5
Microwave Circuit Theory
Microwave circuits are composed of many parts and blocks, such as transmission lines, filters, and amplifiers. It was shown in Chapter 3 how the field distributions within the waveguides could be solved using Maxwell’s equations and boundary conditions. In principle, whole circuits could be analyzed this way. However, this is very cumbersome, and often the knowledge of the field distributions is of no use.
Usually one wants to know how the wave is transmitted through the circuit or reflected from it. Therefore, the circuit can be considered to be a ‘‘black box’’ that has one or more ports. It is sufficient to know the transfer functions between the ports; it is not necessary to know what happens inside the circuit. Thus, the operation of the circuit can be described with only a few parameters. The transfer function of a system comprising of cascaded circuits can be obtained using these parameters. In the circuit theory, Z - and Y -parameters are commonly used. However, the scattering or S -parameters are best suited for describing the operation of a microwave circuit.
5.1 Impedance and Admittance Matrices
The voltages and currents of a low-frequency circuit can be defined uniquely. In case of a high-frequency circuit this is not necessarily true. Voltages and currents can be defined uniquely only for a transmission line that carries a pure TEM mode. For example, the voltage and current of a rectangular
97
98 Radio Engineering for Wireless Communication and Sensor Applications
waveguide can be defined in several ways. In such a line, the electric and magnetic fields and the power propagating in the waveguide are more fundamental quantities than the voltage and current. However, it would be useful if the operation of a high-frequency circuit could be described in terms of voltages, currents, and impedances because then the methods of the circuit theory could be used.
Figure 5.1 shows a circuit having n ports. There is a short homogeneous transmission line at each port. The reference plane (within this line) of each port has to be defined clearly because the parameters of the circuit depend on the positions of the reference planes. Let us assume that only the fundamental TEM mode is propagating into and out of the ports. The discontinuities of the circuit generate higher-order TE and TM modes, which usually are not able to propagate. The reference planes have to be far enough from these discontinuities so that the fields of the nonpropagating modes are well attenuated. For each port i, an entering voltage wave Vi + and current wave Ii+ as well as a leaving voltage wave Vi − and current wave Ii− can be defined.
Figure 5.1 Circuit having n ports.
Microwave Circuit Theory |
99 |
Vi +, Ii+, Vi −, and Ii− are the complex amplitudes of these sinusoidal waves. These equivalent voltage and current waves are defined so that (1) the voltage and current are proportional to the transverse electric and magnetic fields of the wave, respectively, (2) the product of the voltage and current gives the power of the entering or leaving wave, and (3) the ratio of the voltage and current is the characteristic impedance of the port.
The total voltage and the total current at port i are:
Vi = Vi + + Vi − |
(5.1) |
Ii = Ii+ − Ii− |
(5.2) |
For a linear circuit, the voltage at port i is a linear function of the currents at all ports:
Vi = Z i1 I1 + Z i2 I2 + . . . + Z in In |
(5.3) |
The whole circuit can be described with the impedance matrix [Z ] as
3Vn |
4 3Z n1 |
Z n2 |
. . . Z nn |
43In |
4 |
|
||
V1 |
|
Z 11 |
Z 12 |
. . . Z 1n |
I1 |
|
||
V2 |
= |
Z 21 |
Z 22 |
. . . Z 2n |
I2 |
|
(5.4) |
|
A |
A |
A |
A |
A |
A |
|
||
|
|
|
||||||
Correspondingly, currents are obtained from voltages using the admittance matrix [Y ]:
3In 4 |
|
3Yn1 |
Yn2 |
. . . Ynn |
43Vn |
4 |
|
|
I1 |
|
Y 11 |
Y 12 |
. . . Y 1n |
V1 |
|
||
I2 |
= |
Y 21 |
Y 22 |
. . . Y 2n |
V2 |
(5.5) |
||
A |
A |
A |
A |
A |
A |
|
||
|
|
|
||||||
The impedance matrix [1] is an inverse matrix [2] of the admittance matrix, [Z ] = [Y ]−1. The elements of the matrices, Z ij and Yij , are called Z - and Y -parameters. For a reciprocal circuit, the matrices are symmetric:
Z ij = Z ji and Yij = Yji .
A two-port is the most common of the n -port circuits. Many different equivalent circuits can be used to realize the equations of a two-port [3, 4].
100 Radio Engineering for Wireless Communication and Sensor Applications
Figure 5.2 shows an equivalent circuit for a reciprocal two-port. The three Z -parameters for this T-circuit, Z 11 , Z 22 , and Z 12 , are calculated or measured. If the load impedance Z L = −V2 /I2 is connected at port 2, the input impedance measured at port 1 is
Z 1in = Z 11 |
− |
Z 122 |
|
(5.6) |
|
Z 22 |
+ |
|
|||
|
|
Z L |
|||
By measuring the input impedances corresponding to three different load impedances, a set of three equations is obtained. The Z -parameters may then be solved from these equations.
The properties of a circuit composed of two-ports connected in series can be calculated from the transmission matrices [T ] of the two-ports. A transmission matrix is called also a chain or ABCD matrix. A transmission matrix ties the input and output quantities as
FI1 |
G FC D GFI2 G |
|
FI2 G |
|
||||||
V1 |
|
= |
A |
B |
V2 |
= [T ] |
|
V2 |
|
(5.7) |
|
|
|
|
|
|
|
|
|||
The transmission matrix of a |
two-port |
can be |
calculated from its |
|||||||
Z -parameters: |
|
|
|
|
|
|
|
|
|
|
[T ] = |
Z 11 /Z 21 |
Z 11 Z 22 |
/Z 21 − Z 12 |
|
(5.8) |
|||||
F 1/Z 21 |
|
Z 22 /Z 21 |
|
|
G |
|||||
|
|
|
|
|
|
|||||
It should be noted that the positive output current I2 is now defined to flow out of the two-port so that I2 will be the input current to the following two-port. The transmission matrix of a circuit consisting of several two-ports in a series is obtained by multiplying the transmission matrices of the two-ports in the same order.
Figure 5.2 Equivalent T-circuit for a two-port.