1. EKT 441 MICROWAVE COMMUNICATIONS
CHAPTER 3:
MICROWAVE NETWORK ANALYSIS (PART 1)

2. 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.
Convention for positive
polarity current and voltage + -
Port 1
Port 2 R C V1 I1 I2 V2

3. NETWORK ANALYSIS 2 - Ports
For this 2 port circuit we can easily derive the I-V relations.
We can choose V1 and V2 as the independent variables, the I-V relation can be expressed in matrix equations. C I1 I2 V2
jCV2 R V1 I1 V2
Network parameters
(Y-parameters)
2 - Ports I2 V2 V1 I1
Port 1
Port 2 R C

4. NETWORK ANALYSIS 2 - Ports
To determine the network parameters, the following relations can be used:
For example to measure y11, the following setup can be used: or
This means we short circuit the port
2 - Ports I2
V2 = 0 V1 I1
Short circuit

5. NETWORK ANALYSIS 2 - Ports
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 ports, called N - port networks.
2 - Ports I2 V2 V1 I1 V1 V2 I1 I2
Linear circuit, because all
elements have linear I-V relation

6. ABCD MATRIX (4.1a) 2 -Ports (4.1b)
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
direction of positive current!
2 -Ports I2 V2 V1 I1
(4.1a)
(4.1b)
Open circuit Port 2
Short circuit Port 2

7. ABCD MATRIX
The ABCD matrix is useful for characterizing the overall response of 2-port networks that are cascaded to each other.
I2’ V2 V1 I1 I2 V3 I3
Overall ABCD matrix

8. 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.

9. 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.

10. THE SCATTERING MATRIX Consider an n – port network: Port 1 Port n
Each port is considered to be
connected to a Tline with
specific Zc.
Port 1
Reference plane
for local z-axis
(z = 0)
Zc1
Port n
Zcn
Port 2
Linear
n - port
network
Zc2
T-line or
waveguide

11. 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 (-).
z = 0 V1 I1 +z
Port 1  V1
V1+ + -
V1-
V1+ V1-
Port 1
Port n
Port 2
Linear
n - port
Network

12. THE SCATTERING MATRIX (4.3a) (4.3b)
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
(4.3a)
i = 1, 2, 3 … n
Normalized
reflected waves
(4.3b)

13. THE SCATTERING MATRIX Thus in general: Port 1 Port n Port 2 Linear
V1+ V1-
Port 1
Vn+
Vn-
Zc1
Port n
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 propor-
tional to transverse H field, see
[1] for details).
Zcn
Port 2
Linear
n - port
Network
V2+
V2-
Zc2
T-line or
waveguide

14. 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
Zc1
Vn-
Port n
Zcn
Port 2
Linear
n - port
Network
V2+
Zc2
Constant that
depends on the
network construction
V2-

15. THE SCATTERING MATRIX (4.4a) (4.4b)
Considering that we can send energy into all ports, this can be generalized to:
Or written in Matrix equation:
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
(4.4a) or
(4.4b)

16. Figure 4.1: An arbitrary N-port microwave network
THE SCATTERING MATRIX
Consider the N-port network shown in figure 4.1.
Figure 4.1: An arbitrary N-port microwave network

17. 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:
[4.1a]

18. THE SCATTERING MATRIX or
[4.1b]
A specific element of the [S] matrix can be determined as:
[4.2]
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.

19. THE SCATTERING MATRIX (4.5a) (4.5b)
For 2-port networks, (4.4) reduces to:
Note that Vi+ = 0 implies that we terminate i th port with its characteristic impedance.
Thus zero reflection eliminates standing wave.
(4.5a)
(4.5b)

20. THE SCATTERING MATRIX 2 – Port 2 – Port V1+ V2- Zc1 Vs Zc1 Zc2 Zc2 V1-
Measurement of s11 and s21:
V1-
V2+
Zc2 Vs
2 – Port
Zc1
Zc2
Zc1
V2-
Measurement of s22 and s12:

21. 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)

22. THE SCATTERING MATRIX

23. THE SCATTERING MATRIX

24. THE SCATTERING MATRIX (4.6a) (4.6b) Reciprocal and Lossless networks
Impedance and admittance matrices are symmetric for reciprocal networks
A symmetric network happens when:
It is also purely imaginary for lossless network (no real power can be delivered to the network)
A matrix that satisfies the condition of (4.6b) is called a unitary matrix
(4.6a)
(4.6b)

25. Transpose of [S], written as [S]t
THE SCATTERING MATRIX
Transpose of a Matrix (taken from Engineering Maths 4th Ed by KA Stroud)
Transpose of [S], written as [S]t

26. THE SCATTERING MATRIX
Symmetrical Matrix (taken from Engineering Maths 4th Ed by KA Stroud) If
It is symmetrical when aij = aji
When a [S] is symmetric, it is also reciprocal

27. Used to determine reciprocality for a 2 port network
THE SCATTERING MATRIX
Reciprocal and Lossless networks (cont)
The matrix equation in (4.6b) can be re-written in; OR
For i = j
For i ≠ j
(4.7)
Used to determine reciprocality for a 2 port network

28. 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.

29. 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);
The equation becomes;
On port 2

30. THE SCATTERING MATRIX (Ex)
To calculate Zin(1), we can use the following formula;
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:

31. 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.
Where = (141.8//58.56) is the combined resistance of 50 Ω and 8.56 Ω paralled with the Ω resistor. Thus, S21 = S12 = 0.707

32. THE SCATTERING MATRIX (Ex)
A two port network is known to have the following scattering matrix:
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?

33. 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;
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 = So the return loss is;
Used to determine reciprocality for a 2 port network

34. 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:

35. THE SCATTERING MATRIX (Ex)
The second equation gives;
Dividing the first equation by V1+ and using the above result gives the reflection coefficient seen as port 1 as;

36. THE SCATTERING MATRIX (Ex)
The return loss is;
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