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.
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.
The preceding conditions may be inverted to determine the amplitudes a and b .
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 .
Now substitute for a and b the relations to v 2 and i 2 .
Collecting together the terms associated with v 2 and i 2 respectively reveals the form of the transmission matrix.