Whites, EE 481/581
Lecture 14
Page 1 of 8
Lecture 14: Impedance and
Admittance Matrices.
As in low frequency electrical circuits, a matrix description for
portions of microwave circuits can prove useful in simulations
and for understanding the behavior of the subcircuit, among
other reasons.
Matrix descriptions are a very convenient way to integrate the
effects of a subcircuit into a larger circuit without having to
concern oneself with the specific details of the subcircuit.
We will primarily be interested in ABCD and S matrices in this
course, though Z and Y matrices will also prove useful. The
ABCD and S parameters are probably new to you. As we’ll see,
using these matrix descriptions is very similar to other two-port
models for circuits you’ve seen before, such as Z and Y matrices.
Z Matrices
As an example of Z matrices, consider this two-port network:
I1
+
V1
-
I2
Z 
© 2016 Keith W. Whites
+
V2
-
Whites, EE 481/581
Lecture 14
Page 2 of 8
The Z-matrix description of this two-port is defined as
V1   Z11 Z12   I1 
 
V    Z

Z 22   I 2 
 2  
 21 

(1)
 Z 
Z ij 
where
Vi
Ij
(4.28)
I k  0, k  j
Example N14.1: As an example, let’s determine the Z matrix for
this T-network (Fig. 4.6) shown below:
I1
V1
ZA
ZB
ZC
I2
V2
Applying (1) repeatedly to all four Z parameters, we find:
V
Z11  1
 Z A  ZC
( Z in at port 1 w/ port 2 o.c.)
I1 I 0
2
Z12 
Z 21 
Z 22 
V1
I2
V2
I1
V2
I2
 V1  I 2 Z C (think of I 2 as source)  Z12  Z C
I1  0
 V2  I1Z C (think of I1 as source)  Z 21  Z C
I 2 0
 Z B  ZC
I1  0
( Z in at port 2 w/ port 1 o.c.)
Whites, EE 481/581
Lecture 14
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Collecting these calculations, then for this T-network:
ZC 
 Z A  ZC

Z
   Z
Z B  Z C 
C

Notice that this matrix is symmetrical. That is, Z ij  Z ji for
i  j . It can be shown that  Z  will be symmetrical for all
“reciprocal” networks. (See p. 8 for a definition of reciprocal.)
What’s the usefulness of an impedance matrix description? For
one thing, if a complicated circuit exists between the ports, one
can conveniently amalgamate the electrical characteristics into
this one matrix.
Second, if one has networks connected in series, it’s very easy to
combine the Z matrices. For example:
I1
+
V1
-
I1
+
V1
I1
+
V1
-
 Z 
I 2
+ 
V
- 2
 Z 
I 2
+ 
V
- 2
Z 
By definition
I2
+
V2
-
Whites, EE 481/581
Lecture 14
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V  
 I  
V  
I 
1
1
1
    Z     and     Z    1 
V2 
 I 2 
V2 
 I 2 
From the figure we see that I1  I1 , I 2  I 2 , and that
V1  V1  V1 , V2  V2  V2 . So, summing the above two matrix
equations gives
 V   V  
I 
I 
1
1
1

   Z       Z    1 
V2  V2 
 I 2 
 I 2 
Also from the figure, note that I1  I1 and I 2  I 2 . Therefore,
V1 
 I1 


(2)
 Z    Z    I 
V   
 2
 2
Z 
From this result, we see that for a series connection of two-port
networks, we can simply add the Z matrices to form a single
“super” Z matrix
(3)
 Z    Z    Z 
that incorporates the electrical characteristics of both networks
and their mutual interaction.
Y Matrices
A closely related characterization is the Y-matrix description of
a network:
Whites, EE 481/581
Lecture 14
Page 5 of 8
I1
+
V1
-
I2
+
V2
-
Y 
By definition:
 I1  Y11 Y12  V1 
 I   Y Y   V 
21
22   2 
 2 



(4)
Y 
Yij 
where
Ii
Vj
(4.29)
Vk  0, k  j
Comparing (4) and (1) we see that
1
Y    Z 
(4.27),(5)
The Y-parameter description is useful when connecting networks
in parallel:
I1
+
V1
-
I1
+
V1
I1
+
V1
-
Y 
I 2
+ 
V
- 2
Y 
I 2
+ 
V
- 2
Y 
From this diagram, we see that
I2
+
V2
-
Whites, EE 481/581
Lecture 14
Page 6 of 8
 I1   I1  I1 
V1 


Y





I 
V 




 2   I2  I2 
 2
(6)
Y   Y   Y 
(7)
where
Z and Y Matrices for Microwave Networks
We can easily generalize these Z and Y parameter descriptions
for microwave networks and multiport networks.
Consider an N-port network connected to transmission lines
(Fig. 4.5):
V1 , I1
V1 , I1
t1
1
3
Z 
V2 , I 2
V2 , I 2
t3
2
t2
V3 , I 3
V3 , I 3
VN , I N
N
VN , I N
tN
The locations tn , n  1,, N , are the terminal planes for each
port. These are the positions on that TL where the phase is
arbitrarily chosen equal to zero.
At these terminal planes (which are also called the phase
planes), zn  0 so that the voltage on the nth TL
Vn  zn   Vn e  j n zn  Vn e  jn zn
Whites, EE 481/581
Lecture 14
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becomes
Vn  zn  0   Vn  Vn
(4.24a),(8)
Likewise,
I n  zn  0   I n  I n
(4.24b),(9)
Since the telegrapher’s equations are linear, any N linearly
independent combinations of the 2N quantities Vn and I n may be
chosen as the independent variables.
For an impedance description, we choose I n as the independent
variables. Then,
 V1  z1  0  
 I1  z1  0  

Z
Z






11
1N
0
0
V
z

I
z





2
2
2
2

      

(10)
 

 




  Z N 1  Z NN  

0
0
V
z

I
z





 N N

 N N

or,
(4.25),(11)
V    Z    I 
For an admittance description, we choose Vn as the independent
variables:
 I1  z1  0  
 V1  z1  0  

Y
Y

  11

1N  
0
0
I
z

V
z





 2 2
       2 2

(12)











Y
Y

  N1

NN  
 I N  z N  0  
VN  z N  0  
or,
(4.26),(13)
 I   Y   V 
Whites, EE 481/581
Lecture 14
Page 8 of 8
Global Characteristics of Z and Y Matrices
Finally, these are two extremely important properties of Z and Y
matrices:
Z ij  Z ji and Yij  Y ji
1.
(4.36),(14)
That is, the matrices are symmetrical about the main diagonal
for reciprocal networks. (We observed this characteristic in
the Z matrix of an impedance T-network earlier in this
lecture.)
A reciprocal network is one where a source instrument and a
measurement instrument can be exchanged between two ports
and the measured quantity remains unchanged. All passive
(and some active) circuits you’ve encountered in circuits and
electronics courses are reciprocal networks.
2. For a lossless network
e Z ij   0
i, j
From (5), this implies that
e Yij   0
i, j
(4.39),(15)
In other words, for a lossless network the Z and Y matrices are
purely imaginary.