Smith chart

The Smith chart is a graphical tool used in transmission-line theory and RF engineering to read off impedance, admittance, and reflection coefficients. Mathematically, the chart is the image of the right half of the impedance plane under the Möbius transformation

$\Gamma =\frac{Z-Z_0}{Z+Z_0}.$

Here $Z$ is the load impedance, $Z_0$ is the characteristic impedance of the transmission line, and $\Gamma$ is the reflection coefficient. Normalizing $z=Z/Z_0$:

$\Gamma =\frac{z-1}{z+1}.$

For passive loads ($\mathrm{Re}(Z)>0$), $|\Gamma |<1$, so $\Gamma$ lies in the unit disk of the complex plane.

The transformation

The map $z\mapsto (z-1)/(z+1)$ is a Möbius transformation. It maps:

• The right half-plane $\mathrm{Re}(z)>0$ to the open unit disk $|\Gamma |<1$.
• The imaginary axis $\mathrm{Re}(z)=0$ to the unit circle $|\Gamma |=1$.
• $z=1$ (matched load) to $\Gamma =0$ (no reflection — the center of the chart).
• $z=0$ (short circuit) to $\Gamma =-1$.
• $z=\infty$ (open circuit) to $\Gamma =+1$.
• $z=j$ (purely inductive) to $\Gamma =j$ (top of unit circle).
• $z=-j$ (purely capacitive) to $\Gamma =-j$ (bottom).

The curved grid

Because Möbius transformations send circles and lines to circles and lines, the rectangular grid of constant-resistance lines and constant-reactance lines in the $z$-plane maps to a curved grid in the $\Gamma$-plane:

• Constant resistance $\mathrm{Re}(z)=r$: maps to a circle in the $\Gamma$-plane, tangent to the unit circle at $\Gamma =1$. The tangency at $\Gamma =1$ happens because the open-circuit point $z=\infty$ lies on every vertical line $\mathrm{Re}(z)=r$ (each such line is unbounded in the imaginary direction), and $z=\infty \mapsto \Gamma =1$ under the Möbius map. So every constant-resistance circle passes through (is tangent to the unit circle at) $\Gamma =1$. Larger $r$ → smaller circle inside the unit disk, concentrating toward $\Gamma =1$ as $r\to \infty$ and expanding to the full unit circle as $r\to 0$.
• Constant reactance $\mathrm{Im}(z)=x$: maps to an arc of a circle passing through $\Gamma =1$, for the same reason — horizontal lines $\mathrm{Im}(z)=x$ also pass through $\infty$. The arc lies in the upper half-disk (positive reactance, inductive) or lower half-disk (negative reactance, capacitive), depending on the sign of $x$.

The two curved families of the Smith chart grid: constant-$r_L$ circles (closed loops all tangent to $\Gamma =1$) and constant-$x_L$ arcs (open arcs running from $\Gamma =1$ outward to the unit circle). Their intersection is the impedance point.

The two families of curves intersect at right angles, by Conformality (Möbius transformations are conformal away from their pole; the pole is at $z=-1$, outside the right half-plane).

What the chart lets you do

• Plot a load $z=r+jx$: find the intersection of the constant-$r$ circle and the constant-$x$ arc.
• Read the reflection coefficient $\Gamma$ directly: $|\Gamma |$ is the distance from the chart center; $\angle \Gamma$ is the angle.
• Trace impedance along the transmission line: moving toward the source or load along a uniform line rotates $\Gamma$ around the chart center by an angle proportional to the electrical distance traveled.
• Design impedance-matching networks graphically: where to place stubs, series elements, etc., to move from a starting impedance to a target.

In modern RF engineering, the Smith chart is largely used as a visualization built into network analyzers and simulators, but the underlying mathematics — Möbius transformation, conformal mapping, circles-to-circles — is exactly what makes it work.

Applied workflow (Electromagnetics)

In a transmission-line context, the chart is used as a graphical calculator for impedance, admittance, Reflection coefficient, SWR, and matching network design.

The chart has three scales around its rim:

• Inner scale: angle ${\theta }_r$ of reflection coefficient, in degrees.
• Middle scale: “wavelengths toward load” (WTL) — distance from a reference point, expressed in $\lambda$, when moving from the generator toward the load.
• Outer scale: “wavelengths toward generator” (WTG) — distance when moving from the load toward the generator.

Why the rim scales: moving along the line shifts the phase of $\Gamma$. Moving a distance $d$ toward the generator multiplies $\Gamma$ by $e^{-j2\beta d}$ — a clockwise rotation around the chart center by $2\beta d=4\pi d/\lambda$ radians. One full lap is $\lambda /2$.

Application 1: Find Γ from a load

Given normalized load $z_L=r+jx$:

1. Find the intersection of the $r_L=r$ resistance circle and the $x_L=x$ reactance arc. Mark the point on the chart.
2. $|\Gamma |$ is the distance from chart center to the point, divided by the distance from center to the rim ($|\Gamma |=1$ circle).
3. ${\theta }_r$ is read directly off the angle scale.

Example: $z_L=2-j1$ gives $|\Gamma |\approx 0.45$, ${\theta }_r\approx -26.6°$.

Application 2: Find wave impedance Z(d)

Given $z_L$ and want $z(d)$ at distance $d$ from the load:

1. Plot $z_L$ on the chart (point $A$).
2. Draw the constant-SWR circle through $A$ — a circle centered at the chart center passing through $A$. All wave impedances along the line lie on this circle.
3. Rotate clockwise (toward generator) by $d/\lambda$ as read from the outer scale. Land at point $B$.
4. Read $z(d)=r^{\prime }+j{x}^{\prime }$ from the intersecting resistance/reactance curves at $B$.

For input impedance, rotate by the full electrical length $l/\lambda$.

Application 3: Find SWR

1. Plot $z_L$, draw constant-SWR circle.
2. Where the SWR circle crosses the real ${\Gamma }_r$ axis to the right of origin (with $r>1$): that value of $r$ equals the SWR.

This works because on the real axis, $\Gamma$ is real, and at the right intersection $\Gamma =|\Gamma |$ (positive). The normalized impedance there is $r=(1+|\Gamma |)/(1-|\Gamma |)=S$.

Application 4: Find voltage maxima/minima positions

The voltage maximum on the line occurs where $z(d)=S$ (real, equal to SWR) — the right intersection of the SWR circle with the ${\Gamma }_r$ axis.

To find $d_{\max }^1$ (distance from load to first voltage maximum):

1. Plot $z_L$.
2. Read the WTG scale at the angular position of $z_L$.
3. Move clockwise to the right-${\Gamma }_r$ intersection; read the WTG scale there.
4. Difference (mod $\lambda /2$) is $d_{\max }^1$.

Voltage minimum is exactly $\lambda /4$ farther: $d_{\min }^1=d_{\max }^1\pm \lambda /4$.

Application 5: Single-stub matching

1. Plot normalized load admittance $y_L=1/z_L$. (On the chart, $1/z$ is the point diametrically opposite — rotate 180°.)
2. Move along the constant-SWR circle (toward generator) until you intersect the $g=1$ circle. Two intersections; pick one. Read off $d$ from the WTG scale.
3. At that intersection point, the normalized admittance is $1+jb$. Need to cancel $jb$ with a stub providing $-jb$.
4. For a short-circuited stub of $Z_0$: enter the chart at the “short” point ($y=\infty$, the rightmost point on $r=0$ circle), move toward the generator along the rim until you reach a normalized admittance of $-jb$. Read the distance traveled — that’s the stub length $l_s$.
5. For an open-circuited stub: enter at the “open” point ($y=0$, leftmost on $r=0$) and proceed similarly.

After this construction, placing the stub at distance $d$ from the load with length $l_s$ ensures the feedline sees $Z_0$ — matched.

Worked example

A 50 Ω line terminated in $Z_L=(100+j50)$ Ω. Normalize: $z_L=2+j1$.

• Plot $z_L=2+j1$ on the chart at the intersection of $r=2$ and $+x=1$.
• $|\Gamma |\approx 0.45$, ${\theta }_r\approx 27°$ — read off.
• SWR $\approx 2.6$ from where constant-SWR circle hits real axis.
• $d_{\max }^1\approx 0.037\lambda$, $d_{\min }^1\approx 0.287\lambda$.
• For a $0.1\lambda$-long line, rotate clockwise by $0.1\lambda$ (i.e., $0.1\cdot 720°=72°$ on the chart). New point gives $z(0.1\lambda )$.

In context

The Smith chart is the most-cited engineering application of Möbius transformations and conformal mapping. It’s the reason transmission-line theory benefits from complex analysis: the impedance matching problem in $z$-space is hard, but in $\Gamma$-space it’s geometric — move points around inside a disk, with rotation along the line, and read off matching conditions visually.

For the underlying transformation theory, see Möbius transformation. For the broader idea of “transform to an easy region, solve, transform back,” see Conformality. For the applied use in matching, see Impedance matching.