Introduction to the Smith Chart — Reading the Transmission Line Calculator
The Smith chart is a graphical tool for solving transmission line and impedance matching problems. It was invented by Phillip Smith at Bell Telephone Laboratories in 1939 and published in the January 1939 issue of Electronics magazine. Despite being nearly a century old, it remains in daily use by RF engineers and is embedded in every vector network analyzer manufactured today — every time you press the "Smith chart display" button on a modern instrument, you are using a direct descendant of Smith's original paper chart.
The chart's power lies in a transformation: it maps all possible complex impedance values — from short circuit to open circuit, from purely resistive to wildly reactive — onto a bounded circle of unit radius. This transformation converts the unwieldy mathematics of transmission line impedance transformation (which involves hyperbolic tangent functions of complex numbers) into simple rotation around the chart. Moving along a transmission line becomes moving around a circle. Adding a shunt component becomes moving along an arc. Impedance matching becomes geometry.
Normalization — The Foundation
Before plotting anything on a Smith chart, you must normalize the impedance. Normalization means dividing every impedance value by the characteristic impedance Z0 of the transmission line system — almost always 50 Ω in amateur radio work.
z = Z / Z₀ = r + jx
where Z = actual impedance (Ω), Z₀ = system impedance (50 Ω), r = normalized resistance, x = normalized reactance
Examples at Z₀ = 50 Ω:
Z = 50 + j0 Ω → z = 1 + j0 (perfect match)
Z = 100 + j50 Ω → z = 2 + j1
Z = 25 − j25 Ω → z = 0.5 − j0.5
Z = 0 Ω (short) → z = 0 + j0
Z = ∞ (open) → z = ∞ (right edge of chart)
Normalization makes the chart universal — the same chart works whether your system impedance is 50 Ω, 75 Ω, or 300 Ω. The chart has no built-in system impedance; it works with normalized values, and you supply Z₀ when converting back to actual impedances.
The Reflection Coefficient Plane
The Smith chart is actually a plot of the complex reflection coefficient Γ (gamma). The reflection coefficient at any impedance discontinuity is defined as:
where Γ = Γr + jΓi (complex number with real and imaginary parts)
Key properties:
• |Γ| ≤ 1 for all passive loads (no negative resistance)
• |Γ| = 0 for a matched load (Z = Z₀, z = 1)
• |Γ| = 1 for a lossless mismatch (short circuit, open circuit, pure reactance)
• Γ = −1 for a short circuit (Z = 0)
• Γ = +1 for an open circuit (Z = ∞)
• SWR = (1 + |Γ|) / (1 − |Γ|)
The chart plots Γr on the horizontal axis and Γi on the vertical axis. All passive impedances plot inside or on the boundary of the unit circle |Γ| = 1. The center of the chart (Γ = 0) represents the perfect match point, z = 1 + j0. The transformation from z to Γ is a bilinear (Möbius) transformation — it maps circles to circles, which is what creates the characteristic curved grid of resistance circles and reactance arcs.
Resistance Circles
Every impedance with the same normalized resistance r plots on the same circle. These resistance circles are the main grid lines of the Smith chart:
| Normalized resistance r | Circle passes through | Physical meaning (Z₀ = 50 Ω) |
|---|---|---|
| r = 0 | Left edge to right edge (the outer circle boundary) | Pure reactance — no resistance at all |
| r = 0.5 | Tangent to left edge | R = 25 Ω (any reactance) |
| r = 1 | Passes through center | R = 50 Ω (any reactance) — this is the matching circle |
| r = 2 | Right half of chart | R = 100 Ω (any reactance) |
| r = 5 | Small circle near right edge | R = 250 Ω (any reactance) |
| r = ∞ | The rightmost point of the chart | Open circuit |
All resistance circles pass through the right edge of the chart (the open-circuit point, r = ∞). Larger values of r correspond to smaller circles that huddle near the right edge; smaller values of r correspond to larger circles that fill most of the chart.
Annotated Smith chart. Resistance circles run from left to right; reactance arcs run from bottom to top. The r = 1 circle (matching circle) passes through the center. The upper half of the chart is inductive; the lower half is capacitive. The boundary circle represents |Γ| = 1 (lossless mismatch).
View LargerReactance Arcs
Lines of constant normalized reactance x are arcs on the Smith chart, not straight lines. They run from the right edge (open-circuit point) through the chart:
| Normalized reactance x | Location on chart | Physical meaning (Z₀ = 50 Ω) |
|---|---|---|
| x = 0 | The horizontal center line | Pure resistance — no reactance |
| x = +0.5 | Upper half, moderately curved arc | X = +25 Ω (inductive) |
| x = +1 | Upper half, enters center region | X = +50 Ω (inductive) |
| x = +2 | Upper half, small arc near top | X = +100 Ω (inductive) |
| x = −0.5 | Lower half, moderately curved arc | X = −25 Ω (capacitive) |
| x = −1 | Lower half, enters center region | X = −50 Ω (capacitive) |
| x = −2 | Lower half, small arc near bottom | X = −100 Ω (capacitive) |
All reactance arcs originate at the right edge of the chart (open-circuit point) and curve upward (positive/inductive) or downward (negative/capacitive). The horizontal axis (x = 0) is where all purely resistive impedances plot.
Constant-SWR Circles
Any circle centered on the chart's center point represents a locus of constant SWR (and constant |Γ|). If you know the SWR of a transmission line, every impedance on that line traces a circle on the Smith chart as you move along the line. This is one of the most powerful properties of the chart:
Radius of circle = |Γ| = (SWR − 1)/(SWR + 1)
SWR = 1 → |Γ| = 0 (center point, perfect match)
SWR = 1.5 → |Γ| = 0.2 (small circle near center)
SWR = 2 → |Γ| = 0.333
SWR = 3 → |Γ| = 0.5 (halfway to the edge)
SWR = ∞ → |Γ| = 1 (outer boundary, open or short or pure reactance)
The constant-SWR circle passes through both the maximum-impedance point (Zmax = Z₀ × SWR, a purely resistive point on the right part of the horizontal axis) and the minimum-impedance point (Zmin = Z₀ / SWR, a purely resistive point on the left part of the horizontal axis). These are the points where voltage maximum and voltage minimum occur on the standing wave — the classic VSWR standing wave pattern.
Key Points and Regions
| Location | Γ value | Impedance | Physical meaning |
|---|---|---|---|
| Center of chart | 0 | Z₀ (50 Ω) | Perfect match, SWR = 1 |
| Left edge (x = 0) | −1 | 0 Ω | Short circuit |
| Right edge (x = 0) | +1 | ∞ | Open circuit |
| Top of outer circle | +j | Pure inductance | |Γ| = 1, no resistive loss |
| Bottom of outer circle | −j | Pure capacitance | |Γ| = 1, no resistive loss |
| Upper half (x > 0) | Im(Γ) > 0 | Inductive reactance | Phase angle between 0° and +180° |
| Lower half (x < 0) | Im(Γ) < 0 | Capacitive reactance | Phase angle between 0° and −180° |
Moving Around the Chart
The most powerful property of the Smith chart for transmission line work is this: moving along a lossless transmission line from load toward the generator corresponds to rotating clockwise around the constant-SWR circle. Moving from generator toward the load corresponds to rotating counterclockwise.
One complete revolution of the chart (360°) corresponds to moving exactly half a wavelength (λ/2) along the line. This is labeled on the outer ring of most printed Smith charts, where the outer scale reads "wavelengths toward generator" (clockwise, 0 to 0.5) and "wavelengths toward load" (counterclockwise, 0 to 0.5).
Moving λ/2 along the line: one full revolution (360°) → same impedance
Moving λ/4 along the line: half revolution (180°) → impedance inverts (Z → Z₀²/Z)
Moving toward generator: rotate clockwise
Moving toward load: rotate counterclockwise
The |Γ| (distance from center) stays constant while rotating — only the angle changes.
This is why a quarter-wave transformer works: it rotates exactly 180° on the Smith chart, which takes the point ZL to the point Z0²/ZL. If you start at z = 4 (a 200 Ω load on 50 Ω line), rotating 180° takes you to z = 0.25 (12.5 Ω). Use a quarter-wave section of Z0,new = √(50×200) = 100 Ω characteristic impedance, and the input presents 50 Ω to the source.
Reading Impedance from the Chart — A Practical Walk-Through
Given a point plotted on a Smith chart, here is how to read the impedance:
- Read the resistance: identify which r-circle the point sits on (or interpolate between two labeled circles). This gives normalized resistance r.
- Read the reactance: identify which x-arc the point sits on (or interpolate). Points above the horizontal axis are inductive (+jx); points below are capacitive (−jx).
- De-normalize: Z = z × Z₀ = (r + jx) × Z₀ to get actual impedance in ohms.
- Read SWR: measure the distance from the center to the plotted point, and use the radius scale on the chart (or the scale below the chart on printed versions) to read |Γ| and then compute SWR = (1+|Γ|)/(1−|Γ|). Alternatively, rotate the SWR compass to the point and read SWR from the horizontal axis where the constant-SWR circle crosses on the right side.
- Read reflection coefficient angle: the angle of the line from the center to the point, measured from the right horizontal axis, is the angle of Γ. This angle equals twice the distance from the load in electrical degrees.
Reading example: A point on the Smith chart lies in the upper-right region, at the intersection of the r = 2 circle and the x = +1 arc.
Normalized impedance: z = 2 + j1
Actual impedance at Z₀ = 50 Ω: Z = (2 + j1) × 50 = 100 + j50 Ω
The point is at Γ = (z−1)/(z+1) = (1+j1)/(3+j1) = (1+j1)(3−j1) / (9+1) = (3+j3−j1+1)/10 = (4+j2)/10 = 0.4 + j0.2
|Γ| = √(0.4² + 0.2²) = √(0.16+0.04) = √0.20 = 0.447
SWR = (1+0.447)/(1−0.447) = 1.447/0.553 = 2.61
Return loss = −20 log₁₀(0.447) = 7.0 dB
Frequently Asked Questions
Why does moving along a transmission line correspond to rotation on the Smith chart?
Moving along a lossless transmission line multiplies the reflection coefficient Γ by e−j2βl, where β is the phase constant and l is the distance moved. This is a pure phase rotation (multiplication by a unit-magnitude complex exponential), which corresponds geometrically to rotation around a circle centered at the origin. Since the center of the Smith chart is the origin of the Γ plane (Γ = 0), moving along the line traces a circular arc around the chart center at constant radius |Γ|. The factor of 2 in the exponent means one full line wavelength (βl = 360°) corresponds to 720° = two full chart revolutions, or equivalently, a half-wavelength of line corresponds to exactly one full chart revolution (360°).
Is the Smith chart still useful now that computers can do the math?
Yes — for different reasons than in 1939. Modern RF engineers use the Smith chart because it provides geometric intuition about what happens to impedance as you add components or move along transmission lines. When you see a complex impedance move toward the center as you add a shunt stub, you are building a visual model of the matching process that no numerical table provides. Vector network analyzers display their results as Smith charts because an RF engineer looking at the chart can instantly see whether a DUT is capacitive or inductive, how far from matched it is, and what kind of element would move it toward the center — information that requires interpretation when reading from a table of complex numbers. The chart survives because it encodes geometry, not because it saves arithmetic.
What is the admittance Smith chart and when do you use it?
The admittance Smith chart is the same chart, but the grid lines are relabeled as constant conductance circles and constant susceptance arcs. An admittance chart is obtained by rotating the standard impedance chart 180° — so the short-circuit point (left edge on the impedance chart) becomes the open-circuit point, and vice versa. When you are adding shunt (parallel) elements, admittances add directly, so it is more convenient to work on the admittance chart. Many engineers flip between impedance and admittance representations during a single matching problem: impedance for series elements (which add to Z), admittance for shunt elements (which add to Y). A printed Smith chart often includes both grids superimposed — the impedance grid in red and the admittance grid in blue — called a ZY Smith chart.
Test Your Knowledge
Answer the questions below to check your understanding. Every answer can be found in the lesson above.