty of Science,
University of Durham, for the degree of
Master of Science
Department of Applied Physics and Electronics,
School of Engineering and Applied Science,
University of Durham, UK.
Apri 1 1987
19 .iLN !987
ABSTRACT
Low Noise Figure communication receivers require more
efficient frequency converters. Frequency conversion and
multiplication processes cannot take place without the
existence of harmonics in the system and the inherent
property of a nonlinear element is to generate a harmonic
spectrum. Such nonlinearity, in general, may be provided by
semiconductor diodes. This research project deals with the
theoretical analysis as well as the properties of a
nonlinear reactive device, i.e. Varactor Diode.
The power series solutions for the exponential diodes
do not normally converge quickly enough to be of practical
value for numerical evaluations. A different approach is
proposed for the evaluation of harmonic amplitudes and
phases. The harmonic generating properties of four diodes
of the same type were examined using two different
approaches and a good agreement was found between the two
methods.
Many analyses published over the years have tended to
introduce severe approximations which were only valid in
practice over limited ranges of operation. However, it is
believed that the new sampling method presented here
evaluates fully the capabilities of these diodes in
practice.
ACKNOWLEDGEMENTS
I would like to express my gratitude to my supervisors,
Dr. B.L.J.Kulesza and Dr. J.S. Thorp for their guidance,
enthusiasm and continued interest throughout the entire
period of research. I am especially indebted to Dr. B.L.J.
Kulesza for introducing me to the fascinating subject of
nonlinear systems and his active supervision. I am also
grateful to Dr. J.Welford who was very keen to help in
every possible way in writing the 8 F.F.T• program.
My appreciation is also due to my colleagues in the
High
Group,
Frequency Measurements and Applications Research
and the technical staff of the department for their
generous co-operation.
Last, but not least, I am most grateful to my parents
whose love, encouragement and support throughout my work
made it possible for me to produce this Thesis.
Ekaterini Fifa
Durham
April 1987
CONTENTS
1.0 INTRODUCTION
2.0 CONSTRUCTION AND PROPERTIES OF PRACTICAL VARACTORS
2.1 Introduction
3.0
2.2 Theory
2.3 Electrical Characterisation of
practical Varactors
2.4 Applications
SPECTRAL EVALUATION
3.1 Introduction
3.2 Review of Fourier Analysis
3.3 Fingerprinting of Nonlinear Devices
3.4 Varactor Analysis
4.0 COMPUTER AIDED EXPERIMENTAL SPECTRAL CHARACTERISATION
4.1 Introduction
4.2 The Method
4.3 Practical System
4.4 Testing
4.5 Summary
5.0 MEASUREMENTS AND PROCEDURES
5.1 Introduction
5.2 Choice and Measurements of Drive Level
1
7
8
16
21
27
28
34
35
40
41
46
55
65
67
67
5.3 Measurements with Slotted Line
5.4 Spectrum Measurements
6.0 RESULTS
6.1 Introduction
6.2 c-v, Q-V Measurements
6.3 V-I of the Fundamental
6.4 Sampling Method Results
6.5 Spectrum Analyzer Method
6.6 Discussion
7.0 CONCLUSIONS
7.1 Introduction
7.2 Device Characterisation
7.3 Future Work
REFERENCES
APPENDICES
Results
68
72
76
76
103
103
116
117
121
123
127
1
CHAPTER 1
INTRODUCTION
Generation of microwave frequencies at milliwatt power
levels has in the past been
klystrons and other electron
generally achieved employing
beam devices. However, in
many satellite, missile and portable equipment applications
the weight, size and power consumption of these devices
needing high voltage supplies have proved to be a severe
handicap. To overcome these problems solid state devices
are being increasingly exploited.
Many high frequency systems extensively use nonlinear
semiconductor devices for various essential purposes. The
design of circuits incorporating such components requires a
thorough knowledge of their electrical characteristics and
behaviour. The basic principles of amplification with a
nonlinear reactance are not new. Just to mention one
example, nonlinear inductances have been employed in
magnetic amplifiers for many years. As a consequence, the
nonlinearity
mathematical
has
and
always been a subject of great
physical interest especially in
communications and electronics fields.
Many microwave semiconductor devices derive their
usefulness from the nonlinearity of the current-voltage
characteristic and are called varistors <variable
resistances). Nonlinearity of the charge-voltage characte-
2
ristic, 1.e. the nonlinearity of a capacitance. is the
major feature of the diodes which are called variable
reactances or varactors.
The process of diffusion has also been important in the
development of junction diodes. As a consequence the
fragility, burn out and frequency limitation disadvantages
of the point-contact diodes have been eliminated. The
improved diodes have led to advances in the design of
crystal mixers and detectors, especially with respect to
noise figure and frequency limits.
When operated in the forward-biased directions,
junction diodes behave as nonlinear resistances and hence
have low Q values with relatively high RC products.
However, they perform as voltage-dependent or nonlinear
capacitors (varactors> when operated in the reverse-biased
directions. Their nonlinear capacitance characteristics can
be especially useful in low-noise parametric amplification.
The varactor diode has also been important in raising
the efficiency of microwave harmonic generators, modulators
and switches. The power levels of the diodes have been
raised through the use of higher temperature materials such
as GaAs and higher voltages can be applied when thin layer
of intrinsic material are used in their construction.
Conversion efficiencies have been improved simultaneously
so that harmonic generators could supply milliwats of power
up to X-band frequencies and microwatts in the millimeter
region.
3
A semiconductor diode has a junction capacitance
associated with it and while it may present an undesirable
feature in more conventional uses of the diode, it is an
important mechanism in varactors. This capacitance actually
varies with the applied voltage if the diode is
back-biased. Shockley showed that the •Depletion Layer
Capacitance• for a planar geometry diode is given by
K C<v> =
When the junction is biased in the reverse direction
the depletion layer widens and the capacitance value falls.
The conduction taking place across the junction is due to
the motion of hole and electron distributions. Its exact
dependence on the applied voltage varies with the nature of
the junction.
The quality of the varactor operation is affected by
the series resistance to a great extent and a lot of
trouble is taken in the design in order to reduce its value
to an absolute minimum.
Although the theoretical foundations of parametric
amplification and harmonic generation using variable
capacitance elements had
1948 by A. Van dar Ziel in
been laid as early as November
•on the Mixing Properties of
Non-Linear Condensers• [1] very low-loss diodes were not
available and hence the initial experiments did not produce
predicted high gains, high efficiencies or low-noise
4
figures.
A major breakthrough came in July 19~7 when Bakanowski,
Cranna and Uhlir reported the discovery of a technique for
making low-loss silicon and germanium nonlinear capacitors
[2J. The fabrication process used was solid-state
diffusion. The resulting graded junctions were of planar
structure. The important advantage of the •one-dimensional•
or •planar• geometry is that, since the depletion layer
capacitance and series resistance are respectively, dire
ctly and inversely proportional to the area, the RC product
and hence the cut-off frequncy are independent of it.
A •hyper-abrupt• varactor diode fabricated by an
alloy-diffusion process was reported by Shimiza and
Nishizawa in September 1961 in •Alloy-Diffused Variable
Capacitance Diode with Large Figure of Merit• [3J. The
resulting junction in which the impurity concentration
decreases with distance from the p-n boundary showed high
voltage sensitivity of the capacitance.
Results of further investigations were reported by
Sukegawa, Fugikawa and Nishizawa in January 1963 in •si
Alloy-Diffused Variable Capacitance Diode• [4J. An
experimental varactor was obtained. Its capacitance for
only a few volts of reverse bias reduced to one-hundredth
of its zero bias value, a variation not realisable with
ordinary abrupt or graded junctions.
Since their losses are much lower than in varistors,
varactors are preferred elements for harmonic generation,
5
modulation or up-conversion and low-noise amplifications.
It is therefore advantageous to know the full extent of
their capabilities for an efficient performance. At the
present time, the extent and the method of characterisation
of these devices, especially at high frequencies, is
inadequate for many applications. Normally, the device
parameters available are for the static characteristics and
these are usually obtained from low frequency measurements.
If the dynamic characteristics are given, they are
generally at one particular test frequency and drive level.
At the present time there is no detailed quantitative
analysis available to give exact values for the harmonics
generated within the junction diodes. In order therefore to
assess the properties of a nonlinear device at high
frequencies, there is a need to determine experimentally
the generated frequency spectrum. Such a spectrum must
logically be
nonlinearity.
a unique representation of the device
The main objective of this work is to characterise a
number of
generated
devices, the varactor diodes, by means of the
spectral components. This characterisation
provides information useful
particular applications.
in the device evaluation tor
In chapter 2, the theory of construction and properties
of the practical varactors are reviewed. The practical
devices and their applications are also discussed.
Chapter 3 outlines the meaning of spectral evaluation.
6
A review of Fourier Analysis is given and the device
characterisation, or ufingerprinting• using the method of
the spectral representation is presented.
The theory of the new technique developed for the
spectrum measurements is described in chapter 4. The
nonlinear device driven by a sinusoidal signal produces a
distorted output response which contains all the necessary
information for each evaluation. Via a computer this output
is Fourier-analyzed and the resulting frequency spectrum of
the device is calculated using an appropriate programme.
The equipment and the experimental arrangements are first
described followed by the outline of the intended test
procedures for a practical system.
In chapter 5, measurements and experimental setup are
covered. Other detailed aspects relevant to the technique
such as the choice of drive level and measurement of C-V
and Q-V characteristics are presented, followed by the
experimental procedures.
The harmonic measurement results of the four diodes are
displayed in the form of graphs in chapter 6. The
properties of each diode and the resulting spectral
representation are interpreted and discussed.
Finally, in chapter 7, an assessment of the sampler
method dealing with the accuracy, significance and possible
improvements is presented. Suggestions and recommendations
for future work arising from these investigations are
proposed.
7
CHAPTER 2
CONSTRUCTION AND PROPERTIES OF PRACTICAL VARACTORS
2.1 Introduction
A varactor diode is a semiconductor device which is
principally employed in electrical circuits and networks as
a nonlinear, variable-capacitance element. Such a diode
consists of a semiconductor wafer on which a junction is
formed normally by a diffusion process and a suitable
encapsulation. The depletion capacitance of the junction
provides the variable element which is a function of the
applied voltage.
Because of its nonlinearity, the varactor will generate
sums, differences and harmonics when two or more
frequencies are applied. Manley and Rowe [5] have derived
a set of equations relating the powers flowing into and out
of a nonlinear capacitance. The basic idea behind the
equations was to show that , if the device is lossless, the
sum of the powers flowing
frequencies will be zero.
into and out of it at all
The nonlinear capacitance of a varactor is used in many
applications. Low-noise high frequency parametric amplifi-
cation is probably one of the most important. Additionally,
they may be employed in the design of other frequency
converters, frequency multipliers, dividers and modulators
for frequency control and comparison systems. For such
8
circuits based on variable capacitance devices, the Manley
and Rowe equations also provide the information for the
gain and stability. Furthermore, the conditions, under
which the production of harmonics and other frequencies is
possible are given.
Semiconductor rectifiers in the past frequently
exhibited a noticeable nonlinear capacitance effect under
reverse bias. Before 1950's those variable capacitance
diodes were manufactured mainly for voltage tuning
applications. However, such diodes were relatively lossy
and therefore were highly inefficient especially at
microwave frequencies. Low loss variable capacitance diodes
were developed in the middle 1950's under the direction of
Uhlir [6] of Bell Laboratories. They were first reported
in a paper by Bakanowski, Cranna, and Uhlir in 1959 and to
dinstinguish these devices from the more lossy diodes then
available,
varactors.
2.2 Theory
they were named, variable reactances or
2.2.1 Physics of Varactors
Varactors are basicaly semiconductor junction devices
which exhibit a nonlinear depletion or barrier capacitance
variations with applied voltage. This effect is enhanced
and predominates when the diode is reverse-biased; the
junction
behaviour
becomes a variable-capacitance element. Its
depends on the so called impurity profile
9
achieved by introducing appropriate dopants on each side of
the semiconductor junction. Ideally, the diode impurity
profile can either be abrupt or graded as shown in Figs.
2.1 and 2.2, respectively. In practice the devices will
fall somewhere between the two extreme profiles.
At some high reverse voltage the junction breaks down
which is caused by an avalanche multiplication process when
holes and electrons are generated rapidly resulting in a
large reverse current. This effect, must be taken into
account and establishes the reverse voltage limit on the
magnitude of the driving signal. There may also be other
effects present producing excess reverse currents sometimes
even at lower voltages. This may happen in diodes with very
large doping densities which results in the so called Zener
effect. The breakdown voltage limit encompasses all these
effects and is designated as v. in the analyses [7].
For the varactor to function properly no forward
current must flow. The small potential barrier on the
junction does not cause appreciable conduction and hence
does not interfere with the varactor operation. Under
forward conduction, the junction can be represented by a
diffusion capacitance whilst for the reverse bias the most
important depletion layer capacitance predominates. When in
such a condition all the mobile charge carriers are
withdrawn from the depletion region
charge density of donors and acceptors.
leaving a net fixed
The depleted width
decreases with the reduction in reverse bias to a finite
10
n
p
Fig. 2.1 Abrupt Junction
Fig.2.2 ·Graded Junction
11
width because of the potential barrier voltage which would
still exist across the junction.
The fundamental relationship between the charge,
voltage and capacitance is usually expressed as
Q = v c ( 2. 1 )
The depletion capacitance of a varactor as a function of
the voltage v behaves according to the formula [6]
C<v> =
where
Co =
~ =
~ =
and v =
total
= dQ
dv
capacitance at zero bias,
barrier potential,
elastance-voltage exponent,
applied junction voltage.
<2.2)
Combining these two above equations it is possible to show
that the relationship between the charge and the voltage is
[SJ, <see appendix A>
(1/1-f>
v
= ( -:-:--:-.-) (2.3)
v.
12
where
v .· :::... 0 '
q ::;;_ q-..
and
v = 0 ' q L q-..
The q-.. (positive) is the charge stored at v = ~' and QB
(negative) is the charge stored at reverse breakdown, v = VB
The symbol ~ represents the impurity profile of the
varactor, being 1/2 for the abrupt and 1/3 for the graded
junction.
In practice, many varactors are neither exactly abrupt
nor exactly graded. A typical diffused varactor has a
doping profile so that, at low voltages, the capacitance is
inversely proportional to the cube root of the voltage. At
higher voltage it follows a square root law [~].
Variations of the depletion layer width, also affects
the series resistance r., outside the bulk material. It can
be shown [8] that
r. = r. ... :1." + f ( s ..... " - S) <2.4)
Where
S = elastance.
f = resistivity of the lightly doped side.
13
2.2.2 Design Considerations
Varactor diodes are designed to exploit the property
that the capacitance of a reverse-biased junction depends
on the applied voltage. In practice there are three types
of diode that exhibit this useful property i.e.
(i) simple p-n junctions whose operation is based on
minority carrier conduction,
<ii> Schottky barrier or Hot carrier diodes based on
the majority carrier operation and
(iii) Step recovery diodes [BoffJ exploiting the
minority storage effect.
In the design of varactor diode there are certain
objectives which should normally be achieved. The depletion
layer capacitance should decrease from a large value at
near zero bias to a small value at the breakdown voltage.
The value of the zero-bias capacitance is ordinarily
specified according to the intended application. The v.
should be high enough to ensure that in use, the varactor
is not driven into avalanche. Finally, the series
resistance should be low enough to avoid dissipation
losses.
Next, the following conditions of operation must be
followed for a satisfactory performance. It is desirable in
most cases not to allow the bias to swing past zero into
14
forward conduction. Since the driving current flows into
the junction capacitance of the varactor diode and into the
external circuit, the external resistance must be minimised
to avoid losses. The larger the junction capacitance, the
larger the driving current and the higher will be the
circuit losses. As the quality of the varactor diode is
obviously affected by the external circuitry, it is
important in most applications to use lossless reactive
components for filtering purposes.
One important design variable is the impurity
concentration near the junction. If the concentration is
very much larger on one side than on the other, the
characteristics of the device will not be sensitive to the
doping level on the high side. In such a case only the
doping level on the low side has to be carefully
controlled. This is the way p-n junctions are usually
manufactured, heavy doping on one side, and light uniform
doping on the other.
The doping level on the lightly-doped side of the
junction determines the avalanche breakdown voltage of the
junction [6]. As a practical
difficult to make diodes
consideration, it is usually
with breakdown voltages
consistently at or near the theoretical value. Depending on
how well controlled and reproducible the diode processing
will be, the design V3 should be chosen above the largest
peak reverse bias voltage to be applied. The lightly doped
layer is then adjusted to be just thick enough to
15
accommodate the depletion layer at breakdown.
The design area A is determined from the following
expresssion [ 6 J
1/'2
A = Co ( 2Va )
q fo €.- N ( 2. 5)
where
Co = total capacitance at zero bias.
q = charge at the junction,
Eo = permittivity of free space,
f,.. =relative permittivity of the lightly dopped layer,
N = lower impurity concentration <numeric>,
and V0 is given by
kT ln (2.6)
q
where
nn = electron density at the n side of the junction,
PP = hole density at the p side of the junction,
and n .._9. = the product of p and n
For a varactor diode with a Schottky barrier junction,
the epitaxial layer thickness is made equal to the
thickness required for the lightly-doped layer. The
thickness of the heavily doped layer is around 0.2 pm.
16
2.3 Electrical Characterisation of Practical Varactors
2.3.1 The Equivalent Circuit
A varactor diode is fundamentally a semiconductor wafer
which contains a junction of well-defined geometry. The
equivalent circuit of the wafer includes junction capacit
ance CJ, junction resistance RJ in shunt with CJ which are
functions of the applied voltage. Finally, the series
resistance ra which may also be a function of bias. This
comprises the resistance of the semiconductor bulk material
on either side of the junction through which the current
flows and the resistance of the ohmic electrical contacts
to the wafer.
Varactor diodes are normally operated under reverse
bias, where the parallel junction resistance, which is
usually 10 Mohm or more, may be neglected in comparison
with the capacitive reactance of the junction at high
frequencies. Therefore, the equivalent circuit of the
reverse-biased varactor at microwave frequencies reduces to
a capacitance and resistance in series. A forward-biased
varactor on the other hand is more complicated, because it
should also include the diffusion capacitance of the
injected carriers, and their effect on the conductance of
the semiconductor material.
At low frequencies, the varactor has, in general,
forward and reverse characteristics of an ordinary
17
semiconductor diode. In the forward direction the diode
current increases exponentially with the applied voltage,
whilst for reverse bias theoretically a small saturation
current, Is, should exist. In practical diodes this does
not happen because of surface effects around the junction.
At breakdown voltage the diode reverse current increases
rapidly, and is limited only by the resistance of the diode
and any external resistance which may be present in the
c i r cu i t [ 9 ] .
Although the variation in junction capacitance is the
most important characteristic of the varactor diode
presence of parasitic resistances, capacitances, and
inductances due to encapsulation and external circuitry
may affect the operation.
2.3.2 The Depletion Capacitance
We have established that the most important operational
parameter of the varactor diode is its nonlinear depletion
capacitance and the series resistance and the breakdown
voltage
The
are the limiting factors [5].
back-biased diodes do not conduct appreciable
current because the mobile charge carriers are drawn away
from the junction • Suppose that the voltage applied to a
varactor is changed by a differential amount dV,
respective change in stored charge dQ, then
with the
18
dV dQ = A
where
111
A = design area of the varactor,
w = width of the depletion layer,
and E = permittivity of the space-charge layer.
Now the incremental charge is given by
where
( 2. 7)
<2.8)
N. and Nd are the impurity concentrations on each side
of the junction
dx1, and dx2 are the changes in position of the edges
of the width on each side of the junction.
and e is the charge of the electron
As a consequence, the incremental depletion
capacitance can now be expressed as
dQ c = = A
dV w
~a~ here
C = the incremental capacitance given by eqn. <2.2> i.e.
c = (2.9) 1/2
19
2.3.3 The Series Resistance
Another important component of the varactor diode
equivalent circuit is its series resistance. Physically, it
represents the bulk resistance of the semiconductor
material and the resistance of the contacts. Its value may
be calculated [8] by integrating the bulk resistivity of
the semiconductor over the path of the current i.e.
1 R. = (2.10)
A
The p-type and n-type resistivities are functions of
the acceptor and donor impurity densities. The series
resistance is thus a function of x1, x2, which are in turn
functions of the bias voltage. Hence, ra is also a function
of bias voltage. An increase in bias voltage causes a wider
depletion layer which effectively lowers the series
resistance.
It is important to point out that the a.c. series
resistance is different from the d.c. resistance which is
usually lower. The equivalent circuit of the varactor is
shown in Fig. 2.3 in which R_, can be ignored because at
radio frequencies it is shunted by the reactance of the
capacitance C_,. Also in practice the inductance of the
leads, L, and the case capacitance may be neglected, not
20
Lleads
- Clasa
Fig.23 Varactor Equivalent Circuit
~o----7~~~--------~~------~o
F i g .14 S i m p l i f i e d . E q u i v a l en t C i r c u i t
21
because they are unimportant but because their effect may
be absorbed by external circuitry. Therefore, the
simplified equivalent circuit is as shown in Fig. 2.4.
2.4 Applications
2.4.1 Introduction
Varactor diodes, because of their nonlinear low loss
have become extremely useful elements in capacitance
microwaves. A view of the varactor diode is presented in
Fig. 2.5. Some of
applications are in
the most
low-noise
important and extensive
parametric amplifiers,
frequency multipliers or harmonic generators and, in
general, other frequency-converting circuits. The operation
of these circuits is dependent on the nonlinear phenomenon
effect. The following paragraphs will outline the general
principles of harmonic generators and parametric
amplifiers, the two most useful applications.
2.4.2 Harmonic Generation
Harmonic generators and frequency multipliers may be
used as direct sources of signals at high frequencies. Fig.
2.6 presents the diagram of a harmonic generator.
The nonlinear resistance of the semiconductor diode has
been used at microwave frequencies as harmonic generator in
the past. However, its efficiency according to Page,s law
[10] cannot exceed 1/n2 , where n is the order of the
harmonic. Greater efficiencies might be expected and
~
Junction
Substrate
22
Metal -+-- case
--+-- Metal case
ceramic tube
Fig. 2.5 Varactor Diode Pill
~ ' ~ r1V
Wo nwo
Fig. 2.6 Harmonic Genera tor
.,.>-.,.> .,.>
23
achieved from a nonlinear capacitance element used for
harmonic generation. Theoretically, efficiencies of 1007.
are possible and values above 807. are achievable in
practice for the second harmonic multiplication.
In a practical arrangement a sinusoidal generator
drives a circuit, containing a nonlinear capacitance, at
the frequency Wo through a bandpass filter. A load is
connected to the output of the circuit through a bandpass
filter around nwo. If the circuit and filter networks are
lossless, the input power introduced at wo frequency may be
transferred to the load at nw 0 frequency with high
efficiency.
Pumping the nonlinear capacitor with the charge at the
fundamental frequency, w, harmonic voltages are produced
across the diode. These harmonic voltages are approximately
proportional to the amplitude of the driving charge raised
to the power of the harmonic number [101
n
<2.11>
where
Qo = average value of the stored charge,
and Q1 = charge at the fundamental frequency.
As an example let us consider the simplest case when the
varactor is abrupt with = 112, and the only two
24
currents flowing in the varactor are the input current at
the frequency Wo and the output current at the frequency
nwo. The total charge and the current are then given by
q = Oo + 2Q1sin<wat> + 20nsin<nwat> (2. 12>
or
dq i = = 2woQ1cos<wot> + 2nwoQncos<nw 0 t) (2.13)
dt
The instantaneous value of the junction voltage may then be
obtained from eqns.<2.2> and <2.3) leading to
¢-v. v = ~ - (2.14)
(q~-Q-)2
2.4.3 Parametric Amplifiers and Converters
The varactor diode is one of the most promising
nonlinear elements for low-noise parametric amplification.
A parametric amplifier is a circuit based on the principle
that when a resonant network is suitably coupled to an
energy storage element, energy may be extracted from the
source. This energy through the mechanism of the nonlinear
capacitance is transferred to the fields of the resonant
network. Thus in the parametric amplifier energy is
normally taken from a driving source at high frequency to
amplify an input signal at a lower frequency. However,
25
since a practical varactor diode has a small series
resistance, this limits the minimum obtainable noise
figures.
In the amplifier, the capacitance is modulated by the
application of microwave power at the pump frequency,
1 1 = ( 1 + 2 O cos2TTf t > <2.15)
CJ Co
where
f = the pump frequency
Co = capacitance without the pump power
which is a 0 = diode capacitance modulation coefficient,
function of the voltage developed across the diode at
pump frequency.
If only the pump and signal frequencies are present,
there cannot be a steady flow of energy from one to the
other, except when the pump frequency is exactly twice the
signal frequency. For a steady flow of energy, it is
necessary for the voltage across the capacitance to be
maintained in the correct phase, and this can only be
achieved by permitting power to flow at a third frequency.
This the •idler• frequency which is equal to the difference
between the signal and pump frequencies. The amplifier
therefore contains three frequency-selective networks or
filters which will permit power to flow only at these three
frequencies, and the nonlinear capacitance forms the common
26
element between them. Manley-Rowe relations [5] represent
the theoretical power into and out of an idealised
nonlinear capacitance, i.e.
~. 00
L L m Pm,n
= 0 m=O n=-():1 mf1 + nf::z
(2.16)
CfJ 00
z= L n Pm,n
= 0 n=O m=-oo mf1 + nf::z
where
Pm,n is the power flow into or out of the nonlinear
capacitance, and f1, f::z the frequencies at which power is
fed to the capacitance.
Some other varactor applications require three or more
idlers frequencies. If a large current at frequency f1 and
a small current at frequency f::z are put through a varactor
sidebands with frequencies of the form mf1+nfz, for m, n
positive or negative integers are generated. If the power
flows at one of these frequencies to a load, the varactor
is said to be a frequency converter. Frequency converters
are classified according to whether the output frequency is
greater than or less than the input frequency. They are
known as upconverters or downconverters, respectively. Also
according to the integer m in the formula mf1+nfz for the
output frequency, if m = 1 the device is called upper-
sideband converter and if m = -1 lower-sideband converter.
3.1 Introduction
27
CHAPTER 3
SPECTRAL EVALUATION
In order to analyse frequency converting networks there
is a need to have theoretical relationships, for the
nonlinear behaviour based on the physics of the devices
employed. Such relationships may be easily obtained if the
precise laws of the devices are known. In many cases the
laws are approximate due to the lack of knowledge and
because other mechanisms affecting the performance may
exist. As a consequence, to achieve closed-form equations
for the devices we normally introduce severe approximations
in the analysis, which makes the final relationships very
often inaccurate. An
representing a nonlinear
alternative practical way of
device which will include all
these effects may be by
components [11].
its generated spectrum of harmonic
It is well known that if a nonlinear element is
energised by a single-frequency sinusoidal source (V or I>
the response will be a spectrum of harmonic currents or
voltages. It is also logical to assume that the generated
spectrum will contain all the necessary information to
model the device. The resulting frequency spectrum is a
unique representation of the device behaviour for the
specified drive level and device parameters. It may be
28
considered as a device fingerprint which represents its
characteristics and parasitic effects present implying as a
consequence, that any changes in the law of the device
characteristic will be reflected in the measured spectrum.
It can therefore be useful and appropriate to employ
generated harmonic spectra for the assessment of devices
before they are used in frequency-converting applications.
3.2 Review of Fourier Analysis
3.2.1. Fundamental relations
An arbitrary function repeatable with a period of T
seconds may be represented by the relationship f<t>=F<t+T>.
Let us consider that such a function is single-valued and
has a finite number of discontinuities and a finite number
of maxima and minima in one period of oscillation. Under
these conditions, known as Dirichlet Conditions, the
function f<t> may be represented over a complete period and
anywhere between t = - oo and t = +oo, except at the
discontinuities, by a series of harmonic terms. The
harmonic frequencies in the resulting spectrum are integral
multiples of the fundamental. The series is called a
Fourier series and may be obtained for any periodical
waveshape using numerical or other analytical methods.
A general trigonometric Fourier Series for a periodic
function may be written as [12]
00
f(t) = aa + [ a ... cos<nwat> + n=l
(:1)
L bnsin<nw0 t) n=l
( 3. 1)
29
or simplified to
oO
f ( t) = L Cncos(nw.,.t + ~") (3.2) n=O
where
b!> 'l/2
Cn 1 (3.3) = <an +
and
bn
"'" = tan-• (--) (3.4) an
The coefficients Cn and e" give the amplitude and the phase
shift, respectively tor the n~h harmonic and 111.,. is the
fundamental frequency. The Fourier coefficients are
obtained using orthogonality conditions and are given by
T/2 1 s a a = f(t)dt (3.5) T
-T/2
T/2 2
~ an = t<t>cos<nw.,.t>dt <3.6) T
-T/2
T/2 2 ~ f(tlsin(n~otldt bn = <3.7) T
-T/2
30
Fourier Series, can also be expressed in an exponential
form thus
00
[ Jn~at 0 t:
f(t) = F.,e t1.<t<t2 <3.8) n=-oo
wt-:.are
F ... is given as
1 dt (3.9)
The Fourier series as represented (3.8) is
applicable to a periodic signal. The transiti~n to an
aperiodic signal representation is obtained by making the
period approach infinity. This can be achieved by
considering initially the exponential form of the Fourier
series of a periodic function fT<t>.
00
L Jn.., 0 t:
fT(t) = Fne <3.10) n=-co
where
T/2 1 5 -Jn..,ot
F ... = f<t>e dt T
(3.11)
-T/2
31
In order that IF ... I is convergent when T is increased, the
following conditions must be satisfied, i.e.
w... = nwa (3.12>
F<w..,> = TF.., (3.13)
Eqns. (3.10) and (3.11) may now be written as
<3.14>
and
T/2
F<w ... > = 5 -J~Afn ~ fT<t>e dt (3.15)
-T/2
If
ZIT T =
then
(3.16) n=-cr~ 2TT
In the limit, as T tends to infinity, b. w _. dw, the
infinite sum in eqn. <3.16) may be written as
lim T~
1
2TT
32
00
[ l!.w n=-oo
and therefore becomes the Riemann integral [tiJ i.e.
1 f(t) = dw
2TI
Similarly eqn. <3.15> may be written as
(3.17)
(3.18>
(3.19)
and is called the Direct Fourier Transform whilst eqn.
(3.18> is called the Inverse Fourier Transform.
The Fourier Transform pair is one of the integral
transforms that are commonly used in operational analysis.
The frequency distribution of harmonics in a Fourier
series is a line spectrum whereas in a Fourier Integral it
is a continuous spectrum. Therefore, spectra of
mathematically undefined and defined waveforms can be
examined.
3.2.2 Sampling in Frequency Domain
In the frequency domain the sampling theorem states
that it, a function f(t) equals zero outside the function
period T, [13] i.e.
33
f(t) = 0 for t>T (3.20)
then its Fourier Transform F<w> is defined from its values
at F<nTT/T) equidistant points with distance TT/T apart. As
a result F<w> can be expressed as
00
F<w> = L n=-oo
sin(wT-nTT>
wT - nTT (3.21)
One method tor evaluating the Fourier Transform is
based on a Fourier series approach. Considering eqn. <3.19)
T/2 1 s -Jn .... ~ F<nw.,.>
Fn = t<t>e dt = (3.22> T T
-T/2
The eqn. (3.22> can now be expressed in terms ot the
coefficients ot the Fourier expansion i.e.
00
F(w) = L TFn n=-oo
sinCwT/2 - nTT> (3.23)
<wT/2 - nll>
As a result the evaluation ot F<w> is reduced to a simple
34
problem of determining F". This method can also be used to
give the Fourier transform, F<w>, of an arbitrary function
f(t) not satisfying f<t>=f<t+T> i.e. a non-periodic
function [ 12 J.
3.3 Fingerprinting of nonlinear devices
Many nonlinar problems can be solved through the use of
nonlinear differential equations. However, as their
solutions cannot often be written in a closed form,
development of other methods becomes necessary. A physical
system is usually too complex and its analysis is not a
simple process.
In the case of devices it is often necessary to
establish a relationship between parameters and
characteristics. A device is normally modelled into its
simplest form and with an acceptable accuracy so that the
performance of a particular circuit using this device,
operating under specific conditions, can be predicted.
Modelling is a procedure which leads to the equivalent
circuit of a device representing its approximate behaviour.
A given device is initially expressed in terms of physical
variables required for its design and the circuit
applications. There are suitable techniques available to
solve these problems and
parameters. It is important
result in specific device
to know the physical mechanism
of the operation of the device or a system so that
satisfactory modelling can be realised.
35
The harmonic frequency spectrum generated by a
nonlinear device is a unique representation of the device
behaviour and therefore may be considered as its
fingerprint. The basic difference between fingerprinting
using harmonic spectrum and modelling is that the former
the device under will give the true nonlinearity of
particular working conditions. Modelling will always be
approximate while the harmonic spectrum will presicely
represent the device and its mechanism for particular
levels. Such fingerprinting, within the measurement errors
and perturbation effects, could provide a practical method
of device indentification.
Most manufacturers supply data sheets with inadequate
information. Quantities like noise figure, r.f impedance
are usually given at a particular test frequency often
without specifying the operating levels. In many
applications sets or a matched pair of particular devices
are required. By comparing the spectra, •fingerprints•, of
the devices, the required matching for particular
applications may be achieved. It may also be possible, from
such spectra, to predict the performance of a circuit.
3.4 Varactor Analysis
For a back biased diode the incremental capacitance as
a function of voltage is given by [5]
36
C< v > = v )~
where
~ is the potential barrier,
~ is an index whose value depends on impurity profile
of the diode junction,
and v is the applied voltage.
The current through the diode may be written as [5]
dv i = C<v>
dt
or
Co ~~ dv i =
Hi - v)' dt
where
or
and
v~ = input voltage
Va = output voltage
vb = bias voltage
then
i = = K <wV~coswt> <1 - vd
where
I
Vt. =
or
I Vt.
and
K =
I
= V-..sinwt
<jll -
37
sinwt =
Expanding binomially gives
i
Now [ 14 J
1 sin 2 "x =
and [ 14]
1 sin 2 "- 1 x =
I ~(1+1> 2. ~ ( 1 + "6 ) ( 2+ lS )
I '3
I + Vt. + Vt. + Vt. + ••• 00 J
2~
[
n-1 L (-1 )n-k
k=O
[
n-1 L (-1)n+k-1
1<=0
I
3~
2n-1 } ( k ) sin(2n-2k-1lx
which on using for powers of Vt. in the equation results in
~ coswt ,
i = KwV t. + Vt. ~ coswts i nwt +
,2 ( 1:1) 1 + Vt. <1-cos2wt>coswt +
.L. 2
,3 ( 2:1) 1 + Vt. (3sinwt - sin3wt>coswt +
4
,4 ( 3:~) 1 + Vt. <3 - 4cos2wt + cos4wt>coswt +
8
38
1 5 I
+ v .. <10sinwt - 5sin3wt + sin5wt)coswt + 16
Using the following trigonometrical identities,i.e.
1 cosA cosB =
2 ( cos<A-B> + cos<A+B))
and
1 cos A sinB =
2 ( sin(A-B> + sin(A+B>)
and collecting the coefficients at each frequency finally
produces
for the fundamental current i1
5 3 I
1 I
vi. + -- v .. + 23
1 + --
2~
7 , vi. + ... <0] coswt
for the second harmonic i2
L 1
( ll ) ~
I
Kw < <'-Vb) vi. + 2
4 I
v .. +
1 r:lJ ' I
+ v .. + 2~
sin2wt
for the third harmonic i3
Kw<~-Vb) l -_:_ Cll ,3
v .. 22
3
for the fourth harmonic i.-.
Kw<!/)-Vb) l- 1
(:~J '1
I
v .. 23
5
39
for the fifth harmonic i~
[ :4 ( 3:~ ) v:' 1 5+0' \ I~ ... ~ 1 Kw<4J)-Vo> + v .. + cos5wt
I
2h 6 J
for the sixth harmonic ih
[ 1 ( 4:~ ) v :6 1
(6:~) 8 .. ·W} Kw<~-Vo> I
+ v .. + sin6wt 2~ 27
The objective for the above analysis was to show that
even if the assumptions in the driving voltage function
are made it produces still infinite series
euations for the harmonics. These equations, depending on
t he dr i ve 1 eve 1 , provide approximate expressions for the
actual current amplitudes of the frequency components.
40
CHAPTER 4
COMPUTER AIDED EXPERIMENTAL SPECTRAL CHARACTERISATION
4.1 Introduction
There are number of ways for evaluating nonlinear
devices and the choice will normally depend on their final
applications and operating conditions. Testing is
time-consuming and the required equipment is usually
expensive and therefore there is a need to develop quicker
and cheaper techniques for device characterisation. In
addition, to collect a comprehensive data over an extensive
frequency range, requires a lot of measurements which is
even more time consuming. The method of Numerical Fourier
Analysis under consideration in this project offers an
inexpensive means of assessing the devices in much shorter
time [ 15 ].
On energising, the signal applied to a nonlinear device
produces a distorted response which reflects the
nonlinearity of the device characteristic and contains all
the necessary information for its evaluation. The distorted
voltage developed on the output can then be Fourier-
analysed and hence provide the frequency spectrum for the
device identification. In order, however, to increase the
41
speed of analysis computer facilities were employed.
Furthermore, since the analysis in our project had to be
carried out at high frequency there was a need to use a
fast sampler for the distorted waveshapes with a low
frequency display for our guidance as a bonus. In the
following paragraphs, the method and the steps necessary to
achieve these aims are discussed in some detail.
4.2 The Method
4.2.1 General
The method followed here aims to examine a distorted
resulting response when a nonlinear system is driven by a
single frequency signal source. The main objective was to
determine the complete harmonic spectra of four nonlinear
devices, varactors, by means of Numerical Fourier Analysis.
Although, the amplitudes of the harmonics of a distorted
waveform can be measured using a Spectrum Analyzer there
are no high frequency instruments available which will
determine the relative phases of the generated frequency
components within the spectrum.
To carry out directly Numerical Fourier Analysis, on a
high freque~cy waveform becomes difficult because such
waveforms cannot be displayed. The method ivestigated in
this project was made possible because a high frequency
sampler manufactered by Bradley,s was available. Further
limitations would obviously be due to sampling techniques.
The accuracy of the sampling process used depended on
42
the risetime of the sampling step generated by the
instrument used [16]. The sampling points in one period of
the displayed waveform were read by a data acquisition
system and the information was fed into a microprocessor
for further analysis. The microprocessor was programmed to
work in •Fast Fourier Transform• algorithm obtained from
the computer library. Its purpose was to determine the
amplitudes and phases of the harmonic components of the
sampled data. To achieve this, the samples proportional to
the analogue input signal were digitised using an
analogue-to-digital converter, AID, before being fed into
microprocessor. It was also necessary to produce the
external scanning voltage for the sampling unit and this
was done by a digital-to-analogue converter, 0/A,
controlled by the processor. This divided the time-axis
into an equal number of intervals and synchronised the
reading of the sampled points. The resulting harmonic
spectrum represented the nonlinearity of the device and
could be used for its characterisation.
4.2.2 The Theory of Numerical Analysis
The response of a nonlinear device driven by a
sinusoidal signal will be a periodic distorted waveform.
This waveform can be subdivided into M equal sections over
the period, T,
represented by
along the time axis. If each section is
~t, then [121
43
T = M 6t ( 4. 1 )
and if t~ be the time at the end of the m~h interval, then
t ... = m 6 t for 1{m~M (4.2)
and the ordinate at t ... , may be denoted as f < t,..> or f <m 6 t >
In order to proceed with numerical analysis, the
distorted waveform is sampled at the points 6 wt, 2 6 wt,
3 6 wt, • • • , m 6 wt, ••• , up to M A wt = 2TT. Thus, there
are M points and the spacing between adjacent points is
uniform and equal to 2TT/M.
The approximate Fourier coefficients for a periodic
case can now be deduced from eqns. (3.5), <3.6) and <3.7)
and may be written as
M 1 [ a a
,.. f <m 6 t > (4.3> ,... M
m=l
M 2
[ a., :::1 f < m ll t > cos[ n < m ll wt > l (4.4) M
m=l
M 2
[ b., .... f ( m /). t >sin[ n ( m /). wt > l (4.5) ..... 1'1
m=l
44
where 2TT
L\ lift = M
T and L\t = T is the period in seconds.
M
It can be seen from the above equations that the variables
involved are M, m, L\ wt, n and f<tm> with m representing
all integer values from 1 to M. The Fourier coefficients
numerically determined using the above equations and
measured ordinates are finally used to calculate the
amplitude and phase values of the harmonics from
1 b!> 1./2
An = <an + and
bn ~n = tan- 1
an
It is obvious that the higher the value of M, the
closer is the approximation to the true values of the
coefficients and hence the waveshape. The evaluation of the
Fourier coefficients may be accelerated by using a
microprocessor facility and an •F.F.T.• algorithm.
4.2.3 The Aims of Programming
One of the advantages of digital electronic circuits is
their ability to store and retrieve data easily. The
45
experimental research techniques are frequently computer
controlled in order to exploit this facility and minimise
human errors.
It is convenient for a computer to use a binary, or
base two, arithmetical system because only two digits are
needed, i.e. one and zero. Since digital electronics have
only two defined logic states, i.e. high and low, it is
possible to let one state represent a binary one and the
other represent a binary zero, respectively [18].
The largest binary number that may be accepted here has
a decimal value of 256, which is the equivalent of an
8-bit word i.e. the computer is capable of reading from 0
to 255 points maximum.
The process of describing a programming procedure, to a
computer is therefore based on a binary number sy~ The
commands of a Fast Fourier Transform programme, •F.F.T.•,
which was the only technique available tor Fourier
Analysis, was obtained from a computer library, and used in
this project. The programme was stored in RAM memory and
was executed when needed by entering the data. The
procedure, initially, was for the programme to read the
data, i.e. 256 different sampling points of the waveform,
which were next stored in memory in the form of an array.
These sampling points were needed to operate the •F.F.T.•
algorithm and produce the fourier coefficients, aa, an and
bn. From the coefficients the magnitudes and phases of the
harmonics were then calculated [19].
46
4.3 Practical System
The block diagram of the practical system used in the
project is shown in Fig. 4.1 . It was decided, as the
impedance of the sampling adapter was 50 ohm, it would be
advisable to insert some attenuation in series with the
device under test to achieve a better match. The device
was placed in the inner conductor of a coaxial holder. The
experiment consisted of measuring the distortion of the
output waveform, caused by the device nonlinearity.
4.3.1 Sampling Adapter
For the sampling process , Bradley Sampling Adapter
type 158 was used capable of accepting signals with
risetimes just below 1 nsec. or an equivalent passband
from zero frequency to just over 1 GHz. The output from
this unit can be displayed on a monitor [16].
In a conventional oscilloscope, the waveform of the
signal, which is observed, is drawn during a single
horizontal sweep in a time equal to the period of the input
signal. The sampler, on the other hand, builds the input
waveform up as a low frequency display by sampling its high
frequency value many cycles apart. The input signal is
applied to a sampling gate which opens for approximately
350 pS in each cycle by a sampling gate pulse. Each time
the gate is opened the sampler measures the input signal
and then generates an equal value at about the time the
-Printer
C. B.S. 4022
I
I .....,.
Bias 1 Unit I I
I
r Signal -Genera tor
Diode Sampler
-Holder
I
-
Fig. 4.1 Block Diagram 01 the Sampling Ha thod
4 0 32
IEEE488
I
_I
- 0/A
--
X-Y
. I
J_EEE 488
I
I I
l
A/0
r-
0 i splay
Unl t
Tape Reader
-
.,. ""
I I
I I
I I I I I 1
II
11 If II II
---" a I
I I
I
"'---'~ I I I I I I I I
_j1 n n n n n n n L II I I I I II II I I I II I t I 1 r I I 11 II 1 I I I lA :K--r--tl I I I I I ~ ;1 ft I I
___,J) ,t rj jl II II II tl II
" 11 II I J II II .. tJ It Jl It It It tl It· II II II 11 I 11 11 I I 11 II 11
J ~ k ~-.J K ~ K k-'- •I II· t' II I I II . II .. .,. I I
Ia •• •' n II • H tl •• II ,e---•---·---•, 11 II I 11 "' ' .. ~ . tl / 'J 1 I I . al
11
- .)it' ... - ..J ... - _. ._ - --
Signal 1/P
Signal at delay line 0/P
Fast ramp
by trigger circuit
Memory gate pulse
Staircase prodused for each gate puts e
Sampling gate pulse
VerticatO/P signal
Fig .4.2 Sampler Pr inc i pte
~ (.£1
Flg. 4.3 Block Diagram of 1M Sampliftg Adaptor
TriggerL.-_______ I ---~----, deLay Memory
..rL -u-
Ex! Tri~ ~=~~a Qa te I· ..fl... --~~ro~r • '
f'1. romp. ...n_ - l-.r
Ext Sea
~Fast ..n... r1g. ramp
triooer
Reset l ....I '-
'J
Fast gate r tri
n_ T Reset
/1
t • 1 Gate generator1 • t ~Zz. 0 w ~...:::..........--::::::...___ +
_n_
/L~ sweep 0/P X
50
sample is taken. Aliasing errors could occur in this
process bue since we do not consider high order harmonics
then six the effects would be negligible according to the
formula n < M/2.
The sampler has a memory circuit so that each sample
point, dot, is displayed during a small interval of 2 pS
before the next sample is taken [16]. It also provides a
scanning signal which places each dot at the correct
position on the horizontal axis. The gate opens at a
frequency lower than the input frequency so that each
sample represents a different and latter part of th• input
signal. Therefore the waveform is rebuilt from a number of
samples taken over a period equal to many cycles of the
signal. The principle of the reproduction process of the
waveform is illustrated in Fig. 4.2. The block diagram of
the sampler is also presented in Fig. 4.3.
4.3.2 D/A, AID, Converters
Analogue signals via A/D converter were used as inputs
to a computer. Since a microprocessor could only handle
digital information, an analogue-to-digital conversion was
required for interfacing. Similarly, the digital-to-
analogue converter, may deliver a current or a voltage
output that is proportional to the value of the digital
applied input. The information can be in a coded form and
may represent positive or negative quantities or both [20].
The ZN425E is an 8-bit D/A converter, shown in
+ sv
0.01 II. p
Bit 1 Bit 8 ~LN4ZSE ~Bit 2 IZN~ Analogu• output Bit7 Bit 3 · 6.8 K n Bit6 Bit 4
it s I I I q R1 SKO l1l _.
ov -- - sv
Fig. 4.4 D/A Converter
u.. ::&.
0 0 -
> "' +
Ill
ID V)
z:
52
CD V)
...J
0 -., C:::::::J ·- . --'"\ IW"C
0 l!J ,.
-., O::::::J
-
a ~·< Ln
c.. Cl -c.. • > c 0 LJ
Cl ...... <t
~ • en ·-u..
53
Fig. 4.4 , whose major components are an <R-2R> ladder
network, an array of precision bipolar switches, an 8-bit
counter and a reference voltage unit of 2.5 V. It gives an
analogue voltage output, and thus the usual current to
voltage converting amplifier is not required. The accuracy
of the D/A converter depends on the components used in the
converter, and also on the electrical noise and leakage in
the circuit. A reason and an advantage of having D/A
conversion is to keep information in digital form, so that
it is independent of time and temperature [21].
An A/D converter will read at various points the
amplitude of the signal and its value will be allocated in
a particular code using binary system. For this purpose
the ZN427E was used, shown in Fig. 4.5 which is an 8-bit
bipolar Successive Approximation AID converter. The term
Successive Approximation means that the operation of this
converter is based on •n• successive comparisons between
the analogue input, v~"' and the feedback voltage, v~. The
ZN427E consists of a fast comparator < R-2R) ladder
network, analogue voltage switches, successive approxi-
mation register and 2.5 V reference voltage circuit. It
provides a fast conversion at a speed of 10 sec [22].
4.3.3 The Microprocessor
The initial waveform generated by the nonlinear
element, energised with a sinusoidal drive, was fed into
the sampler, and converted to a digital form. The
54
microprocessor, loaded with the "F.F.T.a programme (see
Appendix B> was then able to read the information and
produce an output of amplitudes and phases of constituent
harmonics in the distorted waveform. Here, a Commodore
Business System Computer was used , and peripheral devices
i.e. a printer and a tape reader were connected through the
IEEE 488 port. This was a bidirectional port which allowed
the peripheral devices to be interfaced simultaneously to
the computer.
The computer consisted of three self-contained units.
The first, was the central processor with all the
necessary arithmetic and logic operations. The second, was
the memory where the programme was stored and gives the
computer the ability to, temporalily, save the information
for future use. The third unit consisted of peripheral
devices attached to the computer.
4.3.4 Diode holder
As the project involved the investigation of diode
properties, the construction of a proper diode holder was
essential. The holder was manufactured in a coaxial form
and adapted for use with General Radio <GR> system having
the characteristic impedance of 50 ohm. The main
requirement for a properly designed holder was to ensure
that it retained the characteristic impedance of the line.
It had to meet the test that when the diode holder was
short-circuited and terminated with the characteristic
55
impedance it produced no or negligible standing waves.
Under these conditions, the dimensions of both the inner
and outer conductors of the diode holder would meet the
specifications required for matching purposes [11].
For a coaxial lossless transmission line, the
characteristic impedance is given by [23]
1 fi- In E.
a (4.6)
2TT b
where
p = permittivity of the me~ ~m,
t = permeability of
a = internal diameter ut the o~~2r conductor
and b = diameter of the inner conductor.
It was found that the diode holder had a very satisfactory
VSWR of 1.02 at 16Hz.
4.4 Testing
It was the intention that the sp~-~rci~ , ,, i -.Jrmat ion
acquired obtained by means of this method will be helpful
in the selection of nonlinear devices for the best
performance in particular applications. Although, the
method was simple in principle, many practical difficulties
had to be overcome before reliable results were obtained.
56
The experimental set up consisted of both analogue and
digital units. Its accuracy, therefore, depended on
precision of the measurements at the analogue stage and on
the performance of the intervening digital units. The basic
quantity to be analysed and displayed for information using
the sampling unit, was the distorted output waveform from
the device. A low pass filter was inserted between the
source and the diode to ensure that the distorted
displayed output was the true waveform from the device. The
accuracy of the sampling unit was of the most importance
since the validity of the method depended to a large extent
on the sampling process.
that there was no loss of
and D/A converters.
Furthermore, it had to be ensured
information because of the A/D
The software, i.e. the programme, was tested by feeding
in the theoretical values of a known waveform and
comparing them against the practical observed values on the
output of the system. The theoretical values of an ideal
waveform were generated by a subroutine of the programme.
The accuracy of the complete system was also tested by
feeding directly known generated signals, square pulse and
triangular repetitive waveforms, and recording their output
spectra. The computed readings were then compared against
the expected theoretical values.
The validity of this technique was also partially
confirmed by means of another method. The spectrum analyser
was the alternative since it can measure the harmonic
v
s art end -t
w~v•form shown on display unit
xxxxxxxxxxxxxxxx
XXXXXXXXXX XX X X X
0 127 255
F i g.4.6 Computer Reading
58
amplitudes, however not the phases.
An ideal square wave pulse is shown in Fig. 4.6 .
The first 63 points of the pulse represent the maximum
value 10, from the 64th to 190th the minimum value of 0 and
from the 191st to 253rd the value of 10. Since the ratio is
one-to-one the d.c. value should be half of the height of
the pulse. The •sinea components should be zero because of
the even symmetry and the ucosine• components only for odd
harmonics [24]. Its Fourier Coefficients given by the programme
are shown in table <4.1>.
59
Table 4.1
a co = 5
Harmonics an bn Cn ~" degs.
1 6.39 0 6.39 0
~. 0 0 0 90 "-
3 -2.13 0 2.13 0
4 0 0 0 90
5 1.28 0 1. 28 0
6 0 0 0 90
7 -0.91 0 0.91 0
8 0 0 0 90
9 0.71 0 0.71 0
10 0 0 0 90
11 -0.58 0 0.58 0
12 0 0 0 90
13 0.49 0 0.49 0
14 0 0 0 90
15 -0.42 0 0.42 0
A theoretical square wave is represented by the following
equation [ 25 ]:
1 2 (cost -
cos3t cos5t
2 TT 3 5 - ... ) (4.7) f(t) = +- +
60
and the expected values XlO are presented in table <4.2>.
Table 4.2
Harmonics a ... b ...
1 6.36 0
2 0 0
3 -2.12 0
4 0 0
5 1. 27 0
6 0 0
7 -0.91 0
8 0 0
9 0.71 0
10 0 0
11 -0.58 0
12 0 0
13 0.49 0
14 0 0
15 -0.42 0
61
The values of an ideal sinusoidal waveform generated
within the program. are given in tables <4.3> and (4.4>.
Table 4.3
a.,. = 0
Harmonics a" bn Cn 0" deqrees
1 0 1 1 90
2 0 0 0 90
3 0 0 0 90
4 0 0 0 90
5 0 0 0 90
b 0 0 0 90
Table 4.4
a.,. = 1
Harmonics a" bn Cn 0n degree_'!:
1 0 1 1 90
2 0 0 0 90
3 0 0 0 90
4 0 0 0 90
5 0 0 0 90
b 0 0 0 90
62
The values of an ideal triangular waveform generated
within the program are given in table (4.5>.
Table 4.5
a a = 0
Harmonics an bn Cn 9n degrees
1 25.7 0 25.7 0
2 0 0 0 90
3 2.9 0 2.9 0
4 0 0 0 90
5 1 0 1 0
6 0 0 0 90
7 0.5 0 0.5 0
8 0 0 0 90
9 0.3 0 0.3 0
10 0 0 0 90
11 0.2 0 0.2 0
12 0 0 0 90
13 0.2 0 0.2 0
14 0 0 0 90
15 0 0 0 90
63
A known sinusoidal waveform was fed into the system
and the results are shown in table (4.6>. The tested
frequency is 0.5 GHz. The maximum value of the waveform is
2 and the minimum 0.
Table 4.6
a.,. = 1.004
Harmonics an bn Cn ~" dearees
1 -0.0122 1.0058 1.0058 -89.3
2 0 0 0 90
3 0 0 0 90
4 0 0 0 90
5 0 0 0 90
All the harmonics should be zero except for the fundamental
because if the waveform is a pure sinusoid. Also, a~,=O
because it is sinusoidal signal and hence cosine components
should not exist. The overall percentage error was found to
be less than 0.5%.
To test the error of the diode holder a generated
sinusoidal waveform was applied through the empty holder at
a frequency of 0.5 GHz • The maximum value of the waveform
was 0.738 and the minimum 0. The results are shown in table
<4.7).
64
Table 4.7
a.., = 0.3362
Harmonics a ... b ... Cn 0 ... degrees
1 -0.0248 0.3565 0.3573 -86.0206
2 -0.0209 0.0189 0.0281 -42.12
3 -0.0152 -3.2E-03 0.154 11.8886
4 1.2E-03 6.3E-03 6.4E-03 79.12
5 -3.4E-03 6E-03 3.4E-03 -61.6
A square wave generated from a source was analysed with
this method. The results are shown in table (4.8>. The
frequency tested was 25 MHz, the maximum value of the
waveform 10.336 and the minimum 0.
65
Table 4.8
a a = 5.18
Harmonics an bn Cn 0n degrees
1 -0.36 6.59 6.59 -86.88
2 -0.09 0.02 0.09 -12.53
3 -0.35 2.21 2.23 -81.01
4 -0.1 0.01 0.1 - 5.72
5 -0.38 1.32 1.37 -73.95
6 -0.41 0 0.1 0
7 -0.1 0.88 0.97 -65.02
8 -0.14 0 0.14 0
9 -0.4 0.65 0.76 -58.-4
10 -0.12 -0.05 0.13 22.61
4.5 Summary
It was shown that the determination of the harmonic
spectrum generated by a nonlinear device, energised by a
high frequency fundamental source was possible. The method,
based on Numerical Fourier Analysis used a closed-loop
system incorporating a sampling unit which also translated
high frequency distorted waveform to a low frequency
display. The process of the analysis would have been
tedious and lengthy but with the aid of a microprocessor it
was reduced to a simpler and quicker procedure.
Nonlinear devices are extensivly used in high frequency
66
applications and it is an essential phenomenon to the
operation of many circuits. In these applications the aims
are usually to accentuate the nonlinearity of the device so
that the basic harmonic-generated process is initiated. In
addition, the generated spectrum offers a means of
characterising the device,
lose
as fundamental V-I plots, at
Spectral these frequencies their meaning.
characterisation represents the actual nonlinearity of the
device under normal working conditions. The method also
offers a possibility of comparison between the devices of
the same type
applications.
and their capabilities in particular
It was found that the errors in the harmonic
measurements were lower at frequencies below 16Hz. This was
because the sampling unit limiting frequency of operation
was about 1.2 6Hz caused by a risetime of the sampling
pulse of about 0.5 nsec. Even in the case of a low
frequency, the errors were in the order of lOX. To improve
the accuracy it was necessary therefore, to use a high
precision equipment and higher power source to cover a
bigger range of drive levels.
The resulting spectra or •fingerprints• of the devices
introduced a new method of characterisation. Evaluation of
devices may next be carried out by examining and
interpreting the patterns of the spectra.
5.1 Introduction
67
CHAPTER 5
MEASUREMENTS AND PROCEDURES
The testing of the system and the behaviour of the
diode holder were considered in the previous chapter. The
measurement procedures described here are connected with
the computer-aided experimental technique used in the
determination of generated spectra of nonlinear elements.
Since the nonlinear element response is dependent on the
input level there is a need first to decide, which
parameter should be chosen to represent the drive. The
difficulties involved in carrying out the measurements of
the input quantities forced us to consider the current at
the fundamental frequency measured in the load as the drive
parameter. The values of this fundamental drive current are
important and should be stated with any measurements of C-V
and Q-V characteristics. The available instruments and the
experimental arrangements for the measurements and the
results obtained will be discussed in this chapter.
5.2 Choice and Measurement of Drive Level
The characteristics of nonlinear diodes or elements
are level-dependent and therefore the actual drive
conditions must be specified. The quantity selected for the
drive level must be reliable and repeatable. Applied power,
68
voltage and current at the fundamental frequency were
considered for the representation of the drive level.
Voltage could not be used because no voltmeter of [ 11 ]
sufficiently high impedance was available. In addition, the
measurement of the voltage at the end of a mismatched line
creates problems because of the standing waves. Power was
not a suitable quantity because the power-meter normally
employes a 50 ohm thermistor and there are no frequency
selective instrument available of high enough impedance.
It was finally decided that the drive level should be
represented by the current, I 1, at the fundamental
frequency. The components of the generated harmonic spectra
can then be recorded with reference to this level. It was
also noted that the upper limit of the drive current should
be restricted to avoid the burn-out level of the diode.
The measurement of the drive current was carried out
by measuring the voltage on the outpute by means of a
spectrum analyser, assuming that the current at the input
of 50 ohm load will
diode.
be the same as the current through the
5.3 Measurements with Slotted Line
A slotted line is a section of uniform loss less
transmission line with a longitudinally-oriented slot which
provides access to a pick-up probe that detects voltage
variations on the line as it moves along it. The voltage
induced in the probe circuit is proportional to that
69
existing between the inner and the outer conductors of the
line at any probe position [26].
One of the most informative measurements that can be
carried out on a slotted line, is the measurement of the
Voltage Standing Wave Ratio <VSWR>. The block diagram of
the slotted line assembly is shown in Fig. 5.1. From the
value of the VSWR, the capacitance of the diode can be
deduced, following a series of calculations. Knowing the
capacitance which is a function of the applied bias the C-V
characteristic of the diode may be obtained.
Using the VSWR, the impedance of a load terminating the
line may be calculated. In this case the load is the
diode, and its impedance is given by [27J
ZL =
or
ZL =
f3 =
where
1 +p ZCJ
1 -p
1 - jS...,tan<(iJXmt.n> ZCJ
5..., - jtan(~Xmt.n>
2TT
A
P = phase constant,
S..., = VSWR,
( 5. 1)
(5.2)
.
source I I
VSWR meter
robe
---- ------slotted. lint ---------
varatJI.r --·r ~.,...
--------I 1 diode
Fig. 5.1 Block 0 i a g r a m of S l o Hod Lin a H • as u r e m e n t s
r
d. c. Supply
bias unit
-J I 0
and
A =
f =
71
wavelength,
reflection coefficient of the line due to the
diode.
From the impedance of the diode its capacitance may be
calculated [27].
or
c =
1
we
1
2TTfXc (5.3)
Knowing the capacitance and the applied bias voltage on the
diode its c-v characteristic can be drawn.
Using the obtained capacitance from eqn. <5.3) and the
following relation
Q = cv (5.4)
the Q-V characteristic of the diode can also be obtained.
A DC4303A GaAs No.3 diode was tested at 0.45 GHz
and different bias levels. The C-V and Q-V characteristics
are presented in Fig. 6.1 in chapter 6.
The measurements were carried out according to the
72
following steps:
(i) The frequency was verified by measuring the wavelength
along the slotted line,
(ii) Using a VSWR-meter the VSWR of the diode was measured.
(iii) The position of Xm~n was recorded,
(iv> Varying the applied bias voltage, via the bias unit,
values of VSWR were measured, together with the
positions of minima,
pattern.
5.4 Spectrum Measurements
Xm~n, of the standing wave
As the aim of the work is to assess the harmonic
gen~~ating properties of a diode, the method of spectral
charo,terisation becomes appropriate. The spectra of
interest will be t~ose of the amplitude and relative
phases.
The quantities that can ~e determined are the voltage
measured across a known impedance at n~h harmonic 6nd the
drive level which was stated earlier to be the curre1~t at
the fundamental.
For the Sampling Method Measurements, the experimental
block arrangement for the spectrum measurement is shown in
Fig. 4.1. It consists of analogue and digital parts. The
analogue part includes signal generator, bias unit, diode
holder, sampler and a display unit. The digital part
includes AID, D/A, converters, microprocessor and the
73
peripheral instruments, e.g. printer, tape reader.
The signal source type HP3200B VHF oscillator used in
this method, covers the frequency range from 10 to 500 MHz.
Its frequency accuracy is within 27.. Across the 50 ohm
external load it can deliver a maximum power output greater
than 200 mW in the frequency range 10 to 130 MHz, greater
than 150 mW from 130 to 260 MHz and greater than 25 mW
from 260 to 500 MHz. Initially, the frequency of the
oscillator was set accurately to 475 MHz by calibrating it
against the spectrum analyzer.
The bias unit type GR874-FBL was connected between the
signal generator and the diode holder. This unit has
impedance of 50 ohm and a satisfactory VSWR of less than
1.25 between 300 MHz and 5 GHz. It provided a reverse bias
voltage, necessary to set the operational point of the
varactor diode and comprised a blocking capacitor in series
with the line, an isolating choke, and a low pass filter.
The bias unit was inserted at the source end of the line,
in order to avoid reflections at the measurement
terminals.
Driving the signal from the source through the diode
holder to the sampler as it was described in chapter 4 the
amplitudes and relative phases for each harmonic of the
diodes were obtained. The V-I plots of these results are
presented in Figs. 6.6 6.45 where I1 is the drive current
and V is the voltage output of the diode developed across
the known impedance of the sampler which was 50 ohm [11].
74
In order to confirm these results, another method had
to be used. In this method a spectrum analyzer was
connected after the diode holder, in place of the sampler,
and the same sets of measurements were carried out under
the same conditions. The Spectrum Analyzer <type TR4131/E)
used in this method covered a wide frequency range from 10
kHz to 3500 MHz. The results are presented in Figs. 6.46 -
6.65.
The above results were obtained for the 2nd to 6th
harmonics. The experimental set-up for the V-I characte
ristics of the fundamental, is shown in Fig. 5.2. A high
impedance voltmeter was connected after the generator to
measure the voltage at the input to the diode. Measuring
the output voltage through the spectrum analyzer and
subtracting this value from the input voltage, the diode
voltage was obtained. The drive current was measured by the
method described in para. 5.2. The results are presented in
Figs. 6.2 - 6.5.
u
I I I I r I .,. _, I .. ·-· ·- c.
I CD => 1 I
75
QJ u ':::J 0 Ill
-0 >
'QI
+GI
E
-
•
6.1 Introduction
76
CHAPTER 6
RESULTS
The results of the spectrum measurements carried out on
four diodes are presented in this chapter. As it was
mentioned before these results are shown in form of graphs,
from which the diode harmonic generating properties can
be found. The experimental methods and procedures were
already discussed in chapter 5.
In order to overcome the need for a large number of
plots, an alternative method of representing the amplitudes
and phases of harmonics against the drive level was
introduced. This was done by combining the harmonic spectra
of a diode at different drive levels on a single plot. Each
complete display contains therefore the •fingerprint• at
every energised level which can be extracted at will. In
addition, the dynamic behaviour of the diodes may be
compared for different levels and particular harmonics.
6.2 C-V, Q-V Measurements
Fig. 6.1 presents the C-V and Q-V characteristics of a
DC4303A GaAs Varactor diode which was tested at 0.45 GHz
and different biases levels at a drive level of 4.2 mA. The
lowest value of the capacitance at -10 V bias was 0.48 pF
and at -1 V bias it reached 1.07 pF.
77
:: : ' : : :~: : ::: ::~: ... : : · :v: : ~ ,, . :::= ::::: -· ···· ·-· =~~= : =~· l: :'\l --· ;_::.:, .. _ =-> : : ::: ~ ~::: =: : : t 1 ·: : ::::'. = ~ ~:~: ~~E~~~: = :;:;~::: = ~~ ~ ~== ==.=e==: := = ==~~· = ,,, ~ ~·:~ •;•>• •: == : · =· ~~: =~: r· = :::~: :::~::.:: ~:h: ~: ::~ :=:. ~-:= 1 ....... ~~c-~= ~· ~n-=
:m;~~ :;=~~- - ~!;;;; u-~ _ _;- --_ ~- r- _-__ : ____ :::_ :- ·;;i;; · .:·- :=:~ : = - 10~: ; ;;:- ~;::- ~ - -t:.-n~~: - : ~00 sz-7r+U· ~ - -i li ~:~iit
:-:::!":::= -:: .. ! : :..: - - -- ·- - -
:: ::r:::: :..: ::1 :::-: -- ·- .- -·- --- ----.- - - . . +· . -.. _.._ _ -- . -. . . - .. . .. - .. · · -· -- - . -- - .-.-
: :: : :: - . - - ··f · ·:: : : -: : :: · -- - - --.-
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! = ~.~~ ~:; ;;~L :==y-: i : ~L =~~;_:, _ Jn:~~ ~- ~1::~= ~-- ~ :::; > : · :1: .· : ~ · -: :_: ~ ; :-;.~~A< ~ -·= = - ~ ;: ! · ·· ~ ;~ ; _ ;~~~ J2_I ·· . •7 iJ\T f: ·· :· · I : . - :i : : ~::=r: : l
: : :: : : . = ~ ! . ~ -- • • 1 •• : : · · r · 1 • •· i . : , --i~~i·--~;- \~'0-=~~--- 1~·= ----···---r.+.-:- . ~ :'J .. • T . . .. ! ..... . ·_;_ :_;_ :_: :_·;_~_·: tj :_• -_; - .I :. ··· ·· : :;q~~_i •~1 -81\· . . : . l_~: : :: ~ :::: : ·.ii-• : u I - ::: : : . .. . . . : . .. . . : : . . . · . . . : ]}f .-~ : . . I : : : : .. . I . :1\:.] : : ,~I :: :
-- ~ _ .• _· ___ : ·. : _ _t_ .. : ;_. _: . - • ; . : : . -Yf~ : . : ; : . i . .• : \ : . j__ . i :V,~ : : . . -- • ..J.__ L-..C... · · . !, ___ L....:.....;... _ ------- ----o----- - - 1 - -- ---1~--T- --------- - ~ l :_:,: ... . .• ! : .... : . . ... : . .. : ·: L : i : ' • .• . • . I . : . • • •. : • i •. I ~ .
1) 4 2 0
Fig. 6.1 C- V, Q-V C haract eri~ tics Bia~ Voltage 1n V
78
6.3 I-V of the Fundamental
The input voltage, v~"' of the diode is sinusoidal and
at the fundamental frequency, but in actual fact it may
contain some of the harmonics. On the other hand the output
was at the fundamental frequency because it
was measured selectively using spectrum analyzer. The
voltage across the diode is the difference between the two
i.e. v~n - v1 assumed to be at the fundamental frequency.
The plots representing the fundamental V-I relationships
are shown in Figs. 6.2, 6.3, 6.4 and 6.5. When the diode is
driven higher the error of the measurements increases as
the assumption that the diode voltage is at the fundamental
frequency may not strictly hold. The curves at higher level
may be extrapolated.
6.4 Sampling Method Results
The fundamental drive frequency chosen for the
mesurements was 0.475 GHz in order to maintain the maximum
po~r
diodes
output
of the
from
same
the particular source used. Four
type, DC4303A GaAs, were examined
against the drive current up to 16 mA, for three different
bias conditions, i.e. -2, -4, -6 volts. Comparing the
amplitude spectra it can be seen that the changes due to
different biases do not vary significantly. It was decided,
therefore, that only one plot will be adequate to show the
diode behaviour.
The patterns for the second and third harmonics appear
.. I ;·: ) . ' • '
2.0
~~
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. !
!'l!li l!ji ·! ' P'Z' ' • . . .. , rrrfli: ll,!! ':i! ;:TII I j, : . !!,•.l•):i'il'i;!l'l! · .. , :!;: !.:'' ''. /: ;,r ., :. ·: . ';1,!:;1 :i'il' 1 j'' r I ,.: ,.j: .j: i'" l'illi ii lilll ''
:' :; ;• !' I' i" : ' • ' '1: :11: ii·i :·: !1: I -:. II ltl ~ · rn.'rl"'ililJ , .. .... ,,,. !'"v ... .. ild· ·· · .,. · ': '· : 'I' · ,, .,. ~ · , ·l , !fi ·::· 1:· 1· · ·1
, t mtH!i' 1 :: i::: ';· i'li"' ;• ,, . . • 'l '·m·:· ,_,, !''. :' ·:11··1 , !!. 11', 1 :'·' 1,. 1 ; ~~~ 1 : : . .. ·';jj4li / '! . ~ ~~~~ ?! .. :. ... .. -~r:Ht rj ~i ~; "ii· !:1,11, i!r I :. ', i: + !j~ ~1:'! jiTI ! -~~ !U.l I :, :: . ... ~ :r·l ' !: · ~ li''I'!Ji'i'liJ··· .. ijl lj . .'i·i ' ' •:!Jjj l' ll j!IJ I .:• !' :J 1:!' 1' : ; i! i .. . .· :· •·' _,!I i "; ; ,.: :!1! I ' t .... • • 1:: ;:;1 I ·i I. ,. ,, I I v. {*.,, . ,, i' 'i · , ;! · ' 11 .. [II , .• ! · 11r1u •· ·I,' " tlll' . .u1j ::, · ·'
.... / .. :. i . ... -l•i ,;:, ' :. . 1. :j-f ' tt!li ' ""- II [li! ·~~~!'!" t-'
~. ,,,,:::::::::: · .;~: ::it~!: i;i,:::' i!f1 ri! 1 il!: : ; ; .; : ;.::~ :,:,1 i ' !l·,.,:;:l: 1,· ·;. ii'i ·tw ~~ :+ ~: : , - ... .... : . .. .:: , .. i~·· ··. ··;i ,, ~~ ~ ~~ · : 'rr· ~w .. ~m+WMi:n+.t+4 /; ]· :: 1~ J!ii l! l! i!ii ,:;:' ..• :.: ... :1!! It!! 'I'' !J;: .;,: iijl ~~I I !!1: ;i;; /· 'I!! il!l~.ll :iii ~·;
~ . ;~; ~ ili ~4 Lt: !;t;k ... .... ... ;: .;J ~! :,li :ti ;;~: i I· ' I ;J. ',l: ·;; ;!i+ llill '' ~ t.
. !. i!ii l!!li :;j ;li, Jii: :: 1 , ::./:111 '!1 : , [ i!.1 :i! j I 1 !I ' .·1• ' , ' :!~: I !I!J 11 : '\1 . 1 I :: )11 1 iii :;: .11;: ' ii: . I :::: ji ,!;· ·jl ' ·! :. :!l j 1 1 '! ':I ! :!: J· ! 1., I! '! ,, ; I'' 'I''
;) i ·': t!ttfltii 'i< ~i'+fh:: ·· 1' · : : 'm tci itrl lp' ·; ::r:: . · :;, pi:;~ ' ,'. · · . ' .
1 ::q 11 1' !111 ;,;: ":1 i'1'IU · ·. · .... ,,, Hit w n~~ T !It! :I!! !I 11' " . : ·i tt ·1 'i 11'1 !li t
• 1
::,: :;-: ;':: :::: :: :· ;::l[;;F :. I ... , ... < :i~: . : ~' '1,' ·: ' : < ;tjj ~ Kf ~ .! ! ; · 1. :1
\ :4t : .j_!!i! !~., :j,: 'II i:l ; !! I ~~- ' ii;; "" 1'1' :· d I ·j' .1.: ''I' 1:, : ,!!! ' I II !'Il l
I I ' i .. .. :: !· ·· I f; , , ;; .:~ ! --~ ..... ' _, ,,,~ i ll! ''I' u; ~!. q . ' II!: w,: :!': w , II ' ~~ lU! . [ L~r-4~"-Tfi i: ;i :,ji i;!! : ! ' r " ~~~~ il~~~~ !jl. i! ;:~ :::: il:\! i1 ii1i !p! ~~ !; iJj l ji'IJlliij,'jJ UIH/: ... -+-:- ! . .:U.lliill.h:1;:;: lit' I UUill lUi~; ',i, ~llllli I ~L 'j !: .', !:;; ·UJ l!lliJ.Lli
0 1 3 4 ) 6
F ig.6.2 Diode Drive Voltage v Fundamental Curr<!nt l1 in rnA.
. . . !. I
··!
... I I
• r
····i
... , , : · .ll~~~ · :::; ::: ;il:l
.. :, '
0 2
Fig. 6.3 Diode Drive Voltage
I
I I
I. r .I + I-
I I.' I Y
l j
! : , ,
v )1' I l.·'! 1 ' I
'i i i'
;ji
l l :
3
v Fundamental Current
""!"" i ; i j ! ~ l :
.. .. 'I
I I.
I ·· I·
I
I ' 1 " j ! fl li~ ' "
-i i '
.. ,.[ ! ' '. i
!··
~ ~~ ... ~: :.
t; l
~J h ~--
.. . 1: ..
I •
. I I .. , .. .
. . I . . . . I . .
I .. i
.. I ! . ____ :t .. , ........... :
~ ·
I l
Iiiir:r (l. : 1 .. ~ " . •. I .. I I.
. wn;q :T:J: .. : : · ~ :r -~] • • r !
.. r
': : J. ·
. I
·::; ;;j:j;:: -!>1.:: !"-: h" : .!..: .:t:l.l ::l' :· :::: : ... . ! .
'"1'''' : ,~ : .. ,,,. j .... .. ; , :.1. I :
• : 'i' , · I : · 1 ,, ... ''"'""!"' ":·:: .. , .. , i!.r ':;::: :!·· :•1
.. ::
I 4 5 6
!1 in rnA •
• 80
{SWJ} 'i+JO/i.
-- - --- -\-~--~--~----------~·-_-_· -~--------~··-_·_- -4~~- ---N
If)
d I
\
c: QJ '':;1
LJ
11) ...... c: QJ
E 11)
"'0 c: ::J
u..
>
0 > CJ > '-0
-c: Ql .... .... ::J
LJ
>
Ql 01 11)
0 > Ql
> '-
0
81
to be similar for the four diodes as shown in Figs. 6.6,
6.11, 6.16 and 6.21. If the level is increased the input
pulse becomes narrower and most harmonics tend to occur in
the first lobe of the spectrum envelope. The theoretical
plot of the envelope is shown in Fig. 6.66. It was found
that the second harmonic has appeared in the first lobe for
all the used drive levels. Slight variations with bias are
observed for the second harmonic in all the diodes and the
plot drawn attempted to show the overall average tendency.
Similar behaviour was observed for the third harmonic,
which increased with increasing drive level, as shown in
Figs. 6.7, 6.12, 6.17 and 6.22. It was also noticeable that
the slope for the third harmonic in diodes N° 3 and 4 was
smaller than in diodes N° 1 and 2. The third harmonic for
all the diodes was of a lower level than the second
harmonic and still occurred in the first lobe.
The plots for the fourth harmonic amplitudes, in all
the diodes, had a dip close to around 8 rnA of drive current
as shown in Figs. 6.8, 6.13, 6.18 and 6.23. The amplitudes
were found to be lower than those of the second and the
third for all the diodes. The dip indicated that the fourth
harmonic below 8 rnA drive level occurs in the second lobe
and with increased drive level shifts into the first
lobe.
Similar pattern was also observed for the behaviour of
the fifth harmonic for each of the diodes. This is
presented in Figs. 6.9, 6.14, 6.19 and 6.24. The values of
i I ,, ~( ..
o• -10 i
en e ... en .....
-1o 1 1~
-102
-3 .,
-10 j I
I
I I
f
1-4; ,~o . 0
J
I
i
!
4 ~ :
Fig. 6.6 2nd Harmonic Amplitudes v Drive Level
... J.
... LI. I . I
·• ' I. I
::::t:: :: i ::
·I: r: ·. : ... 1·: I . . . . . I·. I ... ,~. I
I.
. ']'""
, ... f .....
:.:t::
. i I
·:-·r··-· I
. ... L.
~1 .m:
I I
1~~ . . . . ~ : : I . .. :r::::
~:~
.... , ... [:
... I ! I
"[l l· I. .
k. Q
o'
I
I I I
4
1; 2fiJ~~-! :*:f. 'LI ·- r I .. : I i .. :_,t.. •<:; i .. ~~. ;~c .. ,:f. ! .. : '- ---i-. : .. i -+· ·'+: ;~ I .: . I .
... ,.. . .. , .. r.,., ....... j .... ,...j, .: ii];;:IZ'l\1 I
:· --r·+- : .... --~liJ · --'i ~ ,:_, __ : -~~ 4 V ! (Vib] · . : ....
1: 1 . I
· · : · : i ;;..: 6 :VI I : ~-- :1 : I j'" :.::; .. . :. :!:::. ::::E::ll -:: ! · ·-·• · · · ' ·-· - t·- .... , ..... · ·- .., I · .. ·t··l · · _, __ ......... -· ·-· --f... --1 I
--~ - ·- l ·: ..... , I ~- ·, • - .. ! I . : ... ·: I I ..,. ·; . I ,.... ..,., ,_. ~
l·+j·. '''l"" ........ ; .. f :··:
- ! ., ~ , l -- .$ .. L . .J .. a .. i . I I
tl: ::·:J::~:::t : .l::::.::.t . .......... ). .~- ... I I .....
1
., .. 'j'''········•:;· I \ ... ·fit" ·r·--•::i : ~·· I~ 1: :: :
.. ~ ~ l : I -·-·-····~·-··
... 1
1 .I ··:+_~ I
. I --l .. · r_'---j !.•· .. t.-1-,. I i liiiili I! 1,;.,rr. I . ., .... I ! I . ·: :--· -+- ~Eifb'f-'+:-f
1•1 11! I
f6 .I
Fig. 6.7 3rd Harmonic Amplitudes v Drive Leva-l l1 in rnA
1.
00 N
:l ' .
o: '-10 :
"' E .... ~ ·-
"' ··-r1o11g
-2 -10 '
-3 -10
' 0
I+
,. -1 ---~ .
.... ! .. .
·t·---1:-t
..
•: j • J .J I
I I
r . t:. ~:jJ:
i ! +· .. , ... I ; ·: !"'
r:· ! -·
LL ...
:.!
: -· . ~ - ..
: I .,. !
I .. : . !. ..
··i .•. I I ... i •
I·
I -~r j.T :-j + i
l . I . ... 1
4
l ····j ·(
I :-- ~ - i I
i I
!
.. , ''1. • ···
-- · 1·.
i .
l ·.··· ·· ItT I I l l tf l [ :· I ··t· .
.. ' · ~ -J~ ' I ' 'I'' ~---~-+~ I
T
I
. ..
.::J I; ' .
! . r:·· I.,
I I·
··• i I· +j ·
I '
' ~ t ;
1l 1i' l l l' ~ I , , , ~ , )
l· .
I
·· : :i> · • . . :::-::· <r:: >1 : I : . '!'''! .. 1 •. • .• : ••. . •• ~·' · · - --: I .· :·
...... .......... ... --· ... I
:;·r r ; ~' . -- -~- - . .. . . . . .
. : I ~ . . I : 1
·· 1 • I I J2 , .. : . . 16 I
i · J ' '
Fig. 6.% 4rthHarmonic Amplitude!> v Drive Level '1 in rnA
.1 I
.i:L. . . . -•. }. ' . j . .. J .. .. ..... . + ·
, ... .. :. ·! ·· ...... i
I : : -~ - ·:--·-· ' .
. ·- ........ .
~ i ol ~ ~ ~~ . :J. I I
. :----1' . I , I - ~ -- -- .. . : I !. .
·· :I ~ I: I ,. , .. .J .. , ~ .,. . :
' "' .. ... , =· I ' ... ,I g 10 . ... .... : :: l - 1-
. . .j. . . ,.,_ . .... ' I . ! .. .
) . . I · t ' I"
' ., i ""!'"' .. ... . . i
. . . . I . . . .
: I
: ~ ~---;- : ··· /:' ' • • I ' ""'
' .. I + I . . i :: I . ' I .
. .. .. ! I . .. .. . . T ! I
!
1~2
I
:I ~ . 1
:_31' ' . ! 10 . II . i . : ~ / . :
:· :t
I
6 v : : \
\ ~t.
: ~ r- ::::1· ___ :: ·=t· :: .. : .J. --:: I . .. _ • ••• • • •·•·• -· J . ••• '
' • -~~~~ . :--:· ---~ --- :=r-+ . .: .... 1 .
t~----· · ·· 1 · 1 ~ 1
: .
I } t ... . .
~~:±~ - :-- I. : r:t::_ ::·,-· -T·-r·-:--·
'
·I
i !
j · · I
1·..... i
. I
i I 1
. , , . 1 ... , .. .
: : ... ' . i t I : . . . ~ I ' . , I :I 'i : ..... · ; . 11 I .. . I : I ; ; I : ;
I
I I
:~;~ -1.:..;+: 1--~ : . r ~ -.' ~- 1---. '
. I "T" " ' 1 .:. i \
.. ! ! +·· .,. :
I 16 0
I 4 a
Fig. 6.9 sth Harmonic Amp li tudes v Drive L~v~l
1'2 ' 11 in mA
0: "-
' ~
I 4
·' l t
-10°
-101·
-102
-1o3
I 0
"' · E ' c.... !
"' ..... i
0 >
I
I
I
i I
! •. • j. .
i .• •.
I:. i i
· ·· ··· t- I . I
·· I : I
•·· · . i .
;· I: : ., • 1 .....
I · I I
I I : · I . I
:~r : . , .. . ,
1.,. ·r 1·1
. ! .. . -- -1-- I ·-· ' '' ' I I . I I . : .
I
I 1 · · I :-.. ! :"I " I I '
i . I..
I .. .
1 : ·
I ' I
•
' I
I I I
.. , .. ..
I "!" . :
I ; .. ,.
I "'~- " ''"
I
i=a L=i · .... , ~, , + · I ... ,
.. ! .. ;i·, j' I . . I [. ' . I . . ~ .. - ..... .. I I .. : .... -: ... I I ! . ' . .
1. :: .' I .. , '' I
i ·, . : 8 4 I 1
Fig.6.10 6th Hannonic Amplitudes v Drive Level
·· r . ....
' I . . I . ! . . . .
. I . . . .
I . I
i •
I
, ... ,: .j , .. '
I I rj . 1 .. . .
-··· I· ! : ..
: 1l6J 11 in rnA_
I I I . •=t=-f'E"" :I~-J ...... 'I uaf:\s ~ara~~of .• . ~a~ 1 N-~ 1 :w~t ~r I ::::i=f+~r: : ... ~::=r:::~:: : ·
I . : .. L\~- ~ :=:L~;,):=~' ~:~ ·
·· ! ···• · · r· ··r· ·
I I . l d l~ \ · : · I .. . . I I · .
I i .• I I I . I I
' j _ l · I e 1
I I: I l 1
... I -1 0 . ro > . • . : J .. I
"!"' "
. . . i ' . ! . .
'-3 W I! : . , . I
· ····t · · · · • · · I
, .. ,, I .. l .J. ': ,.
I
I : ..
+ II
T
. I .
4
·· i·
I
I I
'
!
i
i
I J
I I
I :
·i
: ! • ; - ~+ --1 ·'+··+ 1.-- ': -~- .. r··: -- ·a·;--· '"'F''" . , ... 1.2 V-I-
I • l;i .... I , ,, t I i I
.. 1 .... : .... ~J\ ~at-~ -: "' 14V J
I : I ! b I ! !" I + I _:6 v I . ! : : . :1_ . : :1' :" ::. : .. : - : :!
I ' I I I l . f ...... , .. ' i I
. . ! ~ ,__. • ' . !
I ... I . I I I . • . T ' L- I ' .. : .::-:
I
. . .. .. . .. j I . . ... ...... .. .. : ... ... ' . • I l I i. : i + ' ' : -:~f-_ ~: :;J,
: ! I : - ~· .. +.. - -~-r--·-: I t t ··(• ' ·i I i-- 'i'
... .. I ~ _ __: __ ... :..- .....
I .. .. ... , ....... I .
... .. ....... . -. ---· -· . l
... ,.. -I I
t. I I I
I I
1 I I i , I I I
I ... ' l I
' , I
8 I . h I_ . I
Fig. 6.11 2nd Harmonic A171plitudes v Dr ive Lev!l 11 in rnA _
00 +:-
: I
-10°
v; · e ! .!: I ..., I ... ,
-101i lg
-102 .
-1o3:
I I 0
I
·r- ·t . . i .
.c:c:: l ::L~
I I
I ~ . . . ' i ~- - · t
l l : I
~
' . ,
I.
Fig .6.12 3rd Harmonic Amplitudes v Dr i ve Level
I -I
I I i ' 16 I_
11 in mA •
I>'Y']P l 9' qa :''I' '~IS /Ill' ~~ - l • : :::·f=~: _:~~- --;~:;~0=~~-:!~-~ I dp'I!Ja1 ' k ' I _,_:+··-·--4--1 __ :.., __ _ .... :-·~1 ~-$-+--····---1 I I · l··~·· ··--- .:.: ... j . ,
.. 1..
i .. ·r· · I
... ! ·
! •e l I C...
.... --~I) . . v- . I .: 111 0 . ·i-·, ··· .. .. 1' """ I
l-2 . }~ · :· .. !
···T··· ~- , ·-·
···!···· ' r ·
., .. 1 ....
": '.
; I ' '' '1' ''
I I
.:: ·!····· ·- ...
I
1 ·J ! I I I . . ! . .
I I · I .
.. I I r · . . ~-~ t ....
I I ! I II i
I 0 4 8
; • I :::.i :rJ~. ~.i::t+J ID·-··tkJ I I 2 . J "''~s '1'-- -T _ · · · .:- Vii
•. ""1.
Walt · T I :
tvbJ : ~- ··· :-\ 4'· V·
I . I . ' I
• I : I . ~ : 6 ·v ~ I . :I! .. :! .. -~~ : '~:: ~: :
:··r:'~t: . ••·· I :-r! +i·- :t
I I . ~-- I
I : I ' . I •-;
I
! ..... ! .. : .. \ .: ···t· I i I ·· ·; --1
r I .
... 1 ... i I : I I : ! . . i .'::::) ~:!:::1::::-.:~j:· :::2~ :- .. 1
. . . .... j' -··-·[- ··-~ +--; .... :..:11 . .. . .. . :~~:r-' +··: .. . ·· : - ···· --~L -- :· I
. . .. ... . . . -~- .. . ; .. . ... :
····t • I • . _ .. . :··· , ..... I I
I . t :·- '-y -·~·-· -·-:· I . I .. ··!· ... : .. :
I • J __ , R·--_,---·--r ~-- - -····-. . . '
1---:::;:~r
··r l · .... :~t-~:r-f-:_ ~r l r ···1 I •. I . I I . .. .-· -·-+ ; ... l : .... !.. ! ' I ' I
~ I 1 i I '
1
:
. ' . I I '
:~~=r· :=::: __ : :~-- . i=- .1¥,- ~--- .. : !t,JL ••·~··r{ ' ! :----, ..... ; ... !._._: i . :+:
_, ' .. L " 't' I I t : I
,t2 I I ! I 16
Fig. 6.13 4rth Harmon ic Amp titud~s v Dr i ve -L~v~ l ~~
Q)
Ul
~ v
. ., 1· 11 p .
{::i t .l 1
bl
- -·er-r ····,-lf'"t
,;,il!;;; j ,;;: j ,.,: l. : -- -
•••ll•••ltl qi l li' •llllll :i
:~:~. ~~ ~': 1' ' ' ' ' 1 ' ' !" "" "" 11 I j ' 11 'I" 1.; 1 I j , ;; , 1,,, I '' ; ;; :· 1· • "'' j ~ ll · •
~ -- Fl= ··r: · ; ~~ ' :iii i ' ::; ,' ' •ii i : • '! !L! ;,, ! ' ' !Ill lUi :'! ·!!! !11: 11 !1! i:l lil! i ii!II!I J :: 1' iii Hi! ' 'i! '
p; .j ;i rl ] •;; ,l!ll:l; rl·
:--- .. -~~~~-~ -, '' • ' .•. , : ',' , ·· i ; . I·· ·: ~~i~i: ,J ~~~~~~'ii! • · ••:n \ ' ~ ';:'[ , ::t . .. -i -c:+:-:'- ·-:- -'c--1- f-t··· - --~1,.. ~1. . .::. :::1· :: . .. .. : :: .. ::.:: :. - ~ · ·- -•-~rf!t. ~f:r.7jl!i • . .. ' ". ' .... , .. ... .... ' " '
I ... .. : : : ) : .... ·_ : .. ... ·I· 1 . - ~ .. j:::: :::: ::: ~v ~): :::: .... 1 .. :. .. . I .. .... .... :• J::: I
. .. . . ' I .. . .. . ... ...... '4'" .. . ( . 6V' .. . ' i I I : .. I ! I ··• L :: ::: ... : . . ~ i - . .. ·
-[ :~ : H~ ~~: • ::f-~f q ~t~ ~~fjt·: :: ... :• '' :.~~B~t~ ·-J~· ~S2~f~ ~ ·-·· .. " ... :-T·- ·-·-[_-·=t.· + .. :r __ .-.. . ·1- : . , .. . ·.·· r-.,... : .. , -·!-· ·-·-·-- - -~.,.,. ·+_ ·_· •
~ 10~ I I I ~ ! I
I -;;; )
E i .,. .. .
I ~ I 0
I . , ,.....
; 1-011 :::>
: i I I
i -21
' 10 ' i . !
. :·: ~:~ :=L: ·:: · ..w.~ -:l. :::t=: J~- ·· : -·•• ·••· ·••• · ~~- ·: J-: • =t=~t:~=F· : ... ... : . ,:.j. f-· ~+ ' '"i. ...( .. .. : ... [1 .. I.;CT"" ·-r- -~ .. ~ .J ... ! .. .,.! ... --j-· .. ,. . '
~-~ f· .. !" .:.L. ·t .) . -·: ~- ""f: .. _. : .. f- :.; ·· .. _J. . ... j . ..... / ··- + .. .... ; I ' ' i . ! I ! . ·· ·I .. ·i···· ... . I .. ··I·· i i . I : . I i : .j.. :: . ... I : . I l .. 1
.•••5E:~ ~~=I~f1,~ ~~~~:-:~24:1 _I~ttj. : I : I . . ' :'' . : i I . ..;- ""T· .... , ... w ...... .l . · .. I .... · 1-+-·-k t-""r ;...j. .. ..+- · ·- ·· · · · ~- ·-· · -· • :- - ~ . I , ·i··!~ v· " ! I ... r- .... [ .. I ··r· - ... .J. .... ,. ~- ,...~ t--r:_ -~· , ... ... ·-+-- --~·- . , • , 1 t i · :, . •. ; 'I : r , · ........... -f'-·l ... j ........ , .. .... . •· .. ... . I-·• .. --! ... /.. . ... f-L .... j ...... ...j . ... ,'
i ! : i : .... : i v ! i ! : . v· I - · I ~ : - t=7 ~~ , ·:::t_ -=:Lc~· · · ~: ~•1-:-:::f=:b. · •:::!::~1 · · ) : . :~•t :: ·:~:L~ I :
·-'~--- .... , . -· .... .. ·r-~-r=zr- . -.... I.. .. . .. , . ·- ·l _ ... + .. .
liE- ~ - ·-1 .. : ~ . • . .... i ... j·- . ... , .. .... ............. ' I . . . r- · ... r- ' . .. . , ... I ... + -.. , .. .. , ---r,··
.. ! . ·- ! t . ... .. T - . - ! .. .... l . I ' ·r·- .. . :. f-· i .. - j' .j. . -I .- ,. t--;· .. - . ! .. , --r-- L.. I . L ' I N I ' ' I I , I r.s,- ~·-pt· -r T " --r· . ,... ··-1-l "17 '"\ . "I .....
, , l f 1 • ·r 1 ' I ,/ .. l .. ·rF+-c- ·7 - 1- ... : ... - ·, · J f-T .. i .... r- ·-·b~t
i I I i I I I '1 I I I ' I f t= , . I I ' I -3 I ' ' ' I I ... I ' I 10 ' .... .. r,..,,.,.. - . ... -~ - ..... +.,-, ..... b - ;,.1 .. J . ..... ··r- fib - ' r ~ ... · "+ -·~ g · · I · i·- · .. / . ...... ;,... · "-+·'·t::E - 1 .. .. ... · ··1;,.-- ~"b=T-'-~ • f I t ····· T - +"- · · · H ·- .. ,. r-· ·-r · t-r-t- .. - .... -- b ~ " T" '
. . . ""ft'-1'-'-t-c . H·- .... . . ··f-EE ....... : ., .... !- - f- +- . ' t ' I r --+...,., ;ff-i--Fl=t--l . .. . . ... r- ~ ..... f-.. ..,. .._ PDf- . • I I I I I I J:·:: I
I . "T"" ·- . +-· 1--, -·'- ' :; I ' ·-p-, I ... - •
I I .... .. .. t-,-.f-,.j. .. . I ... ,_ ,- ... ·- -ki= - --- I I i I . . -4h ... -hi--+ . I .. I . f- .. ,.. . T . -~!·- -- ..... 1 ' """' h'r7;t-•· --;-' '
j ':'" ·' .. ; : I . . . ' t ' I I . ' .: , · : j ' -· --· ... ~- ~f:- .. !-',.-- r;.: ..... .. ;1 .. -+.. 1 ... - ·. . · · . ... ::·.·r· - _.. . J._ ,.,.. :"'_· --· .... 1 • 'j . I I .. ·" I .
: : : ·· • · · ·· I ... ~ I · • ·-·· ·1 ·· •· i ·· 1 r ·r· : b I . . . .. L ..... .L. ~l.. . ... : I l . __ L .. ~ 1'2 ..1 .. .... ! .. .. I .. . .. .. 1 16"'
Fig. 6.14 sth Harmonic Amplitudes v Drive LGVQI 11 in rnA_
~~;; ~_l~~~~::;~~i :;r: ;;. : :-: : , •••• !1\ ' :~ -: :} :. . . . . .. . . ~-... r-.--~-- . .., .... -+- . -r-.-T.. . . . Jl
;;: ~· I - ~~- t-··_· .... .. ... .... . ; .... 1. .. : 'I .... : ~~~~ ::, ... ' ' I· ..:ir_ ~ ·· .. .. . :... . : ·'- ._ ... !. .. : ·+ :.... ! f"' vc:l r •ollll~ : i .. ~ v1· .
I' . ' I ·f_ . I ! . . I VbL ·· ···· . : : ' . ' ' , +- -i 6 v
1
: ::: 1 :j • . J ·r-+· . - I ++·1-i ·· J · ,
T I ' I
1
1 T
' ' ·; ""!
_!1 : .: 4 ..... l....t... ---
Fig. 6.15 6th Harmonic Anplitude5 v Drive · Level l1 in rnA
l ao ! (])
i · I !~·~~ ~ H11 t~l
.. ~-~- ---
:-;;:~tf~~i7f- -~- -- r::-:: i.:::-r•""71"7"C '"·::!l:::t:i'l '' '''''' '''
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" • -r • ~· ·r· :;-- .... ·-·~ f--- t l l I I ·r- I I 1 t
T .· : · · ~ - - · ' f···+ ., ... :·! 1 !\,,~ ~~l~t~~~ ' ' l CI ' . '"';'·f'' - · --- - . r·. ·- ··-· ... . , .. --r-·'--T - ~ -.l ····•11Mf - --< '
i .. I' + ... i i l I . · - ~b 1 ~ 1-' ' I i ' ~,oo i ·- ·-··- .J ... _; .... +- ..,L. --- -~; ;, .. .. .... j: .... .. ..,...,r-c:-: r---•.. ~;,~ . --~~- ~ : ::~ __ Ll , ; : . .. .. I.----_ ( .. . . . c ··+··· -- ........ '·- .. .. ·-· . . . .....:. *'!-:-::· .. .. ...... I :± .. 1 . ... •
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, I ' · ·' - -"-" " ' · ·- .• . -'- ..... .... l.-. ··T·-'- --'-'-'---·- .. . '--'----· ···r· .. . -- - "". · ......... • . I I r-.: :1-:-r=~_r::: 7:~_ -f .. L . .... :....:: ·:-..::=':::!--: . .r. .. ~:.·.::.. ··_-r-___ :J .. - _ '•_ -: .... J, I I I • ---:,·-,-:w-- ·· ~~~E··r .. ··:r··· -·c', · -- ~· -- .. . : : .. ·j• . , .•• bi-- ,._, ______ ,; .• :- . ' I . ..... f.~ , .. ..... ··-" ·:· · ...... -- -, -- , -··· · ··.-·- · · '"'' : . · ... .. jJ.U..l:H-··+~-t·· --!4'- . ' ' b2 I 1·": . I , . ·· J 1:·.
i . . . -· ji-t .- ·t·' - -+- ... :-- 1--- -f-1-"- - f-- '-t r-+- ~ -·· - . -+- .. 1 .... • I ; . I I ! : ! ' : : : I _ ,~ · , -- ( -r ·;: --T ---l-- T -:- ~r .. LL ::: •••• : -- i ;- . ~ :·r--- T. :: ...• ~:- ----· ·
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-~ . I ~· . ; ·: ' I ' : ' ·~ · l ' ' -10 l I I . .... ----c .. ... . --:-· 1--+--- .. .. j .. l , ... :: ~:·~ .... , ..... ,. f- ··:t .. ·l-······· .... , .. .. .... ... .. +- -+·- ' • · ·· -~-- ., .- ···-r-- ---·f·-· -· ·· !·· ·· ----r~ - - -- ···, --- -.. ~. ... ·· ··t···· ···""j···· --1--- ·····r--· •
I I .. .. ···i-t--:-rt-'- - -~7·· ... ,. .... .... .. . . : .... --··: - -- ---·· • .. -· !-. ... I .. -- ··[ ·:· " "I' ...... , ...... .. T ... . .. . , .... •
I ... -·t ::-:t·"" ..... --+ I ; ... "'"' I" . +--: -· ...... :' -:·1-- .. , . .. !· " " ' ,·- ' 1 . ·-rl-7 ·· -: h-+ · ' I ·- .. . ;-· -- [Tlr· -- ~ ·· jm- .. ---t ...... , .. . ··-r ... ,t '
I 'I/ i [_~~;:: · ,:;;:~i~;:~A=~ -~~ : : ,:.: , ~~ .. +}~:1~: ~(' : 1
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"]" " :·;i "l'l,o [[li . I IJ •-•[ I! il .: ; ; :, :•· ' ' ;l '1 1[1 ''"''• I!' '!I'll' •' ;r, I'. ! io l l l' l ' ] I ... : :r : - : ~ ~ !:· . q~- :!~- :!:1 :: ·: .::;: . . 11 • • ; :L l.l. f· 1 .. : 1 :~ . . 1 •• 1 • :: 11: : ••
;.. 0:: .::; : : ;: : _. . . -· ·· · ··I-- 1 ':: ;" ,
1 _ _: ··-,-·:- -l,~ ....... ,
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010l 1 .:.. .. . -~~4 ·:l..'. ... .:... .:.:.... ........ t
8. t ..... L
121 .t ... . l. '- ;~ · · !J
Fig.6.16 2nd Harmonic Amplitudes v Drive Level 11 in mA -
' .. ·'•'i i'·'' ''I 'T l - .,, •·:i ':',j;: .. ..... .. "'
·· -"'-
1
. .l l 1m,:!:! • .. '.. I
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Fig. 6.17 3rd Harmonic Amplt+udes v Drive Level 11 in mA
. i'
-
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11AO I : '··· 1
. ·· · ·· . . .L -c·~- -~~ · :: :::: ,; : ' 11
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I It·.. ·----~--r-r-+-· .. """..! . .. .... ~ --~- ..• ,. .... .. .. . . ,.. .... .. . . ... . . - • : : ~ ·::_ .. ~ .·;·--· :cr~~~=sr- .. :::;.: ...... ! •. ·F · (~:T 4.~ · I . i 1 . . .. ·I . +8:-r.J ; r r ··=+t9':~r !•. 1 j '~ :l ~ ' ~: : ~: ! . ~ --!-J lll: I ~ ·····•·· . ""I : . I I ' I
~1~1 i ~- : :~-· :~-c--+: : - : I .. :d~ ::::L )-: .•• '· . . . . ... . .. i . _, ::~ :J.::: : . ! .. .. -~r+-· .c.J;...... r--~ - - .. + . --·-· ::t.. ... . --r-:t:- -·-' . r··+- 1 I
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. I I · I I ~ I I ,
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1 _ _;_ ____ :t . L_ .... ! .. 1 .. .... ----r-. +- - --+-- -1-- •-- '"T ..... r.
f: : ! i i i I . i ! ! r : I I . Ji I . I I' I I . . 1 I
: 31 , : . I . I . .1 • r- .. : _ .. ..... . : . .. . +.-- .. .j.--16 ' "I I .. --:r--s -· ' . ---r ·-r :! : ·-: . I =1= -=t-· _; - 1-- c- ~-- - . --- ·--t- .... ~ -l · . ! := ·~::-:-= =+:· : ~~:: :1?: : -; .... .. , - ::r=. =t:-:..· ..=·i_ .. :::1=~=~ ::· -· -·! = =r= ~+= :
I __ I __ ., . - . .. .. r' ~~-- · ~ · - - ·-~ -- ·- ... T , .-- j--- · - - ~ -r+i: --r • I b± I +- r---f--- ·-1-----T'-: ·--'-- ' I .... --~- . .. - -'---- - .... -- -·:· . , .. ... ·: I ., I I I I . I . I i .. . . ! __ . . . . .+- ....... f ! ., -- -; ·- - -~ - --· 11 ·-- . i - i - -f-- ·t -· •
i I ' : ! I I I I , , I -- ~--~·t· -- - -+- ':j - ~ - - . l ---r·--· - --- ... ,_ .. -,r-- - ~ -!--- - ! -I I I I j .... I . -- ·- - .. . . I . ---~ · ..... I . j ..... , ... 1-- . I' -- r-- .... ; . .. ·r·r--1 • i - ' i I. ! ' I
I ! ... , 'I · I .. : ··I ··· ! I : I I ' I I I . I I I ' J : . : _j_
4:J.. .~ .. ~-L _ .
0 .. I J ..... ... ___
1 .. I _ _ .... _ I. . .
1(i" I .. . ,
Fig. 6,1S 4rth Hannonic Amplitudes v Drive. Level 11 in rnA .,
Fig. 6.19 )th Harmonic Amplitude~ v Driva L ~:~vel 11 in rnA -
~ ,.
j;llll •1: 1 t 'l d . VI
r1
;·~; t~~~i~i if~ :: ~, , i, : ;; ':::: 'i:' :: : F~~;~l·~ : I r , , , :r 1 , + , , e:~l~b.;; 1 l · f' , __ ~- t -:- --1 · ·7 T -r-·· \h.~ · . -'
1
~- : - -t.~ ·--- - I·
I I I I I ' I ., - '~ I I 6V ' ' ' ' + 1-:_ o I I I • I ' I r' -- . I I I I -!-- I 1 · · · · _j, I •
'-10 I f. ·::.: =:!--: .:r--· ·--:r: - ---==-=~~::. - ·--- ·::r·'- -+-=-:.r= l ::-::.::-c:::- . :::'~- , . _ ·-- --·! _ -~ ---~·-: ·· -+ ~ .. ··+··· : .. --~-+- _ -· --- r----r -- ~--- .. ..:.. • ' ; I r -·;· ·j - ·--;- + "} · -· ··· -:j- .. · · ·-t-· '
! _ 1 .- :-:~- -~t -+= ~~:=r: --+~ -~- ~ · ;, ;: : , · ·· -- -· ~T=:=~ ~=-~ -- : ! - -I . 1- --;- --+- - !-c --f-..l.. . i - ---- ---,j-,J-, . . • .. - !--· ~.,;.t- ~ -- •
-e· -----: - --J·· -- . I. . .. -r- -- - - - -- r~ .. --- ~ . - -.- __ -1 -- """T"·+-- - --l-- -~--- ---- ---. I ~ I : I ·y- i ·· I . . . .... I
I -=. ! I : : .. - I . ! ' !4: ,; :[ ;::: •• T: ; . -· :: --·- I .. : ; • ' -. I I ---~_ - ,....t-+- - ----- -·r·---- - ~ - ---:·+-;--- - -- · - --!- - --r· -- ~1--- ----1
-- --- --- J -1 •
I 01'1 I I I I : I I I I . l I ' - ~ . , ' ' . I. ' I , ' I , I ' - i 0 : ' ' : i I I I I I I ~ 1011 :> ' . ...:. ..... :.+.. ·f-.+- • ··+ . 't-· -·-t· ... .t +-bl-~---+-f-- 1 ... ' ±--· +·j· I
- •· - i · - ~-t-7 · -:r~~·:=··-f -- c~ ~:c -:j:: ~s~r-- =::·::~~ 1 ~ _ :--::·=l~: : · _ ~~~ : - . r--- --r-- --,- -~- ---r ' ! -- r---r-- 1---- · .. ... , . .....,... 1---·t .
I_. ~ r-- -- · · -t i' · - ~ --- --t., f,~ - -- ~ - - --r-- -- -1 ---~ - j-·· -- -- • --'-1---· -t · : · -r· ·-r · · r- · I r-- -- -t --j --~-- , -: rJ: - ·
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I 0
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:1·· r- II JI:Irl r, , i! I + -·- -'?T·-- ' r :~ I I I I t -~· r- ~·~Mw.r 'I I I •
,.. ,. - -T- ... --- ..,...,. ...... - - ~~....,.. .. ...... r-~~f-i-f . I ' J ' :.1 i ' . ·' __ , . -rP~--' ---1 _ ~ --- ___ _ _ ~1-'"--t---r 1'-!'-'1---" ________ r'H _ •
- ! I - . - : I
• . -·~ -- , .: :.,! · ~··· · ,- : . · :: ;! !· , ' · ··-- ··• ' i~ ~· ··-~ 'I I , · ·- t-1~~ ·- .-. -•· -~ttr: '--- l : j;i ::;! :: .. ": l .. l - • - . . I hl-- .... rt-- --: - .• . ~ : .. . :-- 1;;: - : I
: . ··· -- ~ --.: ~~- ._, 1: .. ::. __ . --- -- t t'J -- ·_y:: :.: __ T - ~ ~:J17. ; ---- .. ,. --: ~ ·- t _, · I I - .. : . . 1 ... 4 ..:- J..:~ ~J- --- I ! .. .. .J I 2 I_ :_ ~.: ... i . ... ! b .. .. _ __/ ___ , '
r1 in mA Fig. 6.20 6th Harmonic Amplitud(!s v Drive Level
FF~i~a~ at: I or · dliTIB4 - N~ t.=+$~T~:~ -= ~~~~~~j- : ht+H-c:t.f>r-+ r r , ++T r rT -rg:::F·r-r:f::t:f.:. · ~ ~ ~,.. --~--~ - ~
I --r-- I I · - , ' I • -~ -
Fig. 6.21
- ~ ~-1~ · - . .
--
~,.,...--t-.....-r---f-~---1 2~-
U\ 'J!I'I"K~ :41.-+--4 4-iv·• ..
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----r ---- . .. , .. .l,_ .l. :.~ :~ :~r= : ··f ·F=l::l l I ~~ I ~.~ I I
--1·-·· .... I
· - - ~--
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/ _: I I
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2nd Harmonic Amplitudes v Dr ive -Level 11 in mA _
q:J '-!)
~- .. I ,, .
j " c:.:
I -.6 A ,-{i~J::~ 'tJ --,-~_,,-~·~rrr·ITI"'I"!'Ii!'WIT:.:m:E:R· • • •."···-·~F<001t0=~--.r_. -~;-·ls ~<=· '""':•:.:s-~~-: o : --~ .II ·I 1 ..... · ·• .... "" " ... "" ... ·" '" ... ···
........ ,. i .. :···
. I I o· I L1o : ! I ' .
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i I
i
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I I
v ~
I -~~ L,Q(!
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[4.;.·+·+····
t. 16~ " 4
Fig. 6.2 2 3rd Harmonic Amplitudt! s v Drivt! Level 11 in mA
... -·r····•·· i .... 1 ••.
. I
I I :1
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0 4
I
j . !
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I I ' .I
·'. .. 'I . :·-i-+--1
Ill --·· ·-r I ' I I
: I I ; I 1
i 1 L.l_ '_LL_ J .. 12
Fig.6.23 4rth Harmonic Amplitud2s v Drive· LovQl
I 16
~ in rnA _
' I
10 0
J
G .. A,·· ·• · .~r ..... -:-:: ... =~:. · ,.., 1 _:~ .... .. ·~.:· -~- ,:.:r'··· - ... ... ... .. . . , ..... · -·· ,-· . .. ·.:::..:...;_; , .~ ... . .,.. , . . .. . ..1 · n . ·r 1 . . .. . . • . . .• ,; .. ... . . . ... . . , . . . . • "'' '' I ..Jd. ,~ :··· · ut ! · · uu: 11:::: · · · .. - ~cut · II:!· · ··• · ·•· · ' · .. · ·· .... " · ·· 1
~ 100
I . ,. I ;t l
g_ l
-~±~ f SF ~r , F .=\= •• ····: ···•. f:~iQ•iji:b,~ .:± ~ ···~f---tt - - . --~ .. . . .1. -· .. .. .. . . . ...,.., . ... - ~ - f-41 -':"~;.,1 •t. .,.. ..... ,
I " i<' ::-r :: ,· ·: ~ " ·.· ) .:· [ ''i~-.
! ·· , ! · • I ! · 1- ¥r;;._~l1~~i 4~ J ' ~; : ' "
I ~,J L-l= ! 1
, :i , i l ,~ L~L:,~ec',: :~]- : · -1-r-t· · ···r ·· r- -.... . · _ .. _ .. -1·- - -··· - -..1 · • • - .. .. ..;... · - • -~ ·· ;·- · tLj- -,.- r- +· -1 · : . +-· ': . -- ~-+- ·-t·-· --1--- ~-- f .. , -:r~:l~ =t=;r .::! ·. · _L =t-· ···· ,. · · ,- ~ t· , • · .--.!. r-~ -+ : :
' . I I ' ! I ' I I I ' ' · --·r- -t·· ··t- f--j ·+· -: · [ - ;-- ·+· -1 ~~ - • .: • --~ -- - '
···r · :~J~ -t· f-- r·· · · ··: l t --r----.. ·-••· ·····q-.. -~•: .. · :-· --1'-;;-~r -·:j . ·r -- .. :.L . L -I ··· :
1
· · I · .... ! · · . ~ ~,.- r-- ··· .. , H"' ··1 - - ~-~ ·'·· --+·· .... ) ... : . I I ' I .. . ! : i .• I I. I ; I
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. ! 7 · · ~ ~ -· - · ·····.j :·····i l·· r· ·-······i ·· ! ~ff.!. - ,.... -·1 . . . . r\ .£--L, ..-!I- -·i .. -+.. . .. :.. ... ··· ·~· r,t-:- __ :. ... I ;~ ·-- . ·;: . . ::J.. 1 :: ' ····:_ :-~ :~r:::=•-:::.:::t:::.~ :-:- f=t:=:~ :!:~
V 1-- ---~. -.. -r- ·! . • ··· .. _ · :1 ··· -r·-···1 : ·· :-_ ---- r-:+~ .... !. ... •
i-1-·l- - t·· · ; 1 • • · · -+ t -· ·· r f-· 1· ·· ·-.. ~ --- --~-+~il.;,. fr ,.l · • · ~ :-+· '· ···I · ' t ·· ·· : ·: .. , _:-: ·· I -~:,.. ........ -~ ..•. ~ et:· ·- ,. ' .. .... .. . L. 1.. . . !· .. , 1 . . : . Pr .. 1~- .. ... -1·· .... . -+' .A -- ·
I ' I l ' I \: . . "j I I ., . I ' . h-- -- .. - -· ... .. . ; . ..,.,.,. .,1'·- ... .. ' ' .. ,. .. ' · I I . ' 'I .,. · t- "11 J ·. .. . ~ _ JL , + + ,
' / , .. ] I .I : I I .. 1 I I \' I
-il :b I I I I I., ... I - I . . . I I I , 1'
I I . . • . ' ' I I ' I· , .. ' .... 1o' : I ~::.~~ .:::~t:~t::: :::1_:~(: ::::t::: ..... ~'"t .:.: =r .. ·: :: :::: :::: ~" ::i: : I ' I . -f l.....; · ·rl~r-'-·· · . ·-r .. __ .~. . l-- ··-=H:...- --- .. . ,
1~~ ~ -1~ =1 ! r l: •. ···· .. · r~c: ;,. :J::':l~~ ~ / :
I I
I I 0
r -+.··J··· +_.--· -· ! , -j· . =r __ -_ : _. -. -_. f-.-- ~'· ·_: ·:·_ : 1_-::·_··.+_-.. . ~-~_ : __ ~"'~- . r··-·· · ··· j···- ·· ! t" ''"' -1= + -1 -t·-rH I ! ' . . .. : i ···l··· ···· [··· ·: 1.·· .'.{' II : .. ·t · ·_ . ·;; . . I
....... 4 . ·- .... ~ ; . - .- "12 -- - ·' ........ -- -16 . --· .
Fig. 6.24 5th Harmonic Amplitudes v Driva Lavel 11 in mA -
·~ ' I . 1. - : ... r· ·· ..
: · :i::~: •t i ··: ~rl .. L .L
!
' .
:_ ::q I I ~ ,.=! 1~ ··1
I .. '! .... i . ~
~f : . ... ! .... . . . . i . . . . ... , .. - .. ""["" ' ., ..
I .
i
I
I
:r--i ·I· ..... ..
I : :: .:-. I !
. r··
t ·t ~- ~~~: .... '' '1
I' I
!
o· -· I I :. I
I
I
I i
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I j.
·r
, ..
i'' I
8 i
.l- .
Fig.6.25 6th Harmonic Amp! i tudes v Ori vQ Lev al
'I
I I
IL
11 in mA _
92
the fifth harmonic amplitudes were slightly lower than
those of the fourth and there was a dip at around 10 mA
indicating the transition between the two lobes.
In the case of the sixth harmonic, presented in Figs.
6.10, 6.15, 6.20 and 6.25 it was found that two dips occur.
The first was between 5 and 6 mA and the second around 11
mA. This implies that the sixth harmonic at low drive
level occurs in the third lobe and then as the level is
increased moves into the second and finally into the first
lobe. The amplitudes are lower level than those of fifth
harmonic
The variations of the harmonic phases with respect to
the fundamental are presented in Figs. 6.26-6.45. If the
distorted waveform was symmetrical, this would suggests
that the relation between the fundamental and the harmonics
can only be 0 or 180 degrees, as only cosine components
would be present. It, however, the waveshape was
asymmetrical the phases would be within 90 degrees and sine
and cosine components would be present.
In the case of the first diode, the second harmonic
relative phase was found to increase with the drive level,
then had a peak and later decreased. On the other hand, for
the second diode, it started to increase and then remained
at the same level as shown in Fig. 6.31. The third and
fourth diodes behaved in a similar manner as in Figs. 6.36
and 6.41.
The remaining harmonics behaved in the same way and
c 0
,,
,~' II •1.1 t;,, J::l (,;o
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11 ;! 1 lid
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1
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1 [1[ 1· 111
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60 H · ·· · .. 1' 1 1 ... < .•• I ~! ... ::, . , : ::c ' ', 1' ~~ '!4 !' ' 11 !lri 1 ~ 1 ~ 1 .I' I' i 1 j: ll ·l ifi tl ,l • · ·= : ... 1 :·1!1: · '. ·: . I r:d · ... :: · .... ; , :::: · I · ·~ · T~ rrr il· l t .l : ~ · · ; · : : ~ · ! ; 1..
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1
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, !11 : II I " ' l'· r 1r ' · · ' 1 ~ c I'· · · · r• i 111 ' .. ~ · •1 1 1"
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1 , rl, lil : I r 1!
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..
1
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1
.,,i "! ,,, , .. H ·; .. ;i·~~ .. trt r-1 ~. 'iiJ'· .; , · "'I '1
-1, , ·-1,· .,.: r· .-1
1 I 1 ! · r . I .. . .. ' ~" ! "•I 1 .... ", I. ' II I J , I I I l l ' I n il l, I I . i :r jjj : 11 il~' iili llll!ilY lrl:::l I :!; ,;,, I'' I, ), I !i jqll!llll 1 ,1 , ' li I ill ,L· :;. ,11, II' I I mi
- -~ · - ···· ·:·: ::i : i ll ~ :1 1 ~: i;li :;1i ~ - :1~1 ~ 1 · 1,, :· 1 ~y 1'!! i~~ ·:· ~: .~ ~-: I; ::. ith :;1: I 111 :. ;! .!! . ::: · :~n . · I , 1 . · · ' :: : :~' '1' p: I: !' ":' i"l dti llw ·· :' .. , ,· ~ · ,.,i !1:';':: :w! !i! ;r:. ) r 1 I :~I .1, : ,,:, ~ 1 ,. 1'11 !l: 1!!
1.:1
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I ... '[' r·l 1 .. . ill' , ... I·' !:! i: I ' ' 'I! 'I !I I' ill Ill rml ' I ' lj 'I I I mn I . : ~; lr .ii !:J ;:;: :':' !il: 1;11 II:: : .: ,• ' ,, ·, 'I' 1!1' ·, II! :!.· :l!.d! ! , lj l: '/ i I !,jl ! 'I I ' ... , .. '" I .. . ,. I' ' 'I· . ' ' 'I I I I I :1· II I I' I I I, I ', lij: " II I i . I I ' . ..:.. f!+ii§ ::. : ·' ·'1'r.ri/f' .. I ' ~I.L ~~ ' I ·I' Iii I ' l ., ,I I ' ,, ' I i 'il : !
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1': liil 'Ji ... . .. . ··: :i j :!;~ llij ;;!; :!:: ,;L jJ: i.; .::. . ... , .: ..... ·'WH 'j! l i ' I ' ~ H~ ~[~ ' 14] i![ . ;!11ltlr \ ii i: @
I . . !: :~ i: li:: :J! ;s ·t:: lL~ , .... , ...... ~ ~~: ~~~ ~:: :!:: 1![ 2 iCI ~t: : i! !i :1: >: z :li: ii', :'m~ ·lii
t I :: :: lilltl!!r '"' '"· :;1 : ·l'' :! . ' . : .!· r: .fiiln, tlj' l '"' I , IJ''rl•~~~ :p . rH lrli l!jl! I ljl 'l r · : : : :1 1 1 i !t ~ ! : ! i : : : 1 i ; : ; , ! 1 , , 1 , r . i : : ; l1 ! I jl 1 i : , j i , :j 111 1 , ! I " : ' ! . ' Ill ! 1 1' j 1 j ! J. 1 I j ·: t'il:@'"j' ,.,. :: :: '" · ,r:: ,.. , 1'·:· 1>· ,1 :: tlr : IP ':. ''I II' II :' 1: ,,, lj l Il l It ' , tll· lll, . .... .. . ·:: .qll , ;,1 ",. ' ... . " . , ... . 1·'- ' h· f~ " J! " . " ,:· ' ' •: : ·f:4 •:i " ' ', I , ' Ji l· ! .:,; i' l' I I 1::. ::1: : !11 I'" . 1:. :· . . , c; ;: ~ 1 t•!J rij '1 ' 1:j' ', ,f: II jill f: d I I ' ·ill l'i
[ ·. : u:!~!i iJ UUJ!L ill: ilL ;, J , '; . ,i ]ili tiUll!ii!lli i~f l!l' 'I!; ilii 1)1: rW rut r! 1 ]ilii!Jlli!
I
1- 0
11 in rnA -Fig. 6.26 2nd Harmonic w.r.t. Fundamental Phase Relationship of
~' ' r , rn·l n1~· : :~;·· ~ ~·;~,, ,., , ,,E., :1.: " ~· -r .I 1' :·1!r :.
1r1
iii;;:' ui·1' ;;I 1· ,! i' '8; , e 1·
. 11 I ..._ ~ ' ! . ! uJ -! : I ~ :' ' ! !I ' :'Fl. I ... : ,,, . .
I I i :· :. '1". \ ; ... · ; : ·. .. . .. ' :. · .. 1· .... ... , .. . .
I • I I.. . . I ' I I '" ' : •. ·i: · .. : I • ' III I I
.. ·'·! : :+· ... ;_ .. : .. I ·~~:.!~! ~j ~ l
. . : , .... :., ,.. I · ~' ' · , . . vLI ·' ::1·· I '· I . ·: .... I I .... "' ··· : 1:: .. ... : '. ·, ; i .. :L .. ' ... ! . I ' I
, i . II .IL.. .;,f ... [ l ... ;" . '· .. -.II .. , ... .. !· ' ! '::j:.L Jr: . · '· .j : ..... ~i···l .. ·j···· ..:.{,' ' j I
: ,. ' I ' , ' ; , I ! ·1 : 1 .. . ~-. :L I : ,
. . . . ' I' I I:!·' I . r ... j I· .. f ... _, .,· .....
\I ~ -\ I" ·.}:;l:u~~ \r1
! -t· 1... ·t·- .J.. i ... I ... . I I ' I ' I : I' , .. " '+- I· ' .... I ~ . l • ' I I I I I • i ... , .... r· .. ·. ·.- ·- . .,.: .. ..
. 'I' L i . ... . .• .• j • .
i :. :~~ +. •·:: t t:l :l:l :: ::l i I . I.
I .r..
I I .
i
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·r
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il'
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Fig. 6.27 Phase Rglationship of 3rd Harrronic w.r.t. Fundamenta l
11 in rnA _
'F'i-~ •I! I i -: r:i'l t:l
:!:. •: ' l~ i ::,, :li 'H' : ' lji! tf I -i: liUi !H i : j,~l~ ! !: i! I I II ' '.:1 ill! , ' I ... ,1,., .... ,. ,,, , ... !" "fl , I ... . , , Jl l' '~" 'I I r n i~f l . nl :::: •:: llJi ' I::OS :,. j:(j ff ~~ " ; '~I I J ! :ii: : ' I ' 1: : llll I i IT I
r. ·' ,1;., ; .. . ,1 ,_ ... :. !: ·t11i'' 11' ·I 1 r I!Ji j'r 1- '·' - ·i l!' I , - ,.,,, 1 • . ';I; r• 1 1 ,. -
1-, ::: ~ :''' , ·' ,: ~L ' , •. , , ,, :'. n
1nh1 1 1- :, "~~~i 1 1 . . "
1 .r 11• 1 ,1, I· , : '" " ' . ,, " ,,r. ii: : 1 11" -111 " !1: ,, ., I .1 1,. ,._ .... ·. :; i pj: ' .-·w· 11 ' 1 Jl' r;:: ~:: .. ·; -~ "";' 'ii I I ri ' i i ,; :;r!: I 'l :' ' :.f il i 'I '
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Fig.6.30 Phase Relationship of 6th Harmonic wrt Fundamental 11 in mA ..
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Fig. 6.31 Phase Relationship of zr.d Harmo nic w.r.t. Fundamenta l 11 in mA -
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Fig. 6.32 Phase Relationship of id Harmonic w.r.t. Fundamental
11 in rnA -
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F ig.6.34 Pha!.e Relationship of 5th Harmonic w.r. t. Fundamental -
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Fig. 6.35 Phase Rli!lation!.hip of 6•h Harmonic ~t. Fundamental 11 in mA - -
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: : ' ;: : · :1,:n r:: :: :~ r l · ' 'l ,r, '·r ! rill !'i 1 1 1 :rl,:: q[: 'I ,r n1 1: i ·Iii !' l''ll ; .. _,,. S ·'r ·:•·,.-·· ' -. ~ ~.-- ' . .. :~. ,. 1 ·r, ,,,. ' " . ,, .. lj·i ljll' . r rl 'l/.1 · rl',rrlW ' lj ;~ •- I• ' ' ' !: . , , I ' rl " I'l l·· , ' '~ ·r , : '·: i. , I ' , ! . ~ , , ,, , r ... .. , r,, .. , ;- ,, ~ ,, - .. , ...... .... .... ·" ,J. "= ~·· 'I '· r ·~ t'·· fS.. .. ·'"" . . + : : ~ 1 : : . ' I I rl I I I ; I I I 1 l I : I I I J : I I : • I t I ' I I II I I I I I
-• ,. 1: ·: : :; :'' J ' i···· .· · I' · :!I ':i· 1: : :l'r r·': ·q It! , .. . :: :t :· ... . II !'Jr ,jl q!: ·· ···· ' i ·'' IH :c: ;jJ ··· ·· · .... ' ·' :.:. :. !; ~l if:ffiij: .;+l: 1! ~ P1 4;if4: \ ,:"· :1 1
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l~ ., - ~-"' ... , l ~ . j L;.1
: . : .... ! . . :~ ; · :;t~[ W,r ~i ;J; ' j!;~ ; : M :!; ·+:) J,li I:: :!!! 1!1! ;I!: : ~ ,,) :; :; :;L . .. I <;:: : : :~~: I : ·_I ·: : . .. ~. :~ ! i!li ;;;! ll! t ~ :' ::: : !i!i ~L~;: .1:: :iii 11' ! 1/ ~ 1! : 'i: !§o i l'i;: li ; I: :; : : . ·I I . I I il!· Ji! ! IJ'I · ,ft11· : ~h ,!1 ':' :H! !!: I !'' II i !'I t....-
0 ;:' :;: rl:.i ·' li'i !!:: i'' ·:': · .. , .... t 1 td,:~. J:: 1 :r· · 1:, Jti" ·'l l :; ·i Pl'i l"l 1 ii!l '· r- I : U ,, · .:: ' ' I':.: - - ·, , . 1 l :1: '· .. , " I lrl It" li ·J 1 g 11 : ; :1 11 11
~~ I .. lm ;:·:•j : .. ·::-: !; ;! :' .. f.' :: '·:': . .... ; . ·-,: r;:· 1 I[: i tm ~tt~ ::I ;.;: rr1 i"' ' rl! ~ ~~. f ;, Ill! I r 1 , ,:1 ::;' j:1 : •, . · · ·:: 1
r: 1! :n! l ,l' '·; :! i!i[l· ·. Jr ' ·r: !l:~i 'I 1il !IJ! II11' ;.;;; j ,: .. .. ::: · ... :. 1'--:· :,. :.. .. .. . ' ... ,,, . I t·u 'I· :,., .lj. jlr ' !1 '"' .;,; . ,:. jl.·' r ft+tlllil • .,.. , . ' . : 1· : . 1 • 1 I . r I' 1 '· I ,
1 '1 i 1 •1 111 :r :
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I .. ~ It I ' I I ~ 11' I I t I I : I. I ' I I ) I t
1 · '' .. ' I· J .•. v( I J · i!· !• i:·' ; ~ ;t~~~~" :: ; ~~ ~~ ~:;::jl
~I '?~ d •·· r: ;: ~ « •l ·T 'TI ffi, fli !if !!' ~!! m· .~~ ) ;: ~~ ~~l *' ~~~ -- •:\ :!!, ;; -- :f ' : ; · ~~ ·-·: ·: ,:: :.q, , ~ : ~: j:: ~~: ~ :: ::r : i~~ ~~ ~ H ~ ~ ;: ~w ~:; ~~ : , ~ :: · . ~ ~
:; :: '!! . ; l i ! ~- · :_·. ..•. • . . • .• ,, ·.· / :: :: !lii !!!! :: : :::: :i:' iii! .• i!!] :::; ~; i ! ri'! ! i ' i ili ll1l!: : · · ~ 18J,m : · : ,j;;l~~i "' ' ~ - · ,, . ·'" .. "' 'mTI T!Ti 'T iT' .,,!ii i• :; •:Hr 'rr mml i'!l lflii
-•r " .. ~: .. :: ... ... :•· ~Y~ ' ;L ...... : . "-f ij [! ul! lu1 !g! !!: !!]! !!L i ~ . JU~ ! l.lt @, . ~ : ~-~ ' " I . '~. ~ · ·•• :;:, '' ··· ·· · 1· 1 'i 1
, ,, , :: : : :y: : · ~ ~ ~~: :r :: •• ::~ :!!: I iit ·! iri /::i :i ii 1!11
I .::: '1"' !i-. . : ,::, .,: "I .. ,; !"!! ,,,, .: , ; 1: 1: di: ri:i , .. 'T ,,., ' ij rr18:· ti#~f
i i i:: ... •· ·. t I .. · ~ ;~! 'i[ !' .•• ''ii :· ! ' ' ' ,, " I'" "" "'I'
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f;p I : i i ~ ~j ~ ~~: ~ _! -'l, ,: .· ,1
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I '·t·~~ :L,:·.·:.- .. l. :::l: ,. :': :.J! lllr r".: ·- · :~: !·~. :,;.1,:.1·_ . __ , ,· , · !+~~~~~~~~:,~itill~ji: !,[ ! I . .... I I" , I ·I ' I :i •I·· . . [ .. .. :!Ill [ I I I .II 1 :~--'-'- _.::_, :,_c : ~ .. i:' iil "cc :,;~j · ' " !.'1 1l: .. :i ~ ~ : ~l~fl !J,i "iii!
F ig.6.44 Ph a sa Ra lationship of 5th Harmonic w. r. t. F u ndaman tal 11 in mA
t!lli lil jt ,r ;: :]•! .:! !r il; , 11!1 i 1 rr: '" "TI'TG' : -- .. . , .. .. ·:: ':":. i; ; r!i'1i1!, li ;;:: .• • 'i i !·;:'I 'if: iil l !i ! i Wl' ;:. ::: '. I tltl rf! ilffil i WU~' 11 ~.'11'-i.! l!.iii ! l!:.;., ~ - ;: ~ ~ ·x .. ~ -- '• :i!l i ·::; jil:
11h •! :• :li1 :: .i ii rP' I ]'! I !:. dl: ;f' l ;t,! I , '" , ~ , ~mma .ntut:·. - .di ,,~n.~ I ,,, .+<:14. ;.J.Jug;Af!j;, ''"e : ,., . ,!! , , :1 r, ,, , ,! ,:rl 1: "" :"' 1 IJ! JIIIIIr
1·! ! ; :, !i ,1, I!'' !j ' r: ,.;· 'i•' :t • ,:• , ::: 1 -,,:: ·l!l jl''! iii! rt;i ::~ ;•;: ' 'li :qr ''; ·ih:lir' 'i: .fi! !': 1
!m itt!! I ' I : 1·11-nl I" l'i' . ~ 1 : ·!: · .. , . :· ' ' .I .. . I. [" .. :: : .. : ;.; :: fi' \!:: ::1 : "" '"I 'i-t' '1-H t''"Hil+ lii l f!! ·;:: i: , I I' II IJ44li: Ii i. !~ -l~ Ur.; !! U!: / :: j_ :< .. .. .... : ; ... ]: ·i ii i! I !!!: ~! . L: ji ' , !!!' . I ::r' tli! :::: I ' ,';i '!li i :·,J: ; .. '
utn ,m v ;i::tm ji i ~ ~~ ~ :1:i ;::: ~··· : . , , ~ 1 1. 1. ''i' ii!! ,;,, !ii '' '~' ' Ti' i! i' !J ij!JI ii i " · ,, ~~ , ,., ~: 'f .
, , ' ' I . : : . : : ! ... . .... ! .... h~- f~L.t ,f; :'., ~.~ .. ---Vl .. :
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(
pt ~ i'' ' ··i1 r: · t: . U:~ ;~, I .,.;.. ::-:f. ·· mrr:: ... '~ : .... ·l iL l~ i!r ~:; ::c '"'~I"' ... •'. ': .-- - ~- --- , ' : ol , I ~ ~ \ ~ I :: ! ' ' .1 · W:. · · · · 1
1 • I • ·" • · " · • " · " ·
::I i.l . Iii.' ·.· ; .,... · .• ~ ~. . . . .::: '·: ·.~'. i ' i':: ,: ... ' ·.··.· . :' il" :.-.•. ' · •. · · ·· t'.•''.~ · .. ;ii .. .. ':': ..... ' li ki! I .... !"'"" ' . :' I' . :! · ...•. , . , ., II: ! : . I l "' " ~n :'·:' · ... ·· ( 1: :' -··· .... '1- i'IT '-··r:-- ... .. .. , - ,~f:,.C. M ;-:: ---·--!-- --
,; ·· · I· i i · ·· . ! •. ,.. .. . :; +-- .. . .. I " ... :: :' :· .. .. . ·i · . . : ... .. ,. ... ·:+-- ""!'" I
·; i i ;· 1-- "" ! "· ··. I·· lg·· I I ·:'·· ' .. ..... I ·· ·'1 '. I r .... . : ... ,. '--I::·-· ·H · :-·:.l-- -- -- · :· I': . I"' . : : I I ·:' . I ... :. ;' ' ! l . :·I : ::r ' ., . . . .. . ·h "]· : ... - ~ - ...... J,,. ·'-!--- ----~ -,.:. S· .. , --1 -: I : ' . ·:::: . :::: ·. . ·• ::: li .. . I :· I-- I I" ~ ... ........ I. I. ····!· .... .... ""!". "" "1 I . L I : . ! I 'J '" ... :' . . .::. ; ·~· . ()"' . 4 '· .. : .. ...: ,;; . . 12! ~_;_t=·; :_ - .:c ::L.L .. , 0 ...
11 in mA Fig.6.45 Phasa Re.lationship of 6th Harmon ic w.r.t. Fundaman tal -
103
most stayed in the area betweeen -10 and +45 degrees.
6.5 Spectru~ Analyzer Method Results
It was mentioned in chapter 5 that there was a need to
confirm the validity of the sampling method. To achieve
this spectrum analyzer measurements were used. The results
obtained using this method were carried out under the same
circuit conditions and are presented in Figs. 6.46-6.65.
However, the verification could not have been complete
since spectrum analyzer can only measure harmonic
amplitudes and can determine their relative phases. The
resulting graphs show a reasonable agreement between the
measurements obtained using sampling method and spectrum
analyzer. This confirmation justifies the use and the
application of the sampling method.
The shapes of the plots for different harmonics of the
diodes were similar to the once obtained in the sampling
method. Point by point comparisons has indicated in many
instances a very close agreement between the two
measurements as expected. The positions of the dips or the
transition point for higher harmonics have also occurred at
similar drive levels. All the harmonic amplitudes are
presented in Figs. 6.46-6.65.
6.6 Discussion
An attempt has been made to give a simple and useful
picture of the behaviour of varactor diodes which may
1·:1 j,l,; to'
-T-..,.,-,..-:-~ _;_j' r- r-r-rr_:: -=t- "'1 : - -- T::I + :- ·a--1..,;---r-->~ ::_ ···r ' . . . ' ~ = : .. ' .. . ! ·ti .... t. ' .- .:.:r: L .- +--r: .... . . ' ... , ,"'. J ·. -4 '"'LL ""' ~c. ~·• •- ·-rr•·• , ,. ,,.. . .. . ,., •, , .-_[Jr . .il:l. . . . ~·'"-"· , , , J 1.. + ··"'Cr !.Ja.B;, ""'~ . , ·· • ! I . 1 j · 1, ' .. ,_ .. , • I .. H . -TC ·o:-· ' ;I 'I LJ:t r f'ij . i : . I ~ . :q: . :-!'<' ~ -r~ ,.,.:= ~~IT+- '
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' ::: + ' .... ' : ":' = . : .;k··i ... ~--- -p. ' --< ' · .... • ··· · · l·, , I !. · . .. .. ··· .. ..,,l I . , ' ' .. : ... ~,rT ::+ .. :co;-. . .... I
I _·.· · ···~·- ·~~·~· · · · ··~~~·····S~":•21 ;•~ ~1P~,···· !ili~· :;::~ . V " ~+. - -- ""' F_T I tt! .. . ,. 'llH !II i !. ~ I:IL d II' f+li ]1 '' •. .... ..,.,..~>,.·: • 11-'< . ---~ . ...... ; . . : .· :;: :•:1 ::;, hi: •) i!)f.'I Ht ''T' ' :•T ! II !, r:r. :.:.~ ' ! -- ··---r ' ... :'t" :::: ':T' •• ·I ...... ' .•• .. ,, ! ,,, .'" ~+~+ ;::-T- I I' ' : :: ::i; !. .
' · 1 !ti !:J. !' · • • ·. . .. ·· ··· .... -- · ·. • · • · • i ·.. . -· ·1·· r :1:: ·r!.l •· ..
r-t-1" ·;c; - r 1 : I ...• .... J 'LL+ '-~- r1 : · . . , I . . 1 .... .,_,,_.,. ;==f: I"C; . . T . ., . . ___ '
;J:./ t P ..• [C:l , i ~( f:.- ·~~ .d?;~· •. ·• ~~~ : :·i : I , I
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Fig. 6.46
t=3;E~ ! il--~~~;- i?~~:··~~~~~t£ : --~- IS]~· :._.; ;1.J_ ··· ~ t _ I i·; ,.·. · .. 11:: i•!: : : !li ii:; i: i :j:[ il;i .::! ijl: · ---- ~ ~~- ~lf~ ····· -_ --~ · --r--~ ·-·· ·· · ···~+: ~ · . ''! , ···•• ., .•... 1, ··· L. . .L, • .. ·•· L . J • L- - I 1 1? -=~-, ·-~~6 mA
. .. .lL . - - • I I --4 -
2ndHarmonic Amplitude~ v Drive LQv!!l
T I I
I
·\ll--..rlr4-rlr~·::4ad¢..Jf;.l;:+~¢±o~q~Iij~~Y=t=i
---~ · -· : --~-- , -r · j -: .. .. .. ! ........... .
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~-:ft J_,: :r~ ~T- - -·r ... . ... I
j:J·;r··. :rq- ·r _L _ i .. 1 __ !_ .I ... .. _ L ..
0 4
Fig. 6.47 3rd Harmonic Ampl itudes v Dr iv Q L·QvQl 11 in rnA -
IV:,
. ~ =· = ~H '=-= =.::v >:-:-: c ·1 I : ~=~· · · = t:: -- ii.:,. i.+ i'O"' •. '-... __ '. ~ .· · .. ·. ·11. i : t :· · ... ,!. --~~· · · · :,:. ·1 .. :- .- :.· ~ ..
1! •.·. t .. _! .. _ .. - :, . lc '
1 • . • • ,· ·. ; ·<<==i·· •:: :: • :1 I -· - · - ! ! i - - · "
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:1
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Fig . 6.50
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I ' I .. .... . . I • '·· •.
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•· I 1 '' I I !· ' : .. , I I ., '"' ... ,
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.... ~ ~4lL. .. : ·~~ ... ... . - .... .. L~-- . ..:: :1 :: li :; ~..:.: Ji 2 ..... :: . ·~ r-~ :::: ~ :
6th Harmonic Amplitudes 11 in mA -v Drive Lave l
0
I~ ! ... Ill , ......
·o ::;> .
4
..... r ... :·- · .. . ;
i -·· i I '_j_ ' ' . t • .. .: ...
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1-\ ~~ ~ :y : . 'J '· .... .L J :
! , i'~ ' 1 ••• • ~ · :j I : ... +.l-:t i:
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:::r~~ ~ ' ! " I L .. ·+· .... : .. j. L.t-.. 1.- H-:--:- . ......L . ... L. .. I
l·' i ! := -:!. . . _: : ~--H- . ; I : .. :"<: :. ' i I .. ; I ' .. . I I _!_f-.. 1- '~l ·, J 1 ; ::ilT .. . I +-:~--~- ~ : . :. ,.... . . ~ ............ .. - , ..
+ I
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+
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I I I ' I
i ::::: ::::::::['- : j ~=bi--§~2:::::-~ - : -· . I .:... ... ·: ··· ... .: · j ·· -i=h···.r .. -· -:::·\: ... : j r:;=j=:::: :~ L 'l ' ;I . ·i · j~:~--+· ... 4 .. 1 . : .. ·-~ - 1, • I 1 .!...: .. I I .· 1
• ·.• I
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I ' 1 ---- ... . ... ::~ · . :: ': .i .. --~---rl .. I Q)
I l -f-.~:· . :.... . I ... .. • . • . . • -.. ~-· ... ; .... 1 . . 1 . . + - ... ·• .·: .. T -r· ... r . 1 • · -'- - ... -] - · ! r·+~p..:...- ·+ I' t
-r· r=::ct-·t:~:t I ··· · · ·••• ~:: :=:::: ··
- ..i- _._ - -~ · -1 :· · : I · ! ! .. : I :
.... ··- ··-h·7'"t--,. -~ --t··· 1• 'I I II I 'I ' I ,, • --r-· " "'I I ' r T ..... .. · .. . !... • ·l-H · r· ' ·· · t: ·t · r· .. .. L . I •.. -+ ........... 'I .• · . .... _ , I i I . .. ..,.::: .. , ....... I
. . 1 . ., ' I . . I .: :
·-= I ~ ~ - l:: ~- f.:.. f.- .. . ' :··· ~~t.:: I t-- ·-- .. --1 . -1-- -· .. .,..r-+rt .. - • I
.. ,.,_I - ,- . . . ; I +-f,..c '··· . ·~· ' : I · · I • I·· • . .I . .. t:·f- - I .. I ·:" i
... ... ~ . ~ . .. . ·· I' -+ ... · ·· o~···· ·· · · · : : l ::·, ,. --+--·1 r ! · :or , ··· .... ·· ··· ···· .. : .:: · · · I ' ··l·+· .J .· I . -+·· . : ... . . . . I . --, I I · ··•· ·. · ·-~···· : ·t·~: ~ ::: : ....... ·-··n·l · 1 T ":, ! ,, .. :: : :: , ::, '1: ' , 1
I :::: .. I I . : : .. :: : .. ,' . .L.......L.. .. L. . .. J "i ...... .L........L .... L .:., 10
Fig. 6.51 2nd Harmoni c Ampl i tudgs v Dr ivCI L ~ vg l
11 in rnA
. I
-1 --:71 · ~·: · -, -""TTI' .. ;;::::! ; ' r"'l n-:E::::::J · ..... "1 : .. 1 . . .. . , . , . , .... .. • ' -,. '"T R'' d .. :i ~d k' : .. ;f.~. ; :· : :<: :· .. :. :: : ~l: ::\:~; ' .:+ :
Sil it I ,1 +'fn- . 3'ii' r . .. . . . r' ·d'il ''14 . .. " ,,. ·". . 4 ' is ·'" ·T .. , ,,, '"' · ..... "'H""; ~1 '' • \1' • 1 - ·:±±rr · :::r fi!+.- I ·-r · ~·· · · . · · 1 2l. ~· 1 · ~-'7 • · • ~ . , : , •. · : -P. +. ·r :- ·-: , ··++ · :•c .. J T · .J-..
1• .. ~: ~~ :j: · · ... r ·· ·-+·-r:+
1 1 ... . · .. · - .. ·' ·· J . ·, ... ; "
1, , ... F
. . ' I I rl , .. . · It 'T ............. ~ .... ·• .,, ... T ' 'Ju '
• Q ~Pf _£1:JIIT:l:t:J:e:1:~~~ : "1G ' ,,, ., · ~.::i=t- ·+'c T -' 1-+Tt . '• '.... - ' c ' .. ' I ~ .. L,.. '
! r ; :f~f :f~~ i ] ··~· ';:'·' 7 ~]~::~ts~r : , r . . J ... , . +·· ·~ .... .. . .,,, ... + ... .. r·· . ! . .. r:m~. ~ . ; : n· ···· .. ·· i L[ .. L . r ..:,: · "• .···r ·. · ' , .. : -," ":"::: ·
•- . L - f--·--r .. .. 1 r 1 ' • ·j ' ~rr I i
- : I I ' I I i I ' II 1-H L. . - ....... .... ! .... ! ' ' ~ ' ""- . ' ..... ·-·· ........ . I
. =1 • I : 1 .. 1 . . .. L .. ,..,:p:,~•'=• l=:or'::t::j::: ... ,., . : -11 >
0
.. ~ _I "'. ~~·. ' t.· •. t =:: rr- ..... ,... .. , .. ,.,.,.... .. • ' . ' - - - . ... ,- ' --: .l '
,... 10 : I r..:::· ..J.. 1 :. ; ; r; . I·, , " .re -m;,bi''i~ T ' d . -2 - 10
I - .:~ H·--: · · T ··· j .: J · )?. .. ·r- 4 ··r ~f'-:-..: · l~ .. -- r- ·f.ir:T:' · · -- · , n . , .. .. T . , -+ I •l , I I .. . . . r :· +-· -· .... . I '... ' . . I __ _._ ·-r~:rt· •... f~ . ~J z tl t'··: :::T~~ : · ·t t}[I. I ' .• I 7 . ' .. ; .. b p:·H ·. ,;ck- :\:::. =l:: f=: ::: .... :1.-1: I ' .. h 't· c. :it :i::: ; . ' : ::eft . I" " TO ,:-.c ... ' .. .. ~ "'' .:: ~·· I . I ,_ff /1::. _, ' _, .. + J"fc-;:j; ~ic"l" ' "'• ,,, ,,,, '· "'
i •·. L~kf l:f.. · · · f-i..,..r• 1 :=~~!i . ,,.+7 ·;~'t, ·,: I . ·:7 t·· t-~ - l" . I "1: .: .: J:): :: :I • ' · ' . J .. '" ...... : ... 1 I
I . ~~ ~ ~·· · L : i +I ,~~~'/: 0'? 1: T ••·•··•···· ~ ! .I I I ' i 1' .,! •• .. I . . I. , ij;; !,~ : i I I · ..... c;. .. "' : .. ...... ,. •• '2:18 .C1.; ; · . . . ,. : -3 .. f-..~. ·:·:t:·. :.1- ·--i ~~t: :~a··· =::: ~ ~ .....• ·- =~·~. c;·ii' "-F :~c' ·"·"'" -. -. -.. 10
' '~ -·. .. .. !. . -r- + . .. . . . I II.' c ' I= ... .... " " ..... . ... c . .. , , '" ' '''ll ' " ' . ~ ' I . :: ._ - 4- · f ·- ~:. :. ·! . - - · , .. ~ !;' I. _ -'j· •::, .• .. :..: f : i 'ill !i i :.: .. .. .. , ' I i ·bH . -r. i ..... ··Tr . ' . +I+':-!- ·r: ~-r.-. ~;r : I
I :+::cc ~ I i .- r~L . r .. :; ~ . : ,, -! .. . ,: . .. :-. ·. !.~ .... "'! ·, :r :: r -l I tf ij ·::: ~L~: :. : .. :. ·-r·~ PE . t·· l I
I . . . .. --;--c-b[i I t I I .. ... . ' I I I I ! ··i'"'" i" ·.1· L J . ..:.... ·: ·.L i ! .li.. ..
: . . . 1
; I .. ·" - - ·· 1 I · : . ~ . L.1....:.l . S ·· · . . 1 I I 4 0
Fig. 6.52 id Harmonic Ampli tudlls v Dr ivll Level 11 in mA _
. !' ''
I r · · : I .
I ' I
I . : I . I
! ; : I I
.c .-r•J~r-rj••:•.•r.
-, ·- •·· I
1··· .. ~= --"'" ... !-- -+- ; . .. T ... ... , r~ -- .. ~1 ··- -r-"'t"" .. i .. :
l !! .. !
.. . 1. I ! :_·) I. i-.. u I !
. ' 0
.. i 4
... I ..
· t· · · . 1 . . . .
-~ ·-·!- .• . ' ..
. - ,---·-' ! ...
' ' j'' '
'
I ' ''1
I
Fig,6.53 {th Harmonic Amplitudes v Dr ive Level
r--·-r-; -· -'-·f. ·-' ... : : I
·· ·- ·-· ... .l . . j I
-··-·[-,- ., .. : .. J.:: .!
: I i ' .
.. j ....... .. ,. I . I . . ·· .; : I ' ' L' , .. .. ''"'16 .
11
in mA _
- I , •
1 . ,, { ..
1G ' · !~· ~ \ - :t::::=; :: + ·l rtP:: ~fiJ:~:~ :: ::· :· ..... """' ~':::·~·. ~1 . :: ~: :.· : ~ : a _: · 'd~ ·. _ . ·- ·· _ -.. · .. . ..; :~~ .} :;; · ~: .. _ j~ . , ::;-: ~~ :· ;
I :- ±::" $7 ~ : I L t::~B~"c:+f--1 : -~It1 : E~~J~ :
~ 1o0
' _I_ +--- --+- i ; ., ) . -- I '· • • : • !_ . . I , -- - ~ ~ : - - -2-l¥ ~+n· ' I 1 EI : I I ~ T~,;·i)e:~~~~~ ·:! ~- ~ ! ·r· : --r-· ~-- --~ - f- r · -- : 1 -- -: - --- . .. 1 •"I' •. ·-: . · 1 : -~ -: l·:·l · 1 I ! : : · :
1
-· · ... 1 •. ~, ·f : : ·:•• ·•· ! • · 6 !v . ·· I I : : ' ' 'I ' : ... . I : : I' I
. I ~q- . ~i=- ~~:~-~- ~-::: . ..0 .. ~J~ ::-t:t±:. .: · i ~~~:~:~~ ~~f~~ =t= ,
;;;I I! ' ... I - i
I vt /
. I 1-F-F ... .,... -T-+- " T'" ' . -r= -:1--- -E ..., .,..,.,. ·--~-- ........ . . ~ .
' -1 1 I ~ I -10 ' >i
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1
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I
-i -10 !
I
I I I I
____ ., .. ·:-t= ... - f- --·· , . . -/-· .c;..f-;.:..+-_- ·_+ . . ·-f--r· p.;.;. . ..:.~ .... ' ·r-f-- . I~ .. ... --i r -T - .., - . I . . • .. riTI - ·i"'"~i:.;rf·---·r:r If+- --f% =: J: T ::. ::,: H·::,; ,;t~m: ~~~ ~ :~~~~;r : 1d .. J I . I ···[ ·--]·· ····-- ·········· )7. I· . , ........... 1
~: i : . ~ · .. ··1 ! .1 1. · : i : -· . :1:::: j · ·· · · ~ 1· · •r .... · 1 • .. · . .. I · - -:- . _,r-. .. .. .. ~· . - - - .. ... --:-+--, ~7 ·:t-' - - -- ' :::+. .- ·~~- ::t+-- _- --:: 1
:: : : : ~ -r:. :::t= ::::r=' y-;--:::t:: ::-+::: --~-.:.: ~~~ ; :".:F : ., ... ~ - - -- -- _J_ . •. ·I' .. , ... f-.. •-·- -+-- - - + ..... ··+· ... f-. ... ..... '
*.,., -[--· ... l · 1 ~ ·-~~ --: ~- 1f · -- ~ ;.r·"' ~ -1· .. ·-r~·· ., 'mir ·'Tri · ' .,.,. - ·:r--· - - 1 · --· -- -- ..,. .. -it .,;-~- - · -- ...
1 .. "' fllic -- ·r·- •
I l I . ! . I ' "
f~ -.. -· .,. --; r --T --......... ,_ .. -'--!-· -- - ---~""1-F· ++· .
. I I , . I . ,.. I . I I t :'· :·!· I
Y: ~f,-- -t, .;,...ij'- --· ~- ,-.j , . J, : ,. I .1':,' 1!: ' ' --r-' . .. _J,_ . I J. .. ... I ' , '' " ' . ~ .. 1--l '
I -3 i !"' ·. \ l : $1 i \" .·) ;. ! . I :· ':;: :J ;; . I '-10 ! ·~ :t==rT "T'"'-+ " .. . .. f.... .. , 1 ,
1
. • ·1 ·· -·: ··. 1• ·· ·:; • • • --~ ~~~ ~- --~: :~.; • . -~- = ~--:· 4
- . 1. •I•: -· .1
.. • '"' r • P • ~; · , • · , · :
.,' : . .:::~fj::;:::: ·~· · •l=t:-~~~ ::.:; ---r ·· -=1: - ·' ' ·. •r~;_. + ·,. - . : :; ;; ;: ;· ri( ... .... : I . ..,.r ,.,... · -F~~ -- ·: ---t---- ~+· ·· ~ . .•. ... . - ~ --·--r" 1 .. . !: :.,;" .-i = ,
: / !~, ~p~ -Ft : . ~~~:=r ~-E7t.i:-Lt~' ::111+ ;1;; ~~~ l,.:: Q ;I ... i:...J1 ... ~::.:J... . . .. .. .... L . l.....l~ .:.;_L1r .. ..w ' : .• - ~- _ ·· .
6 . .
Fig.654 5th Harmonic Amplitudes v Drive Level 11 in rnA •
I · · -I- -
•.. • :·. i : ... i ll : :_ I
1
_ . . ~ ·- i
j .. T ' ' t , __ J ~ ' i . -I --1--_--- -- -;;r--;-·- - I . . I . I ....... 'e ., i.. I , I I - ~ -- -- ---~-- .. ,_ +- --+-· -- .... • 1 ! . , ; __ ,
i_1 ; I : i I .... i
~~ ~ ~~ tt l . I -~:~:·~ .
.;];!]"'"' ... !_f·· : : I .. ..
": I . ..
t· ·t·· l '
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ll:: -. I ' --T~ . " F"" ~~:. :
I - ·· t - .. .. I
. - - I J J .... , 1 ··r-· ·- -t
·T- ' I .. ' ' I I I" ..... , . I )' - Jl· - I : ... -- I
l ! . :I .. !. I
-! .. . . - - . . I : ., . .. .. .
1-· :1 : : , ... I
. l ,I I I : 0 . . .... J
4
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.1 ... 1 .. L 1
8
.. • , , . • dl •i· t • ·• • l •·
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i
Fig. 6.55 6th Harmon ic Amp litudes v Dr ive Leva l 11 in mA
c ('
~ I ,, , ;::i i f I ..a :· l (. . ~ r l
~1~ \~ l il~~L ~- .... . ·· ·:-:- .. . j ~r '~ - ~ .L:~ 1..:~~ ~ c::-~-'- -~ i- : : .:: .. :=f·- if{:. ,=~h+ :::; : 1- I ........ ,. ~:r--· .... L · :r = ""' -, - ---'t111rrrTt 1 :~ ~.!1:,.....")1~ll . . . . .. 1 • ,., ... r. •
-10°
I I
~ 1o, I
'1 ~ ' . . ·.-- 'T + ·· "T · ·j · ·.,-- -,.,.r --~~ "~R~ ~r" ~ ,.,.. ~ ~CTI~ ' .. 1-;;., .:. ~.,.,- ~r - t - c-j-·· ... , · -- t· - ;.t .... v~tJl 'l; . :.+~ ' lo: :1!11 ":11 il' !.:1: 1: • 111! 1!11 '
· . ·::trr -- J~~;- - ~ - -r~ .. : ., -r-p - ~-- 1 - - j-+~1~:: ~ _]·~~ : I 1 j .r r ! 1 . . .. i . , :,i, ~; :;; ~!~.~~~~~ l . i ~ ~ ~~ : . . t~
·j . - ~- l.-...... I . : · .. --l--· ·-'"""!--, . M' . '' ..... , ~ - ~ - ... 4N· l-1' : I • ~..l ) . :' I •• ; ' • I I
\ ; i ' I , I ! i l ' .: - 6lv . I
, .... --- t-- t::--· ... I .. : .... ). .. - - ,· - .... : - ·-· ·+·-- - ~ · ~ ' .. .. . "'T ' .... I ' .... ·:·-- + .. ..... 1-· t -'----- f... . f... _ _, _,_ ·-t·· .
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1
.
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. . . ·r· :. --'- ~+.:: =-f== -=r~ :- -r=- :..._:~= -- - r::' .: --+-~ ---t ::. : f ·- .... -~- ... -+~'-+.. :- .. ·; ' ; . :-1-J--, - .::r ... '
. : ---- r'C. ht .. --f-. T"'·-· 1 1- .. r ' LT _...,.,. I_, :J-·1--cinc ,.. f.. • I l .. . ...... "',.! --:- .. -'-: -· -·+ .. . -- 1---- T.f-J I --1- ' . . t ' I 'I , I ~ ' I .... .f. ;... , ... 1 ' ·-: . ~1-- ... , .... --~-- ·-i· · ,; ~ I ·, li-f-· !--,+:"; ·* ' I' . : • . :: : ~ . ' • . • I . ' . I " I '
' ' T · ' ··T·· · :: , t• · :;: ;.~~ ' : · ;~r·- --· ... ~ - -r• ·- 1 II I I ' ; i h Tir n I '
·.1 '. . .. , liL !J·.· · .. : ... •·"1·-- I +- i · .. ,;,' 1' ... ·:: : ~~ 11!! lj!f j 1 ~.! jill H+~ • . ·•·· ; : : · ; I .. I . :: . t ·· · .... . ..).. .. .. ' 1 I.J . .. j.. . .+.'-l;.J..::. jJ? I . ... . .. 'I ' ' I I I I · '
. .. ... . i , . . .:: I . ,. ·:· .. :. : ... I -J) I ' I I I i I I . :k': ' :: ·:: . : - 10: ' . I j ' ~ L I ·I L .... : :. ' ' I . I ' ' ;; l!'l : : '
' ·-: ~~~- :=t~}~:: j :::1::: ::~~~~ ~: I '· . . .. :~· : .:: ' L~;, ~ .. 1#1~ ) ~~ i · - h;;~--c - ·t--j~--- .. r .. t. -·r:-- -t-- l!lTIIim ,J • . -""'· - .. : -l-- ... . _.jj,: -- - --1 - hf; l+lli -1- _,) ! ::! ·: , ... '
, I I + Ji'Jiil j !·'· 'l •'i' ~·· ;..,.;..= . ... - ~-- - l r ... ! ... - r .. i 'I 1:' F-+-· . I .: II • 1- "f' .,.-1 : 1-· . : l . ' .. . 1----- .. ,· ... fm:"'n r- -- 1- -1· ' .. '
I ! I I : I !tl I I I ' . I . ' ' ., j -1 --! ! I - ~ . f-·- .. J ... L= -- ---· "II ... F'r'-- ·· =•' • 1 I , r , I I ! I I ' ~ ; . i ' !
.J. .... .... ~ 1 L .... . L11
.t.. L...L. L :. 1'o .. I 0
Fig. 6.56 znd Harmonic Amplitudes v Drive LevRl 11 in mA-
[,i~~LGl-~ rr~ : ~- , . · · N~ 34 g:..p r+rrin .. ' Ab~Jj£f; ~:r... 2f:rrt?,:J_1ft~~t-J .t .. #t · 1 .. ; . : Lfl9:Jf .~ji£Ji£if t·r ., ' . , . . I
I : I . I ~ I 1 I I : . I I : .il : · :: i • I : I
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·lr:· . r--·· :: '···!· I I .
.1 i . i ... :J ~- l 0 4 16
11 in mA
fig. 6.57 31"d Levtl Harmon ic Ampl i tudes v Or i v e
I
~ ~ ~ I
·I ...... I I I·.:
, I
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..., ~ -1 i l ~ , -1() I 0 I > I
. -21 ;...,o 1
-3 -10
b
Fig.6.58
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~ I H;H! ~ ' I : t : I : ; ~ : 11 : ~
I :· 1 I ' i. , ,::: , i : /~l : : ~l(ij : ;:: : ~ r · ~ ~z~~li' l l ' ' I .. I I I . I I : .. i .••.• :: : : : . ~t ... ·:: I j I I - 6 rV , .. I I ' I I . .. .. ··!· ' ! + I I.
t= :::t:: I • ::t :· -~ -~~+~~r::: :::: ·:~~:! .. ~ ·:. ::: :~r.::- :~:: ·•·· . ····' f · ............ '--+-+- -l·-..,-~ ·-·· ·r·· -t· .... J
+F f+ - i -fi~f!~F~~~ - : : ~-t~~~ : ;~- -. f + ~ ! I ; ! . -~~ret' ec;· "i· ·-:- ,:+ . ·: ... -t- . !·· I ,
I I 'I I ' I I . . I I ' . , ' · ··· .... ... . . ·· ·· · · I · I .. .. ..... 1 . .. ··· 1 .. ' . ... ··1 · .. ' ... , .. .. 1
·· ·· ~ I I . ·· ·· ·~ . , · ~:; u: : .· ! ' ' I I .. . .. I I .... ' . I I ... .. _+......... ·-j-· .... ,. ·+~ .:c-b . ... .,.. .. -·-:--t- ·- . I 1 .... 1 .... "-+- ......... ~- ., 1 ... . .. 1 .. .. .. .. ,-. :1·· I , +- .. · ... -·+ J · · ·i · ·-:~H.,..r-~·- ·,1
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I • ... ,I • _ ....., .-t· --....,.. I . r - ·~ . I ~ . · - r ;:: ~ ~ ~ ·:~ ·. . !. ~- : - ~ ~~ - '
• ..;_ . .l .. . . . L,tt= .:[ ---~f+ JL. . • ! I ! I • I ! + I . 't" j:-, ·9 . -t·-TT~·--+ '
.. I 'I p· ' I . 1, . I' ·-·f· I • ~ -~· - .... . ll ..... ,~r- j.. .... l I
! I I " I : I I : I . I
11 i .... . .! .... ....,. ..... } __ .... r- ... :~ ±] . .. t...: ::1. ·-r:-· = .:.. ... : ~: ... ~-r-: --~:::::.::1:.."-~bc ··- .~ : . ·t= ~·- :_: ~ ~ ·~ ~ ..-. - ' I' I I • ~
I 1 I ' I ' ' ' ' .J 'T -- ... - F""'' ~· ..... 1"' ' ' I '"T' '
1. ..1 . • .. ~.:., .,- ..... · -:-;--r•"'-- - """ ... ~~~ ... I .. • ; ~ I I :
: ; . ····o::: TJ,·:·~· I;::: ~:·~~~ ~ .. :~~· : ''' I " ' .... , .. 'I . -·r l ....... .. .... ~- · ;T ........ ';" ·: · , .... !'""( .. . i '
. ! ... .... ... . ' ·· · ~·· ···~ . : .•.. ,. 1, I ' I I . . I ' .
.. . ! . . ..... 1.. 11 : . ... L._ ... ... ! ill :.. ... .:
4rth . A . Harmon1c mpl1tudes v Dr iva LevQ\ 11 in mA _
~-~~· : . ~ · :: ~~· ff~t.. : ··:_. -·~ t1 ~~-~ I .~~ -~ -~ - ~~- ·r· .:: .: • ': < .• :: ::::::: :? - ~·+j·.;~: rtrtm::·l , .. , .1, "' I''·'... . ' h· 1 r - ,~f- I ' '"• ·' I· ., . ·'Ji !ill I·' ' .1 ' ' ! ! r . · · · · · ' !; I ;~~ , : ;~ ..... ·r "'i ..... ·· ·· · ; .. 1 ~ ~ ~~ : ·r- 1:.: ;> <:i :;ij ::~ :1 11 ;;;i :;!: i!;: ~-~~ ·
• ·t i- ry'+i ·J"~"'T .. : -·+ -- .. . ·--· [..:.:r-:[-r- i· .. ·rj .... :1 •. •• ; :: :: ;: ~· ,,•• i ., . f,;.:F : -~, .. .. · · ... , ~:t· .. - , . . ... : . f - "·- r-t-·+-- .. . . .. . .... . : . -I .
: ~~~ ... . "1.: .. L ~- . · .. :.. 1 : 1 ! .. JJ .. - ~ .. -~Atr•~~~-o,: ',E ~~: · !. ! : I : . I . I INbk 1 ! i ! . ~ i I : I : I I : : I : I ; :6:V I
~t .. : . : I i . i • I I :f[l "~: ~E ffl! )' i
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)- : ··· ...... i 1 , , ·: · ... l .. : : rTr
U1 := I I ' . . . I : I . : .. · .0 I-- · ., ... ... j. .. · .. ,.. I I .. .... t-:-,- .. . : - · r- f--; . > . ... ~~:~ :.:-:! . I ! ; : . I :.: ... ¢ ~ .-:- :::! ..: . L.L.t-·++ 1·= l ::-::::]
.. .. ... . . . · • . j . t I • • 'l·- ... .. .. ..:- 1-:-T-+-: .. ..... ... .. .... . . ·· ·· ·· : · 1 · · I : . , · r.: .. + .... j:: .. . : .. L ... r:: ~I .. ~=~J: ~~ ::· .. .
.... . . .. 1.. .... .. . . • .. } .. ..,. - I. : I ... I : ·- .....; j. ·· I ; ... H .... r-O:I=: . :.J- 1 . I I i I I I : '-~ .. ....... , .. · ..... .. .. · .. · + · · ... --f .... l., · , rb~'- ., .,. ... , • 1 • • i 1 , ! vr :~ i
1 I
1,2 . . . : }---: I • ' .. ij : f( F:-·. 1·: I
.. ~ --- .. . ... .. . .... : .... · ... . :. .... .... .. 1-· I ... 1-t-=· _ - ~~:::::: . t·~· ..... , .. .. · -·~· - -~· ' ....... .. . ~- --· j . -1
... 1..... · : .... .. T .. . · " T-· .. .. .. .. . - '
~f ~:F t•J· . ~ ~ : ~ .. rf .•~ .~·· · ·s. :_ ; .. ,-:!l ;. :_- -+ .. ~ ~: . ~ :· =t=.*+-j.· . __ II'... I • i .... .. ~ ! ·+ i -~." v ! l- "''" r ~-~
I ,, I . ! ' . I ! : ... ;. : ::. .. ' I I 1--r .... 1 .. .. -·r-~· ...... ...... ;... . r- . r ·"" .. .. . . I I ,. . .. . .
, I ' . . I f-T r T "T ... ;- "'; . . ; I ..... ~- r • ..... .... i +-h-· ' : . ! , I . I. .. . i·· j· . 'ld ~3 ; ! I I : : : • . : ' I :
~J~:: -:::: I :::)::::- -:+-:-:: ::::1 . ... : :~: :: -:f-.,.,.-.,- - ~-:.; ~~- . . .. ..: f-.. ; .. . ::1 -::~ :. : .. 1 ...... .... :f . ·-t· .. "T- .... , . .. ... l - ~-· -;-· ·+" .. . --~ ... : ···~~~:: , · ~ ·. ~r~ ~-T:-~~- : . :. -~=-~ -=-t=. .. t • 1 .. ,=t-· : :.::. '"t'" .... ·-·:· · ~f .. .. j .. ·i ,... ~~· ···. l 1 I ··fi ... ,. r. L ··: ··· i "'' ; I : ·-- .. + I . ...~·-r · .. , .... ... .
! f i ... .J ., ... ,. : .. ,~ .. ....... ·t -- 1- l .. ·r. II '""r·· f- ..; ·- 1-i-· ... ..; ..... , .. ... : . i... .. . . .. .. ,. ,:.: : .. , .. i .... i.. I! I . I I I I . ,., ' I . I
.. I .. .. I .. .... . .... . ·-·t-·· · .. . ... ·-r rf'-- ..... ... ...... .. .......... ... .. : ! . . ! : I .. I I : • I
I ' ... : I . ., :: .. !" .. I i i
0 : II
Fig. 6.59
I.
sthHarmonic
~
Amplitudes v
I ' .
-··-· .. .... , .. L......_t .J. I 16 . .. • I I .
Drive- LQvlll 11 in mA -
c
! . . I {': ,, J . t .
-r~:J~4~Xffitirf~~ ~ ,y:~lJ~· - · -- ·m::r-~,- .. -lP· =¥¥·. ~~- . ' ;"""'' ' ' 4 , , " • ' •. ,. , . , . .. :
' ' ' "' . T .. " .. . - " . j _,.. .. . ·- · _. .. :; .. :;: : :·.J- · - - ~! - - : - --·r - _:1; .. : . . "'~- · ~~ ::; . . _.; . . : .. : . · .. 4' _. • ______ .j____ __;__ _ , , ·iii ... ,_ , , •• £ r .. _r+ _ --+- -·r-- I . .. . L :- ·! ·· :q_ il:: .,. j; l ... .. ,, .. .. , .mr
i ......... --;~ : __ ::~ =i ;:: l··- H ···• ''· ~~ ~~~bi~:J:, : ~~~ ·; :~ , 1
• , :
I i T---·:1:- '- i ....... i' ., ·: '- - - ~~ ~t- ' l ' ' I• , __ LL!.uf:-'...-1 I ' : . . . . , "l,b· . I . I "' !r y- .
0 I I : ! - : ' -; I : -\:'\ -" : ' I + ' - 61V !
-
10
• I I . . I I {' . ' .. • ••. i I < -t }-~::I i : t·t· ~ 1
: · 1-- -- .. 1- --~ -- :. - ·· · : · --- - - ' -- .. L. __ j_ -,-~ · -~-- --- i ··· I ···· J__,_ -_ -1 j •
-1 -10 :
I I I -2 :
-10 l
I i
I
I
' -31 f- 1 0 !
I
I 0
- : ! : ' ! I i I i ' i 1- I : t• i ., .. , .. .... . .. , •. --·-· .. -- -I - · .. ,.. f, ... ; .. ----_r- -r-~t ~--~ .,.- f- · --- • .':-·r.-- ' ~ I -· ·j ! I l jl • • ' ' '·: • ! ., . •.: :·' ;: I - +·- --- -- ... j- - ' ' .... , .... - - · ·--!- ·---., .... ---! -- - J. .. -+-"---- - ' , : ' : l I I : i I' I
- I I : I ' I I I ': : 0 ' ' ' l I ! i i I ! :> - : :E:-=:-t.::::=f~- ::=: ~: ! ' I ' . ! . :: ~:J=::f::~~·--=11· = ~J:- ~~ ~ : -:::..:r.:~ .::I :
, 1 --- ··+ -, ... - +·- I --; -- .Ll;-- ::--r J ·· ·- . ...: · I -- --; · t .. -+--- • I :: -~r ;: :: ; , -I ~: ~ ::~ _::-:.- -~ l ::_l[ -: :-r r· : :t~ :·:;::: :
-,-- --!-- 1- 1--.. --1 - , ... +-- _J_ .. k-'- r-~-- ...rrr ---~ . _ _ r-- _ ...... -t· _ • : : , : . · i rr· : ! .
- r-:: ~1-- -T ::l · --T T: :~· : - .. . : - .r-~- 1 ~~~-; - --i : ; t~!) :; ·- ~ · :-- -- •-r ... , ..... -, - I ...... J... . L.. ---- ·-". '!"' i)r~,.,.., __ [_I, ' . i I i ! : i . - ' : .. i :i: ::::! ·v: ~ . I :•:j I
I . I j_ ' ' I I .. . . .. . .. . -- - " ,. . - ' - .. .,. ........ I - . .. - --... ' ' " - ... . - ., ' ' ... - - r,, ' f:,-' ·..,- I
; ... _: ~ ~: ~·:=t= . :· ·:·· + : 1 . - .•• •• • . - -1 ~- . : ~-- ,_ : :: ::-: . ..... : -~- . :~t : v~-. .... ,..rn;h-i---- ... ~· , ___ ·· ·•·· . · --r, r.r. : j · m ~ t-- - .....
1
.,..hhi-r ··J-". . ··t ? : .·. : <: !: .·-· . -t--· - t ·· - ~~ 0 ' . : :.:; .. ! ~ :: · ~ ' ; ~: • ..•. : ~ - •. :. : · ~pt :!:: J:; ~
V --l·· : -'1-!'~ --§f--'- j--· V -:- -----·~- ... .. · • 1\ . ! "f~ ~-. ~~ --- ·. --- - f-. -~-..~, .. ---· -" • " 1'.' . LJ_._ I ' .. :! : . :
4·.·:· ,i:: ,:; : . : --- . , ' -t L l .. . . .. 1- - -- 1---- ... ' ·. · ', '· .. .. ,;n. rmi r-~--·-~· ... ..... .. , ~ --- ,
! . ' 1/ : •. ;!: :: ~~· .. / ··~ ... - ':" : -~' ".,. -Lf--'- ..... r il -- - ! .. - ... . /.... :: .. - .. · -r: I .. ..; ' • ...... ... : .. --- · · - ..... --- -J .... !'
: : : , ~ 1
• • :::: ; :: . · . · · ·i < ~ - ~ ~ ~ ~ ! :.· . :: ~ ~:: : I :
1
1. 1• .. I · I I · - .... ------ -- .1. .1 .. · ., , . ----1 - - -- .... "" .....
' ' I i : : .. .. : . . :I . ·: ', I . : . . : :.i: : 1 . . I I .,. J I ~ .. .. .... .. . J, , .. l"' · --' :"' . ... ,:~,··· ..
: · ·· ~~=;f ~ =~~~~ -i· l 8- • ,::· · : , .: ·:: ;,,: ··:" ~ ; I. ~ . .f..;-_,.. ~_ -r· r+ 1 , .. Jt~ · , , :::. · . . . ,11, ... -1· -;;r. - ;~~-;-, .Ll .
- ·-:·-, - - ·c+ .. ·: +-., . .' .. . , ;:: ' . 1:: 1 ·;7; ; Tf.@-r-·I ,J I.,, ..... ;,,.. , .. , ... '"f·' !' I " -I ',: ;,.., .. - ·l:ll'-1 ·'·It'·'- .. .. . II,T I .; .... rc · : .. . .. :,: ~: -- --·r . -i ... ·--1---- .. :: ·; ;;:, : . .. 'I II: f---'-':.. I I '!I j . • . . ' . I ' ' ~ ! ; ' ' I I I ' l I ' 'j r:·: ··· · ·· ···•·· ··· ·
1
.. ... ·1 'ii .- . I -- • • :i ·. · 1 1 ' I "
I I ' I ,. ' I . . "'4" ' . ··- .. . . . . ·- ..... 1? -'- . 10 .. . '· ..
Fig.6.60 6th Harmonic Amplitude~ v Oriva Laval l1 in rnA
-- r· -:~r
:: . : 1:~ - lr: :., ...... : ... ! ~--!- ' l
4
-----· --I '
., i
I - i·
i I
- ~ ----· --- i -- ' ., ·-+- t-
.. .. , --- ·-
. " t"
j
I . :~ i:+--r·--· .. r t
I
i .'
Fig. 6.61 2nd Harmon ic Amplitudes v Or i v.e Level
. . i .
-~------ - · -1 :
·--'----J ·r-1 ' ' I ' ' '
-~I i6' 1 '
11 1nmA •
~ .. I r (. :
1-.·- ;~· .~ ·-::-c::::;:::; .. . : ·C:.: :::: -· " -:.i.~- ~J Irk": '"T ... :r : :;: ; :r:r '"' ' •'I ... . "". " . . .. ::.c.: .. : .. ·: ~r.,; !;.: _ ~ -. . : . .: : I_ ·" .;; •. ' " ,, .. ·: 1." ar ·: ;;;· .. . "" .... . ,, ... "- :r;-; = r.tT •:i• l . .. .. r\.JCJ "' . ., .. . . . - I ~!!'' ... , . .. ' "
:: _-F~E:: ~ ; ;;: : ; ; ~ ~; :,1: : 11 ::-r~~~~~ ,, , :~ .:"!.: ;· :·': ·,i :i:: . .u~: : :i: , : ~, .: ~ ::~ ~::: .•• :' '::
. j . . 1 . . j ... ... .. 1.1 ..... 1 .... . 1:j i. , .. l .. . .. . ... ~ . ,
. .. I ! . ~ ~t~ ' ·~ ··: =t: ~. r~r ~ :: 1 · · -· ~··:·~~~~ . :;; iH :;J I ~::~:'~~ !! :· ... ~= ~=~: I. .. ,' . I · , . ... .. . . " .. ' I I I I ' I . "" . "r!i ),· "" 'I " . 'I I ·· I 1· : .. 1 -1 . : I 1- . .. ....... . : .. :·:·:: ... : .. -: .... .... .. :v ~-
-1oo • ~§ -h=-r ~' L' ,-!:' .. C+-~c~ --_; ;::l:ii~~I- : ' - r h+.. .. .. I .. ·- · +· -· ... r · r""' • , - --r rm:'" 'i ··· --r • ·..,- ·-, , ":±·· · 1
· - t-tt .. ~ -- --+ -~-- T - ··· ·t : · · r .... !- r---·.,.f-T· •
I.. ·f-i---·! .. . . ·-·- - -- - - · -r· •
~ i ' I . • I : · · e -IIb -- -·-- -i i I--- , - . r=~l--- _:.. . .. , .... ( .... . '- I k 1 I I i . . I • ·,iij'T ' I I .. I - - k2--- H-+·t I :i' I ~..:,. +-~. - , • , , • I . .. • · - 1 • 1=1 . ..;J...IIo-~ I I ·r I I I . I ·• I I I ol I 1 1 , ~f:--"..--r t 1 -1o 1: 1 > ~-Ff--: :r'-- ~-l- _ __[ 1 . ... ! ·- _ ~~ ~ •_r_ .... : = ... L. ..
. . .. - :-c- ~~: --- . -- r-r-- - .. 1- · ---; . -- ·---~--""-rr-··-f-"" -+-- • I • I -i~-=tr ·r::r-= -~ ·I · -1--~ ·r:_--1-· -}. ~-:-[-----~-~·~· -r-- "'- -+· : r- - -r r--, . ~ . . , -- .· ... .. ~- ." i"' . ·c- •
I- . .I I • I . y ' ~:....,. ------·! - ·j+--". - - .... ,.Hl ... .f-. ' ;-,. I· ~. V~ •. : 11 ... - r-, .. 1~ .. ·- .,.-... .. ---~·--!IT: . I I II
II - - - I ./. i - . ' --· ' ,-j I ·-1~ TT'T '"".,.;,.. ' . ! V' ·~ i I I . . • • I II ! 1
11: ~ ~- ' ·'
1, I :
I .. : kr' I , " --· I· I 1- -· . . L.- ,,. fu;...c,- - ·-+-- ' . . . ' i • . . • I : : ::: : I .. : , , " '/!: .. 1 : . I ! .. ...... : .. 1 .. ..... ·.1··. 1:: . :::: : • .... r .. : : :. : ::: . .. !::.
' . I ' ... •' • ' ' . , " • ~
!/• .f::: .. :::J:: ::·:.::::-::..:tJ r . .. i . ::J.:L::::. ~~"-+"~--~:: · ~~~. •-: :_ ,l_:: ~J~ p:f,~~-:·· · ;
1-c---:- . - --:-f-: ·- i" " .. ·- ---r-- ..,- . _[ .. -·+- .... ~ ··¥· 0
--~c--:- - -- ~r ··-- ----- ... . --- .L._.;;.;;~:. : .... ..... ~ - - ~--...;..r-: . . . , L.;. ' .. I 'I • .. . .. . ... . --~ ., - --+- ·•-·-·· -. .. ~:r~hf', ,.,,., ...... T,..bm:"'" r"' .
/ --~;: t j " t -t- _: ',] ,., .• !!! ! ' ·= ;~- :: ,;, ' -: I I It I I I ·: , .. , .,., .. :' .,. t ""' ' :i'! !; · ; ... ! ·: .. ;r ;::: ,;J:,. :I t- ·- ·+----I I T i . . .. "1- "'I"' +: T: .... ·:·!-- ;_: ;., ~,".~ '
1 I ., , ,., 1 , I ,. I i I : . I . ... . .. .. .. .. . '' I"' :::· ....... :·:: ::.;; .. ... ..
-2 i -10
I . I I I I . .. .. . .. I . . I ~ I • I . . I I -1 r I , . I I I ' '
;...10 i --·i-=r-+-- -+-- ---·-· - .. , ··:tl:""..,.,-1-- - i-=~~~_,,__;. .. +;,~~1
' ' , ... 1-, .. J-'-:: ...... C+-· ;·"[- . .. .. . · .,-,· - .,-,· ·:-""'f+S · -i '-'·i·-of::'i2~,. 0
I · --~-• • ··. ·. r- --+·· -----1 •. ··· --f-.. .... .l'::-:- - .:l-.,.,~~ ·- --q· H ~ · -· · I
. .. .. - ·- ..... T ....... ,. .. ' - 1---1-- ... I - ..,.-r:;· -· ·- ·- ..... f: -- .. .. ' I .. .. ·"- M .r · ···-r .. f-·t-.. .... i " " .... ··+ . "'f' -'7"~, '"'"!"' :~h4nr I ""' '
. ::·-c~;T - ·~ ·-'- . 1·--' I ·i · . . --·---:-; .. --' q-l'frmt l• ' --·;·-' H-r-+ .. T .. --. I· - ·· : .. --~.-- - r·-!--· .. +;"' +f ...... •
·t· ·--L. ··.·.·. ...f:: . :-..... !· ... . + --·i+r .. r~-. .. ' I · ' 'I'" -~- -'
I I 0
,
Fig. 6.62
i · .. · ·· I , I · . . . : - ---[- :·1···· .:..L. I · I .. i ' . -~r'c::~~-; •. I
I ! 4 I .·· ~ ' ! ! . I ~ ,,J : u l : I 3rd Harmonic Ampti tudQS
11 in rnA v Dr iva LQve t
.. i . ···r· ··
... t L.l i
, ... 1 ... , ' .... :. I
I . 1 i I
I 0
Fig.6.63
·-T --+ •·
-- ·-:- ·
-· ·r··.
_,_l _, · ··-
I I
. I I i
~ t[thHarmonic Amplitudes v
..... r ·-··· 11 ··· ·· .... .. ... ..
. .l.
Ii II .1' 1 i!li ~L -- -j ·: •... : .: .:'~-~- --~- .. . ·· I· · .
+ :·-! ' I I
. . • . I .. : I : t! ll:tstt
·~' .... r.,~;~J~ :+.. .. I I j I I I I r- .... , ...... , -
.·I : I i , I ,
rrL.LL . . L.. . - ~ ~
Ori ve l Qvet 11 m m A
- I
I
i. i I '
i I
i'
I :'
....... :-' 1rv '
I'
I ·
\: · t '
,. I fl 'I I
.I. .. I (, :I I
GaAsh~i~:frf:d·~tdtidkN~9Xi~c£; l :it: J.~J iWWlilffi ! l~!l::l vr :: jii: I~.
I I
ol -10 !
,, ' .Ill
I .E 1: .... -1 : 0
'-10 · =' ,
I I
I I 0
Fig. 6.64 5th Harmonic Amp I i tud&s v DrivE! Lave I
r=t=61: .. "1
~--- : ,-.. -~ T""' -- : · - '&aJ. 1 [ . 1 ::r-·-.......,-c· -,......- · ... ,. ,,, ,.. .. .. . .. . .. . .... .. .. r ·. . :T :-r~.:- -~ '::i - -· ... .. io ~ ~ r . ·-hf:- l ·+ - . ... . . ,. ' .... ... . .... .. .. .... ::::i. :~ ... ·j. · d · I Ul · n.:1+ bl1~ t · Ill ~~~ - · !Lit-' ~ ·•· --1 . ~~-+--- ·
~·~:-r_~~- : ' ~ · El t'I-id'4,J't 1J I . ! . . I ' ! : l ~!~f·l~~ ·. . ·T ' ~ :
----j--·T ·: ... _ _; .. .. t. ... --- t--· - ,-· -~·-r--;- - l ,.,, tr-V . : i !. I I (Vb) ; ., I ' I
I ()i I : : ! : T i 6 v f- -1:(} I . ....... .. . ..... . I . . . - ' . I . ..!.. - 1-· I • - I . , __ __ , ... .. . .... . . . I I ·· . .. . I :f-::: ... : .. _;.:=r::: .. :r: ... ... : I 'I ... . :.· ! .. ...... I ' ... I..:.:.::...:::- ... L:::~± ::...: I
'I ~ ·· ····· · ·· . 1 1 -T - J - r -:t=- TIT' '·f:--· -- 1 · 1 . r. . . . J -~. ' ~ .... ~: .. ~: -1-~ ~-- . i ' .. . . . I -~~ ~.!~ -·I : ---t-+ --, • 1
.. • ~~ ·;
I .-:: : I I . I : ! ! . •. I l ' -r· -- i~ -T--- ~-~ ··· : · • · ,. ···1 • · r -+--:-T:: ~"'T ·-r- ·1 . .•. .. ; ~ · .. . . I .. i . • • ! .. ! ... '! ,-·:: l ·: -~~ .. - . . . : - .. I, + '"j ' II .. . . j''" ..... :--,--·:·-.-- --~·- . I . . I ... ----~-,-! . I ' . I . . . I .. ' ' II) . . : .. I ; I ' ' . . . ''I' 'I"" .. .,. : I !-1 i~ I ; 1 i 1 : : .. I , :
t~ : :g ! :~:: -i:~:~: : :~~,. .. . .. I ·: ;:~ - =::i :j. ~:: 1: .;..- -..p- ~~ -c:: ;~:-: ~ ~ i ; . -:T= -·· · - ·· I .... ...... , .. ! ·j--;···:-··.~- . --r-----+t- ~- 1
·::-j·-- , .. .., ... ,. [ I I ··· - ·: -· ·· · T 1· :~ -r · ---;-,-- ., .. 1 r - ~ - ---- -- .... .. . . I . .. .... . . ., . .. ----'--- · .. . : ............ ..
,, I ! ' :. .. I ..; • ...c --:--- : ... .... i. . I
... ,.... '.l I .. , . T" -+ ·--j -~+ I · I ' ' . ! I I i :
I : I I •
.. _--1 · · : - - - : - · ! i ·,-·: · ----rn·-; ...... 7 --! , _ -; +.2 , , , . , I , f: · I ~ ' I. 1 ., I I I
f- .. .. :::j::: ... :· .. . : ~;;.;:,; --. _. : ::::::.:: : . : :-:r=; .-+..:·::-::: .. .. · . .. : . .;... j .--·l .. . I. ... ... ·- ··· · h -L::-i' .. .... -- . • •· ~ ---j- .,.. ·--r· I ~.t::~_::;· :~t·=::: ·: :±:~~::~~-~=:: ~- t--~ :~ .. 1 , : ~ :; :;; ,il , · : :--- --~-~ :. :·+- ( ... .. -f+-+,:t·t :\ .. , ... ! / l +- ·-i-- "1'\j--- . ) : 'l;r:;; :: ' .. I . .... L . I
~~~~-- T . · · · ·J~n- .. ' .. ..,. t i/ i .. · ~ - --t- · ... : :: : ;; :;>: ~---;- ·1----- ; J :;:· .--j-- /l'T' -·-1'" m1:··t ·-·· ·· · ·~··· ... .... ·r T ...-11·- i4 rll.l_'' :i:: :: •. ·:I·- r;Tj 11
::1 I I I I I I l'i' "i .... .. ·. I I 1-'--~---- I I- - --r -· . f. . !.. --~ -, ... .j ... l • . . ,. .. ,_ .. ----;·.
I I I ! . ; : : I _ > :: " . ' I I :I · [! ' ·j:· " "'" '"l I 3 I I ' ' I . .. ' I ..; I . I ' I"' . '
-d .... ' L . -+-· ' ... f.-- . - ~ - -- - .. f--·1-· . ' I . . . . . .... . I ~fE 1- -· r-: . -' .... ,. .. .. .. ··· j•· ... 1·- - r--- ,... ' -:~ . - · · · · t- · • ~ . . .• 1 ....... _ ...... · - ~ .. . ,- . .. ;· .... ···· · ·- ·. .jl. .,. . • •• • -· r-'--· . .. .. -- -----+r ···f= .... .. .. ... .. · · · · ---- -+- - · - · _ --r-~·-l
0 I . I I 4 I I . --' ' 1> - -LLI Ll i~ · I
Fig. 6.65 6th Harmonic Amp I i tudl!s v Or iva L<!v&l 11 in rnA-
-G.
114
reveal a wealth of information necessary for their
applications. The results of the sampling and spectrum
analyzer methods have illustrated the general
the diodes.
effects of
The curves of the harmonic voltages have been explored
over a wide range of applied drive levels which served as a
guide to experimental optimization to harmonic amplitudes
and relative phases and a satisfactory agreement between
the two methods was obtained.
An outstanding feature of the sampling method is that
the computation of the Fourier Coefficients may produce
information for many parameters of the diodes.
" ' ' \
1 1:.
\ ~·--...
'
r ~ i \ ,.-........,_
' I ' \,,i \ ! . \ ~ ~, \ --
f( w)
Fig. 6.66 Typical Spattrum of a Varactor Diode
7.1 Introduction
CHAPTER 7
CONCLUSIONS
The techniques and the experimental procedures of the
sampling method and the direct measurements using spectrum
analyzer were described in chapters 4 and 5, respectively.
The amplitudes and phases of the generated spectra for each
varactor diode were presented in chapter 6, then followed
by a discussion of the harmonics, behaviour. As the
measurements were made for four different diodes but of the
same type, the comparison of the results was appropriate.
The validity of the developed sampling method was confirmed
by the direct measurements for the same diode circuit
conditions using the spectrum analyzer. Typical estimated
errors for the amplitudes between the two sets of results
were about 3 to 4% for most of the measured harmonic
amplitudes with a maximum deviation of about 10%.
The harmonic spectra of amplitudes and relative phases
were presented as functions of the drive level. The
spectrum of generated frequencies at any excitation level
of the varactor and bias can be obtained by extraction from
the plots. The changes in the amplitudes of the harmonics
and their relative phases for different energising levels
represented the changes in the generating properties of the
nonlinear capacitance of the varactor diodes. The points of
117
transition between the lobes of the harmonic spectra are
clearly seen to occur at particular drive levels. Such
knowledge of the diode behaviour and actual amplitude
values of the harmonics generated within it can be very
useful in the design of numerous frequency-converting
circuits which play an essential role in many applications.
7.2 Device Characterisation
It is evident that in any planned design undertaking
one requires as much measured data on the devices to be
used before a satisfactory performance can be achieved. It
is also necessary to establish relationships between the
device parameters and its dynamic behaviour. The
information supplied by the manufacturers has always been
limited and usually carried out only at certain test
frequencies and at one or two points of the device
characteristic.
In order to overcome these serious deficiencies there
was a need to introduce a suitable method of classifying
the nonlinear properties of devices for frequency
converting applications. Spectral characterisation dealt
with in this project
these requirements.
is such a method that can satisfy
It is based on the fact that the
nonlinearity of the device must be reflected in the
spectrum of harmonics it generates. Such •fingerprinting•
of the nonlinearity of the device is only valid for a
particular drive levels which does not, however, diminish
118
its apparent advantages.
There are many benefits in being able to identify the
nonlinear devices by means of its generated spectrum, i.e.
its •fingerprint•. The main ones can be in
(i) the comparison of the devices of the same type,
e.g. quality test in manufacture,
( i i ) the assessment of the device capabilities and
hence possible early prediction of its perform-
ance in a circuit by the customer,
<iii) for matchin~ purposes in the design of multi
device circuits,
and <iv) for the replacement of devices in complex systems
without the need of circuit redesign.
A practical device will normally be encapsulated in a
type of package
components. As
which will inherently have parasitic
a consequence of this the device nonlinear
characteristic under dynamic operating conditions at high
frequencies will depart widely from the static d.c.
characteristic. To represent such a device by an equivalent
circuit will be difficult and therefore the equivalent
•fingerprinting• can be considered as an alternative and
appropriate representation.
At high and microwave frequencies it is the harmonic
spectrum generated by the device that will give indication
of the degree of nonlinearity and device capability for use
119
in frequency-converting circuits. For experimental
verification and theoretical analysis it was decided to
consider current driven varactors. It was assumed that the
energising level of the device may be accurately
represented by the current at the fundamental frequency
measured in the output load. As a reference, V-I d. c.
measurements have also been carried out on each one of the
diodes under test.
The theoretical analysis carried out in chapter 3 had
shown the difficulties involved and the need for many
approximations to be introduced before the values of
individual harmonics can be found. This conclusion
supported even more the necessity for an experimental
method which will overcome these problems. None of the
results of the analysis produced closed analytical
solutions that could be used in the prediction of harmonics
generated in the varactor. The reasons for this were
because the resulting expressions were of the series type
and sometimes did not converge rapidly. However, on
comparison it was found that the calculated values using
limited number of terms agreed within reasonable
tolerances with the results obtained experimentally.
Such verification was a proof of the method validity
and accuracy. The processes used in the sampling method
can be easily computerised and can provide a basic
technique for a wideband high frequency testing.
The availability of the sampling method or even a
120
limited (because of lack of phase data> direct-measurement
method using spectrum analyzer tor the purposes ot spectral
characterisation of nonlinear devices could be very useful
when circuit designs are planned. This could be on the
grounds of economy as in many cases the prices of the
devices are high. Selecting nonlinear devices for a
particular frequency converting circuit is important if the
predicted performance is to be achieved. Use of unsuitable
devices can increase the cost of the design and
development. It is probably up to the manufacturers to
provide more data of the right kind to the customer so that
a correct choice can be made.
This new measurement method also offers a means of
determining a complete spectrum (amplitudes and phases>
generated within a nonlinear element over a wide frequency
range. The resultant spectra which are the •fingerprints•,
represent new forms of device characterisations. The
pattern may show a regular trend consistent with the normal
behaviour of the device. It may also display an anomalous
trend which signifies the peculiarity of the device under
certain defined conditions. Regarding the accuracy, there
are no mathematical approximations involved except for the
experimental errors. It is hoped that this project will
stimulate industrial interest because of its additional
facility of phase determination, not available in any
spectral analyzer.
121
7.3 Future Work
This sampling method of characterising nonlinear
devices, i.e. varactors in this project, produced
reasonably good results. There is, however, a need for more
work to be done basically, first, to eliminate test circuit
and system imperfections and second speed up the
procedures. The areas needing improvements are as follows:
<1> it would be very useful to have a computer with a good
graphics display facility which would allow more precise
determination of the points on the waveform to be analysed.
As a result a comparison can also be made between the
sampler output and input waveforms in the system and hence
check on any distortions in the sampler circuits.
(2) a modern and higher precision sampler unit could be of
great benefit. The existing sampler we have used was the
weakest link in the system. It was very difficult to
stabilise the signal output because of sensitive trigger
controls.
<3> there is a need for another numerical method and
appropriate computer algorithm for calculating the Fourier
Coefficients and which can also account for the presence of
noise. Alternatively this may be achieved by inputing the
data a number of times and then averaging the values before
calculating the coefficients and hence reduce any random
122
noise in the sampler-interface connection. The result of
this, would be a more accurate determination of the low
level harmonics.
<4> a thorough examination of the input connections would
also be required. High frequency problems were occasionally
encountered when certain cable and generator power level
combinations were used. There is also a need to have a
higher power source to increase the drive levels and hence
measured harmonic spectra of fully driven devices.
An improved system could probably have a very wide
industrial application in microwave testing. This will have
advantages over a single frequency measurements which would
normally be carried out using a slotted line assembly. The
method is much quicker and can produce the direct
information over a band of frequencies.
123
REFERENCES
1. VANDER ZIEL,A., uon the Mixing Properties of Nonlinear
Condensersa, J.A.P., Vol.19, No. 11, 999-1006,
November 1948.
BAKANOWSKI,A.E. et al., •Diffused Ge and Si Nonlinear
Capacitor Diodes•, Presented at the IRE-AlEE Semicon.
Dev. Res. Conference, Boulder, Cole., July 1957.
3. SHIMIZU,A. & NISHIZAWA, J., 8 Alloy-Diffused Variable
Capacitance Diode with Large Figure of Merit•, IRE
Trans. Electron Devices <USA>, Vol.ED., No 5, 370-7,
Sept. 1961.
4. SUKEGAWA,T. et al., •si Alloy-Diffused Variable
Capacitance Diode•, Solid State Electronics <GB>,
Vol.6, 1-24, Jan. 1963.
5. WATSON,H.A., •Microwave Semiconductor Devices and their
Circuit Applications•, McGraw-Hill, N.Y., 1969.
6. HOWES,M.S. & MORGAN,D.V., •variable Impedance Devices•,
John Wiley and Sons 1978.
7. LIAO,S.Y., •Microwave Solid State Devices•, Prentice
Hall Inc.,New Jersey, 1985.
8. PENFIELD,Jr.P. & RAFUSE R.P., •varactor Applications•,
MIT Press, Cambridge, 1962.
9. SHURMER,H.V., •Microwave Semiconductor Devices•,
Pitman Publishing, 1971.
10. LEVINE,S.N. & KURZOK,R.R., •selected Papers on Semicon-
ductor Microwave Electronics•, Dover Publications 1964.
124
11. HASSAN, Y.M., uspectral Characterisation of Devices at
High Frequencies and Measurement Methodsu, Ph.D. thesis,
Durham Univ., England, 1984.
12. PAPOULIS,A., •The Fourier Integral and its Applications•,
McGraw-Hill, N.Y., 1962.
13. AHMED, N. & NATARAJAN,T., •Discrete time Signals and
Systems•, Reston Publishing Company, 1983.
14. JOLLEY,L.B.W., •summation of Seriesu, Dover Publication
London, 1961.
15. PUTEH,R.P., •studies and Computing Techniques in the
Spectral Characterisation of Solids•,
Durham Univ. England, 1984.
Ph.D. thesis,
16. •Instruction Manual of the Sampling Adaptor Type 158•
G. & E. Bradley Ltd., London.
17. PIPES,L.A. & HARVILL L.R., •Applied Mathematics for
Engineers and Physicists•, McGraw-Hill, London, 1970.
18. HEFFER,D.E., KING,G.A. & KEITH D., •Basic Principles and
Practiceof Microprocessors•, Edward Arnold Ltd., 1981.
19. DOWNEY,J.M. & ROGERS,S.M., •Pet Interfacing•, Howard
W.Sams and Co, Inc., USA, 1981.
20. BARNA, A. & PORAT, D.I., •Integrated Citcuits in Digital
Electronics•, John Wiley and Sons 1973.
21. GARRETT, R.H., •Analog I/0 Design•, Reston Publishing
Company 1981.
22. SEITZER,D., PRETZL,G., & HAMDY N.A., •Electronic Analog-to
Digital Converters•, John Wiley and Sons 1983.
23. CHIPMAN,R.A., •Transmission Lines•, Schaum Outline Series
125
in Engineering, McGraw-Hill, London, 1956.
24. HARVEY,A.E., •Microwave Engineering•,
London, 1963.
Academic Press,
25. TERMAN, F.E., aRadio Engineers' Handbook", McGraw-Hill
Company Ltd., London, 1950.
26. DAVIDSON,C.W., "Transmission Lines for Communications,
The MacMilland Press Ltd., 1978.
27. DUNLOP.J. & SMITH,D.G., •Telecomunications Engineering",
Van Nostrad Reinhold, London, 1984.
28. SCOTT,R.E., •Linear Circuits•, Addison-Wesley, London, 1960.
29. KATIB , M.K., •Evaluation of Harmonic Generating
Properties of Schottky Barrier Diodes•,
Durham Univ. England, 1976.
Ph.D. thesis,
30. FOX, L. & MAYERS, D.F., •computing Methods for Scientists
and Engineers•, Clarendon Press, Oxford, 1968.
31. HUNTER,L.P., •Handbook of Semiconductor Electronics•,
McGraw-Hill Inc. London 1970.
32. LIAO,S.Y., •Microwave Devices and Circuits•, Prentice
Hall, USA, 1980.
33. SHARMA,B.L., •Metal Semiconductor Schottky Barrier
Junction and their Applications•,Plenum Press, N.Y., 1984
34. STREMLER,F.G., •Introduction to Communication System•,
Addison-Wesley, 1977.
35. LATHI,B.P., •communication Systems•, John Wiley & Sons,
London, 1968.
36. CHAMPENEY,D.C., •Fourier Transforms and their Physical
Applications•, Academic Press, 1973.
1~
37. CUNNINGHAM,W.J., •Introduction to Nonlinear Analysisa,
McGraw-Hill Book Company, 1955.
38. BARLOW,H.M. & CULLEN,A.L., •Microwave Measurementsa,
Constable and Company Ltd., 1950.
39. EVERITT,W.L. & ANNER,G.E., " Communication Engineeringu,
Third Edition., McGraw-Hill Book Co.Inc., 1956.
40. KULESZA, B.L.J., •Efficient Harmonic Generation and
Frequency Changing using Semiconductor Devicesa, Ph.D.
Thesis, Birmingham Univ., UK., 1967.
41. ARMSTRONG, R., aspectral Identification of Nonlinear
Devices•, Ph.D. Thesis, Durham Univ., UK., 1983.
42. SAUL, P.H., •Evaluation of a Step-Recovery Diode in
a Broad-Band Frequency Multiplier•, Ph.D. Thesis,
Durham Univ., UK., 1974.
43. SCHEID, F., •Numerical Analysis•, Schaum's Outline
Series, McGraw-Hill Book Company, 1968.
44. ABRAMOWITZ, M. and STEGUN, I.A., •Handbook of
Mathematical Function•, Dover, 1964, New York, 1965.
127
APPENDIX A
Let i = I sinwt
i i
5 ~ (I sinwt) dt ( - I coswt) q = idt = =
Ill
i~ i ..
1 C<t> = Co
(1-
dV i = C<t> =
dt
... ~idt=
i
5 idt = ( - I
i ..
= v )1 0
K
((; v>'
dV =
i
coswt)
l 1 Co~ =
S<t> (~ - V>~
dV
dt
K t.-1 ----<0 -V>
1 - -a
I
=
= (- coswt - <-w
coswt
i
i.
K
<0 - V)~
It!-
w coswt >)
= -------<1~- I> = q~- q
. -•• q. - q
where
q = Q.
K t.-1 <~ - V>
and v = v.
Ill
128
i.e.
K q~ - Q. =
1 - ~
or
K Q16 - Q. = 1-l
1 -~ <¢ - v.>
fi 1-1
Q9' - q v =
q. Q. ~ v.
or
(1/1-1>
Q91 - q v =
v.
129
APPENDIX B
•FAST FOURIER TRANSFORM PROGRAM•
130
PEAD'r'.
10 PEM++++++++++++++++++++++++++++++++++++++++++++++++++ 20 REM+++++++++++ FOUPIEP TRANSFORM PPOGPAM ++++++++++++ 40 REM++++++++++++++++++++++++++++++++++++++++++++++++++ 100 DIM A(30)~8(30)~P(30>~PS(30>~SQ(30)~TH(30)~E(30) 110 DIM F(71)~NF(71) 120 DIM L(256>~C(256)~D(256).J(256) 500 REM++++++++++ INTRODUCTION ++++++++++ 510 PRINT" :"J!IIIJIIfiiW!JIJ!tl" 520 PRINT" FOURIER TRANSFORN PROORAt-1" 530 PRI~H 11
540 PRINT:PRINT:PRINT:PRINT 550 PR I tn II (PREss ANY KEY To cmn I t·~UE > II 560 GET A$ : IF A:f:= II II THEt-~ 56~3
570 PR HH 11 :::IJ" 580 PRINT II PROGRAN t-~m·J RUt·n·~ It¥::; II 600 GOSLIB 6000 610 GOTO 1400
"
1000 REM++++++++++ READ IN DATA PTS. ++++++++++ 1010 POKE 59426.0 1020 FORI=0T0255:POKE59426.I:NEXT:FORK=1T01000:NEXT 1030 POKE59426~0:POKE59468~192:POKE59468~224 1040 Z=PEEK(59457)-130 1~350 PRINT Z 1060 IF 2=0 OOTO 1080 1070 IF Z<>O THEN 1030 1080 FOP 1=0 TO 255 1090 POKE59468,192:POKE59468~224:POkE59426.I 1100 K=PEEK(59457) 1110 LCI>=<K-130>+5.251/127+25 1120 PRINT I;L(I) 1130 NEXT I 1200 REM++++++++++ PRINT# DATA READ IN ? ++++++++++ 1210 MF=1000:REN NULTIPLYING FACTOR USED BELOW 1220 PRINT :PRHH 11 DO 'r'OU l.oJI~::;H TO PRitH# DATA HELD It~ ARRA'r' L< I ' -:c·ll
1230 I t·~PUT 11 1 ='r'E::; ~3=HO 11 ~ Z 1240 IF 2=0 THEN 1400 1250 IF Z=1 THEN 1280 1260 PRINT :PRINT 11 I t·~PUT ERROR PLEA~:;E TR'r' AGA It~ 11
1270 OOTO 1230 1280 OPEt-..2 , 4 , 0 : CMD2 1290 FOR Y=0 TO 255 STEP 4 1300 J(Y)•HH<MF+L<Y> )f"MF
_ -._13l.B._J..('t±U~~N.T<MF!t!L<..'i±.UUt1F----- -· ------------ ~- -~--- ----- -·----- ~.- ..
1320 J<Y+2)=INT<MF+L<Y~2))/MF 1330 J(Y+3)=INT<MF+L<Y+3))/MF 1340 PRINT Y::J<Y),Y+1;;J<Y+1), 1350 PRINT Y+2;;J(Y+2).Y+3;:JO::Y+3) 136~3 NE><T 1
T1
1370 PPitH#2 138(1 CLO:::;E 2
ljj
1400 REM++++++++++ CALCULATE COEFFICIENTS ++++++++++ 1410 PPitH 1420 I t·~PUT II t·~UM8EF.: OF HAPt·lON I cs II : H 1430 IHPUT 11 :3TARTINO POHH 11 ~::;
1440 It·~PLIT 11 Et·m POitH 11 ~E
1450 PRINT :PRINT 11 DO 'T'OU WI :3H TCt CHAt·K;E THESE VALUES ·-:-' 11
1460 INPUT II 1 ='T'ES 0=NO II ; 8 1470 IF 8=1 THEH 1410 1480 IF 8=0 THEN 1500 1490 PF.~ I NT II H~PUT ERROR PLEA::;E TF.~ 1T1 AGA It~ II : OOTO 146(1 1500 IF S<l THEN 1560 1510 IF S>=O::E-1> THEN 1560 1520 IF E>255 THEN 1560 1530 IF H<l THEN 1580 1540 IF H>30 THEN 1580 1550 OOTO 160(1 1560 PPI~H :PRitH 11 ERROP ·::TAPT>1 Et·m<256 Et~[C::::;TART+1 11
157(1 1580 1590 160(1 1610 1620 1630 1640
OOTO 141 ~~1 PRitH :PRitH"EPPOR GOTO 141 (1 REM+++++ CALCULATE AO +++++ R=E-S:DT=2+n/<R+1>:A=0 FOR I=~; TO E A=A+L(I) NEXT I
1650 AO=A/R/<2~0.5) 1660 REM***** CALCULATE AN AND 8N +++++ 1670 FOR N=1 TO H 1680 t·1=1 1690 C(S-1>=0:D<S-1>=0 1700 FOR I=S TO E 1710 C<I>=C<I-1)+LO::I>*COS<N+M+DT> 1720 DCI>=D<I-1>+L<I>+SIN<N+M+OT> 1 730 t1=t'l+ 1 1740 NE:=<T I 1750 ACN>=CCE)/R/(2~0.5)/2 1760 8CN)=OCE)/R/C2~0.5)/2 1770 t·~E:>=:T N 1800 REM*********** SCALE COEFFICIENTS ********** 181 0 'r'= 1.-··'2 1:312 T=10 1814 AA=A8SO::AO> 1816 0=1 1820 FOR N=1 TO H 1830 E<N>=A8SCA<N>> 1840 NE::H t·~
1850 GOSU8 52~30
1860 ~OR N=1 TO H 1870 E<N>=A8S(8(N)) 1880 t-.IE:=<:T ~4
1890 GOSU8 52~3~3
1900 IF AA<10 THEN T=100 1910 IF AA<1 THEN T=1000 1920 FOR N=1 TO H 1930 P=1 1935 R=1 1940 IF A<N><O THEN P=-1 1945 IF B<N><0 THEN R=-1
, - ~ ~~~-'!'~=~~~~:fliE_B~<~t..f~>:.:. ~~-----$-..... _...,. ....... ~.--......... ___ ._.....,.,_ ........ __ -· -·--------·-&----..,.., . .,..,, __ .., __ _
1955 ZZ=ABS<T*A<N>> 1960 IF ZZ>Y THEN 19?0 1962 A<t-~>=~3
1965 GOTO 200~3
1970 X=ZZ-INT<ZZ1 1980 IF X>=Y THEN 1990 1982 A(N>=P*INT<ZZ)/T 19E:5 GOTO 20~30
1990 A<N>=P*INT<ZZ+l)/T 2000 IF YY>Y THEN 2030 2010 B0-1>=0 2020 GOTO 2080 2030 Q=YY-INTCYY) 2040 IF Q>=Y THEI'~ 20?0 2050 B<N>=R*INT<YY)/T 2060 GOTO 2080 2070 8(N)=R+INT<YY+1)/T 2080 SQ<N>=INT<T+SQR<A<N>+A(N)+8<N>+B(N)))/T 2090 IF A<N>=O THEN 2120 2100 TH<N>=INT<T*180,~rr+ATN(8(N)/A(N))),~T 211 ~3 GOTO 21 :~:0 2120 THo:t·D=90 21 :;:o t·~D<T t-~
2140 IF A0<0 THEN 0=-1 2150 XX=ABS<T+AO> 2160 IF XX>Y THEN 2190 217[1 A0=[1 21 :?.~3 GOTO 23~3~3
2190 W=XX-INT<XX> 2200 IF W>=Y THEN 2230 2210 AO=O*INTCXX)/T 2220 GOTO 2300 2230 AO=O*INT<XX+1>/T 2300 REM********** PRINT COEFFICIENTS ********** 2:310 PRHH 2320 PR I tH n FOURIER COEFF I C I Etn::;." 2330 PRII'H 2350 PF~INT"AO=" :AO 2360 PRitH 2370 FOR N=1 TO H 2380 PF.: I tH II t-~= II : t-~
2390 PR nn II AN= II .: A n-o : II 2400 PRit·H 241 0 t·4E~n t-~
Bt-~= II :8 ( t·4) : II
2500 REM********** PRINT# COEFFICIENTS ? ++++++++++ 2510 PRitH 2520 PR I tH "DCI 'r'OU ~·JAtH TO PR I tH# ABO'•/E RE:::;UL T::;;·::· 11
25:30 I NF'UT" 1 ='r'E::;; O=t·KI 11 :PO 2540 IF PO=l THEN 2600 2550 IF PO=O THEN 3000 2560 PR I 1-H : PR I tH 11 INPUT ERROR PLEASE TR'r' AGA I t-4" 2570 GOTO 25:30 2600 REM***** PRINT# COEFFICIENTS ***** 2610 OPEN3,4,0:CMD3 2620 PRINT:PRINT:PRINT 26:;:(1 PR I tH II FOURIER CCIEFF I c I EtHS II 2640 PRit·H" II :PRitH 2650 PR I t·4T II II • II AO= II : AO 266~3 PRitH 2670 FOR N=1 TO H 2680 IF N=10 GOTO 2750 2690 PRINT" N=":N. 2700 PRINT" At·4=" .:A<N> .: II Bt-~="8(t·4) 2710 PRINT" II. II CN=" :SG!(N) .: II PHASE=" :TH(N) 2720 PRINT
___ ---~?..~~ --~~KT. N_____ __ _____ ___ ___ ___ _ _ ____ --------- __ ---~-----~-~----_ .. _______ _..., • .,....,.. .. ·---·---•· ,..
. 2+.4a IF H< 10 THEt·~ 2::: 1 (1
2?50 FOR N=10 TO H 2760 PR I t·H II N= II : t-.1 ' 277~3 PR I tH II At-~= II :A' t·~ ' : ,, 2780 P~~HH 11
" "
279~3 PPitH 2800 t-~Ei<T N 281~3 PRitH#3 2820 CLOSE 3
133
3000 REM********** PPINT# GRAPH ? ********** :301 ~3 P~~ I tn 3020 PRitH 11 DO 'r'OU ~·JAtH TO PF.:ItH# GF.:f1PH -;:·II
3030 II'~PUT 11 1='T'ES 0=t--10 11 .:I!JO 3040 IF W0=1 THEN 3500 3050 IF W0=0 THEN 4900 :3060 P~~ I tn : PR I tn II I tWUT ERR OF.: PLEASE TR'r' AOA IN II : PP I tn 3~37~3 OOTO 303~3
3500 REM***** PRINT# GRAPH ***** 351~3 PRitH 3520 H~PUT 11 HOL·J t·lAtN '·/EPT I CAL PO ItH:::; DO 'r'OU ~·JAtH Ot~ THE (iF'APH" : :::;F 3530 IF SF<5 THEN 3570 3540 IF SF>50 THEN 3570 3550 SF=2*INT<SF/2) 356~3 OOTO :3650 3570 PRitH 35:30 PR I tH II ERROR 5 :: '·,•'EPT I CAL PAt·K1E <5~~1 II
3590 OOTO 3510 3650 REM***** CALCULATE F(T)S +++++ 3660 i<=tr+2/70 3670 FOR T=0 TO 70 3680 PS(0)=0 3690 FOR N=1 TO H 3700 IF A<N><>0 THEN 3720 3710 IF 8(N)=0 THEN 3740
3730 GOTO 3760 3740 PS(N)=0 376~3 NE><T N 3770 F<T>=PSO::H>+AO 3780 ND.:T T 3800 REM***** NORMALISE F<T>S ***** 3810 A=ABSO::F0::0)) 3820 G=-:3 3830 FOR T=1 TO 70 3840 IF A>=ABS(F(T)) THEN 3860 3850 A=ABSO::FO::T)) :3860 NE:X:T T 3870 FOR T=0 TO 70 3880 NFO::T)=INTO::SF*FO::T)/A) 3890 IF G<=NFO::T> THEN 3910 :3900 G=NF ( T) 3910 IF NFO::T>>0 THEN U=l 3920 IF NFO::T><0 THEN D=l :3930 NE::<T T 4000 REM***** PLOT# GRAPH ***** 4010 OPEN1?4?0:CM01 4020 PRINT:PRINT:PRINT 4030 PR I tH II GRAPH cm·~STRUCTED FRON FOURIER COEFF I c I Etn::; II
4~340 PRitH 11 II
4050 PF<~ItH" STARTING POitH=" :S;" END POitH="E 4060 PRINT 4070 IF U=0 THEN 4250 4080 FOR L=SF TO 1 STEP-1 4090 IF L<:>SF THEN 4110 4100 PRINT" 1 I II:: :OOTO 4170
.. _4!!0..JF._-h_<><.=.S.._F_,-,..::;3..;,>....,;..T.;,.;H;.;;;E•H.._.4 ... 1.-3,;;;0 _____ ~--------- ·----------
4120 PRINT II F ( T > I" : : GOTO 41 7Q1 4130 IF L<:>::;;F,.·'2 THEt~ 4160 4140 PRINT" 1..·'2 I": 4150 GOTO 4170 4160 PRINT" I": 4170 4180 4190 4200 4210 4220 4230 4240 4250 4260 4270 4280 4290 430(1 4::::1 ~) 432~3
4:3:3~3
FOR T=(~1 TO IF NFIT)=L PRitH" II : GOTO 422(1 PRitH". II : t-~D<T T PRII'H t.fE::.:T L PRII'H" FOR T=O TO IF HF<T>=0 PRitH" _ II . .. GOTO 4:3H3 PRitH". II : t·~E::<T T PRitH PF.: I t-H II
70 THEt~
~3 I" .: 7~3
THEt·~
I II :
4340 FOR T=0 TO 70
4210
4::::~30
4350 IF NFO:T><>-1 THEN 4:380 4360 PF~ItH". II: 4370 GOTO 451 ~3 4380 IF T=17 THEN 4440 4:390 IF T=34 THEN 4460 4400 IF T=49 THEH 4480 4410 IF T=69 THEN 4500 4420 PRINT" ": 4430 GOTO 451 (1
4440 PRit-H"rr/2": :T=19 4450 GOTO 451 ~3 4460 PF~ I t-H II fT II .:
4470 GOTO 4510 4480 PRINT":3rr/2": :T=52 4490 GOTO 45h3 4500 PR It-H "2rr" .: : T=7~~1 451 0 NE:=-:T T 4520 IF 0=0 THEN 4740 4530 PRit-H" I".: 4540 FOR T=0 TO 70 4550 IF NF(T)=-2 THEN 4570 456~3 PF.: HH" ".: : OOTO 45::a3 457(1 PRII'H". II: 4580 t·~E::<r T 4590 PRII'H 4600 FOR L=3 TO ABSIG) 4610 IF L<>SF/2 THEN 4630 4620 PRINT II -1 .. ··'2 I" .: : GOTO 4660 4630 IF L<>SF THEN 4650 4640 PRHH" -1 I": :GOTO 4660 4650 PRitH" I": 4660 FOR T=O TO 70 4670 IF NF(T)=(-l*L> THEN 4700 468~3 PRINT" ": 469e GOTO 4710 4700 4710 4720 4730 4740 4750 4900 491
PRINT". II.: NEXT T F'RII'H NEXT L PRHHtH CLOSE 1 REMII!!t!ll!lt!II!!IE!IE**
N EHO PROG? **********
l"'::l
4920 PRINT 11 DO YOU ~JISH TO HWUT t·10RE DATA OF: TO F:E-u~::;E f=tE:O'·/E DATA" 4930 PRINT II OR TO Et~[l PROORAt·1 -.II
4940 H~PUT". 1 =MC)RE 2=RE-u~:;E 3=Etm 11 ~EF
4950 IF EP=1 THEN 500 4960 IF EP=2 THEN 1200 4970 IF EP=3 THEN 9999 4980 PR ItH 11 HWUT ERROR F'LEt=6E TP'T' ACiA It~" : F·F: ltH 4990 OOTO 4940 5200 REM********** SCALINCi SUBROUTINE ********** 5210 FOR N=1 TO H 5220 IF E(N)<=AA THEN 5240 5230 AA=E ( ~~ > 5240 NEXT N 5250 RETURN 6000 REM PLOT SQUARE WAVE 6010 FOR H=1 TO 63 6020 L(H)=10:L(H+190)=10 6030 NE><T H 6050 FOR J =64 TO 189 6060 L<J>=O 6070 t·~E::<T J 6080 OPEN4,4~0:CMD 4 6090 PRINT:PRINT 6100 PRHH" DATA H~ ARRA'T' U: I ::•=10 FOR 1<=I<=63" 6110 PRUH" =~3 FOP 64<=1<=189" 6115 PRINT" =1~3 FOR 190<=1<=253" 6120 PRitH#4 613l3 CLOSE4 6140 RETURN 6200 REM SINEWAVE 6201 REM 5=1 E=255 6205 DX=n/127:2=10 6210 FOR H=1 TO 256 6220 L(H)=Z*SIN<<H-1>*DX> 62:30 t·~E>n H 6240 OPEN 5,4,0:CMD5 6250 PRINT:PRINT:PRINT 6260 PR I ~n" DATA 1 t·~ ARRA'r' L • I ::. = ll3*~3 IN.: I _:. 6270 PRHH#5 6280 CLOSES 6290 RETURN 6300 REM COS WAVE 6301 REM S=1 E=128 6:31 0 o:x:=n ,.··'64 6320 FOR K=1 TO 129 6330 L<K>=COS((K-1>+DX>+l 6340 ~~E>::T K
9999 Et·m
F.~EAD'r'.
~·JHERE 1 <=I <=255 II