04.Multiport circuit parameters and transmission lines
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THE INDEFINITE SCATTERING MATRIX |
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Z 0 |
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Z 0 |
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+ |
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+ |
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V |
S |
V |
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– |
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– |
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Z 0
+
V
–
FIGURE 4.22 The indefinite scattering parameter circuit.
of the columns D 1. For the first property the three-port shown in Fig. 4.22 is excited at all three terminals by the same voltage value. The output wave is
bj D Sj1a1 C Sj2a2 C Sj3a3, j D 1, 2, 3 |
4.149 |
Under this excitation all the input waves, aj, have the same amplitude, so Eq. (4.149) becomes
bj D Sj1 C Sj2 C Sj3 a1, j D 1, 2, 3 |
4.150 |
p p
Since from Eqs. (4.132) and (4.134) ak D Z0ICk and bk D Z0Ik , Eq. (4.150) can be written in terms of the incident and reflected currents:
IJ D [Sj1 C Sj2 C Sj3]I1C |
4.151 |
When all the terminal voltages are set equal, then all the terminal currents must be zero, since there can be no voltage difference between any two ports. Thus Ij D IC1 , which means that
Sj1 C Sj2 C Sj3 D 1 |
4.152 |
proving that the sum of the rows D 1.
To show that the sum of the columns D 1, only port-1 is excited with a voltage source. This gives a1 D6 0 and a2 D a3 D 0. By Kirchhoff’s current law the sum
of the currents into the three terminal circuit is zero: |
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0 D I1 C I2 C I3 |
4.153 |
D I1C I1 C I2C I2 C I3C I3 |
4.154 |
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MULTIPORT CIRCUIT PARAMETERS AND TRANSMISSION LINES |
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Now, since I2C D I3C D 0 because of a2, a3, |
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I1C D I1 C I2 C I3 |
4.155 |
In addition |
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bk D Sk1a1 |
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Ik D Sk1I1C |
4.156 |
so |
I1C D [S11 C S21 C S31]I1C |
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4.157 |
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which affirms that the sum of the columns for the indefinite scattering matrix is 1.
PROBLEMS
4.1Convert the following scattering parameters (related to 50 ') to ABCD parameters:
jS11j |
6 |
S11 |
jS21j |
6 |
S21 |
jS12j |
6 |
S12 |
jS22j |
6 |
S22 |
0.49 |
29 |
3.25 |
85 |
0.10 |
65 |
0.65 |
33 |
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4.2Given the S parameters, derive the z parameters.
4.3Two transmission lines are cascaded together. Transmission line 1 has a characteristic impedance of Z01 D 50 ', has a length of 30/8 cm, and is terminated on the right-hand side by a resistive load of 25 '. The lefthand side is connected to transmission line 2 whose characteristic impedance
Z02 D 30 ', and its length is &/4 at 1 GHz. What is the input impedance at the left-hand side of the 30 ' line?
4.4The transmission line circuit of length $ and characteristic impedance Z0 is terminated by a resistance RL. Determine the Q for this circuit at the first appropriate nonzero frequency.
REFERENCES
1.W.-K. Chen, Active Network and Feedback Amplifier Theory, New York: McGraw-Hill, 1980.
2.S. R. Seshadri, Fundamentals of Transmission Lines and Electromagnetic Fields, Reading MA: Addison-Wesley, 1971, pp. 335–350.
3.C. T. Tai, Generalized Vector and Dyadic Analysis, Piscataway, NJ: IEEE Press, 1997.
4.H. A. Wheeler, “Transmission-Line Properties of a Strip on a Dielectric Sheet on a Plane,” IEEE Trans. Microwave Theory Tech., Vol. MTT-25, pp. 631–647, 1977.
REFERENCES 83
5.E. Hammerstad and O. Jensen, “Accurate Models for Microstrip Computer-Aided Design,” 1980 IEEE MTT-S International Microwave Symp. Digest, pp. 407–409, 1980.
6.K. C. Gupta, R. Garg, and R. Chadha, Computer Aided Design of Microwave Circuits, Dedham, MA: Artech House, 1981.
7.R. A. Pucel, D. J. Masse,´ and C. P. Hartwig, “Losses in Microstrip,” IEEE Trans. Microwave Theory Tech., Vol. MTT-16, pp. 342–350, 1968.
8.R. A. Pucel, D. J.Masse,´ and C. P. Hartwig, “Corrections to ‘Losses in Microstrip,’ “IEEE Trans. Microwave Theory Tech., Vol. MTT-16, p. 1064, 1968.