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

2. The Scattering Matrix Consider an n – port network: Each port is considered to be connected to a Tline with specific Z 0. Linear n - port network T-line or waveguide Port 2 Port 1 Port n Reference plane for local z-axis (z = 0) Z 02 Z 01 Z 0n

3. 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 (-). V1+ V1-V1+ V1- Linear n - port Network Port 2 Port 1 Port n z = 0 V1V1 I1I1 +z Port 1  V1V1 V1+V1+ + - V1-V1- +

4. 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: (4.3a) Normalized incident waves Normalized reflected waves (4.3b) i = 1, 2, 3 … n

5. The Scattering Matrix Thus in general: V i + and V i - 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). V i + and V i - 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). V 2 +, a 2 V 2 -, b 2 V 1 +, a 1 V 1 -, b 1 V n +,a n V n -,b n Linear n - port Network T-line or waveguide Port 2 Port 1 Port n Z 01 Z 02 Z 0n

6. 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: V2+V2+ V1-V1- Vn-Vn- Port 2 Port 1 Port n Z 01 Z 02 Z 0n V2-V2- Linear n - port Network Constant that depends on the network construction

7. The Scattering Matrix Considering that we can send energy into all ports, this can be generalized to: Or written in Matrix equation: Where s ij 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 Z 0i (4.4a) (4.4b) or

8. The Scattering Matrix Consider the N-port network shown in figure 4.1. Figure 4.1: An arbitrary N-port microwave network

9. The Scattering Matrix V n + is the amplitude of the voltage wave incident on port n. V n - 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]

10. The Scattering Matrix or [4.1b] A specific element of the [S] matrix can be determined as: [4.2] S ij is found by driving port j with an incident wave V j +, and measuring the reflected wave amplitude, V i -, 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, S ii is the reflection coefficient seen looking into port i when all other ports are terminated in matched loads, and S ij is the transmission coefficient from port j to port i when all other ports are terminated in matched loads.

11. The Scattering Matrix For 2-port networks, (4.4) reduces to: Note that V i + = 0 implies that we terminate i th port with its characteristic impedance. Thus zero reflection eliminates standing wave. (4.5a) (4.5b)

12. 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)

13. The Scattering Matrix 2 – Port Z c2 Z 02 Z 01 VsVs V1+V1+ V1-V1- V2-V2- V1-V1- 2 – Port Z 01 Z 02 Z 01 Z 02 VsVs V2-V2- V2+V2+ Measurement of s 11 and s 21 : Measurement of s 22 and s 12 :

14. Reciprocity of Networks Reciprocal and symmetric networks “A network is reciprocal if a zero impedance source and a zero impedance ammeter can be placed at any locations in a network and their positions interchanged without changing the ammeter reading” - Lorentz Reciprocity results with network elements that are linear and bilateral; they have the same behavior for currents flowing in either direction A symmetric network happens when: (4.6a)

15. Reciprocity of Networks Transpose of a Matrix (taken from Engineering Maths 4 th Ed by KA Stroud) Transpose of [S], written as [S] t

16. Reciprocity of Networks Reciprocal and symmetric networks As a consequence of reciprocity, the Z-matrix and Y-matrix for a reciprocal network is defined as: For a symmetrical network, the Z-matrix and Y-matrix for a reciprocal network is defined as:

17. Reciprocity of Networks Reciprocal and symmetric networks Examples of matrices reflecting reciprocal and symmetrical properties 1.. 2.. 3.. 4.. Reciprocal but not symmetrical Reciprocal & symmetrical

18. Lossless Networks Lossless networks A lossless network happens when: It is also purely imaginary for lossless network (no real power can be delivered to the network, e.g an ideal transformer) A matrix that satisfies the condition of (4.6b) is called a unitary matrix, which can be re-written as: (4.6b) For i = j For i ≠ j (4.7)

19. Lossless Networks Lossless networks (cont) Which also can be re-written (for a 2-port network): 1.. 2.. Examples of lossless matrices; and

20. Lossless Networks Lossless networks (cont) Proof? This circuit is symmetrical as well as reciprocal

21. Lossless Networks Lossless networks Example 1 Proof that the S parameter below is unitary (lossless)

22. The Scattering Matrix (Example) Example 2 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.

23. The Scattering Matrix (Example) From the following formula, S 11 can be found as the reflection coefficient seen at port 1 when port 2 is terminated with a matched load (Z 0 =50 Ω); The equation becomes; On port 2

24. The Scattering Matrix (Example) To calculate Z in (1), we can use the following formula; Thus S 11 = 0. Because of the symmetry of the circuit, S 22 = 0. S 21 can be found by applying an incident wave at port 1, V 1 +, and measuring the outcome at port 2, V 2 -. This is equivalent to the transmission coefficient from port 1 to port 2:

25. The Scattering Matrix (Example) From the fact that S 11 = S 22 = 0, we know that V 1 - = 0 when port 2 is terminated in Z 0 = 50 Ω, and that V 2 + = 0. In this case we have V 1 + = V 1 and V 2 - = V 2. 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, S 21 = S 12 = 0.707

26. The Scattering Matrix (Example) Example 3 A two port network is known to have the following scattering matrix: a) Determine if the network is reciprocal and lossless. b) If port 2 is terminated with a matched load, what is the return loss seen at port 1? c) If port 2 is terminated with a short circuit, what is the return loss seen at port 1?

27. The Scattering Matrix (Example) Q: Determine if the network is reciprocal and lossless From the matrix, [S] is not symmetric and not reciprocal. To determine whether it is lossless; 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 = S 11 = 0.15. So the return loss is;

28. The Scattering Matrix (Example) 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 V 2 + = - V 2 - (for a short circuit at port 2), we can write:

29. The Scattering Matrix (Example) The second equation gives; Dividing the first equation by V 1 + and using the above result gives the reflection coefficient seen as port 1 as;

30. The Scattering Matrix (Example) The return loss is; Important points to note: Reflection coefficient looking into port n is not equal to S nn, unless all other ports are matched Transmission coefficient from port m to port n is not equal to S nm, 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