Download Microwave Network Analysis p1

EKT 441
MICROWAVE COMMUNICATIONS
CHAPTER 3:
MICROWAVE NETWORK ANALYSIS
(PART 1)
NETWORK ANALYSIS



Most electrical circuits can be modeled as a “black box” that
contains a linear network comprising of R, L, C and dependant
sources.
Has four terminals, 2-input ports and 2-output ports
Hence, large class of electronics can be modeled as two-port
networks, which completely describes behavior in terms of
voltage (V) and currents (I) (illustrated in Fig 1 below)
Figure 1
NETWORK ANALYSIS


Several ways to characterize this network, such as
1. Impedance parameters
2. Admittance parameters
3. Hybrid parameters
4. Transmission parameters
Scattering parameters (S-parameters) is introduced
later as a technique to characterize high-frequency
and microwave circuits
NETWORK ANALYSIS
Impedance Parameters

Considering Figure 1, considering network is linear, principle of
superposition can be applied. Voltage, V1 at port 1 can be
expressed in terms of 2 currents as follow;
V1  Z11I1  Z12 I 2


Since V1 is in Volts, I1 and I2 are in Amperes, Z11 and Z12 must be
in Ohms. These are called impedance parameters
Similarly, for V2, we can write V2 in terms of I1 and I2 as follow;
V2  Z21I1  Z22 I 2
NETWORK ANALYSIS
Impedance Parameters (cont)

Using the matrix representation, we can write;
V1  Z11Z12   I1 
V   Z Z   I 
 2   21 22   2 

Or

Where [Z] is called the impedance matrix of the two-port network
V   Z I 
NETWORK ANALYSIS
Impedance Parameters (cont)

If port 2 of the network is left open, then I2 will be zero. In this
condition;
V1
Z11 
I1

and
I 2 0
Z 21 
V2
I1
I 2 0
Similarly, when port 1 of the network is left open, then I1 will be
zero. In this condition;
Z12 
V1
I2
and
I1  0
Z 22 
V2
I2
I1  0
NETWORK ANALYSIS
Example 1

Find the impedance parameters of the 2-port network shown
here
NETWORK ANALYSIS
Example 1: Solution

If I2 is zero, then V1 and V2 can be found from Ohm’s Law as 6I1.
Hence from the equations
Z11 

V1
I1

I 2 0
6 I1
 6
I1
Z 21 
V2
I1

I 2 0
6 I1
 6
I1
Similarly, when the source is connected at port 2 and port 1 has
an open circuit, we find that;
V2  V1  6I 2
NETWORK ANALYSIS
Example 1: Solution

Hence, from
V
Z12  1
I2

I1  0
6I
 2  6
I2
V2
Z 22 
I2
Therefore,
 Z11 Z12  6 6

Z


Z
6
6


22 
 21
I1  0
6I 2

 6
I2
NETWORK ANALYSIS
Example 2

Find the impedance parameters of the 2-port network shown
here
NETWORK ANALYSIS
Example 2: Solution

As before, assume that the source is connected at port-1 while
port 2 is open. In this condition, V1 = 12I1 and V2 = 0. Therefore,
Z11 

V1
I1

I 2 0
12 I1
 12 and
I1
Z 21 
V2
I1
 0
I 2 0
Similarly, with a source connected at port-2 while port-1 has an
open circuit, we find that,
V2  3I 2
and
V1  0
NETWORK ANALYSIS
Example 2: Solution

Hence,
V
Z12  1
I2

0
and
I1  0
Therefore,
 Z11 Z12  12 0

Z


Z
0
3


22 
 21
V2
Z 22 
I2
I1  0
3I 2

 3
I2
NETWORK ANALYSIS
Admittance Parameters

Consider again Figure 1. Assuming the network is linear, principle
of superposition can be applied. Current, I1 at port 1 can be
expressed in terms of 2 voltages as follow;
I1  Y11V1  Y12V2


Since I1 is in Amperes, V1 and V2 are in Volts, Y11 and Y12 must
be in Siemens. These are called admittance parameters
Similarly, we can write I2 in terms of V1 and V2 as follow;
I 2  Y21V1  Y22V2
NETWORK ANALYSIS
Admittance Parameters (cont)

Using the matrix representation, we can write;
 I1  Y11 Y12  V1 
 I   Y Y  V 
 2   21 22   2 

Or

Where [Y] is called the admittance matrix of the two-port network
I   Y V 
NETWORK ANALYSIS
Admittance Parameters (cont)

If port 2 of the network has a short circuit, then V2 will be zero. In
this condition;
I1
Y11 
V1 V
and
2 0

Y21 
I2
V1
V2  0
Similarly, with a source connected at port 2, and a short circuit at
port 1, then V1 will be zero. In this condition;
Y12 
I1
V2
and
V1  0
Y22 
I2
V2
V1  0
NETWORK ANALYSIS
Example 3

Find the admittance parameters of the 2-port network shown here
NETWORK ANALYSIS
Example 3: Solution

If V2 is zero, then I1 is equal to 0.05V1, I2 is equal to -0.05V1.
Hence from the equations above;
Y11 
I1
V1 V
2 0


0.05V1
 0.05S
V1
Y21 
I2
V1

V2  0
 0.05V1
 0.05S
V1
Similarly, with a source connected at port 2 and port 1 having a
short circuit, we find that;
I 2  I1  0.05V2
NETWORK ANALYSIS
Example 3: Solution (cont)

Hence, from
I
Y12  1
V2

V1  0
 0.05V2

 0.05S
V2
I2
Y22 
V2
Therefore,
Y11 Y12   0.05  0.05
Y Y    0.05 0.05 

 21 22  
V1  0
0.05V2

 0.05S
V2
NETWORK ANALYSIS
Example 4

Find the admittance parameters of the 2-port network shown here
NETWORK ANALYSIS
Example 4: Solution

Assuming that a source is connected to at port-1 while keeping
port 2 as a short circuit, we find that;
I1 

0.10.2  0.025
0.0225
V1 
V1 A
0.1  0.2  0.025
0.325
And if voltage across 0.2S is VN, then;
VN 

I1
0.0225
V

V1  1 V
0.2  0.025 0.225  0.325
3.25
Therefore;
I 2  0.2VN  
0.2
V1 A
3.25
NETWORK ANALYSIS
Example 4: Solution (cont)

Therefore;
I
Y11  1
V1 V
2 0

0.0225

 0.0692S
0.325
I2
Y21 
V1 V
2 0
0.2

 0.0615S
3.25
Similarly, with a source at port-2 and port-1 having a short circuit;
0.20.1  0.025
0.025
I2 
V1 
V2 A
0.1  0.2  0.025
0.325
NETWORK ANALYSIS
Example 4: Solution (cont)

And if voltage across 0.1S is VM, then,
I2
0.025
2V2
VM 

V2 
V
0.1  0.025 0.125  0.325
3.25

Therefore,
I1  0.1VM  

0.2
V2 A
3.25
Hence;
I
Y12  1
V2
V1  0
0.2

 0.0615S
3.25
Y22 
I2
V2

V1  0
0.025
 0.0769S
0.325
NETWORK ANALYSIS
Example 4: Solution (cont)

Therefore,
Y11 Y12   0.0692  0.0615
Y Y    0.0615 0.0769 

 21 22  
NETWORK ANALYSIS
Hybrid Parameters

Consider again Figure 1. Assuming the network is linear, principle of
superposition can be applied. Voltage, V1 at port-1 can be expressed in
terms of current I1 at port-2 and voltage V2 at port-2, as follow;
V1  h11I1  h12V2

Similarly, we can write I2 in terms of I1 and V2 as follow;
I 2  h21I1  h22V2


Since V1 and V2 are in volts, while I1 and I2 are in amperes, parameter
h11 must be in ohms, h12 and h21 must be dimensionless, and h22 must
be in siemens.
These are called hybrid parameters.
NETWORK ANALYSIS
Hybrid Parameters (cont)

Using the matrix representation, we can write;
V1   h11 h12   I1 
 I   h
 V 
h
 2   21 22   2 


Hybrid parameters are especially important in transistor circuit
analysis. The parameters are defined as follow; If port-2 has a
short circuit, then V2 will be zero.
This condition gives;
V1
h11 
I1 V
and
2 0
h21 
I2
I1
V2  0
NETWORK ANALYSIS
Hybrid Parameters (cont)

Similarly, with a source connected to port-2 while port-1 is open;
h12 



V1
V2
and
I1  0
h22 
I2
V2
I1  0
Thus, parameters h11 and h21 represent the input impedance and
the forward current gain, respectively, when a short circuit is at
port-2.
Similarly, h12 and h22 represent reverse voltage gain and the
output admittance, respectively, when port-1 has an open circuit.
In circuit analysis, these are generally denoted as hi, hf, hr and ho,
respectively.
NETWORK ANALYSIS
Example 5: Hybrid parameters

Find hybrid parameters of the 2-port network shown here
NETWORK ANALYSIS
Example 5: Solution

With a short circuit at port-2,
63 

V1  I1 12 
  14 I1
63


And using the current divider rule, we find that
I2  
6
2
I1   I1
3  6
3
NETWORK ANALYSIS
Example 5: Solution (cont)

Therefore;
V
h11  1
I1 V
 14
2 0

V2  0
2

3
Similarly, with a source at port-2 and port-1 having an open
circuit;
V2  (3  6) I 2  9I 2

I2
h21 
I1
And
V1  6I 2
NETWORK ANALYSIS
Example 5: Solution (cont)

Because there is no current flowing through the 12Ω resistor,
hence;
h12 

V1
V2

V1  0
6I 2 2

9I 2 3
Thus,
 h11
h
 21
2 

14

h12  

3



h22    2 1 S 
 3 9 
h22 
I2
V2

I1  0
1
S
9
NETWORK ANALYSIS
Transmission Parameters

Consider again Figure 1. Since the network is linear, the
superposition principle can be applied. Assuming that it contains
no independent sources, Voltage V1 and current at port 1 can be
expressed in terms of current I2 and voltage V2 at port-2, as
follow;
V1  AV2  BI 2

Similarly, we can write I1 in terms of I2 and V2 as follow;
I1  CV2  DI 2

Since V1 and V2 are in volts, while I1 and I2 are in amperes,
parameter A and D must be in dimensionless, B must be in
Ohms, and C must be in Siemens.
NETWORK ANALYSIS
Transmission Parameters (cont)

Using the matrix representation, we can write;
V1   A B  V2 
 I   C D  I 
 2 
 1 


Transmission parameters, also known as elements of chain
matrix, are especially important for analysis of circuits connected
in cascade. These parameters are determined as follow; If port-2
has a short circuit, then V2 will be zero.
This condition gives;
I1
V1
D

and
B
 I 2 V 0
 I2
V2  0
2
NETWORK ANALYSIS
Transmission Parameters (cont)

Similarly, with a source connected at port-1 while port-2 is open,
we find;
V1
A
V2
and
I 2 0
I1
C
V2
I 2 0
NETWORK ANALYSIS
Example 6: Transmission parameters

Find transmission parameters of the 2-port network shown here
NETWORK ANALYSIS
Example 6: Solution

With a source connected to port-1, while port-2 has a short circuit
(so that V2 is zero)
I 2  I1

and
V1  I1
Therefore;
B
V1
 I1 V
2 0
 1
and
D
I1
 I2
1
V2  0
NETWORK ANALYSIS
Example 6: Solution (cont)

Similarly, with a source connected at port-1, while port-2 is
open (so that I2 is zero)
V2  V1

I1  0
Hence;
V1
A
V2

and
1
I 2 0
Thus;
 A B  1 1
C D   0 1

 

and
C
I1
V2
0
I 2 0
NETWORK ANALYSIS
Example 7: Transmission parameters

Find transmission parameters of the 2-port network shown here
NETWORK ANALYSIS
Example 7: Solution

With a source connected to port-1, while port-2 has a short circuit
(so that V2 is zero), we find that

1 
2  j
 I1 
V1  1 
I1
1  j
 1  j 

and
1
 j  I   1 I
I2 
1
1
1
1

j

1
j
Therefore;
B
V1
 I1 V
2 0
 ( 2  j ) 
and
D
I1
 I2
 1  j
V2  0
NETWORK ANALYSIS
Example 7: Solution (cont)


Similarly, with a source connected at port-1, while port-2 is
open (so that I2 is zero)
1

 1  j 
1 
and
V

I1
 I1  
 I1
V1  1 
2
j
j 

 j 
Hence;
V1
A
V2

Thus;
 1  j
and
I 2 0
 A B  1  j 2  j 
C D   j

1

j


 

C
I1
V2
 j
I 2 0
ABCD MATRIX

Of particular interest in RF and microwave systems is ABCD
parameters. ABCD parameters are the most useful for
representing Tline and other linear microwave components in
general.
Take note of the
V1   A B  V2 
 I   C D   I 
 2 
 1 
 V1  AV2  BI 2
direction of positive current!
I1
I2
(4.1a)
V1
2 -Ports
I1  CV2  DI 2
V1
I1
V1
B
C
A
D
I 2 V 0
V2 I  0
V2 I  0
2
2
2
Open circuit Port 2
I1
I 2 V 0
2
(4.1b)
Short circuit Port 2
V2
ABCD MATRIX

The ABCD matrix is useful for characterizing the overall response
of 2-port networks that are cascaded to each other.
I2 ’
I1
V1
 A1 B1 
C D 
 1 1
I2
V2
I3
 A2 B2 
C D 
2
 2
V3
V1   A1
 I   C
 1  1
B1   A2
D1  C2
V1   A3
 
 I1  C3
B2  V3 
D2   I 3 
B3  V3 
D3   I 3 
Overall ABCD matrix
NETWORK ANALYSIS




Many times we are only interested in the voltage (V) and current
(I) relationship at the terminals/ports of a complex circuit.
If mathematical relations can be derived for V and I, the circuit
can be considered as a black box.
For a linear circuit, the I-V relationship is linear and can be written
in the form of matrix equations.
A simple example of linear 2-port circuit is shown below. Each
port is associated with 2 parameters, the V and I.
I1
Port 1 V1
R
Convention for positive
polarity current and voltage
I2
C
+
V2 Port 2
-
NETWORK ANALYSIS

For this 2 port circuit we can easily derive the I-V relations.
I1  1 V1  V2 
R

R
I1
I1  I 2  jCV2
V1
V
R

 I 2   1 V1  1  jC V2
R
I
I1
2
jCV
C
2
V
2
2
We can choose V1 and V2 as the independent variables, the I-V
Network parameters
(Y-parameters)
relation can be expressed in matrix equations.
1
 I1   R
I    1
 2   R
I1

 V 
R
 1 
1  jC  V
R
 2 
1
R

 I1   y11
I    y
 2   21
I
I1
y12  V1 
y22  V2 
I2
2
Port 1 V1
C
V
2
Port 2
V1
2 - Ports
V2
NETWORK ANALYSIS

To determine the network parameters, the following relations can
be used:
I1
I1
y

y

11
12
I
y
y
V
 1   11
V1 V  0
V 2 V 0
12   1 

2
1
I   y



 2   21 y22  V2 
I
y21  2
V1 V  0
2
or
I  Y V

I
y22  2
V 2 V 0
1
This means we short circuit the port
For example to measure y11, the following setup can be used:
I1
V1
I2
2 - Ports
V2 = 0
Short circuit
NETWORK ANALYSIS


By choosing different combination of independent variables,
different network parameters can be defined. This applies to all
linear circuits no matter how complex.
Furthermore this concept can be generalized to more than 2
I1
ports, called N - port networks.
I1
V1
Linear circuit, because all
elements have linear I-V relation
V1
I2
V2
2 - Ports
V1   z11
V    z
 2   21
z12   I1 
z22   I 2 
V1   h11
 I   h
 2   21
h12   I1 
h22  V2 
I2
V2
THE SCATTERING MATRIX





Usually we use Y, Z, H or ABCD parameters to
describe a linear two port network.
These parameters require us to open or short a
network to find the parameters.
At radio frequencies it is difficult to have a proper short
or open circuit, there are parasitic inductance and
capacitance in most instances.
Open/short condition leads to standing wave, can
cause oscillation and destruction of device.
For non-TEM propagation mode, it is not possible to
measure voltage and current. We can only measure
power from E and H fields.
THE SCATTERING MATRIX
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Hence a new set of parameters (S) is needed which
 Do not need open/short condition.
 Do not cause standing wave.
 Relates to incident and reflected power waves, instead of
voltage and current.
• As oppose to V and I, S-parameters relate the reflected and incident
voltage waves.
• S-parameters have the following advantages:
1. Relates to familiar measurement such as reflection coefficient,
gain, loss etc.
2. Can cascade S-parameters of multiple devices to predict system
performance (similar to ABCD parameters).
3. Can compute Z, Y or H parameters from S-parameters if needed.