Solutions of The Telegrapher's Equations As Circuit Components
The solutions of the telegrapher's equations can be inserted directly into a circuit as components. The circuit in the left figure implements the solutions of the telegrapher's equations.
The right hand circuit is derived from the left hand circuit by source transformations. It also implements the solutions of the telegrapher's equations.
The solution of the telegrapher's equations can be expressed as an ABCD type Two-port network with the following defining equations
- The symbols: in the source book have been replaced by the symbols : in the preceding two equations.
The ABCD type two-port gives and as functions of and . Both of the circuits above, when solved for and as functions of and yield exactly the same equations.
In the right hand circuit, all voltages except the port voltages are with respect to ground and the differential amplifiers have unshown connections to ground. An example of a transmission line modeled by this circuit would be a balanced transmission line such as a telephone line. The impedances Z(s), the voltage dependent current sources (VDCSs) and the difference amplifiers (the triangle with the number "1") account for the interaction of the transmission line with the external 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 to in the sense that, and would be same whether this circuit or an actual transmission line was connected between and . There is no implication that there are actually amplifiers inside the transmission line.
Every two-wire or balanced transmission line has an implicit (or in some cases explicit) third wire which may be called shield, sheath, common, Earth or ground. So every two-wire balanced transmission line has two modes which are nominally called the differential and common modes. The circuit shown on the right only models the differential mode.
In the left hand circuit, the voltage doublers, the difference amplifiers and impedances Z(s) account for the interaction of the transmission line with the external circuit. This circuit, as depicted, is also fully symmetric, and also not drawn that way. This circuit is a useful equivalent for an unbalanced transmission line like a coaxial cable or a micro strip line.
These are not the only possible equivalent circuits.
Read more about this topic: Transmission Line
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