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Theses and Dissertations

1-1-1976

A study of single and coupled non-lineah electricalresonators.Benton Phillips Zwart

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A STUDY OF

SINGLE AND COUPLED

NON-LINEAH ELECTRICAL RESONATORS

by

Benton Phillips Zwart

A Thesis

Presented to the Gradual-? Commit r.-?a

of Lehi'^h University

in Candidacy for the Degr&e of

Master of Science

in Electrical Engineering

Lehigh University

19?6

ProQuest Number: EP76316

All rights reserved

INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted.

In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed,

a note will indicate the deletion.

uest

ProQuest EP76316

Published by ProQuest LLC (2015). Copyright of the Dissertation is held by the Author.

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CERTIFICATE OP APPROVAL

This thesi3 is accepted and approved in partial

fulfillment of the requirements for the degree of

Master of Science.

? '2, '--' /I? Date

Head of the Department of

Ele c trica.l Engineer.\n§,

11

ACKNOWLEDGMENTS

The author wishes to thank Professor Nikolai

Eberhardt for providing inspiration and guidance

during the preparation of this thesis.

I also wish to thank Professor Daniel Lsenov

and Professor Robert Guy Sarubbi, who allowed me the

time necessary for the complation of this study.

V/hile preparing this paper, the following were

most helpfuli Mr. William P» Evans* Fr. Timothy

Spellraan, Mr. James R. Johnson, Miss Gay E. Kedden

and Mr. Christopher W. Zwart.

Most of all, I v/ould like to express my appre-

ciation of my parents for. their continuing confidence

and support.

* * 9 ni

TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS iii

ABSTRACT 1

INTRODUCTION , 2

I CIRCUIT DEVELOPMENT t . k

A - The Non-Linear Capacitor.,..*.. l±

B - The Inductor 12

C - Deriving an Output x'4.

II EXPERIMENTAL RESULTS X6

III RESONATOR ANALYSIS AND PREDICTIONS.,... 21

IV COMPARISON BETWEEN EXPERIMENTAL AND

THEORETICAL DATA 30

V EXTENSION TO TWO COUPLED RESONATORS 43

A - The Analysis 45 B - The Linear Simplification 50

C - Experimental Results 55

VI CONCLUSIONS AND SUGGESTIONS FOR

FURTHER STUDY 63

1

IV

TABLE OF CONTENTS (cont.)

Pao a?e

BIBLIOGRAPHY 66

APPENDIX A Determination of Variables in

Equation 1.1 6?

APPENDIX B Computer Program to Plot the

Theoretical C(v) Characteristics

for the Fabricated Non-Linear

Capacitor 71

APPENDIX C The Transformation of Equation

3.11 into Equation 3.12 72

APPENDIX D Computer Program to Solve Two

Specific Simultaneous Third

Order Equations f . 75

APPENDIX E Computer Program to Plot the

Theoretical Response of Two

Coupled Linear Resonators 76

VITA 77

LIST OF FIGURES

Page

Figure 1 Non-Linear Capacitor Schematic... 1±

Figure 2 Determination of Varactor

Capacitance Characteristic

Figure 3 The Peak Rectifier ... lif

Figure 4 Single Resonator Schematic....... 15

Figure 5 Experimental Determination of

Resonator Response „ *°

Figure 6 Single Resonator Model 21

Figure 7 Resonator Enclosure , ^j-

Figure 8 Coupled Resonator Model hh

VI

LIST OF GRAPHS

Graph 1 Measured Varactor C(V) ............

Graph 2 C(VS + VB) for Various VB .,

Graph 3 Non-Linear Resonant Response)

Experimental ....... o ....

Graph ^ Non-Linear Resonant Response,;

Experimental

Graph 5 Non-Linear Resonant Response)

Graph 6 Non-Linear Resonant Response;

Predicted «, ,.,

Graph 7 Non-Linear Resonant Response;

Predicted r

Graph 8 Parametric Root Locus .............

Graph 9 Non-Linear Resonant Responses

Comparison

Graph 10 Non-Linear Resonant Response)

Comparison

vii

8

10

18

19

26

2?

28

29

31

32

LIST OF GRAPHS (cont.)

Page

Graph 11 Non-Linear Resonant Responsej

Comparison ........... 33

Graph 12 Non-Linear Resonant Response;

Comparison 3^

Graph 13 Non-Linear Resonant Response;

Comparison 35

Graph Ik Characteristic Parameter

Comparison 37

Graph 15 Non-Linear Resonant Response;

Comparison ....*....«.«.•.*....<. jy

Graph 16 Non-Linear Resonant Response;

Comparison ,,...„,..„......,,,.** kO

Graph 17 Non-Linear Resonant Response;

Comparison . »...«.«..... «.«**«»«> **■! > •

Graph 18 Characteristic Parameter

Comparison .................«,.. t . k2

Graph 19 Linear Response - Driven Resonator 51

Vlll

LIST 0? GRAPHS (cont.)

Pa~e

Graph 20 Linear Response - Coupled Rejsojp^-tor 52

Graph 21 Linear Response - Driven Resonator 53

Graph 22 Linear Response - Coupled Resonator 54

Graph 23 Coupled Non-Linear Resonant Response $6

Graph 24 Coupled Non-Linear Resonant Response 57

Gra;ih 25 Coupled Non-Linear Resonant Response 58

Graph 26 Coupled Non-Linear Resonant Response 59

Graph 27 Coiiplec1 Non-Linear Resonant Response 60

Graph 28 Coupled Non-Linear Resonant Response 61

Graph 29 For the Determination of Variables in Equation 1.1 68

XI

ABSTRACT

A theoretical and experimental study of the

behavior of single and coupled non-linear R-L-C

resonators is presented. The non-linearity is confined

to the capacitance and is realised by two varactor

diodes connected in antiparallel to provide a symmetric

characteristic.

Families of resonance curves are plotted for both

single and two caoacitively coupled resonators, showing

the transition from the linear to the non-linear

domain of operation and exhibiting various jump

phenomena.

The theory is based on the first harmonic approx-

imation, and a comparison between the predicted and

the observed response is made. An extention of the

first harmonic approach to two coupled resonators is

attempted. It yields a set of four simultaneous third

order equations which contain the amplitude and phase

response. Since no exact solutions exist, best fit

approximations would have to be made. This has not been

attempted, however.

INTRODUCTION

It may be stated with relative safety that almost

all physical systems are, to some degree, non-linear.

In general, these non-linearities tend to be negligibly

small, thus lending the system to solution by any one

of the several conventional methods which are tivailable

for solving linear equations. Cases do exist,, however,

in which the non-linearity is too large to be neglected

and still retain an accurate model of the system. It is

in these instances that a different approach must be

taken in order to develop a solution.

The first harmonic method^ produces as" its solution,

a prediction as to the amplitude and phase of the steady

state response of any general system with respect to the

amplitude and phase of some sinusoidal driving force.

The response is assumed to be of the form:

B| sin («/t ,f Bz CCJ wt

which is then substituted back into the differential

equations describing the system, yielding in this case,

two simultaneous non-linear equations for B^ and B2«

The higher order harmonics, which are produced by all

non-linear systems, are assumed to be negligible in this

approach, and it is thi's assumption which limits the

precision of the solution. The method may be extended

to include these higher order harmonics , but if the

magnitude of the induced non-linearity is kept rela-

tively small, then the increase in precision gained

does not justify the large increase in algebraic

complexity.

Since the range of validity of the first harmonic

method is difficult to estimate, experimentation is

heavily relied upon in order to delineate the useful-

ness of the theory. Also, this theory does not appear

to have been extended to two coupled resonators, as no

previous experimental results for this case were avail-

able.

CHAPTER I

CIRCUIT DEVELOPMENT

A - The Non-Linear Capacitor

The system to be analyzed is an R-L-C resonant

circuit with a non-linear capacitor inserted in place

of the usual linear one. This capacitor was constructed

by using two varactor diodes suitably biased and connec-

ted, as shown in l?i^ure 1.

Figure 1

Non-Linear Capacitor Schematic

It was desired to produce a non-linearity that would

be symmetrical with respect to V ~ 0, i.e., 0(V) = C(-V).

Providing that | V | ^ Yg , and that the n,~V character-

istics of diodes Dj_ and Dj are the same, the circuit

presented in Figure 1 will fill the requirements.

The capacitors labeled OR are D.C. blocking capacitors

used to set up proper "biasing of the varactors and

still allow the A.C. signal to pass. This requires

that CR be very much greater that the nominal varactor

capacitance? a condition which v.-as easily met. The

resistors labeled R were inserted as protection for

the diodes in the event that a battery should be put

in backwards. With the batteries correctly installed,

tha resistor has no effect on the circuits operation

since virtually no D.C. current flows (with 10 volts

reverse bias, no current was detectable by a micro-

ammeter). The requirement that the . two diode char-

acteristics be the same was the most difficult to

meet, since it had to be satisfied by trial and error.

The method which was used to determine the C-V

characteristic of a given varactor is shown diagram-

matically in Figure 2.

,

L ..JTTL

~2

cp P.lotte r + i

~<

■ h Cap.

^eter ro

'/rt4i,i?\ Y

\

_.J x

Figure 2

Determination of Varactor Capacitance Characteristic

Tho capacitance meter measures small signal

capacitance at a frequency of one megahertz, and

produces as one of its outputs, a voltage which is

proportional to the capacitance under test. The

inductance labeled L i*s one which must be large

enough to appear as an open circuit to the A.O. test

signal, yet allow the "biasing voltage ramp to build up

freely across the diode. Since the slope of the voltage

ramp was very much less than that of the applied one

megahertz test signal, little difficulty was encoun-

tered in meeting this requirement. Crj again represents

a D.C. blocking; capacitor which prevents any D.C. current

from flowing into the capacitance meter. Its value was

chosen to be very much greater than the nominal viractor

capacitance for reasons previously outlined. The measured

capacitance value was then plotted against the biasing

voltage to provide a permanent record of a given diode

characteristic. Four diodes were then found having

characteristics which deviated by les3 than 1% from

the group average over a ten volt biasing range. A

typical experimental plot is exhibited as Graph 1 .

In order to faithfully model the C~V character-

istic for the diodesi the variables v/ithin the thaor-

etically derived equation fox* varactor capacitance-,

given asi

i.i c(v) - K (B + V)8*

were iteratively adjusted until a close match between

the experimental data and the equation was reached

(see appendix A). The values obtained are c< = -,k^r^kk,

K » 56.0195 x 10~12 farads, 3 = .7^9^ volts* where V

The graph as displayed was computer produced to allcv simple scale changes as required by thesis format. It was made by feeding the computer a great number of data points which wore taken from the actual experimental graph produced as seen in Figure 2.

0.1 < K

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r_

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01 si-

V

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_^ ID CO

CSJ OL UJ 51

ID CO Cs.'

o o o a

GGc&UJ-CttJd NJ f A )3 8

is the instantaneous value of the reverse biasing

voltage which appears across the diode. In the circuit

under analysis, both diodes are reverse biased to the

same D.C. level» but are oriented in different direc-

tions. In this way, the applied signal tends to increase

the capacitance of one diode while decreasing that cf

the other. The equation for the overall capacitance as

a function of the applied voltage, C(V) is p;iven byt

1.2 CfVj= KRB + VB+V^ +• CC?*VB -V)^]

where VB is the D.C. reverse bias applied by the battery,

and V is the A.C, signal voltage. It should be noted

that this equation breaks down when |7 | exceeds B+Vg .

This point corresponds with having an infinite forward

current flowing through one of the diodes, and therefore

represents an extremely unlikely condition. C(V) was

plotted for several values of V'B on Graph 2 (see appendix

B) and 9,6 volts was selected to be a suitable value

for the biasing level.

Upon the establishment of a fixed value for the

bias, the equation for C(V) was approximated by a two

term polynomial in V, for which the coefficients were

selected to fit the values produced by equation 1.2 .

The polynomial fit was necessary to facilitate the

to •p

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oocrijs

o

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&y.:ri.L STTtS 9d.r 15 seros;

Sat^yj-03Jd NJ (A)3 10 ^

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exnansion of terns in the first harmonic method, and

produces an error of less than l.A-/.'., provided that |vj

is less than 6r.5 voltn. The polynomial used was

obtained by modifying a truncated series expansion,

and is given byi

1.3 C(V) = C0 + K V2"

where C0 = *39.2M*6 x 10""12 farad, and K = .1568 x 10~12

farad/volt~.

11

B - The Inductor

Since1 the varactor capacitance was measured at

one megahertz, it was decided to select an inductor

which would cause the resonant frequency of the circuit

to he in the neighborhood of one megahertz. Assuming

a nominal capacitance value of 39 pfr "the inductor should 2 2-1

have a value given by L ~ (W f0 C) ~ = .6*4-9 rnhy.. In

order to build as high a Q factor as possible into the

circuit, the inductor was wound by hand using a special

multi-filament wire, each filament of which was individ-

ually insulated from the others in the bundle. The form

into which the wound coil was placed has a variable air

gap in the magnetically permeable core material to

allow fine tuning once the inductor was assembled. When

complete, the coil was found to have an inductance of

.780 rnhy and -; D.C. series resistance of 2-°-. This

would correspond to a Q factor of Q = (L/C) ?/R ~ 2203,

however a. much lower value has to be expected for

reasons such as the skin effect and ferrite losses.

In order to couple out of the resonator without

destroying its high Q factor, the applied load had to

be very small. This was facilitated by winding ten turns

of wire over the 120 turn main inductance coil. The

impedance transformer thus formed, having an impedance

12

transformation ratio of (120/10)2 = 1^4, virtually

assures that the applied load of a peak rectifier and

a ten megohm oscilloscope prooe will have very little

effect upon the resonator's operation. Coupling into

the circuit was achieved in a similar manner by placing

two turns of wire around the outside of the completed

inductor. Due to the circuit's high Q factor, even

this very weak input coupling is able to provide a

sufficient driving force near resonance.

13

0 - Derl'ring in Output

120 turn

C ■R V out

Figure 3

The Peak Rectifier if-

The peak rectifier, which is attached to the

output coil on the inductor, is depicted in Figure 3.

Its components must be chosen in such a way that only

a small load will 'be transformed back into, the main

resonator, yet the decay time must be fast enough to

accurately follow the envelope of the response as the

driving frequency is changed* Again, the relatively

large difference between the resonator operating

frequency and the sweep rate allows an easy solution to

the problem. Choosing the capacitance value to be

.013 /U.farads and the resistance value to be 100K ohms

yields a time constant of 1.3 r's which is sufficiently

fast for the upcoming experimentation.

1**

^

...J"

n 'out

o-

o »«

+ L i

cB

zr Di in

C B

D2

vf3~

J 'J

J21

Figure 4

Sitifrle Resonator Schematic

The completed resonator, as shown schematically

in Figure *+■, was placed into a metal "box in order to

shield it, both mechanically and electromagnetically,

from external disturbances.

15

0'.l\T!LE'J. 11

)'XPj;Rir.'^:TAL FAULTS

Hefore it was possible to begin a meaningful

numerical analysis of the system described in chapter

2, there were certain parameters which had to be

experimentally determined, such as the output trans-

formation ratio and the actual loaded Q factor. The

experiment which was designed to determine these factors

is diagrammed in Figure 5.

Variable D.C. Offset

I N

v.c.o.

Qutpu t

Resonator

Peak Rectifier

Output

Figure 5

Experimental Determination of Resonator Response

16

A family of characteristics wm; :<;renerated by

varying the output amplitude of the Volta.re Controlled

Oscillator (V.C.O.)» and then sweeping the frequency

up through resonance and back down again. The curves

thus generated are exhibited as Graphs 3 anc* ^' It can

be seen that as the amplitude of the driving force is

reduced, not only does the amplitude of the response

decrease, but the non-linear characteristics of the

response also become smaller. This diminishing; non-

linearity is evidenced by the returning; symmetricallity

of the response about the frequency of maximum output.

It is from the curves produced with the resonator

operating with a minimal non-linearity that the Q factor

may be determined.

The Q factor was found by knowing thatr

2a ^ = BW

where BW is the -3d3 bandwidth of the response and fQ

is the center frequency. The Q factor may then be

related to the elements of the circuit through the

equation i

Q. - ( C R" 2.2 Q. = V C K* ,/

Since the inductor and the capacitor were assumed to

17

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have the values which wore experimentally determined

prior to their- installation in che circuit, the only

parameter which coul'i he adjusted in equation 2.2 to

provide equality was R. This is reasonable in term.? of

the model since the resistor is the element which

produces loss in the system, and a low Q factor is

indicative of higher losses.

From one of the more linear response curves, the

bandwidth was found to be ft.299 kHz with a center

frequency o.f 9i0.ftl4 kHz. Substituting this data into

equation 2.1 yields a Q factor of IC9.7 • Solving i

equation 2.2 for R gives R = (L/C)VQ. Using the

previously determined values for Q, L, and C in this

equation produces the value of R = !-VQ*Ch ohms.

To determine the voltage transformation ratio,

the resonator was driven by a fixed amplitude, 900 kHz

sine wave, while the voltage across the primary coil

(120 turn) and the output coil (10 turn) v/ere simultan-

eously monitored. The ratio of these two voltages forms

the transformation ratio, T. This value was. found to be

T = Vout/Vprinary = .53/6.0 = 8,ft33 x 1CT2. T was then

used to calibrate the Y axis of the experimental data

seen as Graphs 3 and k.

20

OHaPT^ III

RESONATOR ANALYSIS AND PREDICTIONS

In order to begin the analysis, the schematic

shown in Figure ^ was first reduced to the model seen

as Figure 6.

1 t

I y-

R

L

VR I'V

Figure 6

Single Resonator Model

C(V0)

Prom this point, differential equations were developed

in keeping with the assumptions set forth in the first

harmonic method; said assumptions being*

V< - A Sin cj-t

21

Vc - Bf S/n OJ r + B^ cos Wt

Proceeding further givest

3.1 Vs « V* + VL - Vc

JcCVc) Vc: - \77vTs 4±

yields*

3^ Vr< - X R

Substituting 3«? into 3'3 and 3.^+ produces*

Differentiation of equation 3.5 givest

3-7 vL- L dc j-t + L C LVc)—j^iL—

Application of the chain rule to the first term of

equation 3*7 yields«

22

Frot-i this point on, in order to simplify the notation,

~~ vvill be replaced by Vc , and • ,-^1 z<- will be replaced

by \/c .

Equations '}*6 and 3»8 are now substituted back

into y»l to producet

3-9 Vs * RCCVJ Vc t L-J^- V/ HCtt)Vc - Vc

Recalling equation 1.3 and differentiating results ini

3 10 JjCCVc) = 9 , 3-10 JVc ^*VC

Substituting 1.3 and 3.10 into 3*9 givess

3.11 VS = R(C.-H<.V*)VL +?J<L\/CVC +L(Co<-KVcl)Vc + Vc

The assumed solution for V is now inserted into 3»H»

the indicated mathematical operations are performed

(see appendix C), and the resulting equation appears as

3.12 . Remember that whereas the operations called for

by 3*11 will produce third harmonic terms, those terms

are neglected by using the rationale that for a small

non-linearity, the third harmonic terms are insignificant

with resuect to the fundamental terms.

23

4 eiJ 5/d a;-t

,2.17 -»

~ ^ + o^j cos c^±

In order to solve this equation for all values of t,

it must be true that the coefficient of the sine term

on the right side is equal to A, and the coefficient

of the cosine term is equal to zero. This leads to the

following pair of coupled, non-linear equations»

4(l-LCoCU~) __ LCJ 3

2*

The simultaneous solution of 7,1'} and }. l!'r should

predict the behavior of the circuit shown in Figure k.

Since the peak rectifier produces as its output a

signal equal to the magnitude of the response,

2 2 ? (Bi^ + B2 ) wag formed and plotted as a function of

frequency for different driving force amplitudes for

later comparison with the experimentally derived data.

The results produced by the solution of equations 3*13

and "},lh are shown as Graphs 5, 6, and ?.

Since analytic methods are not available for

handling coupled third order equations, a computer sub- 5

routine had to be used to solve equations 3.13 and

3.1'f- . This subroutine makes use of Brown's method for

iteratively solving general non-linear systems of

equations. The program (see appendix D) which called this

subroutine was written in such a way that the frequency

at which the jump occurs was precisely determined. Both

amplitude and phase information may be obtained from

Graph 8r upon which the jump appears as a straight line*

25

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CHAPTRK IV

COMPARISON BliTV/SEN EXPERIMENTAL AND THEORETICAL DAT*

Upon comparing Graphs 3 and h with Graphs 5 thru

7, the impression is that the numerical method has been

a fuirly ^ood predictor of the actual response. This

comparison, however, is somewhat misleadin& since the

frequency ranges (X axes) were laid out to different

scales. In order to "better assess the effectiveness of

the first, harmonic method in predicting the actual

response, Graphs 9 thru 13 were prepared. Each displays

the overlay of an experimental curve (indicated by D ),

and a theoretical curve (indicated by O )i plotted to

the same scale and chosen such that the maximum ampli-

tude attained by the prediction corresponds with that

reached by the experiment. ^

Upon examining these graphs, it was evident that

whereas-some discrepancies did exist, the general shape

of the predicted --.urve, including the jump discontinuity

and the center frequency, was borne out by the experiment

Some of the factors which may account for the

observed discrepancies include the Q factor, the true

center frequency, the experimentally derived transfor-

mation ratio T, the accuracy of the method used to model

30

a> CO d o a; Cn CO to C <» o (X

to TJ d) <D K

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"the non-linearity, and the inaccuracies inherent in the

first harmonic method itself.

In an attempt, to remove the error producing; effect

of sons of these parameter.7:, a p-rnph was prepared

having (^f- wi)/wl as the X coordinate, where ^f is

the frequency of trie jump up and c-Jl is the frequency

of the ju.-cp down; and (Amax - --'-(nin-^max as ^e Y

coordinate, where Amax is the l^r^e^t amplitude at

W - O^i , and Amin is the largest amplitude at ^ = c^f ,

It was the intention that such a plot would remove the

effects of error produced hy the plotter scale factors,

the transformation ratio 0.', and small errors in the

center frequency. Graph 1^, therefore, should provide

a more faithful indicator of how well the first harmonic

method performed<■

In the interests of producing the hest possible

model for the upcoming coupled resonator experiment,

the parameters K and Q were altered in order to more

closely match the prediction to the experiment. The -12 , 2

values settled upon are K = .22 x 10 farad/volt ,

and Q = 100. These values are of the same order of

magnitude as those which were theoretically derived,

hut the prediction which they provide agrees much

more closely with the experimental data. A comparison

36

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< M

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of the numerical results using these new vaiua3 with

the experiment is exhibited as Graphs 15» 16, 17 and

18.

38

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CO C O CO & Vi W C CD o cc p

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42

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EXTENSION TO T'.70 'lOV^LED PToOri.Vi'OS.l

With the development of a process which is capable,

within certain limits, of predicting: the? "behavior of

a single non-linear resonator, it should prove tn be

an interesting exercise to test the method's ability when

applied to the case of tv/o coupled resonators.

The construction of the second resonator proceeded

in much the sane fashion as that of the first, with care

beine: taken to duplicate every possible detail of the

first resonator in the second. It was for this reason

that at the beginning of the experiment„ four similar

diodes were chosen instead of two. The second coil was

wound in precisely the same way as the first, in order

that their characteristics match as closely as

possible. Even the physical layout of the first circuit

was duplicated down to the positioning of the circuit

board mounts and the sizes of the metal boxes into which

the'finished resonators were placed. The tv/o completed

boxes ware then joined to a third box containing the

coupling capacitor? the finished structure, then,

resembling the letter H (Figure 7).

^3

Figure 7

Resonator Enclosure

l!+I3

Figure B

Coupled Fesonator Model

W

A - The Analvsir-

The analysi-■ begins wiLh Fl.Tirre 8, arid oroceed>? .In

the inanner developed in chanter III and appendix 0.

It should be noted that since the individual elements

of the two resonators were Hatched, the same symbol

denoting; an element's value nay be used regardless of

which resonator that value came from.

From Kirchoffb voltage law» ;

5-1 _

5.2

5.3

With t

Vs= R*: - KX.+ L^- V,

v, = Vct + v2

V; = RX, + L R±z ~ L 0J-7-

5.b

5.5

5.6

I, - CCV,) V

*2 = c(vo vz

Substituting; 5*5 into 5.2 and solving for 13 ffives

5.? X? - Cc (v, -Vz)

Using; 5*k, 5.6, and 5.7 in equation 5.1 and 5.3

produces, ut>on combination with 1.3s

<*5

•R V. -- fi rc v, - RCC V-, + LCc V, - L cc vL

+ f< ( C0 + K V?) \>( i- L (Co ■* K Vtx) V, '

+ ZKLV, Vtx + V,

In accordance with the first harmonic method, the

following assumptions were madei

5.10 V5 - A Sin cut

5.11 V, = 8, S/n UJ t + 0Z cos c^r

5.12 V « B^'Sin Cv't- "•" 84 cos tu'tr

Usinfl: appendix equations C.l, C.2, and CO in conjunction

with the substitution of 5.10 thru 5-12 into 5.R and 5.9

yieldsr upon simplification:

k6

5.H 0 -

5.16

rv - — s^e, r ( K

5.1^ O

+ -IT" ^ LGv7

' - ^

V '^T^ +(liT^^/JBl

4- ~ B> + (

__ ^LCCCJ y RK B.

R K

fCc p A A RKu/

5.15 0 6, - -1T~B*BX

— B 3 (-

iLtCt+Cc)^ J ^ R i\ ^KW7 D^ B.

+ JLkJkis^ K _ _fi£

O = 3 L tA>»

« 0/

K B3V Bf

4 L a u; Ric -B, -

4 a K S,

It is interesting to note the similarities between

equations 5.13 and 5.15 as well as those "between 5.1*+

and 5«l6 . It may also be noted that in the degenerate

*7

oar.e where Cc - 0, equa"i;ions 5.1'3 an.1 jil!'r reduce to

equations 1.13 ?-l^;'1 l.li: . Thic is exactly what should

happen, since, when Cc = 0, tna sysoern has been

siniplj.f\cd to the ca?° of one resonator.

When the attempt was made to solve equations 5« 13

thru 5.16" via the computer, it was found that the task

wag too complex for the method which previously had

provided satisfactory results. Upon a closer exami-

nation of the type of solution which was being searched

for, a reason for the difficulty was uncovered. If each

of the equations 5.11 thru 5.1^ were put into the

following formi

then the following 'four equations would resulti

from 5.13) B, *f, ( I3L) 8, } B, t lu) 5.17

from 5.1/0 B, " fz ( Bz } B3 j Bi , Cj) 5.18

from 5.15) g, - f3 ( BL , S3 f S^t 0J) 5.19 r

fronjft.16) B, ^ f^ (I3Z , Bit B1 , Cu) 5.P0

For each value of d/' , equations 5.17 thru 5.20 must be

solved simultaneously, and this is where the problem

Jf8

lie:-.. Co. T:■: nin/"': o.-taatio i 5.1-7 wit;-! j.ir.-, for instance,

will produce a particular su/'face of solutions in

Dgt ?.^j, ?.>; space. 3in.iln.rly, the pairir t?: of equation

6.17 with 6.19 and 6.20 will produce two more surfaces

of solution in Bj>, B3, B>|. apace, for a total of three(

surfaces. It must now be realized that the intersect i.01

of any two of these three surfaces- will produce a curve

in T?2» B3, B/; space, and that the root finding routine

must look for the simultaneous intersection of the

three resulting curves. Si.-;ce the curves lie in three

space, there is no apparent reason to expect that all

three curves will intersect at one point. It could, in

fact, be the case that no curve intersects any other

curve at any point, in which instance a simultaneous

solution of equations 5-13 thru 5.16 simply does not

exist. This, of course, is just a restatement of the

mathematical theorem which stipulates that solutions

to n simultaneous non-linear equations in n unknowns

only exist in special cases. Very likely, however, thru

the approximate nature of the first harmonic approach,

the three curves come close to each other at one or

more points of the variable space, and if these points

could be located, an approximation would result,

comparable to that produced for the single resonator.

^9

)

B - The Linear Simplification

In order to provide some form of comparison for

the experimental data gath vred from the coupled res-

onttora, a computer program wao written. (see appen-

dix E) to plot the magnitudes of Vj_ and Vg versus

for the circuit of Figure 8. The difference here is

that the capacitors no longer have their non-linear

characteristics. The Graphs (19 thru 22) thus

provided by varying the coupling capacitance should

resemble the experimental curves, but for small driving

force amplitudes only. For larger amplitudes, the non-

linearities become significant, thus invalidating the

simplified linear model.

^

50

o-.

599' 6 ttOL

'■3un

lit"l tQL-

IJldlvH I A

51

/.or

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en O r— CU.

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a:

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LLJ CO

CJ

oo

CE: LLJ

_J

sec 2 lit1 I /St"

52

o

o a

LO

a a a

e

LO

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LLJ

(X

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U.J

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ce: 1

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cr-

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r

y

3an.LJidwy IA 53

P.

O O

Cvi CM

< PC o

T^S'lc

5^

C - Kxperirnen fcal I:o:-,ults

The experimental setup is very similar to that

shown in Figure 5i the only difference being that th*re

are now two output points, one on each resonator. While

one output was-providing data for the plotter, the other

was connected to a dummy load of 50K ohms in order to

simulate the plotter loading. In this way, the possible

effects of moving the plotter loading point from one

resonator to the other were kept to a minimum.

The experimental data, seen as Graphs 21 thru 28,

was obtained by fixing the coupling capacitor's value

( Cc ), and then sweeping thru the indicated frequency

range for several different driving force amplitudes'.

Because the second resonator's transformation ratio was

slightly different than the first ( .10666 compared to

.08833 )» the plotter scaling had to be changed when the

output point was changed from resonator 1 to resonator 2.

For a given value of the coupling capacitor, care was

taken to insure that the frequency ranges spanned were

identical, thus facilitating a direct comparison between

Vi and V2 *

As can be seen by a comparison of Graphs 19 thru 22

with the lower amplitude curves of Graphs 23 thru 28, a

definite correlation between the linear and the non-

5 c

- C tl c,c = :>o PI

^7.805

GRAPH 23

51.581

Radian Frequency

COUPLED NON-LINEAR RESONANT RESPONSE

56

55.356xlo5

Cc '= 2.8 p

51.728

GRAPH Zh

v

5^.78^

Radian Frequency

COUPLED NON-LINEAR RESONANT RESPONSE

57

57.3 39x10'

O

r-i CM

VP\ > >

CM C) to O a)

w -F -p P* d cd < o o PC •H •H o •a Ti

e c M M

'CM

o o

00*^

8pnq.TT.fiWV asuodsay

58

x CM

CM W o 01

•\D *?»■

• O VO a, U"Y

w

>> EH V ^ c <r.

t- O <-• 3 o cr4 co CD W *-< K

p^

o co w CO .-i F—■

---,-»• T3 H • & M vn tc c vr»

o

w a. o o

OC VA

-CA

^J- VA

Cr- = 1.0- of

53.^87

GRAPH 26

55.510

Radian Frequency

COUPLED NON-LINEAR RESONANT RESPONSE

57.53bx.l0-

59

VO-

Cc = 0.8 pf

-P •H rH

E <

CM

-=i--

C\J-

GRAPH 27

CD •cs

-p •H t-\ P- E <

VO

•=}-"

CV-

52.139 5^.229 Radian Frequency

56.318x10'

COUPLED NON-LINEAR RESONANT RESPONSE

6o

Cr = 0.2 pf

GRAPH 28

53-715 55.590

Radian Frequency

57.^65x10'

COUPLED NON-LINEAR RESONANT RESPONSE 61

linear casaa exirjta. The decrease in the separation

and eventual elimination of the double peak3 in Vj_ and

V2 as Cc i-j reduce-."!, matches the linear prediction

fairly well. For the lar^-vr amplitude responses,

radical deviations from the linear case are observed

and may be seen quite clearly in Graph 25.

At thi3 point no discussion of the particulars of

the non-linear resonance curves will be attempted.

They speak for themselves and indicate that much has

yet to be understood.

62

CHAPTER VI

CONCLUSIONS AND.SUGGESTIONS FOR FURTHER 3: JOY

The first harmonic method has shown itself to be a

valid and fairly straightforward technique for the

solution of simple non-linear systems. The method pre-

dicted the appearance of the (jump discontinuity, thus

demonstrating that the equations developed during the

analysis do indeed model the system. When the ?;y3tern

bscomes more complicated, as in the coupled resonator

experiment, modeling equations are generated, but their

solution would be a study in itself. Previously men-

tioned was the problem of verifying the existence of

any solution at all. There also exists the problem in-

herent in most numerical root-finding routines of sup-

plying a first guess. Numerical methods often require

an initial value, from which the actual solution is.

iteratively derived, Upon reaching a point of discon-

tinuity in the non-linear response, two solutions exist

simultaneously, and this is usually enough to upset,

standard numerical routines. Also, at this point of

discontinuity, there is ouch a large change in the true

solution that the previous solution, which is usually

U3ed as a first guess, is actually a fairly poor guess.

63

V

Upon receipt of such a poor initial value, it is not

uncommon for a root-finding algorithm to diverge.

In the interest of continuing this line of research,

the .following is set forth:

Insight into the backwards-leaning' shape of the

single resonator response can be gained by looking at the

C(V) curves plotted as Graph 2. As the amplitude of the

signal increases, so does the average capacitance of the

varactor diode combination. This in turn lowers the

apparent resonant frequency of the circuit. Assuming

that the driv*e frequency has starred above resonance and

then been decreased, upon reaching the apparent resonant

frequency, any further decrease in frequency will:

1) Decrease the response amplitude, which will

2) Decrease the average varactor capacitance, which will

3) Increase the apparent resonant frequency, which will

U-) Further decrease the response amplitude.

It is this sort of positive feedback which causes the

abrupt jumps in amplitude.

In examinating the coupled resonators, it .should

prove interesting to study the phase relationship between

V^ and V2 in order to gain some understanding as to the

manner in which energy is coupled between the two reso-

nators. Once this has been accomplished, perhaps a

64

theory can be developed which will explain the unusual

shape of the response ^een in Graph 25.

Another possible area for experimentation would

ensider the caw^ of two coupled resonators in which one

resonator has a backwards-leaning response, but the other

has a forward-leaning response. The resulting bandpass

filter should have so;.;a unique lock-in and cut-off prop-

erties.

Lastly, it roay be possible, using computer programs

which are different than those used within the context of

this ]">aper, to obtain approximate solutions to equations

5.13 thru 5»!6» 1^ this should occur, a comparison be-

tween the resultant prediction and the data presented

v/ithin this thesis could prove to be highly informative.

65

BIBLIOGRAPHY

1. Blaquiere, A. Non-Linear System Analysis.

New Yor:<: Academic Press, i960.

2. Zwart, B. P.. "The Third Harmonic Method" A term

paper for Prof. N. Eberhardt, Lehigh University,

EE 407 (December, 197*0.

3. Deboo, and Burrous. Integrated Circuits and

Semiconductor- Devicest Theory and Application,

Page 566. New York: KcGraw Hill, 1971.

4. Dnsoer, and Kuh. Basic Circuit Theory, Page 222.

New York: KcGraw Hill, 1969.

5. Th> International Mathematical & Statistical

Libraries, Inc. "Subroutine Zsystm" The Simultan-

eous Solution of N Non-Linear Coupled Equations.

Texas, The IKSL, Edition 5, 1975.

6. Selby, S.K. Standard Mathematical Tables,

Twentieth Edition, Page 189. Ohiox CRC Press, 1972.

66

APPSINDIX A •>

Do t.ii-.nination of Variablos in Equation 1.1

The theoretical ^quation-^ is:

C(Vr) = K(Vr + B)0<

for which o< , K and B are Unknown. From 1.1 it i3

predictable "hat a plot of Ln(C) versus Ln(Vr + B)

should produce a straight line with a slope of* <=< .

The physical interpretation of the quantity B is that of

the diodes contact potential, and since the varactors are

silicon devices, .choosing B ~ .7 should be a good first

guess.

Ui'ing data taken from Graph 1 and performing the

logarithmic operations indicated above, it was found that

a straight line gave a very good fit to the converted

data points (Graph 29). The fact that the straight line

fit the data well, even for values of Vr which were close

to zero, indicated that the initial guess of B = ,7 was

indeed a good one„

From 1.1 it follows naturally that>

From the straight line fit, the following data points are

obtained:

67

o CM

c «M o o c ni

o\ o 3 C\J •H D

-P o> w Oj PH C C < •H •H oc E o $-. c;

<D o> -H i-t <D ,0

X(

^ su o cd

PH >

y

v^v

>—- C

^1

vn

0*17

<J

(C/)ui

68

0't

Data X: Ln(Ci) ~ 3.801 Lr.(Vri f .?) ~ .50

Data 2: Ln(C2) » 2.992 Ln(Vr2 + •?) = 2.30

Using this data in equation A.l yields ^< = -.449^4 .

Taking the log of both sides of equation 1.1 gives 1

A. 2 Ln(G) = Ln(K) + o<. Ln(Vr + B)

Substituting data 1 into A.2 produces:

3.801 - Ln(K) - . 22*4722

Which solves to:

Ln(K) - 4-, 0257, or K = 56.0195 PA

As a cheek, using this result in equation A.2 with data

2 should also produce equality.

*>

2.992 = -.44944 x 2.3 + 4.0257

2.992 « 2.9919 Checks.

At this point., the only variable in equation 1.1 which

has not been determined is B. From the .data point G =

63-75 pff Vr « 0.0, 1.1 is solved for:

B = (C/K)""1//c< =..7**94 volts.

This value of B is sufficiently close to the value

initially assumedf that a second iteration of the pro-

cedure outlined in this appendix would not produce a

significant change in either o£ , B or K.

69

Equation 1.1 is therefore specified asi

V) = K(V + B)c<

V.'h'-tre i

K « 56.0195 pf

B ~ .7^94 volts /{

V = applied reverse bias in volts

70

APPENDIX B

Computer Program to Plot the Theoretical C(V)

Curve for tha Fabricated Non-Linear Capacitor

This Fortran program was written by the author so

that a plot of equation 1.2 might be displayed.

1.2 ccv;« K[_(8+ve-vf~+(e + vB-v)^

Several plots were made on the same axi3 by setting Vg

to a particular value and then sweeping V from 0 to Vg.

This program has been placed in the Lshigh Univer-

sity Computer Center program library under the name of

Plotter.

71

J

APPENDIX G

Tho Transformation of Equation 3«H into Equation 3«12

First, form Vc, its derivatives, and the products of Vc

with itself and its derivatives.

Vav ~ 3\ S\nUJz +- &L C°S tot

Vc = CJ Bj cos cot - OJ 3L Gin 'cj±

Vc ' ~ OJ &, Sin Cd± - UJZ BL COS OJ-t

It then follows that6!

VCVC = d -y Sm Wt + 2,6^ ess zwt

From which the following are obtained:

C.l . 7 '

^ ccs Wt t COS 3Wtl

72

I p - -■ ~ -' ~ " 2 SI f? £*/ -t

+ A^lA. ("cC5u;t + COS 3 Cut:) i

is* . '

Bi 15/ (^COS&7-ir-C0S3Co>t:)

V

C • 3 -7 7 2 2. Bi + Pitf - —- COS. CAJtZ VCV* =-W-[-^^S.nWt +

1 ,, 3 3.'

4 APx-.0^ ( Sin 30U± _ S/n wtj

— -™A -- [cos 3CJJ-C +■ cos cu-fcj

+ .JlJk. (cos ctyt - Cos 3^r) a

_!_ -PxJ3' fs/n 3Wt + S/n */*} ]

As per the first harmonic method, the third harmonic

terms are now dropped, and C.l, C.2, C.3 are substi-

tuted into equation 3*11 to produce C.4- as follows:

73

c .h

A 5inCJt = B, Stn wt +• B2 cos CJ t ^ R C U 6, C0S CJ "t

-(^CoCJ6/?, Sin wt

_&ZML cos** + ?'-^2& linutt

$,h LJ± _ li£L cos OJ*\

-LCoCJ2-^, 5m ojt - LCoU1 B^ Cos cj±

.- L KCJ - z~~ Sin cut + -^— CoscJt

When the terms of equation C»b are properly combined

and reduced, equation 3*12 results.

74

AFP2NDI* D

Computer Program to Solve Two Specific

Simultaneous Third Order Equations

This Fortran program, written by the author, is

designed to solve equations 3-13 and 3«1^» with a

particular emphasis placed on finding the precise fre-

quency and amplitude of any discontinuity. The program

itself sets up several numerical parameters, $hich are

then used by an International Mathematical & Statistical

Libraries-' subroutine. It is this subroutine which per-

forms the actual numerical iteration.

The author's program, named Solve, and the IIWSL

subroutine, named Zsystm, ave available thru the Lehigh

University Computer Center program library.

:*"t

75

\

APPENDIX E

Computer Program to Plot the Theoretical

Response of Two Coupled Linear Resonators

This Fortran program was written by the author who

used standard impedance ratio techniques to obtain the

magnitudes of Vx and Vg# as indicated in Figure 8. In

order to simplify the formulation of the problem, the

program uses complex variables so that the reduction of

complex fractions was not necessary. The equations

plotted are:

v, =A

j'jJL *■ ft -r

)/W ^ '/jcjc j QJC^_ A lvJ{

1U> L + R +

K-Y (fat. + R)/i^c JUJL + f{ +■ 1/jVJC

juj Cc ■+•

These are plotted for several values of Cc.

The Program has been.Nplaced on file with the Lehigh

University Computer Center under the name of Complx.

76

J

■ ' AUTHUVS VITA

Benton Phillips Zv/art was born in Stafford, Connec-

ticut on October 7, 1952 to Bernardus Mulder and Lila

Rose (Phillips) Zwart.

He attended New Canaan High School, v/hare ha was

inducted into the National Honor Society, and from which

he graaua^d in 197C.

Upon his graduation, Summa Cum Laude, from Lehigh

University in 197^ with a B.S.E.E., Mr. Zwart was com-

missioned a Second Lieutenant in the United Stabes Air-

Force, During his undergraduate career, the author

served as the Treasurer of Tau Beta Pi and Recording

Secretary for Eta Kappa Nu. Mr. Zwart remained at

Lehigh as a teaching assistant while pursuing the degree

of Master of Science in Electrical Engineering. During

the Summer of 19^5 and Spring of 1976, he was employed

by Pennsylvania Power and Light Company as a member of a

university-based research team studying correlations

between daily insolation and average wind speed as

applied to power generation.

Mr. Zwart intends to continue his education by

taking courses leading to an M.D.

77