Microwave Circuit Theory |
101 |
Generally, the measurement of Z - and Y -parameters is difficult at microwave frequencies because the measurement of the total voltages and currents at the ports is difficult, and in the case of waveguides carrying TE or TM modes it is impossible. Furthermore, in the case of some active circuits, the load impedances needed in the measurement may cause instability in the circuit.
5.2 Scattering Matrices
The scattering or S -parameters [1, 5, 6] are defined using the voltage waves entering the ports, Vi +, and leaving the ports, Vi −. If the circuit in Figure 5.1 is linear and all its ports have the same characteristic impedance of Z 0 , the voltage wave leaving port i may be written as
Vi − = Si1 V1+ + Si2 V2+ + . . . + Sin Vn+ |
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(5.9) |
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The whole circuit is described by the scattering matrix [S ] as |
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3Vn− |
4 3Sn1 |
Sn2 |
. . . Snn |
43Vn+ |
4 |
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V1− |
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S11 |
S12 |
. . . S1n |
V1+ |
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V2− |
= |
S21 |
S22 |
. . . S2n |
V2+ |
(5.10) |
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or [V − ] = [S ] [V + ]. The power flowing into port i is | Vi + |2/(2Z 0 ), and the power flowing out of port i is | Vi − |2/(2Z 0 ).
Usually, all the ports of a microwave circuit have similar connectors, such as 50-V coaxial connectors or waveguide flanges, and the characteristic impedances of the ports have the same value. However, in a general case, the characteristic impedances Z 0i may have different values. For example, the ports of a coaxial-to-waveguide adapter have different characteristic impedances. Then, the voltage waves should be normalized as
ai = |
Vi |
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(5.11) |
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√Z0i |
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b i = |
Vi |
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(5.12) |
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√Z 0i |
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102 Radio Engineering for Wireless Communication and Sensor Applications
The total voltage and current are expressed using the normalized voltage waves as
Vi = Vi + + Vi − = √ |
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(ai + b i ) |
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Z 0i |
(5.13) |
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Ii = |
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XVi + − Vi − C = |
1 |
(ai − b i ) |
(5.14) |
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√Z0i |
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Z 0i |
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The power flowing into port i is | ai | 2/2, and the power flowing out of port i is | bi | 2/2. The scattering matrix presentation using normalized
waves is now |
4 3Sn1 |
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43an 4 |
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3b n |
Sn2 |
. . . Snn |
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b 1 |
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S11 |
S12 |
. . . S1n |
a1 |
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b 2 |
= |
S21 |
S22 |
. . . S2n |
a2 |
(5.15) |
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or [b ] = [S ] [a ].
If all the ports are terminated with matched loads, the reflection coeffi-
cient for port i is r i = Sii |
= bi /ai , and the transducer power gain from port |
j to port i is Gij = | Sij |2 |
= | bi /aj |2. |
The scattering matrix of a reciprocal circuit is symmetrical: Sij = Sji . |
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In other words, the transposed matrix is the same as the matrix itself: [S ]T = [S ]. A reciprocal circuit operates the same way, regardless of the direction of the power flow. Most passive circuits are reciprocal; circuits that include ferrite components are the exceptions.
If the circuit has no loss, the sum of the powers flowing into the ports equals the sum of the powers flowing out of the ports:
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∑ |
| ai |2 = ∑ | b i |2 = ∑ |
∑ Sij aj |
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(5.16) |
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i =1 |
i =1 |
i =1 |
j =1 |
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If all the voltage waves ai are chosen to be zero, except ak , then |
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∑ | Sik |2 = |
∑ Sik Sik* = 1 |
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(5.17) |
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i =1 |
i =1 |
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Thus, for any column of the scattering matrix of a lossless circuit, the sum of the squares of the scattering parameters is 1. The same applies for all
Microwave Circuit Theory |
103 |
rows. If the voltage waves ak and al are chosen to be nonzero and other waves entering the circuit are zero, it can be proven that
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∑ Sik Sil* = 0 |
(5.18) |
i =1
For any two columns, the scattering parameters of a lossless circuit fulfill this equation. A similar equation applies for any two rows. The scattering matrix of a lossless circuit is unitary; that is, the transposed scattering matrix is equal to the inverse of the complex conjugate of the scattering matrix.
Scattering matrices of some simple circuits:
• A lossless transmission line having a length of l and a characteristic impedance of Z 0 , as shown in Figure 5.3. Z 0 is also the characteristic impedance of both ports. When one of the ports is terminated with a matched load, the reflection coefficient of the other port is zero. Thus, S11 = S22 = 0. If the voltage wave entering port 1 is a1 = 1,
the voltage wave leaving port 2 is b2 = e −jbl, and S21 = b2 /a1 = e −jbl. Due to the symmetry, S12 = S21 .
• A joint of two transmission lines, as shown in Figure 5.4. The characteristic impedances of the transmission lines and ports are Z 01 and Z 02 . The reference planes of the ports are located at a distance of
Figure 5.3 Section of a lossless transmission line and its scattering matrix.
Figure 5.4 Joint of two transmission lines and its scattering matrix.