Reflection coefficient

The reflection coefficient $\Gamma$ at the load of a Transmission line is the ratio of reflected voltage wave amplitude to incident voltage wave amplitude:

$\Gamma =\frac{V_0^{-}}{V_0^{+}}.$

In terms of the load impedance $Z_L$ and the line’s Characteristic impedance $Z_0$:

$\Gamma =\frac{Z_L-Z_0}{Z_L+Z_0}=\frac{z_L-1}{z_L+1},$

where $z_L=Z_L/Z_0$ is the normalized load impedance.

$\Gamma$ is in general complex: $\Gamma =|\Gamma |e^{j{\theta }_r}$, with $|\Gamma |\le 1$ for passive loads and $-\pi \le {\theta }_r\le \pi$.

Derivation

At the load (position $z=0$), the total voltage and current are:

${\tilde{V}}_L=V_0^{+}+V_0^{-},{\tilde{I}}_L=\frac{V_0^{+}}{Z_0}-\frac{V_0^{-}}{Z_0}.$

The load itself enforces ${\tilde{V}}_L=Z_L{\tilde{I}}_L$:

$V_0^{+}+V_0^{-}=Z_L\left(\frac{V_0^{+}}{Z_0}-\frac{V_0^{-}}{Z_0}\right)=\frac{Z_L}{Z_0}(V_0^{+}-V_0^{-}).$

Solve for $V_0^{-}/V_0^{+}$:

$\Gamma =\frac{V_0^{-}}{V_0^{+}}=\frac{Z_L/Z_0-1}{Z_L/Z_0+1}=\frac{Z_L-Z_0}{Z_L+Z_0}.$

The forward wave gets partially reflected by the load, with $\Gamma$ as the ratio. The remaining $1-|\Gamma {|}^2$ fraction of the power is absorbed.

Special cases

Load	$\Gamma$	Physical interpretation
Matched ($Z_L=Z_0$)	0	No reflection. All power absorbed.
Open circuit ($Z_L=\infty$)	$+1$	Full reflection in phase.
Short circuit ($Z_L=0$)	$-1$	Full reflection inverted.
Reactive ($Z_L=jX$)	$	\Gamma

For a matched load, no energy is reflected — the load absorbs everything. The line carries only a forward wave; no standing wave forms.

For an open, the reflected wave is in phase with the incident: voltage doubles at the open end ($V_{\text{oc}}=2V_0^{+}$), current is zero.

For a short, the reflected wave is inverted: voltage at the short is zero, current doubles.

For a purely reactive load (pure inductor or pure capacitor), the load can’t absorb any real power — all incident energy reflects, $|\Gamma |=1$. But the phase of $\Gamma$ encodes the reactance type and magnitude.

Reflection coefficient table for common loads

A useful quick-reference summary (magnitudes and phases for representative load types on a line of real $Z_0$):

Load $Z_L$	Normalized $z_L$	$\Gamma$	$\Vert \Gamma \Vert$	$\angle \Gamma$
Open ($\infty$)	$\infty$	$+1$	1	$0°$
Short ($0$)	$0$	$-1$	1	$180°$ (or $-180°$)
Matched ($Z_0$)	$1$	$0$	0	undefined
Real, $R_L>Z_0$	$r>1$	$(r-1)/(r+1)$	$(r-1)/(r+1)$	$0°$
Real, $R_L<Z_0$	$r<1$	$(r-1)/(r+1)$	$(1-r)/(1+r)$	$180°$
Pure inductor ($j{X}_L$, $X_L>0$)	$jx$, $x>0$	$(jx-1)/(jx+1)$	1	between $0°$ and $180°$
Pure capacitor ($j{X}_L$, $X_L<0$)	$jx$, $x<0$	$(jx-1)/(jx+1)$	1	between $-180°$ and $0°$
Inductive, complex ($R+jX$, $X>0$)	$r+jx$, $x>0$	general	$\le 1$	upper half plane
Capacitive, complex ($R+jX$, $X<0$)	$r+jx$, $x<0$	general	$\le 1$	lower half plane

Pattern: $|\Gamma |=0$ for matched, $|\Gamma |=1$ for any purely reactive or open/short load, and $0<|\Gamma |<1$ for a complex impedance with positive real part. The sign of the imaginary part of $z_L$ determines whether $\Gamma$ sits in the upper or lower half of the Smith-chart disk. The sign of the real-part deviation from $z=1$ determines $\angle \Gamma$ relative to the $\pm$ real-axis quadrants.

For the Ulaby Table 2.3 detailed cases (real, larger and smaller; capacitive, inductive; open and short with their ${\Gamma }_d$ rotation as you move along the line), these special points anchor the Smith chart: every other load impedance plots between them in a definite way.

Current reflection coefficient

The current’s reflection coefficient has the opposite sign:

$\frac{I_0^{-}}{I_0^{+}}=-\Gamma .$

Derivation: by definition of Characteristic impedance, the forward wave satisfies $I_0^{+}=V_0^{+}/Z_0$, and the backward wave satisfies $I_0^{-}=-V_0^{-}/Z_0$. The minus sign on the backward current comes from current direction: forward current flows in $+\hat{\mathbf{z}}$, backward current in $-\hat{\mathbf{z}}$, but we measure both relative to the same $+\hat{\mathbf{z}}$ axis. So

$\frac{I_0^{-}}{I_0^{+}}=\frac{-V_0^{-}/Z_0}{V_0^{+}/Z_0}=-\frac{V_0^{-}}{V_0^{+}}=-\Gamma .$

Voltage reflects with amplitude $\Gamma$; current reflects with amplitude $-\Gamma$.

Phase-shifted reflection coefficient

At distance $d$ from the load (so $z=-d$ if $z=0$ at the load), the relative phase of forward and backward waves has shifted. Defining

${\Gamma }_d=\Gamma e^{-j2\beta d}=|\Gamma |e^{j({\theta }_r-2\beta d)},$

the wave impedance and other quantities can be expressed in terms of ${\Gamma }_d$. As you move away from the load (increase $d$), the phase of ${\Gamma }_d$ winds clockwise on a polar plot — this is the rotation visible on a Smith chart.

Reflected power

If the incident time-averaged power is $P_{\text{av}}^i=|V_0^{+}{|}^2/(2Z_0)$, the reflected power is

$P_{\text{av}}^r=|\Gamma {|}^2{P}_{\text{av}}^i.$

The net power delivered to the load:

$P_{\text{av}}^L=(1-|\Gamma {|}^2)P_{\text{av}}^i.$

For a matched load $|\Gamma |=0$, full power delivered. For a 50% mismatched line ($|\Gamma |=0.5$), 75% of power gets through; the other 25% returns to the source.

This is why matching matters: any reflected energy is either dissipated in the source impedance (lost), reflected back from the source (creating multiple reflections), or — if the source is intolerant of reflections — potentially damaging.

In context

The reflection coefficient is the central object of transmission line analysis. Almost everything else derives from it:

• Standing wave ratio: $S=(1+|\Gamma |)/(1-|\Gamma |)$.
• Input impedance at distance $d$: $Z(d)=Z_0(1+{\Gamma }_d)/(1-{\Gamma }_d)$.
• Power delivered to load: $P=(1-|\Gamma {|}^2)P_{\text{inc}}$.
• Smith chart coordinates: $\Gamma$ plotted directly in the complex plane.

The Smith chart is essentially a graphical calculator for these relations.