3.6 Smith chart

Definition of Smith chart

The Smith chart is a graphical tool for visualizing and solving transmission line and impedance matching problems. It maps every possible complex impedance (or reflection coefficient) onto a single circular plot, letting you read off quantities like $\Gamma$, VSWR, and input impedance without grinding through complex algebra.

Phillip H. Smith developed the chart in 1939 at Bell Labs. Despite modern simulation software, it remains a standard tool because it builds geometric intuition about how impedances transform along a line or through a matching network.

Components of Smith chart

Reflection coefficient plane

The Smith chart is, at its core, a plot of the complex reflection coefficient $\Gamma$. Every point inside the unit circle corresponds to a passive impedance.

• The center of the chart is $\Gamma = 0$ (a perfect match to $Z_0$).
• The outer boundary is $|\Gamma| = 1$ (total reflection: short, open, or purely reactive load).
• The angle of $\Gamma$ gives the phase shift between incident and reflected waves, measured from the positive real axis.
• The defining relation is:

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

This bilinear (Möbius) transformation is what maps the right-half complex impedance plane onto the unit disk, and it's the entire mathematical basis of the chart.

Impedance and admittance coordinates

All values on the Smith chart are normalized to the system's characteristic impedance:

• Normalized impedance: $z = Z / Z_0 = r + jx$
• Normalized admittance: $y = Y / Y_0 = g + jb$, where $Y_0 = 1/Z_0$

Because $y = 1/z$, switching between impedance and admittance views is equivalent to rotating every point by 180° on the chart. Many printed charts overlay both coordinate grids so you can read either one directly.

Constant resistance circles

These circles represent all impedances sharing the same normalized resistance $r$.

• Every constant-$r$ circle is centered on the real axis at the point $\left(\frac{r}{1+r},\, 0\right)$ in the $\Gamma$-plane, with radius $\frac{1}{1+r}$.
• The $r = 0$ circle is the full outer boundary of the chart (purely reactive impedances).
• As $r$ increases, the circles shrink and shift rightward. The rightmost point ($r \to \infty$) corresponds to an open circuit; the leftmost point ($r = 0,\, x = 0$) is a short circuit.
• All constant resistance circles pass through the point $\Gamma = 1$ (the open-circuit point at the far right).

Constant reactance arcs

These arcs represent all impedances sharing the same normalized reactance $x$.

• Each arc is a portion of a circle centered on a vertical line through $\Gamma = 1$, with center at $\left(1,\, 1/x\right)$ and radius $1/|x|$.
• Arcs in the upper half of the chart correspond to inductive (positive) reactance.
• Arcs in the lower half correspond to capacitive (negative) reactance.
• The real axis itself is the $x = 0$ arc (purely resistive impedances).
• As $|x| \to \infty$, the arcs converge toward the open-circuit point on the far right.

Applications of Smith chart

Impedance matching

The most common use of the Smith chart is designing matching networks to maximize power transfer (and minimize reflections) between a source and load.

1. Plot the normalized load impedance $z_L$ on the chart.
2. Identify the target impedance (usually the chart center, $z = 1$).
3. Determine a path from $z_L$ to the target using series/parallel reactive elements or transmission line sections.
4. Read component values directly from the chart coordinates.

The chart makes it straightforward to compare different topologies (L-network, Pi-network, stub matching) by seeing which path is simplest or gives the best bandwidth.

Transmission line analysis

Moving along a transmission line of electrical length $\beta \ell$ corresponds to rotating around a constant-$|\Gamma|$ circle on the chart.

• Toward the generator: rotate clockwise by an angle $2\beta\ell$.
• One full rotation (360°) equals $\lambda/2$ of line length, because the impedance repeats every half-wavelength.
• At any point along the rotation you can read off the input impedance, $\Gamma$, and VSWR.

This is far quicker than evaluating the standard input impedance formula:

$Z_{in} = Z_0 \frac{Z_L + jZ_0 \tan(\beta\ell)}{Z_0 + jZ_L \tan(\beta\ell)}$

Stub matching networks

Stub matching eliminates reflections by placing a short- or open-circuited stub at a calculated distance from the load.

Single-stub matching procedure:

1. Plot the normalized load impedance $z_L$.
2. Draw the constant-VSWR circle through $z_L$.
3. Rotate toward the generator until you intersect the $g = 1$ circle (for a parallel stub) or the $r = 1$ circle (for a series stub). The rotation angle gives the stub's distance from the load.
4. Read the susceptance (or reactance) at that intersection. The stub must provide the negative of that value to cancel it.
5. Determine the stub length by tracing from the short-circuit or open-circuit point along the chart's outer edge until you reach the required susceptance/reactance.

Lossy transmission lines

For lines with non-negligible attenuation constant $\alpha$, the impedance no longer traces a circle as you move along the line. Instead, it follows an inward spiral toward the center of the chart.

• The reflection coefficient magnitude decays as $|\Gamma| e^{-2\alpha\ell}$ with distance $\ell$.
• After enough line length, any load impedance spirals toward $z = 1$ (the line's own characteristic impedance dominates).
• You can trace this spiral on the chart by simultaneously rotating (phase) and shrinking the radius (attenuation) at each incremental step.

Plotting on Smith chart

Normalized impedance and admittance

Before anything goes on the chart, normalize it:

$z = \frac{Z}{Z_0} = r + jx \qquad\qquad y = \frac{Y}{Y_0} = g + jb$

For example, if $Z_0 = 50\,\Omega$ and your load is $Z_L = 100 + j75\,\Omega$, then $z_L = 2 + j1.5$.

Plotting impedances and admittances

To plot $z = r + jx$:

1. Find the constant resistance circle for $r$.
2. Find the constant reactance arc for $x$.
3. Their intersection is your impedance point.

Admittances work the same way using the constant conductance circles ($g$) and constant susceptance arcs ($b$). On a combined impedance/admittance chart, the admittance grid is the impedance grid rotated 180°.

Plotting reflection coefficients

If you know $\Gamma$ in polar form ($|\Gamma|,\, \angle\theta$):

1. Measure $|\Gamma|$ as a fraction of the chart radius. The outer edge is $|\Gamma| = 1$; the center is 0.
2. Measure the angle $\theta$ counterclockwise from the positive real axis (right side of chart).
3. Mark the point. You can then read off the corresponding normalized impedance from the grid.

Plotting VSWR circles

A VSWR circle is simply a circle of constant $|\Gamma|$, centered at the chart origin.

• VSWR and $|\Gamma|$ are related by $\text{VSWR} = \frac{1 + |\Gamma|}{1 - |\Gamma|}$.
• VSWR = 1 is a point at the center (perfect match). VSWR $\to \infty$ is the outer boundary.
• Once you plot the load impedance, draw a circle through it centered on the chart center. Every point on that circle is an impedance you'll encounter somewhere along the line, and the circle's radius tells you the VSWR.

Transformations on Smith chart

Impedance transformations

Adding a series reactive element changes only the reactance, so you move along the constant resistance circle through your current point:

• Series inductor: move upward (increasing $x$).
• Series capacitor: move downward (decreasing $x$).

Adding a parallel element is easier in admittance coordinates. Convert to admittance (rotate 180° or use the admittance grid), then move along the constant conductance circle:

• Parallel capacitor: move upward on the admittance chart (increasing $b$).
• Parallel inductor: move downward on the admittance chart (decreasing $b$).

Admittance transformations

These follow the same logic as impedance transformations with roles swapped:

• A shunt (parallel) component changes only the susceptance $b$, so you move along a constant-$g$ circle.
• A series component in admittance terms changes the conductance, moving along a constant-$b$ arc.

Admittance coordinates are especially convenient for parallel-stub and shunt-element matching, since you stay on a single set of circles.

Transmission line length vs impedance

As you move a distance $d$ toward the generator along a lossless line:

• The impedance point rotates clockwise on the constant-VSWR circle.
• The rotation angle is $2\beta d = \frac{4\pi d}{\lambda}$ radians.
• A half-wavelength ($d = \lambda/2$) gives a full 360° rotation, returning to the same impedance.
• A quarter-wavelength rotation moves you to the diametrically opposite point on the VSWR circle. This is the basis of quarter-wave impedance transformers.

The outer scale of most Smith charts is marked in fractions of a wavelength, so you can read distances directly.

Lossy line transformations

For a lossy line with attenuation constant $\alpha$ and phase constant $\beta$:

• The impedance point spirals inward (toward $z = 1$) while rotating clockwise.
• At each increment $\Delta\ell$, multiply $|\Gamma|$ by $e^{-2\alpha\Delta\ell}$ and rotate by $2\beta\Delta\ell$.
• For long enough lines, the spiral converges to the center regardless of the load. Physically, this means the reflected wave is attenuated to insignificance and the line looks matched.

Smith chart vs other methods

Advantages of Smith chart

• Provides immediate visual insight into how close a load is to being matched and what kind of transformation is needed.
• Converts complex impedance algebra into geometric operations (rotations, circle intersections).
• Handles impedance-to-admittance conversion trivially (180° rotation).
• Works at any single frequency and for any $Z_0$, since everything is normalized.

Limitations of Smith chart

• Inherently a single-frequency tool. It doesn't show bandwidth or frequency response directly.
• Graphical accuracy is limited by plotting precision; fine differences in $|\Gamma|$ or angle can be hard to resolve.
• The standard chart assumes lossless lines. Lossy analysis requires the spiral technique, which is tedious by hand.
• Multi-stage or broadband matching problems quickly become unwieldy on paper and are better handled computationally.

Comparison with analytical methods

Analytical approaches (impedance matrices, ABCD parameters, scattering parameters) give exact numerical results and can be automated in code. They're essential for optimization and for problems involving many cascaded elements. The tradeoff is that they offer less geometric intuition. In practice, engineers often sketch a solution on the Smith chart first, then refine it analytically or in simulation.

Comparison with computer-aided design

Modern EDA tools (e.g., Keysight ADS, AWR Microwave Office) can sweep frequency, optimize component values, account for parasitics, and generate physical layouts. They're indispensable for production-level design. But most of these tools include a built-in Smith chart display precisely because the visual representation remains the fastest way to understand what's happening with impedance. The Smith chart and CAD tools are complementary, not competing.

Advanced Smith chart techniques

Multiple stub matching

Single-stub matching works perfectly at one frequency but can be narrow-band. Adding a second (or third) stub at a different location along the line provides extra degrees of freedom to broaden the match or handle loads that a single stub can't reach.

Design approach on the Smith chart:

1. Choose stub spacing (commonly $\lambda/8$ or $3\lambda/8$).
2. Plot the load admittance and rotate to the first stub location.
3. Add susceptance from the first stub to move onto a specific circle that, after rotation to the second stub, will intersect the $g = 1$ circle.
4. Rotate to the second stub location and add the required susceptance to reach $y = 1$.

A known limitation: double-stub matching has a forbidden region of load admittances that cannot be matched for a given stub spacing. Triple-stub matching eliminates this restriction.

Broadband matching with Smith chart

Broadband matching aims to keep $|\Gamma|$ below a specified threshold across a frequency band, not just at a single frequency.

• Plot the load impedance at several frequencies across the band. The locus of these points traces a curve on the chart.
• Design matching sections (quarter-wave transformers, tapered lines, or multi-section networks) that compress this curve toward the center of the chart at all plotted frequencies.
• The Bode-Fano criterion sets a theoretical limit on how well you can match a given load over a given bandwidth. The Smith chart helps you visualize how close your design comes to that limit.

Smith chart for active devices

Transistors and amplifiers have frequency-dependent, complex input and output impedances specified by their S-parameters.

• Plot $S_{11}$ and $S_{22}$ on the Smith chart to see the device's native impedance behavior versus frequency.
• Design input and output matching networks to transform these impedances toward the desired source/load impedances for optimal gain, noise figure, or output power.
• Stability circles (derived from all four S-parameters) can be overlaid on the Smith chart to identify impedance regions where the device may oscillate. Keeping your matching network's impedance outside the unstable region ensures stable operation.

Smith chart in microwave circuit design

At microwave frequencies, every interconnect is a transmission line and every junction has parasitic reactance. The Smith chart is used throughout the design flow:

• Filters: Visualize how each resonator section transforms impedance across the passband and stopband.
• Couplers and power dividers: Verify port impedances and isolation by plotting S-parameters on the chart.
• Antennas: Plot antenna input impedance versus frequency to design a feed matching network.
• MMIC design: On-chip matching networks for monolithic microwave ICs are routinely designed with Smith chart overlays in EDA tools, since chip-level parasitics make intuitive visualization especially valuable.