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

2. 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

3. 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

4. NETWORK ANALYSIS Impedance Parameters Considering Figure 1, considering network is linear, principle of superposition can be applied. Voltage, V 1 at port 1 can be expressed in terms of 2 currents as follow; Since V 1 is in Volts, I 1 and I 2 are in Amperes, Z 11 and Z 12 must be in Ohms. These are called impedance parameters Similarly, for V 2, we can write V 2 in terms of I 1 and I 2 as follow;

5. NETWORK ANALYSIS Impedance Parameters (cont) Using the matrix representation, we can write; Or Where [Z] is called the impedance matrix of the two-port network

6. NETWORK ANALYSIS Impedance Parameters (cont) If port 2 of the network is left open, then I 2 will be zero. In this condition; Similarly, when port 1 of the network is left open, then I 1 will be zero. In this condition; and

7. NETWORK ANALYSIS Example 1 Find the impedance parameters of the 2-port network shown here

8. NETWORK ANALYSIS Example 1: Solution If I 2 is zero, then V 1 and V 2 can be found from Ohm’s Law as 6I 1. Hence from the equations Similarly, when the source is connected at port 2 and port 1 has an open circuit, we find that;

9. NETWORK ANALYSIS Example 1: Solution Hence, from Therefore,

10. NETWORK ANALYSIS Example 2 Find the impedance parameters of the 2-port network shown here

11. NETWORK ANALYSIS Example 2: Solution As before, assume that the source is connected at port-1 while port 2 is open. In this condition, V 1 = 12I 1 and V2 = 0. Therefore, Similarly, with a source connected at port-2 while port-1 has an open circuit, we find that, and

12. NETWORK ANALYSIS Example 2: Solution Hence, Therefore, and

13. NETWORK ANALYSIS Admittance Parameters Consider again Figure 1. Assuming the network is linear, principle of superposition can be applied. Current, I 1 at port 1 can be expressed in terms of 2 voltages as follow; Since I 1 is in Amperes, V 1 and V 2 are in Volts, Y 11 and Y 12 must be in Siemens. These are called admittance parameters Similarly, we can write I 2 in terms of V 1 and V 2 as follow;

14. NETWORK ANALYSIS Admittance Parameters (cont) Using the matrix representation, we can write; Or Where [Y] is called the admittance matrix of the two-port network

15. NETWORK ANALYSIS Admittance Parameters (cont) If port 2 of the network has a short circuit, then V 2 will be zero. In this condition; Similarly, with a source connected at port 2, and a short circuit at port 1, then V 1 will be zero. In this condition; and

16. NETWORK ANALYSIS Example 3 Find the admittance parameters of the 2-port network shown here

17. NETWORK ANALYSIS Example 3: Solution If V 2 is zero, then I 1 is equal to 0.05V 1, I 2 is equal to -0.05V 1. Hence from the equations above; Similarly, with a source connected at port 2 and port 1 having a short circuit, we find that;

18. NETWORK ANALYSIS Example 3: Solution (cont) Hence, from Therefore,

19. NETWORK ANALYSIS Example 4 Find the admittance parameters of the 2-port network shown here

20. 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; And if voltage across 0.2S is V N, then; Therefore;

21. NETWORK ANALYSIS Example 4: Solution (cont) Therefore; Similarly, with a source at port-2 and port-1 having a short circuit;

22. NETWORK ANALYSIS Example 4: Solution (cont) And if voltage across 0.1S is V M, then, Therefore, Hence;

23. NETWORK ANALYSIS Example 4: Solution (cont) Therefore,

24. NETWORK ANALYSIS Hybrid Parameters Consider again Figure 1. Assuming the network is linear, principle of superposition can be applied. Voltage, V 1 at port-1 can be expressed in terms of current I 1 at port-2 and voltage V 2 at port-2, as follow; Similarly, we can write I 2 in terms of I 1 and V 2 as follow; Since V 1 and V 2 are in volts, while I 1 and I 2 are in amperes, parameter h 11 must be in ohms, h 12 and h 21 must be dimensionless, and h 22 must be in siemens. These are called hybrid parameters.

25. NETWORK ANALYSIS Hybrid Parameters (cont) Using the matrix representation, we can write; Hybrid parameters are especially important in transistor circuit analysis. The parameters are defined as follow; If port-2 has a short circuit, then V 2 will be zero. This condition gives; and

26. NETWORK ANALYSIS Hybrid Parameters (cont) Similarly, with a source connected to port-2 while port-1 is open; Thus, parameters h 11 and h 21 represent the input impedance and the forward current gain, respectively, when a short circuit is at port-2. Similarly, h 12 and h 22 represent reverse voltage gain and the output admittance, respectively, when port-1 has an open circuit. In circuit analysis, these are generally denoted as h i, h f, h r and h o, respectively. and

27. NETWORK ANALYSIS Example 5: Hybrid parameters Find hybrid parameters of the 2-port network shown here

28. NETWORK ANALYSIS Example 5: Solution With a short circuit at port-2, And using the current divider rule, we find that

29. NETWORK ANALYSIS Example 5: Solution (cont) Therefore; Similarly, with a source at port-2 and port-1 having an open circuit; And

30. NETWORK ANALYSIS Example 5: Solution (cont) Because there is no current flowing through the 12Ω resistor, hence; Thus,

31. 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 V 1 and current at port 1 can be expressed in terms of current I 2 and voltage V 2 at port-2, as follow; Similarly, we can write I 1 in terms of I 2 and V 2 as follow; Since V 1 and V 2 are in volts, while I 1 and I 2 are in amperes, parameter A and D must be in dimensionless, B must be in Ohms, and C must be in Siemens.

32. NETWORK ANALYSIS Transmission Parameters (cont) Using the matrix representation, we can write; 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 V 2 will be zero. This condition gives; and

33. NETWORK ANALYSIS Transmission Parameters (cont) Similarly, with a source connected at port-1 while port-2 is open, we find; and

34. NETWORK ANALYSIS Example 6: Transmission parameters Find transmission parameters of the 2-port network shown here

35. NETWORK ANALYSIS Example 6: Solution With a source connected to port-1, while port-2 has a short circuit (so that V 2 is zero) Therefore; and

36. NETWORK ANALYSIS Example 6: Solution (cont) Similarly, with a source connected at port-1, while port-2 is open (so that I 2 is zero) Hence; Thus; and

37. NETWORK ANALYSIS Example 7: Transmission parameters Find transmission parameters of the 2-port network shown here

38. NETWORK ANALYSIS Example 7: Solution With a source connected to port-1, while port-2 has a short circuit (so that V 2 is zero), we find that Therefore; and

39. NETWORK ANALYSIS Example 7: Solution (cont) Similarly, with a source connected at port-1, while port-2 is open (so that I 2 is zero) Hence; Thus; and

40. 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. (4.1a) (4.1b) 2 -Ports I2I2 V2V2 V1V1 I1I1 Take note of the direction of positive current! Short circuit Port 2 Open circuit Port 2

41. ABCD MATRIX The ABCD matrix is useful for characterizing the overall response of 2-port networks that are cascaded to each other. I2’I2’ V2V2 V1V1 I1I1 I2I2 V3V3 I3I3 Overall ABCD matrix

42. 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. Port 1 Port 2 R CV1V1 I1I1 I2I2 V2V2 Convention for positive polarity current and voltage + -

43. NETWORK ANALYSIS 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 I1I1 I2I2 V2V2 j  CV 2 R V1V1 I1I1 V2V2 2 - Ports I2I2 V2V2 V1V1 I1I1 Port 1 Port 2 R CV1V1 I1I1 I2I2 V2V2 Network parameters (Y-parameters)

44. NETWORK ANALYSIS To determine the network parameters, the following relations can be used: For example to measure y 11, the following setup can be used: This means we short circuit the port or 2 - Ports I2I2 V 2 = 0 V1V1 I1I1 Short circuit

45. 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 ports, called N - port networks. 2 - Ports I2I2 V2V2 V1V1 I1I1 V1V1 V2V2 I1I1 I2I2 Linear circuit, because all elements have linear I-V relation

46. 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.

47. 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. 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.