# Derivation of Characteristic Impedance?

I start from the telegrapher's equation: $-\frac{dV(z)}{dz}=(R'+j\omega L')I(z)$, where $V(z)$ and $I(z)$ are the phasors of voltage and current respectively, in the transmission line model. $R'$ and $L'$ are resistance per unit length and inductance per unit length respectively.

The solution to the wave equation $\frac{d^2V(z)}{dz^2}-\gamma^2V(z)=0$ where $\gamma=\sqrt{(R'+j\omega L')(G'+j\omega C')}$ has the form $V(z)=V_o^+e^{-\gamma z}+V_o^-e^{\gamma z}$. $G'$ and $C'$ are respectively the conductance per unit length and capacitance per unit length of the transmission line.

From the telegrapher's equation we get: $-\frac{dV(z)}{dz}=\gamma V_o^+e^{-\gamma z}-\gamma V_o^-e^{\gamma z}=\gamma (V_o^+e^{-\gamma z}-V_o^-e^{\gamma z})=(R'+j\omega L')I(z)$

$I(z)=\frac{\gamma}{R'+j\omega L'}(V_o^+e^{-\gamma z}-V_o^-e^{\gamma z})$

...and I'm stuck here.

Given that characteristic impedance $\frac{V_o^+}{I_o^+}=Z_o=\frac{V_o^-}{I_o^-}$, how do I arrive at $Z_o=\frac{R'+j\omega L'}{\gamma}=\sqrt{\frac{R'+j\omega L'}{G'+j\omega C'}}$?

I'm not sure how to get $Z_o$ from $\frac{V_o^+e^{-\gamma z}+V_o^-e^{\gamma z}}{I(z)}=\frac{V_o^+e^{-\gamma z}-V_o^-e^{\gamma z}}{I_o^+e^{-\gamma z}+I_o^-e^{\gamma z}}$

This seems the simplest way to derive characteristic impedance. R, L, G and C represent: -

• series resistance of cable/length
• series inductance of cable/length
• parallel conductance of cable/length
• parallel capacitance of cable/length

Therefore the impedance looking into a short "lump" comprising these elements is: -

$Z_0 = R + jwL + Z_o//\dfrac{1}{G + jwC}$

In other words, the impedance looking in to the "lump" is the series impedance ($R +jwL$) plus the shunt components and the shunt components are: -

G and C plus the next "lump" which "offers" itself as another lump of $Z_0$.

$Z_0 = R + jwL + \dfrac{\frac{Z_0}{G+jwC}}{Z_0 + \frac{1}{G+jwC}}$

$Z_0= R + jwL + \dfrac{Z_0}{1 + Z_0(G+jwC)}$

multiplying through by $1 + Z_0(G+jwC)$ gives: -

$Z_0[1 + Z_0(G+jwC)] = [R+jwL][1 + Z_0(G+jwC)]+Z_0$

which becomes this: -

$Z_0 + Z_0^2(G+jwC) = R+jwL + Z_0[(R+jwL)(G+jwC)]+Z_0$

The important thing next is to recognize that $(R+jwL)(G+jwC)$ is insignificant as the "lump" approaches zero length and we are left with: -

$Z_0^2(G+jwC) = R+jwL$

hence $Z_0 = \sqrt{\dfrac{R+jwL}{G+jwC}}$

its very easy.First you put I(z)=Io+e-yz + Io-eyz.substitute Vo+=Io+*Zo and Vo-=-Io-*Zo in the equation where you got stuck .