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Derivation of Transmission Line Equations

By Applied Electronics - Friday, March 3, 2017 No Comments
A transmission line is any wires that carry information from one point to another. Examples of transmission lines are coaxial cables, twin pair cables and others. They are used in TV, radio broadcasting system and others where radio signals have to be transmitted or received using antenna.

Following is a diagram of simple two wire transmission line model.


The figure shows that the two wire transmission lines are divided into small segments of length Δz. Each wire section is composed of resistor and inductor and the wires are connected with conductance G and C. That is the upper wire has resistance and inductance of R1 and L1 and similarly for lower wire which has resistance and inductance of R2 and L2.

The resistance, inductance, capacitance and conductance are in Ohm/m, H/m, F/m and S/m, that is in units of per meters.

The resistances R1 and R2, and indutances L1 and L2 can be combined into the following simplied circuit as shown.

Now using Ohms law we can find out the changes in voltages and currents from z to z+Δz as follows,


Letting Δz approach to zero we can rewrite the above equation in differential form,

 Differentiating with respect to z,


  Let,

which γ is called the propagation constant.

Then the 2nd order differential above can be rewritten as,

These pair of equations are called Transmission line equations or the Telegraph equations. They describe the line voltage and line current as function of distance z along the transmission lines. Note that the time dependence ejwt is omitted here.

Next see How to derive characteristic impedance of the transmission line.

See Download Analysis and Synthesis of Wire Antennas PDF free and Antenna Toolkit by Joseph Carr PDF free download.

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