fundamentals of microwave engineering rf coupling...
TRANSCRIPT
Fundamentals of Microwave Engineering &
RF coupling issues
Luigi Celona
Istituto Nazionale di Fisica Nucleare Laboratori Nazionali del Sud
Via S. Sofia 64 – 95123 Catania Italy
To: Agatino, Giacomo, Nicolò, Andrea and …last arrived Emanuele
CERN Accelerator School and Slovak University of Technology, Senec, June 2012 L. Celona, Fundamentals of Microwave Engineering & RF Coupling issues
2
Contents
Introduction
Equivalent circuit of a TRL
Derivation of wave equation and its solution
The propagation constant
Characteristic Impedance
Reflection Coefficient
Standing Waves, VSWR
Impedance and Reflection
Incident and Reflected Power
Smith Charts
Load Matching
Single Stub Tuners
Impedance and admittance matrices
S matrix
Waveguides Rectangular and circular
WG
Propagation and cutoff
Attenuation
Field lines
RLC model
Resonator
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Frequency spectrum
3
103
3×105 3×106 3×107 3×108 3×109 3×1010 3×1011 3×1012 3×1013 3×1014
102 10 1 10-1 10-2 10-3 10-4 10-5 10-6
Frequency (Hz)
Wavelength (m)
Microwaves
AM
bro
adca
st
radio
Long w
ave
radio
Short
wav
e
radio
VH
F T
V
FM
ra
dio
Far
infr
ared
Infr
ared
Vis
ible
lig
ht
Approximate band designation
L-band 1-2 GHz
S-band 2-4 GHz
C-band 4-8 GHz
X-band 8-12 GHz
Ku-band 12-18 GHz
K-band 18-26 GHz
Ka-band 26-40 GHz
U-band 40-60 GHz
Low frequencies: so large that the phase variation across the dimensions is little.
Optical frequencies: much shorter than the component dimensions.
Geometrical optics regime
Phase of the voltage and currents changes significantly over the physical lenght of the device, microwave components are distributed, the lumped circuit element approximation are not valid.
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4
Electrical Model of a Transmission Line
R= series resistance per unit lenght, for both conductors, in /m. L= series inductance per unit lenght, for both conductors, in H/m. G= shunt conductance per unit lenght in S/m. C= shunt capacitance per unit lenght in F/m.
i(z,t)
v(z,t) +
-
z
z
•
• •
• i(z,t)
v(z,t)
+
-
Rz Lz
Gz Cz v(z+z,t)
+
-
i(z+z,t)
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5
Wave equation – Propagation constant
wave travelling along +z direction wave travelling along -z direction
0 0
0 0
+ -
+ -
jβαCjGLjRγ is called attenuation constant [dB/m]
is called phase constant [radians/m]
β
2πλ2πλβ
)(
0
)(
0
)(
0
)(
0
)(
)(
ztjzztjz
ztjzztjz
eeIeeIzI
eeVeeVzV
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6
Characteristic Impedance
has negative sign +
+
-
- - 0
0 0
0
0
0
0 0
0 +
+
-
-
)(
00
)(
00
)(
0
)(
0
)/()/()(
)(
ztjzztjz
ztjzztjz
eeZVeeZVzI
eeVeeVzV
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7
Terminated lossless transmission line
i(z)
v(z)
z
IL
ZL Z0 ,
0
VL
V0
V0
-d
zjzj
zjzj
eZVeZVzI
eVeVzV
)/()/()(
)(
0000
00
0
0
000
00
00
)0(
)0(0@ V
ZZ
ZZVZ
VV
VV
I
VZz
L
LL
0
0
0
0
ZZ
ZZ
V
V
L
L
][)(
][)(
0
0
0
zjzj
zjzj
eeZ
VzI
eeVzV
+
-
The amplitude of the reflected wave normalized to the amplitude of the incident wave is known as reflection coefficient
ZL = Zo = 0 NO REFLECTIONS (maximum power delivered to the load)
MATCHED LINE
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8
Standing Waves
LZoZ
0z
z
d
At z=0
oL
oL
ZZ
ZZVV
00
Along the transmission line:
zVeVV
eVeVV
zj
zjzj
cos21 00
00
traveling wave standing wave
If ZL Zo
(by adding and substracting: ) zjeV 0
zjzj
zjzj
eZVeZVzI
eVeVzV
)/()/()(
)(
0000
00
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9
Voltage Standing Wave Ratio (VSWR)
1
1
1
1
0
0
min
max
V
V
V
VVSWR
The VSWR is always greater than 1
2
1
Incident wave
Reflected wave
Standing wave
][log20 dBRL
zjzj eVeVV 00
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10
Input Impedance of a Transmission Line
LZoZ
0z
z
d
oL
oLL
ZZ
ZZ
INZ
towards load
towards generator
djZZ
djZZZ
e
eZ
dI
dVZ
L
L
dj
dj
IN
tan
tan
1
1
)(
)(
0
002
2
0
][)(
][)(
0
0
0
zjzj
zjzj
eeZ
VzI
eeVzV
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11
Reflection Coefficient Along a Transmission Line
LZoZ
0x
x
d
oL
oLL
ZZ
ZZ
G
towards load
towards generator
dj
Ldj
dj
L
genforward
reverseG e
eV
eV
V
V
2
)(
0
)(
0
Wave has to travel down and back
][)( 0
zjzj eeVzV
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12
Impedance and Reflection
G
L
Re
Im
d2
There is a one-to-one correspondence between G and ZL
oG
oGG
ZZ
ZZ
G
GoG
1
1ZZ
d2jL
d2jL
oGe1
e1ZZ
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Focus on “special lenghts”
13
For a general transmission line with lenght:
•l=n/2
meaning that a half-wavelenght line does not alter or transform the load impedance
•l= /4 + n/2
•Such lines are called quarter wave transformers since they have the effect of trasforming the load impedance in an inverse manner depending on the characteristic impedance of the line.
ZIN = ZL
L
INZ
ZZ
2
0
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14
Incident and Reflected Power
LZoZ
0zd
dz
The rate of energy flowing through the plane at z=-d
o
L
o
zjzj
Z
V
Z
V
Z
VP
zVeeZ
VzIzVP
2
02
2
02
0
2
0
222*
0
2
0*
2
1
2
1)1(
2
1
)(1Re2
1)()(Re
2
1
forward power reflected power
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15
Incident and Reflected Power
Power does not flow! Energy flows.
The net rate of energy transfer is equal to the
difference in power of the individual waves
To maximize the power transferred to the load we want:
0L
which implies:
oL ZZ
When ZL = Zo, the load is matched to the transmission line
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16
dB and dBm
A dB is defined as a POWER ratio. For example:
log20
log10
P
Plog10
2
for
revdB
A dBm is defined as log unit of power referenced to 1mW:
mW1
Plog10PdBm
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17
Load Matching
What if the load cannot be made equal to Zo for some other reasons? Then, we need to build a matching network so that the source effectively sees a match load.
0
LZsP 0Z M
Typically we only want to use lossless devices such as capacitors, inductors, transmission lines, in our matching network so that we do not dissipate any power in the network and deliver all the available power to the load.
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18
Normalized Impedance
LL
o
LL jxr
Z
Zz
It will be easier if we normalize the load impedance to the characteristic impedance of the transmission line attached to the load.
1
1Lz
Since the impedance is a complex number, the reflection coefficient will be a complex number
ir j
22
22
1
1
ir
irLr
221
2
ir
iLx
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19
Smith Charts
The impedance as a function of reflection coefficient can be re-written in the form:
2
2
2
1
1
1L
i
L
Lr
rr
r
2
2
2 111
LL
irxx
22
22
1
1
ir
irLr
221
2
ir
iLx
L
L
L
rradius
r
rC
1
1
0,1
L
L
xradius
xC
1
1,1
These are equations for circles on the (r,i) plane
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20
Smith Chart – Real Circles
1 0.5 0 0.5 1
1
0.5
0.5
1
Re
Im
r=0 r=1/3
r=1 r=2.5
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21
Smith Chart – Imaginary Circles
1 0.5 0 0.5 1
1
0.5
0.5
1
Re
Im
x=1/3 x=1 x=2.5
x=-1/3 x=-1 x=-2.5
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22
Smith Chart
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23
Smith Chart Example 1
Given:
50Zo
455.0L
What is ZL?
5.67j5.67
35.1j35.150ZL
circleL 5.0||
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24
Smith Chart Example 2
Given:
GHzfZ ro 356.275
755.37 jZL
0.15.0 jzL
cm25.656.2103
1039
8
cml 2
?? SWRZin
32.025.6
0.2l
455.0135.032.0
circleSWR 3.4
Ref. position of the load=0.135 We need to move toward to the gen. an electrical distance equal to the line length
0.2175.1828.025.075 jjZin
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25
Admittance
A matching network is going to be a combination of elements connected in series AND parallel.
Impedance is NOT well suited when working with parallel configurations.
21L ZZZ
2Z1Z
2Z
1Z
21
21L
ZZ
ZZZ
ZIV
For parallel loads it is better to work with admittance.
YVI 2Y1Y
21L YYY 11
Z
1Y
Impedance is well suited when working with series configurations. For example:
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26
Impedance and Admittance Smith Charts
For a matching network that contains elements connected in series and parallel, we will need two types of Smith charts impedance Smith chart
admittance Smith Chart
The admittance Smith chart is the impedance Smith chart rotated 180 degrees. We could use one Smith chart and flip the reflection
coefficient vector 180 degrees when switching between a series configuration to a parallel configuration.
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27
1 0.5 0 0.5 1
1
0.5
0.5
1
1 0.5 0 0.5 1
1
0.5
0.5
1
Admittance Smith Chart
Re
Im
g=1/3
b=-1 b=-1/3
g=1 g=2.5 g=0
b=2.5 b=1/3
b=1
b=-2.5
Im
Re
jbgYZY
Yy o
o
1
1y
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28
• Procedure:
• Plot 1+j1 on chart
• vector =
• Flip vector 180 degrees
Admittance Smith Chart Example 1
Given:
64445.0
What is ?
1j1y
116445.0
Plot y
Flip 180 degrees
Read
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29
• Procedure:
• Plot
• Flip vector by 180 degrees
• Read coordinate
Admittance Smith Chart Example 2
Given:
What is Y?
455.0 50Zo
Plot
Flip 180 degrees
Read y
36.0j38.0y
mhos10x2.7j6.7Y
36.0j38.050
1Y
3
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30
Matching Example
0
100sP 50Z0 M
Match 100 load to a 50 system at 100MHz
A 100 resistor in parallel would do the trick but ½ of the power would be dissipated in the matching network. We want to use only lossless elements such as inductors and capacitors so we don’t dissipate any power in the matching network
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31
Matching Example
We need to go from z=2+j0 to z=1+j0 on the Smith chart
We won’t get any closer by adding series impedance so we will need to add something in parallel.
We need to flip over to the admittance chart
Impedance Chart
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32
Matching Example
y=0.5+j0
Before we add the admittance, add a mirror of the r=1 circle as a guide.
Admittance Chart
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33
Matching Example
y=0.5+j0
Before we add the admittance, add a mirror of the r=1 circle as a guide
Now add positive imaginary admittance.
Admittance Chart
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34
Matching Example
y=0.5+j0
Before we add the admittance, add a mirror of the r=1 circle as a guide
Now add positive imaginary admittance jb = j0.5
Admittance Chart
pF16C
CMHz1002j50
5.0j
5.0jjb
pF16 100
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35
Matching Example
We will now add series impedance
Flip to the impedance Smith Chart
We land at on the r=1 circle at x=-1
Impedance Chart
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36
Matching Example
Add positive imaginary admittance to get to z=1+j0
Impedance Chart
pF16100
nH80L
LMHz1002j500.1j
0.1jjx
nH80
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37
Matching Example
This solution would have also worked
Impedance Chart
pF32
100nH160
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38
50 60 70 80 90 100 110 120 130 140 15040
35
30
25
20
15
10
5
0
Frequency (MHz)
Re
fle
ctio
n C
oe
ffic
ien
t (d
B)
Matching Bandwidth
1
1
vi
Im gammai
11 ui Re gammai
50 MHz
150 MHz
Because the inductor and capacitor impedances change with frequency, the match works over a narrow frequency range
pF16100
nH80
Impedance Chart
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Matching with lumped elements
39
The simplest type of matching network is therefore the L section which uses two reactive elements to match an arbitrary load impedance to a transmission line.
If zL=ZL/Z0 is inside the 1+jx circle on the Smith chart the solution (a) is used.
If zL=ZL/Z0 is outside the 1+jx circle on the Smith chart the solution (b) is used.
If the frequency is low enough (frequencies up to 1 GHz) lumped capacitance and inductance can be used. There is, however, a large range of frequencies where this solution may be not realizable. This is the main limitation of the L matching section technique.
CERN Accelerator School and Slovak University of Technology, Senec, June 2012 L. Celona, Fundamentals of Microwave Engineering & RF Coupling issues
Analytical solution
40
If ZL>Z0 (implies that zL is inside the 1+jx circle on the Smith chart)
BR
Z
R
ZX
BX
XR
ZRXRZRXB
LL
L
LL
LLLLL
00
22
0
22
0
1
/
If ZL<Z0 (implies that zL is outside the 1+jx circle on the Smith chart)
0
0
0
/)(
)(
Z
RRZB
XRZRX
LL
LLL
Positive X=inductor Negative X=capacitor
Positive B=capacitor Negative B=inductor
Two solutions, but…one may result in a significant smaller value for the reactive components and may be preferred due to the matching bandwidth or VSWR on the line.
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Example
41
Match a series RC load with impedance 200+j100 to a 100 line at 500 MHz
C1=0.92 pF L1=38.8H
C2=2.61 pF L2=46.1H
Solution 1 is slightly better than solution 2.
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Single stub tuning
42
An important matching technique uses an open-circuited or a short-circuited transmission line (a “stub”) either in parallel or in series with the transmission line at a certain distance from the load.
Two adjustable parameters: distance (d) from the load, susceptance or reactance provided by the shunt or series stub.
Shunt: select d so that the susceptance seen is Y0+jB and add the stub with susceptance –jB.
Series: select d so that the impedance seen is Z0+jX and add the stub with impedance –jX.
Tuning circuit very easy to implement
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All together!
43
Given: 1015 jZL
2.03.0 jzL
Assuming load consisting in a RL series matched at 2 Ghz, calculate the shunt stubs to have the match.
GHzfZo 250
zL
yL
y2
y1
d1 d2
0.284
0.328
0.171
33.11
33.11
2
1
jy
jy
387.0171.0)284.05.0(
044.0284.0328.0
2
1
d
d
J1.33=0.147
J1.33=0.353
Add a lenght of stub with susceptance –jB: Solution1) l=0.147 Solution2) l=0.353
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Frequency response
44
Two possibilities: it is normally desired to keep the matching stub as close as possible to the load to reduce the losses caused by a possibly large standing wave ratio on the line between the stub and the load.
Solution 1 has a significantly better bandwidht than solution 2.
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Double stub tuning
45
The single stub tuning is able to match any load impedance to a transmission line, but suffer of the disadvantage of requiring a varianle length of line between the load and the stub.
Design of matching with double stub tuner, or multiple stub tuner is far away from the scope of this course.
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The Quarter wave transformer
46
Simple and useful method for matching a real load impedance to a transmission line. It can be extended to multisection design for a broader bandwidth.
Drawback: QWT can only match a real load impedance. A complex load impedance can always be transformed to a real impedance by using an appropriate series or shunt reactive stub.
LZZZ 01
Z0 Z1 ZL
l
At f0 l=\4 At f0 +Δf l \4
ljZZ
ljZZZZ
L
LIN
tan
tan
1
11
𝛽𝑙 =2𝜋
𝜆 𝜆
4=
𝜋
2
L
INZ
ZZ
2
1
To get =0 0ZZ IN
Geometric mean of load and source imp.
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Mismatch vs. frequency
47
LZZZ 01Z0 Z1 ZL
l
00
0
2cos
2 f
fwith
ZZ
ZZ
L
L
For near /2
If we set the max value of Reflection Coefficient mthat can be tolerated, the bandwidth of the matching transformer is:
0
0
20
0
0
0
2
1cos
42
42
)(2
2
ZZ
ZZ
f
ff
f
f
ff
L
L
m
mmm
mm
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Example
48
36.2201 LZZZ
Z0=50 Z1 ZL=10
L, f0=3GHz
Determine the percent bandwith for which the SWR<1.5
For this value of Z1 L=/4 at f0=3GHz. A SWR=1.5 correspond to:
2.01
1
SWR
SWRm
%2929.02
1cos
42
0
0
20
ZZ
ZZ
f
f
L
L
m
m
MHzf
f870
0
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Multisection transformer
49
Z0 ZN ZL
Z1 Z2 Z3
0 1 2 3 N
N equal sections of transmission lines ZL real Z monotically decreasing: Z0> Z1> Z2> Z3>… > ZN
jN
N
jj eee 24
2
2
10 ....)(
We can synthesize any desired reflection coefficient response as a function of frequency (), by properly choosing the N and using enough sections. Commonly used:
Binomial transformer (maximally flat design) Chebyshev (equal ripple) transformer
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Binomial Transformers
50
For a given number of sections, the passband response is as flat as possible near the design frequency (maximally flat design).
This kind of response is designed, for a N sections transformer, by setting the first N-1 derivatives of the reflection coefficient to zero.
Z0 ZN ZL Z1 Z2 Z3
0 1 2 3 N
jnN
n
N
n
Nj eCAeA 2
0
2 )1()(
!)!(
!2
0
0
nnN
NC
ZZ
ZZA N
n
L
LN
where:
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Binomial Transformers Bandwidth
51
If we set the max value of Reflection Coefficient mthat can be tolerated, the bandwidth of the matching transformer is:
N
mmm
NN
mA
aA
/1
2
1coscos2
N
mmm
Af
ff
f
f/1
0
0
0 2
142
42
)(2
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Tables
52
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Example (analytical solution)
53
Z0=100 Z3 ZL=50 Z1 Z2
0 1 2 3
Design the 3 sections transformer, calculate BW per m=0.05
3!1!2
!33
!1!2
!31
!0!2
!3
71.02
1cos
420417.02
3
2
3
1
3
/1
00
0
CCC
Aa
f
f
ZZ
ZZA
o
N
m
L
LN
5.54;00.4100
50ln)1(27.70lnln2lnln;2
7.70;26.4100
50ln)3(27.01lnln2lnln;1
7.91;518.4100
50ln)1(2100lnln2lnln;0
1
33
023
2
33
112
1
33
001
ZZ
ZCZZn
ZZ
ZCZZn
ZZ
ZCZZn
o
LN
o
LN
o
LN
Greater BW!!!
CERN Accelerator School and Slovak University of Technology, Senec, June 2012 L. Celona, Fundamentals of Microwave Engineering & RF Coupling issues
Example with tables
54
Z0=100 Z3 ZL=50 Z1 Z2
0 1 2 3
Design the 3 sections transformer, calculate BW per m=0.05
53.548337.1
71.704142.1
68.910907.1
03
02
01
ZZ
ZZ
ZZ
Greater BW!!!
CERN Accelerator School and Slovak University of Technology, Senec, June 2012 L. Celona, Fundamentals of Microwave Engineering & RF Coupling issues
Chebyschev Multisection matching transformers
55
In contrast with binomial transformers, the chebyschev ones optimize bandwidth at expenses of passband ripple.
)()(2)(
.................................
188)(
34)(
12)(
)(
21
24
4
3
3
2
2
1
xTxxTxT
xxxT
xxxT
xxT
xxT
nnn
For -1<x<1, |Tn(x)|<1. In this range the polynomials oscillate between ±1. This is the equal ripple property and this region will be mapped to the passband of the matching transformer. For |x|>1, |Tn(x)|>1. |Tn(x)| increases faster as n increases. This region will map to the frequency range outside the passband.
Design procedure based on Chebyshev polynomials:
CERN Accelerator School and Slovak University of Technology, Senec, June 2012 L. Celona, Fundamentals of Microwave Engineering & RF Coupling issues
Chebyschev Multisection matching transformers
56
We need to map: m to x=-1 and - m to x=1
)()(2)(
.................................
1)12(cossec4)32cos44(cossec)cos(sec
cossec3)cos33(cossec)cos(sec
1)2cos1(sec)cos(sec
cossec)cos(sec
21
24
4
3
3
2
2
1
xTxxTxT
T
T
T
T
nnn
mm
mmm
mm
mm
Often used for D.C., filters…
CERN Accelerator School and Slovak University of Technology, Senec, June 2012 L. Celona, Fundamentals of Microwave Engineering & RF Coupling issues
Chebyschev Multisection matching transformers
57
)(sec
1)cos(sec)(
0
0
mNL
LmN
Nj
TZZ
ZZAwhereTAe
Now if m is tha maximum allowable reflection coefficient in the passband then A=m, since the maximum value in the passband of TN(secmcos ) is the unity. Then m and fractional bandwidth are given by:
m
L
L
m
m
f
f
ZZ
ZZa
N
42
1cosh
1coshsec
0
0
0
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Tables
58
CERN Accelerator School and Slovak University of Technology, Senec, June 2012 L. Celona, Fundamentals of Microwave Engineering & RF Coupling issues
Example with tables
59
Z0=50 Z3 ZL=100 Z1 Z2
0 1 2 3
Design a transformer with maximum reflection coefficient equal to 0.05 (A=m=0.05).
%10202.14
2
2.44395.11
cosh1
coshsec
0
0
0
m
m
L
L
m
m
f
f
ZZ
ZZa
N
15.877429.1
71.704142.1
37.571475.1
03
02
01
ZZ
ZZ
ZZ
CERN Accelerator School and Slovak University of Technology, Senec, June 2012 L. Celona, Fundamentals of Microwave Engineering & RF Coupling issues
Tapered Lines
60
As the number of the discrete sections increases, the step changes in the characteristic impedance become smaller. Thus, in the limit of an infinite number of sections, we approach a continuosly tapered line.
By changing the type of taper, we can obtain different passband characteristics.
)(zZz
Lzj dz
Z
Z
dz
de
00
2 ln2
1)(
Three special cases analyzed in the following: •Exponential taper •Triangular taper •Klopfenstein taper
For a given taper lenght the klpofstein impedance taper is the optimum in the sense that the reflection coefficient is minimum over the passband.
It is derived from a stepped Chebyshev transformer as the number of sections increas to infinity.
CERN Accelerator School and Slovak University of Technology, Senec, June 2012 L. Celona, Fundamentals of Microwave Engineering & RF Coupling issues
61
Impedance (Z) and Admittance (Y) Matrices
This linear two port “black box” can be fully described if we know how the voltage and the current at port 1 relate to the voltage and the current at port 2.
1V 2V
2I1I
2221212
2121111
IzIzV
IzIzV
zij and yij characterize the properties of the black box.
IZV
2221212
2121111
VzVyI
VyVyI
VYI
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62
Impedance (Z) and Admittance (Y) Matrices
We need 4 measurements to determine [Y] or [Z]:
Open output (I2=0)
1V 2V
2I1I
2221212
2121111
IzIzV
IzIzV
2221212
2121111
VzVyI
VyVyI
1
111
I
Vz
1
221
I
Vz
Open input (I1=0)
2
112
I
Vz
2
222
I
Vz
Short output (V2=0)
1
111
V
Iy
1
221
V
Iy
Short input (V1=0)
2
112
V
Iy
2
222
V
Iz
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S matrix
63
It is helpful to think about travelling waves along a TRL!
gV
2V
2221212
2121111
VsVsV
VsVsVgZ
LZ
2V
1V
1V
j
iij
V
Vs
Means we are sitting and measuring the receding voltage wave at port I, while the driving source is at port j.
02
1
111
VwhenV
Vs
01
2
112
VwhenV
Vs
02
1
221
VwhenV
Vs
01
2
222
VwhenV
Vs
the input reflection coefficient
the reverse transmission coefficient the reverse reflection coefficient
the input transmission coefficient
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64
Normalized Scattering (S) Parameters
Since measuring voltages and currents at microwave frequencies becomes unpratical, the S- parameters are related to the incident and reflected power.
The S matrix defined previously is called the un-normalized scattering matrix. For convenience, define normalized waves:
ii o
ii
o
ii
Z
Vb
Z
Va
22
Where Zoi is the characteristic impedance of the transmission line connecting port (i)
|ai|2 is the forward power into port (i)
|bi|2 is the reverse power from port (i)
CERN Accelerator School and Slovak University of Technology, Senec, June 2012 L. Celona, Fundamentals of Microwave Engineering & RF Coupling issues
65
Normalized Scattering (S) Parameters
The normalized scattering matrix is:
Where:
2221212
2121111
asasb
asasb
j,i
io
jo
j,i SZ
Zs
If the characteristic impedance on both ports is the same then the normalized and un-normalized S parameters are the same.
Normalized S parameters are the most commonly used.
Energy conservation: for a loss less network SUM(Sij)=1
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66
Normalized S Parameters
The s parameters can be drawn pictorially
s11 and s22 can be thought of as reflection coefficients
s21 and s12 can be thought of as transmission coefficients
s parameters are complex numbers where the angle corresponds to a phase shift between the forward and reverse waves
s11 s22
s21
s12
a1
a2 b1
b2
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Why S-parameters?
67
• S-parameters are a powerful way to describe an electrical network • The ease with which scattering parameters can be measured makes them
especially well suited to describe transistor or other active devices. • S-parameters change with freqency / load impedance / source impedance /
network • The most important advantage of S-parameters stems from the fact that
travelling waves, unlike terminal voltage and currents, do not vary in magnitude along a transmission line.
• S11 is the reflection coefficient • S21 describes the forward transmission coefficient (responding port 1st!) • S-parameters have both magnitude and phase information • Sometimes the gain (or loss) is more important than the phase shift and the phase
information may be ignored • S-parameters may describe large and complex networks
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68
Examples of S parameters
2 1
10
01s
1 2
0G
00sG
Short
Amplifier
1
2
01
00s
Zo
Isolator
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69
Lorentz Reciprocity
If the device is made out of linear isotropic materials (resistors, capacitors, inductors, metal, etc..) then:
ssT
j,ii,j ss ji
or
for
This is equivalent to saying that the transmitting pattern of an antenna is the same as the receiving pattern
reciprocal devices: transmission line
short
directional coupler non-reciprocal devices: amplifier
isolator
circulator
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70
Lossless Devices
The s matrix of a lossless device is unitary:
1ssT*
j
2
j,i
i
2
j,i
s1
s1for all j
for all i
Lossless devices: transmission line
short
circulator
Non-lossless devices: amplifier
isolator
CERN Accelerator School and Slovak University of Technology, Senec, June 2012 L. Celona, Fundamentals of Microwave Engineering & RF Coupling issues
71
Network Analyzers
Network analyzers measure S parameters as a function of frequency
At a single frequency, network analyzers send out forward waves a1 and a2 and measure the phase and amplitude of the reflected waves b1 and b2 with respect to the forward waves.
a1 a2
b1 b2
02a1
111
a
bs
02a1
221
a
bs
01a2
112
a
bs
01a2
222
a
bs
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72
Network Analyzer Calibration
To measure the pure S parameters of a device, we need to eliminate the effects of cables, connectors, etc. attaching the device to the network analyzer
s11 s22
s21
s12
x11 x22
x21
x12
y11 y22
y21
y12
yx21
yx12
Connector Y Connector X
We want to know the S parameters at these reference planes
We measure the S parameters at these reference planes
CERN Accelerator School and Slovak University of Technology, Senec, June 2012 L. Celona, Fundamentals of Microwave Engineering & RF Coupling issues
73
Network Analyzer Calibration
There are 10 unknowns in the connectors
We need 10 independent measurements to eliminate these unknowns Develop calibration standards
Place the standards in place of the Device Under Test (DUT) and measure the S- parameters of the standards and the connectors
Because the S parameters of the calibration standards are known (theoretically), the S parameters of the connectors can be determined and can be mathematically eliminated once the DUT is placed back in the measuring fixtures.
CERN Accelerator School and Slovak University of Technology, Senec, June 2012 L. Celona, Fundamentals of Microwave Engineering & RF Coupling issues
74
Network Analyzer Calibration
Since we measure four S parameters for each calibration standard, we need at least three independent standards.
One possible set is:
10
01s
0e
e0s
j
j
01
10sThru
Short
Delay* *~90degrees
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Waveguides
75
The mathematical solution to the waveguide problem is tedious. In order to find the various field components in a rectangular or circular guide six differential equations have to be solved.
EjH
HjE
zj
z
zj
z
eyxhzyxhzyxH
eyxezyxezyxE
)],(),([),,(
)],(),([),,(
^
^
Assuming that: a)waveguide region is source-free, b)z direction propagation (inifitely long structure), c) time harmonic fields with exp(jwt) dependence, d) perfectly conducting wall, the Maxwell equation can be rewritten as:
TEM waves Ez=Hz=0
TE waves Ez=0 Hz0
TM waves Ez0 Hz=0
TEM wave has no cutoff and can exist when two or more conducors are present.
Cutoff!
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Rectangular Waveguide
76
No TEM waves can propagate!
EjH
HjE
Boundary conditions
0,0
0,0
zy
zy
EEbyy
EEaxx
TEmn waves Ez=0 Hz0
zj
mnzz
zj
mn
c
y
zj
mn
c
y
zj
mn
c
x
zj
mn
c
x
eb
yn
a
xmAHE
eb
yn
a
xmA
bk
njHe
b
yn
a
xmA
ak
mjE
eb
yn
a
xmA
ak
mjHe
b
yn
a
xmA
bk
njE
coscos0
sincoscossin
cossinsincos
22
22
22
222
b
n
a
mkkk c
22
b
n
a
mkk c
>0 for propagation !!!
numberwavek
2
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Propagation constant and cutoff frequency
77
22
2
1
2
b
n
a
mkf c
Cmn
Each mode (combination of m and n) has a cutoff frequency given by:
At a given frequency f only those modes having f>fc will propagate! The contrary will lead to an imaginary meaning that all field components will decay exponentially from the source excitation (evanescent mode)!
The mode with lowest cutoff is called dominant mode, if a>b the lowest fc occur for the TE10 mode:
zj
y
zj
x
zj
z
yzx
ea
xA
ajE
ea
xA
ajH
ea
xAH
HEE
sin
sin
cos
0
10
10
10
afC
2
110
mNpkabkba
R
aabARP
bAaP
Sc
sl
/2
22
]Re[4
232
3
2
322
10
2
2
10
3
10
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Rectangular Waveguide
78
EjH
HjE
Boundary conditions
0,0
0,0
zy
zy
EEbyy
EEaxx
TMmn waves Ez 0 Hz=0
0sinsin
sincoscossin
cossinsincos
22
22
z
zj
mnz
zj
mn
c
y
zj
mn
c
y
zj
mn
c
x
zj
mn
c
x
Heb
yn
a
xmBE
eb
yn
a
xmA
ak
mjHe
b
yn
a
xmB
bk
njE
eb
yn
a
xmB
bk
njHe
b
yn
a
xmB
ak
mjE
22
222
b
n
a
mkkk c
>0 for propagation !!!
Fields are zero if the indexes m or n are zero! The lowes TM mode that can propagate into the waveguide is the TM11.
22
2
1
b
n
a
mfCmn
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Field lines for some low order modes
79
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Other parameters
80
wave impedance
k
H
E
H
EZ
y
x
x
y
TE
/
medium intrinsic impedance
kg
22 The guided wavelenght is the distance between two equal phase planes inside the waveguide
kH
E
H
EZ
y
x
x
y
TM
1
kvp Phase velocity
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Table of rectangular waveguide
81
WG EIA-WR IEC-R Fc1, GHz Flo, GHz Fhi, GHz Fc2, GHz A, in B, in A, mm B, mm
00 2300 3 .257 0.32 0.49 .513 23.000 11.500 584.200 292.100
0 2100 4 .281 0.35 0.53 .562 21.000 10.500 533.400 266.700
1 1800 5 .328 0.41 0.625 .656 18.000 9.000 457.200 228.600
2 1500 6 .393 0.49 0.75 .787 15.000 7.500 381.000 190.500
3 1150 8 .513 0.64 0.96 1.026 11.500 5.750 292.100 146.050
4 975 9 .605 0.75 1.12 1.211 9.750 4.875 247.650 123.825
5 770 12 .766 0.96 1.45 1.533 7.700 3.850 195.580 97.790
6 650 14 .908 1.12 1.70 1.816 6.500 3.250 165.100 82.550
7 510 18 1.157 1.45 2.20 2.314 5.100 2.550 129.540 64.770
8 430 22 1.372 1.70 2.60 2.745 4.300 2.150 109.220 54.610
9A 340 26 1.736 2.20 3.30 3.471 3.400 1.700 86.360 43.180
10 284 32 2.078 2.60 3.95 4.156 2.840 1.340 72.136 34.036
11A 229 40 2.577 3.30 4.90 5.154 2.290 1.145 58.166 29.083
12 187 48 3.152 3.95 5.85 6.305 1.872 .872 47.549 22.149
13 159 58 3.712 4.90 7.05 7.423 1.590 .795 40.386 20.193
14 137 70 4.301 5.85 8.20 8.603 1.372 .622 34.849 15.799
15 112 84 5.260 7.05 10.0 10.519 1.122 .497 28.499 12.624
16 90 100 6.557 8.20 12.4 13.114 .900 .400 22.860 10.160
17 75 120 7.869 10.0 15.0 15.737 .750 .375 19.050 9.525
18 62 140 9.488 12.4 18.0 18.976 .622 .311 15.799 7.899
19 51 180 11.571 15.0 22.0 23.143 .510 .255 12.954 6.477
20 42 220 14.051 18.0 26.5 28.102 .420 .170 10.668 4.318
21 34 260 17.357 22.0 33.0 34.714 .340 .170 8.636 4.318
22 28 320 21.077 26.5 40.0 42.153 .280 .140 7.112 3.556
23 22 400 26.346 33.0 50.0 52.691 .224 .112 5.690 2.845
24 19 500 31.391 40.0 60.0 62.781 .188 .094 4.775 2.388
25 15 620 39.875 50.0 75.0 79.749 .148 .074 3.759 1.880
26 12 740 48.372 60.0 90.0 96.745 .122 .061 3.099 1.549
27 10 900 59.014 75.0 110 118.029 .100 .050 2.540 1.270
28 8 1200 73.768 90.0 140 147.536 .080 .040 2.032 1.016
29 7 1400 90.791 110 170 181.582 .065 .033 1.651 .826
30 5 1800 115.714 140 220 231.428 .051 .026 1.295 .648
31 4 2200 137.242 170 260 274.485 .043 .022 1.092 .546
32 3 2600 173.571 220 325 347.143 .034 .017 .864 .432
- 2 - 295.071 325 500 590.143 .020 .010 .508 .254
CERN Accelerator School and Slovak University of Technology, Senec, June 2012 L. Celona, Fundamentals of Microwave Engineering & RF Coupling issues
Circular waveguide
82
TEnm modes
0
)()cossin(
)()sincos(
'
2
z
zj
cn
c
zj
cn
c
E
ekJnBnAk
jE
ekJnBnAk
njE
zj
cnz
zj
cn
c
zj
cn
c
ekJnBnAH
ekJnBnAk
njH
ekJnBnAk
jH
)()cossin(
)()sincos(
)()cossin(
2
'
Jn(x) is the n-order Bessel function of first kind
J’n(x) is the n-order Bessel function derivative of first kind
2'
222
a
pkkk nm
c >0 for propagation !!!
a
pkf nmc
Cmn22
'
Cutoff frequencies
where p’nm is the mth root of J’n(x)
n=0 n=1 n=2 n=3 n=4 n=5 n=6
m=1
m=2
m=3
CERN Accelerator School and Slovak University of Technology, Senec, June 2012 L. Celona, Fundamentals of Microwave Engineering & RF Coupling issues
Circular waveguide
83
TMnm modes
zj
cnz
zj
cn
c
zj
cn
c
ekJnBnAE
ekJnBnAk
njE
ekJnBnAk
jE
)()cossin(
)()sincos(
)()cossin(
2
'
0
)()cossin(
)()sincos(
'
2
z
zj
cn
c
zj
cn
c
H
ekJnBnAk
jH
ekJnBnAk
njH
Jn(x) is the n-order Bessel function of first kind
J’n(x) is the n-order Bessel function derivative of first kind
2
222
a
pkkk nm
c >0 for propagation !!!
a
pkf nmc
Cmn22
Cutoff frequencies
where pnm is the mth root of Jn(x)
n=0 n=1 n=2 n=3 n=4 n=5 n=6
m=1
m=2
m=3
CERN Accelerator School and Slovak University of Technology, Senec, June 2012 L. Celona, Fundamentals of Microwave Engineering & RF Coupling issues
Field lines for some low order modes
84
0 1 2 3
11TE 21TE 01TE31TE
41TE
12TE
01TM 11TM 21TM 02TM
)( 11TEf
f
c
c
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Circular waveguides table
85
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Microwave coupling to plasma source
86
Water-cooled Bend
Microwave pressure window Placed behind the bend in order to avoid any damage due to the back-streaming electrons
Matching Transformer It realizes a progressive match between the waveguide impedance and the plasma chamber impedance, thus concentrating the electric field at the center of the plasma chamber
DC- Break • Low microwave losses - Good voltage insulation up to 100 kV
Automatic Tuning Unit
Adjusts the modulus/phase of the incoming wave to match the
plasma impedance Dual-arm Directional
Coupler
It is used to measure the forward and the reflected power
Water- cooled copper plasma chamber
Waveguide /Coax Adapter Excytes the TE10 dominant mode in WR340 waveguide
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Multisection matching transformers in WG
87
TRIPS ELECTROMAGNETIC
FIELD @ 2.45 GHz, 500 W
VIS ELECTROMAGNETIC
FIELD @ 2.45 GHz, 500 W
10 % ENHANCEMENT WITH VIS TRANSFORMER
VIS TRANSFORMER INSERTION
LOSS 0.0085 dB @ 2.45 GHz
S21 Vs FREQUENCY FOR THE TWO STRUCTURES
ELECTROMAGNETIC FIELD @ 2.45 GHz,
500 W
Four step double ridges
Matching transformer coupled to the plasma chamber
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Microwave resonators
88
Used in a wide variety of applications: filters, oscillators, frequency meters, tuned amplifiers, etc… The behaviour near resonance is very similar to the lumped element resonators.
EmlossIN
ININ
IN
WWjPP
CjLjRIIZVIP
CjLjRZ
2
1
2
1
2
1
2
1
1
22
When resonance occurs the average storage electric and magnetic energies are equal
LEm PWWondlossenergy
storedenergyaverageQ /
sec/
At resonance: R
L
RCQ 0
0
1
LC
10
Fractional BW: Q
BW1
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Rectangular waveguide cavities
89
Usually short circuited at both ends forming a closed box or a cavity. Power dissipated on metallic walls as well as in the dielctric filling the cavity.
Cutoff wavenumber and resonant frequencies
222
222
22
d
l
b
n
a
mcckf
d
l
b
n
a
mk
rrrr
mnlmnl
mnl
Q for TE10l mode
8
22
1
2
2
0
3323322
3
EabdP
addalbdbalR
bkadQ
d
S
CERN Accelerator School and Slovak University of Technology, Senec, June 2012 L. Celona, Fundamentals of Microwave Engineering & RF Coupling issues
Circular waveguide cavities
90
Resonant frequencies TEnml mode
22
22'
2
2
d
l
a
pcf
d
l
a
pcf
nm
rr
nml
nm
rr
nml
Jn(x) is the n-order Bessel function of first kind
J’n(x) is the n-order Bessel function derivative of first kind
where p’nm is the mth root of J’n(x) and pnm is the mth root of Jn(x),
Resonant frequencies TMnml mode
Q evaluation