Solutions of The Telegrapher's Equations As Circuit Components
The solutions of the telegrapher's equations can be inserted into a circuit as components of an equivalent sub-circuit as shown the figure. As drawn, all voltages are with respect to ground and all amplifiers have unshown connections to ground. An example of a transmission line modeled by this circuit would be an unbalanced transmission line such as a strip line on a circuit board. The impedance Z(s), the voltage doubler (the triangle with the number "2") and the difference amplifier (the triangle with the number "1") account for the interaction of the transmission line with the rest of the circuit. The T(s) blocks account for delay, attenuation, dispersion and whatever happens to the signal in transit. One of the T(s) blocks carries the "forward wave" and the other carries the "backward wave". The circuit, as depicted, is fully symmetric, although it is not drawn that way. The circuit depicted is equivalent to a transmission line connected from V1 to V2 in the sense that V1, V2, I1 and I2 would be same whether this circuit or an actual transmission line was connected between V1 and V2. There is no implication that there are actually amplifiers inside the transmission line.
This is not the only possible equivalent circuit. Voltage amplifiers and sensors can be replaced with current, transimpedance or transconductance amplifiers. Series impedances can be replaced with shunt admittances. The circuit can be augmented to account for different "grounds" at each end. The circuit can be made fully differential.
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