EKT 356 MICROWAVE COMMUNICATIONS CHAPTER 3: MICROWAVE NETWORK ANALYSIS (PART 1)

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

NETWORK ANALYSIS Several ways to characterize this network, such as Impedance parameters Admittance parameters Hybrid parameters Transmission parameters Scattering parameters (S-parameters) is introduced later as a technique to characterize high-frequency and microwave circuits

NETWORK ANALYSIS Impedance Parameters Considering Figure 1, considering network is linear, principle of superposition can be applied. Voltage, V1 at port 1 can be expressed in terms of 2 currents as follow; Since V1 is in Volts, I1 and I2 are in Amperes, Z11 and Z12 must be in Ohms. These are called impedance parameters Similarly, for V2, we can write V2 in terms of I1 and I2 as follow;

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

NETWORK ANALYSIS Impedance Parameters (cont) If port 2 of the network is left open, then I2 will be zero. In this condition; Similarly, when port 1 of the network is left open, then I1 will be zero. In this condition; and and

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

NETWORK ANALYSIS Example 1: Solution If I2 is zero, then V1 and V2 can be found from Ohm’s Law as 6I1. Hence from the equations Similarly, when the source is connected at port 2 and port 1 has an open circuit, we find that;

NETWORK ANALYSIS Example 1: Solution Hence, from Therefore,

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

NETWORK ANALYSIS Example 2: Solution As before, assume that the source is connected at port-1 while port 2 is open. In this condition, V1 = 12I1 and V2 = 0. Therefore, Similarly, with a source connected at port-2 while port-1 has an open circuit, we find that, and and

NETWORK ANALYSIS Example 2: Solution Hence, Therefore, and

NETWORK ANALYSIS Admittance Parameters Consider again Figure 1. Assuming the network is linear, principle of superposition can be applied. Current, I1 at port 1 can be expressed in terms of 2 voltages as follow; Since I1 is in Amperes, V1 and V2 are in Volts, Y11 and Y12 must be in Siemens. These are called admittance parameters Similarly, we can write I2 in terms of V1 and V2 as follow;

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

NETWORK ANALYSIS Admittance Parameters (cont) If port 2 of the network has a short circuit, then V2 will be zero. In this condition; Similarly, with a source connected at port 2, and a short circuit at port 1, then V1 will be zero. In this condition; and and

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

NETWORK ANALYSIS Example 3: Solution If V2 is zero, then I1 is equal to 0.05V1, I2 is equal to -0.05V1. Hence from the equations above; Similarly, with a source connected at port 2 and port 1 having a short circuit, we find that;

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

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

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 VN, then; Therefore;

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

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

NETWORK ANALYSIS Example 4: Solution (cont) Therefore,

NETWORK ANALYSIS Hybrid Parameters Consider again Figure 1. Assuming the network is linear, principle of superposition can be applied. Voltage, V1 at port-1 can be expressed in terms of current I1 at port-2 and voltage V2 at port-2, as follow; Similarly, we can write I2 in terms of I1 and V2 as follow; Since V1 and V2 are in volts, while I1 and I2 are in amperes, parameter h11 must be in ohms, h12 and h21 must be dimensionless, and h22 must be in siemens. These are called hybrid parameters.

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 V2 will be zero. This condition gives; and

NETWORK ANALYSIS Hybrid Parameters (cont) Similarly, with a source connected to port-2 while port-1 is open; Thus, parameters h11 and h21 represent the input impedance and the forward current gain, respectively, when a short circuit is at port-2. Similarly, h12 and h22 represent reverse voltage gain and the output admittance, respectively, when port-1 has an open circuit. In circuit analysis, these are generally denoted as hi, hf, hr and ho, respectively. and

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

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

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

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

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 V1 and current at port 1 can be expressed in terms of current I2 and voltage V2 at port-2, as follow; Similarly, we can write I1 in terms of I2 and V2 as follow; Since V1 and V2 are in volts, while I1 and I2 are in amperes, parameter A and D must be in dimensionless, B must be in Ohms, and C must be in Siemens.

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 V2 will be zero. This condition gives; and

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

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

NETWORK ANALYSIS Example 6: Solution With a source connected to port-1, while port-2 has a short circuit (so that V2 is zero) Therefore; and and

NETWORK ANALYSIS Example 6: Solution (cont) Similarly, with a source connected at port-1, while port-2 is open (so that I2 is zero) Hence; Thus; and and

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

NETWORK ANALYSIS Example 7: Solution With a source connected to port-1, while port-2 has a short circuit (so that V2 is zero), we find that Therefore; and and

NETWORK ANALYSIS Example 7: Solution (cont) Similarly, with a source connected at port-1, while port-2 is open (so that I2 is zero) Hence; Thus; and and

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

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

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

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

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

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

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.