Transmission Line Impedance

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This section will relate the phasors of voltage and current waves through the transmission-line impedance.

In equations eq:TLVolt-eq:TLCurr $\tilde {V}_0^+ e^{-\gamma z}$ and $\tilde {V}_0^- e^{\gamma z}$ are the phasors of forward and reflected going voltage waves anywhere on the transmission line (for any $z$ ). $\tilde {I}_0^+ e^{-\gamma z}$ and $\tilde {I}_0^- e^{\gamma z}$ are the phasors of forward and reflected current waves anywhere on the transmission line.

$\begin{eqnarray} \tilde{V}(z)=\tilde{V}_0^+ e^{-\gamma z} + \tilde{V}_0^- e^{\gamma z}\label{eq:TLVolt} \\ I(z)=\tilde{I}_0^+ e^{-\gamma z} + \tilde{I}_0^- e^{\gamma z}\label{eq:TLCurr} \end{eqnarray}$

To find the transmission-line impedance, we first substitute the voltage wave equation eq:TLVolt into Telegrapher’s Equation Eq.eq:te12new to obtain Equation eq:te12new1.

$\begin{eqnarray} -\frac{\partial \tilde{V}(z)}{\partial z} = (R+j\omega L) I(z) \label{eq:te12new} \\ \gamma \tilde{V}_0^+ e^{-\gamma z} - \gamma \tilde{V}_0^- e^{\gamma z} = (R+ j \omega L) I(z) \label{eq:te12new1} \end{eqnarray}$

We now rearrange Equation eq:te12new1 to find the current I(z) and multiply through to get Equation eq:TLImpedanceTE.

$\begin{eqnarray} I(z)=\frac{\gamma}{R+j\omega L} ( \tilde{V}_0^+ e^{-\gamma z} + \tilde{V}_0^- e^{\gamma z}) \nonumber \\ I(z)=\frac{\gamma \tilde{V}_0^+}{R+ j \omega L} e^{-\gamma z} - \frac{\gamma \tilde{V}_0^-}{R+ j \omega L} e^{\gamma z} \label{eq:TLImpedanceTE} \end{eqnarray}$

We can now compare Equation eq:TLCurr for current, a solution of the wave equation, with the Eq.eq:TLImpedanceTE. Since both equations represent current, and for two transcendental equations to be equal, the coefficients next to exponential terms have to be the same. When we equate the coefficients, we get the equations below.

$\begin{eqnarray} \tilde{I}_0^+=\frac{\gamma \tilde{V}_0^+}{R+ j \omega L} \label{eq:i+v+} \\ \tilde{I}_0^-= - \frac{\gamma \tilde{V}_0^-}{R+ j \omega L} \label{eq:i-v-} \end{eqnarray}$

We define the characteristic impedance of a transmission line as the ratio of the voltage to the current amplitude of the forward wave as shown in Equation eq:i+v+, or the ratio of the voltage to the current amplitude of the reflected wave as shown in Equation eq:i-v-.

$\begin{eqnarray} Z_0=\frac{\tilde{V}_0^+}{ \tilde{I}_0^+}=\frac{R+j\omega L}{\gamma} \label{eq:TLimp1}\\ Z_0=-\frac{\tilde{V}_0^-}{ \tilde{I}_0^-}=\frac{R+j\omega L}{\gamma} \label{eq:TLimp2} \end{eqnarray}$

We can further simplify Equations eq:TLimp1-eq:TLimp2 to obtain the final Equation eq:tlfinal for the transmission line impedance. This equation is valid for both lossy and lossless transmission lines.

$\begin{eqnarray} Z_0=\frac{\tilde{V}_0^+}{ \tilde{I}_0^+} \nonumber \\ \nonumber Z_0=\frac{R+j\omega L}{\gamma} \nonumber \\ \nonumber Z_0=\sqrt{\frac{R+j\omega L}{G+ j\omega C}} \label{eq:tlfinal} \end{eqnarray}$

For lossless transmission line, where $R \rightarrow 0$ and $G \rightarrow 0$ , the equation simplifies to

$\begin{eqnarray} Z_0=\sqrt{\frac{L}{C}} \end{eqnarray}$

Equations for voltage and current on a transmission-line

Using the definition of transmission-line impedance $Z_0$ , we can now simplify the Equations eq:TLVolt-eq:TLCurr for voltage and current on the transmission line, by replacing the currents $\tilde {I}_0^+=\frac {\tilde {V}_0^+}{Z_0}$ , and $\tilde {I}_0^- = -\frac {\tilde {V}_0^-}{Z_0}$ .

$\begin{eqnarray} \tilde{V}(z)=\tilde{V}_0^+ e^{-j \beta z} +\tilde{V}_0^- e^{j \beta z} \label{eq:TLIvtl} \\ I(z)=\frac{\tilde{V}_0^+}{Z_0} e^{- j \beta z} - \frac{\tilde{V}_0^-}{Z_0} e^{j \beta z}\label{eq:TLIctl} \end{eqnarray}$