Chapter 3 RF & M/W Network Theory and Analysis Shankar Gangaju Kathmandu Engineering College Kalimati, Kathmandu Department of Electronics and Computer Engineering 1

Microwave Network Analysis A microwave network consists of microwave devices and components (sources, attenuators, resonators, filters, amplifiers, etc.) coupled together by transmission lines or waveguides for the desired transmission of microwave signals through ports. The point of intersection of two or more signals, devices, components, circuits or modules is called port or simply junction. Circuits operating at low frequencies, for which the circuit dimensions are small relative to the wavelength, can be treated as an interconnection of lumped passive or active components with unique voltages and currents defined at any point in the circuit. In the case of microwave, the circuit dimensions are small enough so that there is negligible phase change from one point in the circuit to another 2

So far we’ve used maxwell’s equations and transmission line theory to understand concepts of propagation and impedance. However we don’t want to build only transmission line, we want to build filters, amplifiers and oscillators. At microwave frequencies, we cannot use the KCL and KVL techniques of low frequency analysis to determine the transfer characteristics of a network. Also we don’t want to solve Maxwell's equations for every network. This gives us more information than is necessary and it is too hard. Subsequently we need another set of techniques for the analysis of circuits and systems at microwave frequencies, one that combines circuit analysis with wave theory. 3

Two Port Networks A two-port network (a kind of four-terminal network) is a electrical network or device with two pairs of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them satisfy the essential requirement known as the port condition: the electric current entering one terminal must equal the current emerging from the other terminal on the same port. The ports constitute interfaces where the network connects to other networks, the points where signals are applied or outputs are taken. In a two-port network, often port 1 is considered the input port and port 2 is considered the output port. The two-port network model is used in mathematical circuit analysis techniques to isolate portions of larger circuits. A two-port network is regarded as a "black box" with its properties specified by a matrix of numbers. This allows the response of the network to signals applied to the ports to be calculated easily, without solving for all the internal voltages and currents in the network. 4

5

Limitations of ABCD, Y, Z and h- Parameters At low frequencies, physical length of the network is larger than wavelength (λ) of the signal. Therefore the measurable input and output values are voltage and current analyzed in terms of ABCD, Y, Z and h-parameters with well- defined termination conditions. These parameters are analyzed under short or open circuit conditions. But in microwaves open or short circuit conditions are not easily achievable and terminating active devices, this way can damage the devices due to the total reflection of power back into the devices. 6

Limitations of ABCD, Y, Z and h- Parameters Open or short circuit conditions often results in oscillation for a wide range of frequencies for active devices such as the transistor and negative resistance diode. Physical length of the components or devices at microwave frequencies are comparable or much smaller than wavelength (λ). Hence the voltage and current are not well defined at each discrete point. So a distributive analysis is required. Z, Y, ABCD and h-parameters often change the biasing conditions such as junction capacitances at higher frequencies. Unavailability of equipment to measure RF/MW total current and voltage. 7

Solutions: Input-output behavior of network is defined in terms of normalized power waves. Ratio of the power waves is recorded, called scattering parameters. S-parameters are measured based on properly terminated transmission lines (not open/short circuit conditions) 8

S-parameters The S-parameters are members of a family of similar parameters, other examples being: Y- parameters, Z-parameters, H-parameters, and ABCD-parameters. They differ from these, in the sense that S-parameters do not use open or short circuit conditions to characterize a linear electrical network; instead, matched loads are used. These terminations are much easier to use at high signal frequencies than open-circuit and short- circuit terminations. Moreover, the quantities are measured in terms of power. Many electrical properties of networks of components (inductors, capacitors, resistors) may be expressed using S-parameters, such as gain, return loss, voltage standing wave ratio (VSWR), reflection coefficient and amplifier stability. The term 'scattering' is more common to optical engineering than RF engineering, referring to the effect observed when a plane electromagnetic wave is incident on an obstruction or passes across dissimilar dielectric media. In the context of S-parameters, scattering refers to the way in which the traveling currents and voltages in a transmission line are affected when they meet a discontinuity caused by the insertion of a network into the transmission line. This is equivalent to the wave meeting an impedance differing from the line's characteristic impedance. 9

10

Power, voltage and current can be considered to be in the form of waves travelling in both directions. For a wave incident on Port 1, some part of this signal reflects back out of that port and some portion of the signal exits other ports. S-parameters are measured by sending a single frequency signal into the network or “black box” and detecting what waves exit from each port.

S 11 refers to the signal reflected at Port 1 for the signal incident at Port 1. Scattering parameter S 11 is the ratio of the two waves b1/a1. What does S 11 refers to?

S 21 refers to the signal exiting at Port 2 for the signal incident at Port 1. Scattering parameter S 21 is the ratio of the two waves b2/a1. What does S 21,S 22 and S 12 refers to?

Scattering Matrix Problem arises in measuring currents and voltages at microwave frequencies. However they can be derived from measurable quantities such as VSWR, reflection coefficient, power, etc. The easiest parameters to measure are incident and reflected power. The optimum test conditions are when the two ports are terminated in matched loads. For describing and analyzing a microwave network the input and output parameters are defined by scattering matrix. 16

Scattering Matrix Scattering matrix is also known as S-matrix or S-parameters. Scattering matrices are widely used in RF and microwave frequencies for component modelling, component specifications and circuit design. S-parameters can be measured by network analyzers. For a general n-port network, the s-matrix is given in the following equations: 17 a i = incident wave voltages at port i b i = reflected wave voltages at port i

Properties of S-matrix A generalized n-port has n 2 scattering coefficients. While the S ij may be all independent, in general due to symmetries etc. the number of independent coefficients is much smaller. An n-port is reciprocal when S ij = S ji for all i and j. Most passive components are reciprocal (resistors, capacitors, transformers, etc., except for structures involving magnetized ferrites, plasmas etc.), active components such as amplifiers are generally non-reciprocal. A two-port is symmetric, when it is reciprocal (S 21 = S 12 ) and when the input and output reflection coefficients are equal (S 22 = S 11 ). For any matched port i, S ii =0. 18

Properties of S-matrix 19

Larger networks: A Network may have any number of ports. The S-matrix for an n-port network contains n 2 coefficients (S-parameters), each one representing a possible input-output path. The number of rows and columns in an S-parameters matrix is equal to the number of ports. For the S-parameter subscripts “ij”, “j” is the port that is excited (the input port) and “i” is the output port.

Two Port Network Analysis 21 General Two port Network Two port Network with Ports Terminated in Matched Loads

Two Port Network Analysis Incident and reflected amplitudes at any point gives average power. Hence there exists three power components: Incident Power (P i ) Transmitted Power (P t ) Reflected Power (P r ) Accordingly network can be analyzed with the reflection coefficient (Г) which depends upon point of insights. When looked towards the input port, reflection coefficient is given by Г s. When looking towards the network from input port, it is given by Г s. 22

Two Port Network Analysis When looking towards network from port 2, it is Г out. When looking towards the load from network it is Г L. The same network can be defined by S-matrix. where, S 11 = parameter describing input. S 22 = parameter describing output. S 12 and S 21 = parameters describing the network. 23

Two Port Network Analysis If the output parameters are defined by matrix [b] and input parameters by matrix [a], then [b] = [S][a] For two port network Thus, 24

Two Port Network Analysis 25 Hence, from above equations

Two Port Network Analysis 26 Signal flow diagram In other words, S 11 = Return loss at port 1. S 22 = Return loss at port 2. S 12 = Isolation loss. S 21 = Insertion loss.

Losses in the Network Consider 2-port network as shown 27

Losses in the Network 28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43