Simple Cases Involving Transmission Lines

Figure C.3 summarizes three classic forms of transmission matrices used to define the overall system gain of a typical transmission-line configuration. In the figure Z C represents the characteristic impedance and H the one-way transfer function of a transmission line.

Figure C.3. These three forms of transmission matrix are often used to describe digital transmission circuits.

graphics/xcfig03.gif

The derivations of the top two forms in Figure C.3 are self-evident from the definition of the transmission matrix (Figure C.1). The third form corresponding to a transmission line is developed in the following way.

The general form of solution for the signals on a transmission line is composed of two traveling waves, one propagating to the right and one moving to the left (see Section 2.2.5, Figure 2.6). Suppose that at the right-hand end of the line the signal amplitudes of the two waves are denoted a and b . The currents associated with these two waveforms at that point must then be + a/Z C (a current moving to the right) and - b / Z C (representing a current moving to the left). At the right-hand end of the line, the superposition of these waves must generate the voltage and current extant at that end.

Equation C.4

graphics/xcequ04.gif

The preceding conditions may be inverted to determine the amplitudes a and b .

Equation C.5

graphics/xcequ05.gif

At the left end of the line the same conditions prevail, except that the amplitudes of the right- and left-traveling waveforms must be adjusted to account for their propagation through the transmission medium. The amplitude of the left-going waveform is diminished by H , the one-way transfer function of the transmission line, while the amplitude of the right-traveling waveform must be increased by H “1 , so that after traveling to the right end of the line, it will appear at the correct amplitude a . Summing the voltages and currents at the left end of the line produces a relationship between a , b and v 1 , i 1 .

Equation C.6

graphics/xcequ06.gif

Now substitute for a and b the relations to v 2 and i 2 .

Equation C.7

graphics/xcequ07.gif

Collecting together the terms associated with v 2 and i 2 respectively reveals the form of the transmission matrix.

Equation C.8

graphics/xcequ08.gif


Fundamentals

Transmission Line Parameters

Performance Regions

Frequency-Domain Modeling

Pcb (printed-circuit board) Traces

Differential Signaling

Generic Building-Cabling Standards

100-Ohm Balanced Twisted-Pair Cabling

150-Ohm STP-A Cabling

Coaxial Cabling

Fiber-Optic Cabling

Clock Distribution

Time-Domain Simulation Tools and Methods

Points to Remember

Appendix A. Building a Signal Integrity Department

Appendix B. Calculation of Loss Slope

Appendix C. Two-Port Analysis

Appendix D. Accuracy of Pi Model

Appendix E. erf( )

Notes

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High-Speed Signal Propagation[c] Advanced Black Magic
High-Speed Signal Propagation[c] Advanced Black Magic
ISBN: 013084408X
EAN: N/A
Year: 2005
Pages: 163
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