Wave impedance and input impedance

The wave impedance $Z(d)$ at distance $d$ from the load on a Transmission line is the ratio of total voltage to total current at that point:

$Z(d)=\frac{\tilde{V}(d)}{\tilde{I}(d)}.$

This is different from the Characteristic impedance $Z_0$. $Z_0$ is the V/I ratio for a single forward (or backward) wave. $Z(d)$ accounts for both forward and reflected waves and varies along the line.

Closed form (lossless line)

For a lossless line with Reflection coefficient $\Gamma$ at the load:

$Z(d)=Z_0\frac{1+\Gamma e^{-j2\beta d}}{1-\Gamma e^{-j2\beta d}}=Z_0\frac{1+{\Gamma }_d}{1-{\Gamma }_d},$

where ${\Gamma }_d=\Gamma e^{-j2\beta d}$ is the phase-shifted reflection coefficient. As you move away from the load, the magnitude $|{\Gamma }_d|=|\Gamma |$ stays constant, but the phase rotates clockwise by $2\beta d$.

Equivalent form using $z_L=Z_L/Z_0$:

$Z(d)=Z_0\frac{z_L+j\tan (\beta d)}{1+j{z}_L\tan (\beta d)}.$

Or in unnormalized form:

$Z(d)=Z_0\frac{Z_L+j{Z}_0\tan (\beta d)}{Z_0+j{Z}_L\tan (\beta d)}.$

Input impedance

The input impedance is the wave impedance at the source end of a line of total length $l$:

$Z_{\text{in}}=Z(d=l)=Z_0\frac{Z_L+j{Z}_0\tan (\beta l)}{Z_0+j{Z}_L\tan (\beta l)}.$

This is the impedance the generator “sees” looking into the line. It depends on:

• $Z_L$ (the load)
• $Z_0$ (the line)
• $\beta l$ (the electrical length of the line in radians, where $\beta =2\pi /\lambda$)

A short stub of fixed physical length appears differently at different frequencies because $\beta l$ depends on $\omega$.

Periodicity

Since $e^{-j2\beta d}$ has period $\lambda /2$ in $d$ (because $2\beta (\lambda /2)=2\pi$), the wave impedance pattern repeats every half wavelength:

$Z(d+\lambda /2)=Z(d).$

This is why “moving $\lambda /2$ along the line” returns you to the same impedance — even on a mismatched line. It’s a famous and useful property: a half-wave line transforms $Z_L$ to itself.

Special cases

Short-circuit terminated ($Z_L=0$): The wave impedance becomes

$Z^{\text{sc}}(d)=j{Z}_0\tan (\beta d).$

Purely reactive (no real part). For $\tan (\beta d)>0$: inductive ($Z^{\text{sc}}=j\omega L_{\text{eq}}$). For $\tan (\beta d)<0$: capacitive. The line short-circuit “looks like” different reactive elements depending on its length:

• $d<\lambda /4$: appears inductive.
• $d=\lambda /4$: appears open (infinite impedance).
• $\lambda /4<d<\lambda /2$: appears capacitive.
• $d=\lambda /2$: appears short (zero impedance, periodic).

Open-circuit terminated ($Z_L=\infty$):

$Z^{\text{oc}}(d)=-j{Z}_0\cot (\beta d).$

Also purely reactive, with the inductive/capacitive roles swapped from the short-circuit case.

These are called stubs when used as impedance-matching elements — see Impedance matching.

Quarter-wave line ($d=\lambda /4$, $\beta d=\pi /2$):

$Z(\lambda /4)=Z_0\frac{Z_L+j{Z}_0\cdot \infty }{Z_0+j{Z}_L\cdot \infty }=\frac{Z_0^2}{Z_L}.$

A quarter-wavelength line transforms $Z_L$ into $Z_0^2/Z_L$. This is the basis of the quarter-wave transformer: pick $Z_0=\sqrt{Z_L{Z}_{\text{src}}}$ to match a real load to a source.

Half-wave line ($d=\lambda /2$, $\beta d=\pi$):

$Z(\lambda /2)=Z_L.$

Half-wave lines act as 1:1 impedance transformers — transparent at the design frequency.

Source voltage relation

Once $Z_{\text{in}}$ is known, the voltage at the input end of the line, with generator $V_g$ and source impedance $Z_g$, is

${\tilde{V}}_i=\frac{V_g{Z}_{\text{in}}}{Z_g+Z_{\text{in}}},$

and the forward-wave amplitude:

$V_0^{+}=\left(\frac{V_g{Z}_{\text{in}}}{Z_g+Z_{\text{in}}}\right)\frac{1}{e^{j\beta l}+\Gamma e^{-j\beta l}}.$

Once $V_0^{+}$ is known, the full voltage and current distribution along the line follows from the basic phasor expressions.

Worked example

A 100 Ω lossless line of length $0.125\lambda$ terminated in $Z_L=50-j(1/(2\pi \cdot 10^8\cdot 10\text{ pF}))=50-j159$ Ω at 100 MHz. Find $Z_{\text{in}}$.

$\beta l=2\pi \cdot 0.125=\pi /4$, so $\tan (\beta l)=1$.

$Z_{\text{in}}=100\cdot \frac{50-j159+j100}{100+j(50-j159)}=100\cdot \frac{50-j59}{259+j50}.$

The answer is a specific complex number. The point isn’t the arithmetic; it’s that $Z_{\text{in}}$ depends on $Z_L$, the line characteristic impedance, and the electrical length $\beta l$. Move the same load to a longer or shorter line, or change the frequency, and $Z_{\text{in}}$ changes.

In practice, a Smith chart does this calculation graphically by rotating the load’s $\Gamma$ point around the chart’s center by the electrical length, then reading the wave impedance off the new position.