1. S- Parameters

2. Impedance and Admittance matrices
For n ports network we can relate the voltages and currents by impedance
and admittance matrices
Impedance matrix
Admittance matrix
where

3. Reciprocal and Lossless Networks
Reciprocal networks usually contain nonreciprocal media such as ferrites or plasma, or active devices. We can show that the impedance and admittance matrices are symmetrical, so that.
Lossless networks can be shown that Zij or Yij are imaginary
Refer to text book Pozar pg

4. Example
Find the Z parameters of the two-port T –network as shown below I1 I2 V1 V2
Solution
Similarly we can show that
Port 2 open-circuited
This is an example of reciprocal network!!
Port 1 open-circuited

5. S-parameters Input signal reflected signal transmitted signal
Port 1
Microwave device
Port 2
Vi2
Vi1
Input signal
Vr2
Vr1
reflected signal
transmitted signal
Vt1
Vt2
Transmission and reflection coefficients

6. S-parameters Let Vb1= b1 , Vi1=a1 , Vi2=a2 , Then we can rewrite
Voltage of traveling wave away from port 1 is
Voltage of
Reflected wave
From port 1
Voltage of
Transmitted wave
From port 2
Voltage of transmitted wave away from port 2 is
Let Vb1= b1 , Vi1=a1 , Vi2=a2 ,
Then we can rewrite

7. S-parameters S11and S22 are a measure of reflected signal at port
1 and port 2 respectively
S21 is a measure of gain or
loss of a signal from port 1
to port 2.
S12 ia a measure of gain or
loss of a signal from port 2
to port 1.
Hence
In matrix form
Logarithmic form
S11=20 log(r1)
S22=20 log(r2)
S12=20 log(t12)
S21=20 log(t21)
S-matrix

8. S-parameters Vr2=0 means port 2 is matched

9. Multi-port network
Port 5
network
Port 1
Port 4
Port 2
Port 3

10. Example
Below is a matched 3 dB attenuator. Find the S-parameter of the circuit.
Z1=Z2= 8.56 W and Z3= W
Solution
By assuming the output port is terminated by Zo = 50 W, then
Because of symmetry , then S22=0

11. Continue V1 Vo V2
From the fact that S11=S22=0 , we know that Vr1=0 when port 2 is matched, and that Vi2=0. Therefore Vi1= V1 and Vt2=V2
Therefore S12 = S21 = 0.707

12. Lossless network
For lossless n-network , total input power = total output power. Thus
Where a and b are the amplitude of the signal
Putting in matrix form at a* = bt b*
=at St S* a*
Note that bt=atSt and b*=S*a*
Called unitary
matrix
Thus at (I – St S* )a* =0
This implies that St S* =I
In summation form

13. Conversion of Z to S and S to Z
where

14. Reciprocal and symmetrical network
Since the [U] is diagonal , thus
For reciprocal network
Since [Z] is symmetry
Thus it can be shown that

15. Example
A certain two-port network is measured and the following scattering matrix is obtained:
From the data , determine whether the network is reciprocal or lossless. If a short circuit is placed on port 2, what will be the resulting return loss at port 1?
Solution
Since [S] is symmetry, the network is reciprocal. To be lossless, the S parameters must satisfy
For i=j
|S11|2 + |S12|2 = (0.1)2 + (0.8)2 = 0.65
Since the summation is not equal to 1, thus
it is not a lossless network.

16. continue
Reflected power at port 1 when port 2 is shorted can be calculated as follow and the fact that a2= -b2 for port 2 being short circuited, thus
(1)
b1=S11a1 + S12a2 = S11a1 - S12b2
Short at port 2 a2
(2)
b2=S21a1 + S22a2 = S21a1 - S22b2
-a2=b2
From (2) we have
(3)
Dividing (1) by a1 and substitute the result in (3) ,we have
Return loss

17. ABCD parameters Voltages and currents in a general circuit
Network V2 V1
Voltages and currents in a general circuit
In matrix form
Given V1 and I1, V2 and I2 can be determined if ABDC matrix is known.
This can be written as Or
A –ve sign is included in the definition of D

18. Cascaded network However V2a=V1b and –I2a=I1b then
The main use of ABCD matrices are
for chaining circuit elements together
Or just convert to one matrix
Where

19. Determination of ABCD parameters
Because A is independent of B, to determine A put I2 equal to zero and determine
the voltage gain V1/V2=A of the circuit. In this case port 2 must be open circuit.
for port 2 open circuit
for port 2 short circuit
for port 2 short circuit
for port 2 open circuit

20. ABCD matrix for series impedanceZ V2 V1
for port 2 open circuit
for port 2 short circuit
V1= - I2 Z hence B= Z
V1= V2 hence A=1
for port 2 short circuit
for port 2 open circuit
I1 = - I2 = 0 hence C= 0
I1 = - I2 hence D= 1
The full ABCD matrix can be written

21. ABCD for T impedance networkZ1 Z2 V1 Z3 V2
for port 2 open circuit
therefore
then

22. Continue for port 2 short circuit Solving for voltage in Z2 Hence But
VZ2
Solving for voltage in Z2 Z2 Z3
Hence
But

23. Continue Z1 I1 I2
for port 2 open circuit V2 Z3
Analysis
Therefore

24. Continue for port 2 short circuit I1 is divided into Z2 and Z3, thus
VZ2 Z2 Z3
I1 is divided into Z2 and Z3, thus
Full matrix
Hence

25. ABCD for transmission line
Input V1
Transmission line Zo g V2
z =0
z = -
For transmission line
f and b represent forward and backward propagation voltage and current
Amplitudes. The time varying term can be dropped in further analysis.

26. continue At the input z = - At the output z = 0
(2)
(1)
At the output z = 0
(4)
(3)
Now find A,B,C and D using the above 4 equations
for port 2 open circuit
For I2 =0 Eq.( 4 ) gives Vf= Vb=Vo giving

27. continue Note that From Eq. (1) and (3) we have
for port 2 short circuit
For V2 = 0 , Eq. (3) implies –Vf= Vb = Vo . From Eq. (1) and (4) we have

28. continue for port 2 open circuit
For I2=0 , Eq. (4) implies Vf = Vb = Vo . From Eq.(2) and (3) we have
for port 2 short circuit
For V2=0 , Eq. (3) implies Vf = -Vb = Vo . From Eq.(2) and (4) we have

29. continue Note that The complete matrix is therefore Where
a= attenuation
k=wave propagation
constant
When the transmission line is lossless this reduces to
Lossless line
a = 0

30. Table of ABCD network Transmission line Z Series impedance
Shunt impedance

31. Table of ABCD network Ideal transformer n:1 Z1 Z2 T-network Z3 Z3 Z1
p-network
Ideal transformer
n:1

32. Conversion S to ABCD For conversion of ABCD to S-parameter
For conversion of S to ABCD-parameter
Zo is a characteristic impedance of the transmission line connected to the
ABCD network, usually 50 ohm.