Transcript Microwave Network Analysis Part 1

EKT 441
MICROWAVE COMMUNICATIONS
CHAPTER 3:
MICROWAVE NETWORK ANALYSIS
(PART 1)
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
R
Convention for positive
polarity current and voltage
I2
+
Port 1 V1
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
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
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

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.
THE SCATTERING MATRIX

Consider an n – port network:
Each port is considered to be
connected to a Tline with
specific Zc.
Reference plane
for local z-axis
(z = 0)
Port 1
Zc1
Port n
Zcn
Port 2
Zc2
T-line or
waveguide
Linear
n - port
network
THE SCATTERING MATRIX


There is a voltage and current on each port.
This voltage (or current) can be decomposed into the incident (+) and
reflected component (-).
V z   V2 e  jz  V2 e jz
V 0  V2  V2  V2
V1+ V 1-
Port 1
V1
I1

Port
V1 n
I 0  I 2  I 2  I 2
z=0
Linear
n - port
Network
Port 1
+z
+
V1-
-
I z   I 2 e  jz  I 2 e jz
Port 2
V1+
+
V1  V1  V1
I1  I1  I1

c1
 1 V1  V1
Z

THE SCATTERING MATRIX


The port voltage and current can be normalized with respect to the
impedance connected to it.
It is customary to define normalized voltage waves at each port as:
Normalized
incident waves
ai 
Vi
(4.3a)
Z ci
i = 1, 2, 3 … n

ai  I i Z ci
bi 
Vi
Z ci

bi  I i Z ci
Normalized
reflected waves
(4.3b)
THE SCATTERING MATRIX

Thus in general:
V1+ V1-
Port 1
Vn+
Vn- Port n
Zc1
Port 2
V2+
V2Zc2
T-line or
waveguide
Zcn
Linear
n - port
Network
Vi+ and Vi- are propagating
voltage waves, which can
be the actual voltage for TEM
modes or the equivalent
voltages for non-TEM modes.
(for non-TEM, V is defined
proportional to transverse E
field while I is defined proportional to transverse H field, see
[1] for details).
THE SCATTERING MATRIX
If the n – port network is linear (make sure you know what this
means!), there is a linear relationship between the normalized waves.
For instance if we energize port 2:


V1-
Port 1
V sV

V s V


Zc1
Vn-
Port n
1
12
2

2
Port 2
V2+
V2
-
Zc2
Linear
n - port
Network
22
2
Zcn
V
n

s V
n2

2
Constant that
depends on the
network construction
THE SCATTERING MATRIX

Considering that we can send energy into all ports, this can be
generalized to: V  s V  s V  s V    s V

1

11

1
12

2
13

3
1n
n
V  s V  s V  s V  s V

2

21

1
22
2

23
3
2n
n
V  s V  s V  s V  s V

n


n1

1
n2

n3
3
nn
(4.4a)

n
Or written in Matrix equation:
V

V
 :

V

V  SV


or

21
2
n
 s
 
  s
 :
 
 s
11
1

2


n1
s
s
:
s
12
22
n2
... s  V 
 
... s  V 
 :  : 
 
... s  V 

1n
1

2n
2
(4.4b)

nn
n
Where sij is known as the generalized Scattering (S) parameter, or
just S-parameters for short. From (4.3), each port i can have
different characteristic impedance Zci
THE SCATTERING MATRIX

Consider the N-port network shown in figure 4.1.
Figure 4.1: An arbitrary N-port microwave network
THE SCATTERING MATRIX
Vn+ is the amplitude of the voltage wave incident on port n.
 Vn- is the amplitude of the voltage wave reflected from port n.
 The scattering matrix or [S] matrix, is defined in relation to
these incident and reflected voltage wave as:

V1   S11
  
V2   S 21
 .   .
 
 .   .
 .   .
  
Vn   S N 1
S12
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. S1N  V1 
 
.
.  V2 
.
.  . 
 
.
.  . 
.
.  . 
  
. S NN  Vn 
[4.1a]
THE SCATTERING MATRIX
or
V   S V 


[4.1b]
A specific element of the [S] matrix can be determined as:
Vi 
S ij  
Vj
[4.2]
Vk  0 , for k  j
Sij is found by driving port j with an incident wave Vj+, and measuring
the reflected wave amplitude, Vi-, coming out of port i.
The incident waves on all ports except j-th port are set to zero (which
means that all ports should be terminated in matched load to avoid
reflections).
Thus, Sii is the reflection coefficient seen looking into port i when all
other ports are terminated in matched loads, and Sij is the transmission
coefficient from port j to port i when all other ports are terminated in
matched loads.
THE SCATTERING MATRIX

For 2-port networks, (4.4) reduces to:
V   s
 
V  s

11
1

21
2
V
s 
V

1
11
1


s
s
12
22
V 
 V 
   S  
 V 
V 


1


2
V
s 
V
2

2

21
V2  0
(4.5a)
1
1
V
s 
V

2

22
V2  0
2
V
s 
V

(4.5b)
1

12
V1  0
2

V1  0
Note that Vi+ = 0 implies that we terminate i th port with its
characteristic impedance.
Thus zero reflection eliminates standing wave.
THE SCATTERING MATRIX
V1+
Vs
V2-
Zc1
Zc1
2 – Port
V1-
V
s 
V
Measurement of s11 and s21:

-
Zc2
V
s 
V
1
11
1
V1
Zc2

2

21
V2 0
1

V2 0
V2+
Zc2
Zc1
Zc2
2 – Port
Zc1
V2Measurement of s22 and
s12:
V
s 
V

2
22
2
V
s 
V

1

12
V1 0
2

V1 0
Vs
THE SCATTERING MATRIX


Input-output behavior of network is defined in terms of normalized
power waves
S-parameters are measured based on properly terminated
transmission lines (and not open/short circuit conditions)
[s]  {[ s] }
*
t
1
THE SCATTERING MATRIX
THE SCATTERING MATRIX
THE SCATTERING MATRIX
Reciprocal and Lossless networks
 Impedance and admittance matrices are symmetric for reciprocal
networks
 A symmetric network happens when:
[ s]  [ s]
(4.6a)
t

It is also purely imaginary for lossless network (no real power can
be delivered to the network)
[s]  {[ s] }
*

t
1
(4.6b)
A matrix that satisfies the condition of (4.6b) is called a unitary
matrix
THE SCATTERING MATRIX

Transpose of a Matrix (taken from Engineering
Maths 4th Ed by KA Stroud)
a b 
S   

c d 
Transpose of [S],
written as [S]t
a c 
S t  

b d 
THE SCATTERING MATRIX

Symmetrical Matrix (taken from Engineering
Maths 4th Ed by KA Stroud)
If
a
S   
a
11
12
a 

a 
21
22
It is symmetrical when aij = aji
When a [S] is symmetric, it is also reciprocal
THE SCATTERING MATRIX
Reciprocal and Lossless networks (cont)
 The matrix equation in (4.6b) can be re-written in;
S S 1
For i = j
S S
For i ≠ j
N
*
ki
ki
(4.7)
k 1
N
ki
*
kj
0
k 1

Used to determine
reciprocality for a 2 port
network
OR
| S || S | 1
11
21
THE SCATTERING MATRIX (Ex)

Find the S parameters of the 3 dB attenuator circuit shown
in Figure 4.2.
Figure 4.2: A matched 3 dB attenuator with a 50 Ω
characteristic impedance.
THE SCATTERING MATRIX (Ex)

From the following formula, S11 can be found as the
reflection coefficient seen at port 1 when port 2 is
terminated with a matched load (Z0 =50 Ω);
V
S 
V

i
ij

Vk  0 fork  j
j

The equation becomes;
V
S 
V

(1)
1
11
1

Z Z

Z Z
V2 0

(1)
in
V2 0
Z0
(1)
in
On port 2
0
0
THE SCATTERING MATRIX (Ex)

To calculate Zin(1), we can use the following formula;
Z
(1)
in
141.8(8.56  50)
 8.56 
 50
141.8  (8.56  50)

Thus S11 = 0. Because of the symmetry of the circuit,
S22 = 0.
 S21 can be found by applying an incident wave at port
1, V1+, and measuring the outcome at port 2, V2-. This is
equivalent to the transmission coefficient from port 1 to

port 2:
V
S 
V
2
21
1

V2 0
THE SCATTERING MATRIX (Ex)

From the fact that S11 = S22 = 0, we know that V1- = 0
when port 2 is terminated in Z0 = 50 Ω, and that V2+ = 0. In
this case we have V1+ = V1 and V2- = V2.
41.44  50 

V V V 

  0.7071V
 41.44  8.56  50  8.56 

2

2
1
1
Where 41.44 = (141.8//58.56) is the combined resistance
of 50 Ω and 8.56 Ω paralled with the 141.8 Ω resistor.
Thus, S21 = S12 = 0.707
THE SCATTERING MATRIX (Ex)

A two port network is known to have the following
scattering matrix:
 0.15  0
S   
0.85  45

a)
b)
c)
0.85  45 

0.2  0 



Determine if the network is reciprocal and lossless.
If port 2 is terminated with a matched load, what is the
return loss seen at port 1?
If port 2 is terminated with a short circuit, what is the
return loss seen at port 1?
THE SCATTERING MATRIX (Ex)


Q: Determine if the network is reciprocal and lossless
Since [S] is not symmetric, the network is not reciprocal. Taking the 1st
column, (i = 1) gives;
| S |  | S |  (0.15)  (0.85)  0.745  1
2
11
2
2
2
21
So the network is not lossless.
 Q: If port two is terminated with a matched load, what is the return loss
seen at port 1?
 When port 2 is terminated with a matched load, the reflection coefficient
seen at port 1 is Γ = S11 = 0.15. So the return loss is;
RL  20 log |  | 20 log( 0.15)  16.5dB
Used to determine
reciprocality for a 2 port
network
| S || S | 1
11
21
THE SCATTERING MATRIX (Ex)



Q: If port two is terminated with a short circuit, what is the return loss
seen at port 1?
When port 2 is terminated with a short circuit, the reflection coefficient
seen at port 1 can be found as follow
From the definition of the scattering matrix and the fact that V2+ = - V2(for a short circuit at port 2), we can write:
V  s V s V  s V s V

1

11
1

12
2

11
1
12
2
V  s V s V  s V s V

2

21
1

22
2

21
1
22

2

THE SCATTERING MATRIX (Ex)

The second equation gives;
S
V 
V
1 S


21
2
1
22

Dividing the first equation by V1+ and using the above result gives the
reflection coefficient seen as port 1 as;



V
V
S S
S S
S 
V
V
1 S
1
2

1
11
12

12
21
11
1
22
(0.85  45 )(0.8545 )
  0.15 
 0.452
1  0.2
0
0
THE SCATTERING MATRIX (Ex)

The return loss is;
RL  20 log |  | 20 log( 0.452)  6.9dB

Important points to note:
 Reflection coefficient looking into port n is not equal to Snn, unless
all other ports are matched
 Transmission coefficient from port m to port n is not equal to Snm,
unless all other ports are matched
 S parameters of a network are properties only of the network itself
(assuming the network is linear)
 It is defined under the condition that all ports are matched
 Changing the termination or excitation of a network does not change
its S parameters, but may change the reflection coefficient seen at a
given port, or transmission coefficient between two ports