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Derivation of Characteristic Impedance of Transmission line

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We talked in great detail about Transmission lines in the introductory lecture. Now we will see how to derive characteristic impedance from transmission line equations.

V=V^+ e^{-\gamma x}V^- e^{+\gamma x}

I =I^+ e^{-\gamma x}I^- e^{+\gamma x}

As discussed in the video, if lumped  circuit analysis has to be valid at all frequencies, the length of the sub section must tend to Zero.(Because even if the sub section is very small, after certain frequency, the effect of transit time will be significant)… (It is discussed in great detail in the video lecture)

\lim_{\Delta x \to 0} \frac{\Delta V}{\Delta x} =\frac{dV}{dx} = -(R+j\omega L) I

\lim_{\Delta x \to 0} \frac{\Delta I}{\Delta x}=\frac{dI}{dx} = -(G+j \omega C) V

Substituting the value of V in the above equation, we get.

\frac {d}{dx} {V^+ e^{-\gamma x}V^- e^{+\gamma x}} =-(R+j\omega L) { I^+ e^{-\gamma x} +  I^- e^{+\gamma x}}

==> - \gamma V^+ e^{-\gamma x} +  \gamma V^- e^{+\gamma x} =-(R+j\omega L) { I^+ e^{-\gamma x} +  I^- e^{+\gamma x}}

Co-efficient of e^{-\gamma x}    :  – \gamma V^+-(R+j\omega L)  I^+

Co-efficient of e^{\gamma x}    :  \gamma V^--(R+j\omega L)  I^-

Since  \gamma = \sqrt{ (R+j\omega L) (G+ j\omega C) }

\frac{V^+}{I^+} = \frac{R+j\omega L}{\gamma} = \sqrt{\frac{R+j\omega L}{G+ j\omega C}}

\frac{V^-}{I^-} = - \frac{R+j\omega L}{\gamma}\sqrt{\frac{R+j\omega L}{G+ j\omega C}}

It can be seen that, the above equations has the dimensions of Impendance(Ratio of Voltage to Current) and is a function of primary constants of the line and operating frequency. It is therefore called the “Characteristic Impendance” of the transmission line , often denoted by Z_o.

Z_o\sqrt{\frac{R+j\omega L}{G+ j\omega C}}

The formula of \gamma and Z_o of a transmission line are very important. Many GATE memory based questions will be based on this formula. So remember  the equation and solve few questions related to the characteristic impedance of transmission line.

 

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