lecture 10 introduction to microwave network analysis analysis. microwave networks: voltages and...
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Lecture 10
Introduction to Microwave Network Analysis
Microwave Networks: Voltages and Currents
• the theory of microwave networks was developed to enable circuit-like analysis methods which are simpler than field methods
• it also enables the integrated analysis of microwave structures with conventional lumped components (ICs, chip transistors, resistors, etc.)
• usual low-frequency definitions of voltage and current are not valid
V d
E L
CI d H L
if E is not conservative, voltage integral depends on the chosen integration path and is thus ambiguous
if there are no metallic leads (dielectric guides), the current integral is meaningless and the voltage one is ambiguous
ElecEng4FJ4 LECTURE 10: MICROWAVE NETWORK ANALYSIS 2
• network parameters (Z, Y, ABCD) based on voltages and currents are often inadequate
Equivalent Voltages and Currents
considerations in defining equivalent voltages and currents• equivalent voltage and current describe a traveling wave mode• voltage must be proportional to the transverse E field of a mode• current must be proportional to the transverse H field of a mode• the product of the rms voltage and rms current must produce the
power carried by the mode • in the case of a TEM line, the ratio of the voltage and the current
(V/I) must equal the characteristic impedance Z0
• in the case of a waveguide of uniform cross-section, the V/I ratio must equal to the wave impedance Zw if known
• the V/I ratio is taken as 1 if characteristic and wave impedances are not available (general waveguides, numerical solutions)
• the choice of the V/I ratio does not affect the scattering parametersElecEng4FJ4 LECTURE 10: MICROWAVE NETWORK ANALYSIS 3
Equivalent Voltages and Currents (2)
consider the transverse components of a field traveling along +z
1 1ˆ( ) 2 2
e ee eSV I
V IS ds V IC C
e h z
complex power and normalization:
ˆ ( ) V ISds C C e h z
0E 0H
set normalization condition as ˆ( ) 1S
ds e h z 1 /I VC C
ElecEng4FJ4 LECTURE 10: MICROWAVE NETWORK ANALYSIS 4
physical meaning of e and h: the rms field phasors of the traveling wave in the waveguide such that it carries 1 W power
00 ( , )( , )
( , , ) ( , ) ( , , ) ( , )e ej z j z
V I
x yx y
V Ix y z x y e x y z x y eC C
HE
E e H h
ˆ0.5 ( )S
S ds E H z
(proof on next slide)
Equivalent Voltages and Currents (3) – optional
2
Let ,
ˆ( ) ( )
j jr r
jV I V I r rS S
e e
ds C C C C e d
e e h h
e h z e h s
2( ) 1 ( ) 1 jr r V IS S
d d C C e e h s e h s
real
2
Let | | ( absorbs the angle of entirely)1
||1
||
jV V e
j
j jVV
IV
C
C C e VeCe CC e
E
Loss-free case: φ = 0
(now absorbs the angle of ) eI H
ˆif ( ) 1 then 1/I VSds C C e h zProof:
( , ) is realV
x yC
e
( , ) is realI
x yC
h
PROOF
ElecEng4FJ4 LECTURE 10: MICROWAVE NETWORK ANALYSIS 5
Equivalent Voltages and Currents – Summary
general definitions of voltage and current following from the normalization
0 0
0 0
/ ( , ) ( , )
( , ) ( , ) /
e
V
V
S
e S
Vx y x yC
x y C I x y
d
d
E e Eh h s
eH h He s
0
0
ˆ( )1 ˆ( )
e V S
e SV
V C ds
I dsC
E h z
e H z
⇒
ElecEng4FJ4 LECTURE 10: MICROWAVE NETWORK ANALYSIS 6
wave power ( ) does not depend on the choice of CV
modal vectors e and h do not depend on the choice of CV either
apply normalization
voltage-to-current ratio however does depend on the choice of CV
Re{ }e eV I
Voltage-to-Current Ratio
2 20
0
1
ˆ( )
ˆ( )e
VS
S
Ve
dsVI ds
C C
E h z
e H z
integrals in numerator and denominator cancel; follows from Maxwell’s equations in a source-free medium:
/ /
jj
E H hH Ε e
/ /
jj
e h Hh e E
( )( )
jj
h E h He H e Ε
( )( )
jj
H e H hE h E e
( )( ) ( )( ) ) (( )
j jj j
h E E h EH h E e hH e e H e HH h E e
( ) ( ) (*) E h e H ElecEng4FJ4 LECTURE 10: MICROWAVE NETWORK ANALYSIS 7
PROOF – OPTIONAL
voltage/current DEF 1: using a known characteristic impedance Z0(suitable for TEM TLs)
20 0 0 , 1 /e
V IVe
V C Z C Z C ZI
0 ˆ( )e SV Z ds E h z
0
1 ˆ( )e SI ds
Z e H z
00
( , , ) ( , ) ( , , ) ( , )e j z j ze
Vx y z x y e x y z Z I x y eZ
E e H h
1/2 1/2
1 1/2 1 1/2UNITS: [ ] , [ ] , [ ] , [ ] , [ ] m , [ ] m
V I e eC C V V I A
e h
ElecEng4FJ4 LECTURE 10: MICROWAVE NETWORK ANALYSIS 8
Voltage-to-Current Ratio: Definition 1
ˆ ˆ( , ) ( , )( , ) and ( , )w w
x y x yx y x yZ Z
z e z Eh H
voltage/current DEF 2: using a known wave impedance Zw (suitable for waveguides of homogeneous cross-section)
2 ew V wV
e
VZ C C ZI
ˆ( )e w SV Z ds E h z 1 ˆ( )e S
wI ds
Z e H z
( , , ) ( , ) ( , , ) ( , )e j z j zw e
w
Vx y z x y e x y z Z I x y eZ
E e H h
units are the same as with DEF 1
ElecEng4FJ4 LECTURE 10: MICROWAVE NETWORK ANALYSIS 9
Voltage-to-Current Ratio: Definition 2
• the meaning of Zw:
• in TEM TLs: wZ
voltage/current DEF 3: impedance set to unity (general, used for waveguides of heterogeneous cross-section that are analyzed numerically)
2 1 1eV IV
e
V C C CI
ˆ( )e SV ds E h z ˆ( )e S
I ds e H z
( , , ) ( , ) ( , , ) ( , )e
je e
V
z j zx y z x y e x y z x y eV I
E e H h
1/2 1/2
1 1/2 1 1/2UNITS: [ ] none, [ ] none, [ ] , [ ] , [ ] m , [ ] m
V I e eC C V W I W
e h
root-power wave
e eV I
ElecEng4FJ4 LECTURE 10: MICROWAVE NETWORK ANALYSIS 10
Voltage-to-Current Ratio: Definition 3
Repeat the example for slide 8 this time setting voltage-to-current ratio to unity. Do you obtain the same modal vectors? (homework)
Microwave Network: Voltage/Current Formulation
• at each port incident and reflected voltage/current waves can be defined
N-port network
• at the nth port:
( 1, , )
n n n
n n n
n N
V V VI I I
(omit ~ for simpler notation)
ElecEng4FJ4 LECTURE 10: MICROWAVE NETWORK ANALYSIS 11
[Pozar]
Impedance and Admittance Matrices – Review
1 11 12 1 1
2 21 22 2
2
N
N N NN N
V Z Z Z IV Z Z I
V Z Z I
• relate the total voltages to the total currents at the ports
0 for all k
iij
j I k j
VZI
all ports except port j are open-circuited
1 11 12 1 1
2 21 22 2
2
N
N N NN N
I Y Y Y VI Y Y V
I Y Y V
impedance or Z matrix
0 for all k
iij
j V k j
IYV
all ports except port j are short-circuited
admittance or Y matrix
1Y ZElecEng4FJ4 LECTURE 10: MICROWAVE NETWORK ANALYSIS 12
Impedance Matrix: Example
2
1
1
2
111
1 0
222
2 0
112
2 0?
221 12
1 0
?
?
?
I
I
I
I
VZIVZIVZIVZ ZI
A CZ Z
B CZ Z
CZ
1I 2I
ElecEng4FJ4 LECTURE 10: MICROWAVE NETWORK ANALYSIS 13
Transmission (ABCD) Matrix (aka Cascade Parameters)
• defined for a 2-port network (see Fig. a)
• particularly useful when cascading 2-port networks (see Fig. b)
1 2 2
1 2 2
V AV BII CV DI
1 2
1 2
V A B VI C D I
1 1 1 2 2 3
1 1 1 2 2 3
V A B A B VI C D C D I
ElecEng4FJ4 LECTURE 10: MICROWAVE NETWORK ANALYSIS 14
Reciprocal Networks consider any two ports (e.g., ports 1 and 2) in a network containing
only linear media (no plasma, ferrites or active devices)
short-circuit all other ports – the network is now a 2-port network
apply Reciprocity Theorem in the volume of the network for two possible sources a and b residing outside the network
( ) ( )a b b aS Sd d E H s E H s
Reciprocity Theorem• consider two separate sets of sources (set a and set b) and their fields
in a linear medium so that superposition applies (linear media)/
/a a a b
a a a b
jj
E H M HH Ε J E
/ /
b b b a
b b b a
jj
E H M HH E J E
ElecEng4FJ4 LECTURE 10: MICROWAVE NETWORK ANALYSIS 15
PROOF – OPTIONAL
Reciprocity Theorem in Electromagnetics – Optional • combine all 4 equations properly by using the vector identity
( )
a b b a
b a a b a b b a
E H E HH E Ε H H E E H
( )a b b a a b a b b a b a E H E H E J H M E J H M • reciprocity theorem – differential form
( )
( )s
a b b aS
a b a b b a b aV
d
dv
E H E H s
E J H M E J H M
• reciprocity theorem – integral form
• in our case, there are no sources in the volume of the network( ) ( )a b b aS S
d d E H s E H s ElecEng4FJ4 LECTURE 10: MICROWAVE NETWORK ANALYSIS 16
Reciprocal Networks (2)
a
b• field at port 1
1 1 1
1 1 1
a a
b b
VV
E eE e
1 1 1
1 1 1
a a
b b
II
H hH h
• field at port 22 2 2
2 2 2
a a
b b
VV
E eE e
2 2 2
2 2 2
a a
b b
II
H hH h
1 21 1 1 1 1 1 2 2 2 2 2 2( ) ( ) ( ) ( ) 0a b b a a b b aS S
V I V I d V I V I d e h s e h s apply Reciprocity Theorem: only over port cross-sections
unity normalization holds for both ports, e.g.,
1 11 1 2 2( ) ( ) 1
S Sd d e h s e h s
1 1 1 1 2 2 2 2 0a b b a a b b aV I V I V I V I
short
short
short
assume CV = 1 (result doesn’t depend on choice of CV)
( ) 0S
ElecEng4FJ4 LECTURE 10: MICROWAVE NETWORK ANALYSIS 17
Z and Y Matrices of Reciprocal Networks
substitute Y-matrix equations1 11 1 12 2
2 21 1 22 2
a a a
a a a
I Y V Y VI Y V Y V
1 11 1 12 2
2 21 1 22 2
b b b
b b b
I Y V Y VI Y V Y V
12 21 1 2 2 1( )( ) 0a b a bY Y V V V V 12 21Y Y
• both the Y and Z matrices are symmetric for the reciprocal (linear medium) networks
into1 1 1 1 2 2 2 2 0a b b a a b b aV I V I V I V I
• the ABCD matrix for a reciprocal network fulfills
1AD BC
ElecEng4FJ4 LECTURE 10: MICROWAVE NETWORK ANALYSIS 18
Loss-free Networks
since the N-port network is loss-free, the net real power delivered to the network must be zero
1 1 1express complex power: ( )2 2 2
T T T TP V I ZI I I Z I
assume the network is reciprocal, ZT = Z
1 1
1Re 02
N N
av m mn nn m
P I Z I
consider the case where all ports are left open-circuited except the nth port
10, all Re 02k av n nn nI k n P I Z I
Re 0, 1, ,nnZ n N
ElecEng4FJ4 LECTURE 10: MICROWAVE NETWORK ANALYSIS 19
Z and Y Matrices of Loss-free Networks
now let all ports be open-circuited but ports n and m
real
10, all , Re[ ( )] 02k av nm n m n mI k n m P Z I I I I
Re 0, , 1,nmZ n m N
• all elements of the Z matrix for a loss-free network are purely imaginary, i.e., ReZ = 0
• analogous derivation shows that ReY = 0, too
ElecEng4FJ4 LECTURE 10: MICROWAVE NETWORK ANALYSIS 20
Summary
ElecEng4FJ4 LECTURE 10: MICROWAVE NETWORK ANALYSIS 21
• voltages and currents for microwave networks depend on the field differently compared to static (or low-frequency) networks
• to calculate equivalent voltages and currents of traveling waves, we need the modal vectors: electric e and magnetic h
• e and h are the rms field phasors of the traveling wave in the TL/waveguide such that it carries 1 W power
• the most general definition sets the voltage-to-current ratio to 1: voltage is the same as current describing the root-power wave
• reciprocal networks feature symmetric Z and Y matrices
• loss-free networks feature purely imaginary Z and Y matrices
1/2ˆ ˆ( ) ( ) , We eS SV ds I ds E h z e H z