What is a Smith chart?

A Smith chart maps the normalized impedance z = Z/Z_0 onto the reflection-coefficient plane (|Gamma| <= 1) as a single point. Constant-resistance and constant-reactance circles let you read off any impedance graphically. Invented by Phillip H. Smith in 1939, it is still the standard tool for impedance matching and measurement in RF and microwave engineering.

How are the reflection coefficient and VSWR related?

The reflection coefficient Gamma = (Z_L - Z_0) / (Z_L + Z_0) is a complex number that tells you how much of an incident wave is reflected at the load. Its magnitude |Gamma| determines the voltage standing wave ratio VSWR = (1 + |Gamma|) / (1 - |Gamma|). At a perfect match (Z_L = Z_0), Gamma is zero and VSWR is one. At total reflection (short or open), |Gamma| = 1 and VSWR is infinite.

What do the constant-resistance and constant-reactance circles represent?

All impedances sharing the same normalized resistance r form a circle in the Gamma plane, centered at (r/(1+r), 0) with radius 1/(1+r). Likewise, impedances sharing a common normalized reactance x form a circle centered at (1, 1/x) with radius 1/|x|. Overlaying circles for r = 0.2, 0.5, 1, 2, 5 and x = +/-0.2, +/-0.5, +/-1, +/-2, +/-5 lets you read off z directly from the position of the Gamma point.

Why is Z_0 = 50 ohms standard?

For coaxial cable the characteristic impedance for minimum loss is about 77 ohms and the one for maximum power handling is about 30 ohms; their geometric mean is close to 50 ohms, and 50 ohms became the compromise standard. Loss-critical applications such as TV coax use 75 ohms. Changing Z_0 changes the normalized impedance z = Z_L / Z_0, and therefore the position of Gamma on the Smith chart, even when Z_L is unchanged.

Smith Chart Simulator — Free Online Calculator

Q: How are the reflection coefficient and VSWR related?

The reflection coefficient Gamma = (Z_L - Z_0) / (Z_L + Z_0) is a complex number that tells you how much of an incident wave is reflected at the load. Its magnitude |Gamma| determines the voltage standing wave ratio VSWR = (1 + |Gamma|) / (1 - |Gamma|). At a perfect match (Z_L = Z_0), Gamma is zero and VSWR is one. At total reflection (short or open), |Gamma| = 1 and VSWR is infinite.

Q: What do the constant-resistance and constant-reactance circles represent?

All impedances sharing the same normalized resistance r form a circle in the Gamma plane, centered at (r/(1+r), 0) with radius 1/(1+r). Likewise, impedances sharing a common normalized reactance x form a circle centered at (1, 1/x) with radius 1/|x|. Overlaying circles for r = 0.2, 0.5, 1, 2, 5 and x = +/-0.2, +/-0.5, +/-1, +/-2, +/-5 lets you read off z directly from the position of the Gamma point.

Q: Why is Z_0 = 50 ohms standard?

For coaxial cable the characteristic impedance for minimum loss is about 77 ohms and the one for maximum power handling is about 30 ohms; their geometric mean is close to 50 ohms, and 50 ohms became the compromise standard. Loss-critical applications such as TV coax use 75 ohms. Changing Z_0 changes the normalized impedance z = Z_L / Z_0, and therefore the position of Gamma on the Smith chart, even when Z_L is unchanged.

Parameters

Reference impedance Z_0

Load Re(Z_L)

Load Im(Z_L)

Frequency f

GHz

Frequency is only used to display the wavelength λ = c/f. It does not affect Gamma in this lossless terminal-reflection model.

Results

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Normalized z = Z_L/Z_0

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Reflection coefficient Γ (polar)

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VSWR

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Wavelength λ (free space)

Smith chart (Γ plane)

Red = constant-resistance circles (r=0.2,0.5,1,2,5) / Blue = constant-reactance circles (x=±0.2,±0.5,±1,±2,±5) / Dashed green = |Γ| circle / Yellow = current Γ

Theory & Key Formulas

The Smith chart maps the normalized impedance z = Z/Z_0 = r + j x to the reflection coefficient Γ = u + j v via a bilinear transform, displayed inside the unit circle of the Γ plane.

Reflection coefficient at the load:

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

Voltage standing wave ratio (VSWR):

$$\text{VSWR} = \frac{1 + |\Gamma|}{1 - |\Gamma|}$$

Constant-resistance circle (center and radius):

$$\text{center} = \left(\frac{r}{1+r},\,0\right), \qquad \text{radius} = \frac{1}{1+r}$$

Constant-reactance circle (center and radius):

$$\text{center} = \left(1,\,\frac{1}{x}\right), \qquad \text{radius} = \frac{1}{|x|}$$

The center, Γ = 0 (i.e. z = 1), is the matched point with zero reflection and VSWR = 1. Larger |Γ| means a worse match; the unit circle itself is total reflection.

What is the Smith Chart Simulator?

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A Smith chart looks like a tangle of circles. What is it actually for?

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Roughly, it is a map that turns any complex load impedance into one point in the reflection-coefficient plane. The formula is $\Gamma = (z-1)/(z+1)$, where z is the impedance normalized by the reference Z_0. With the defaults above (Z_0=50, Z_L=75+j100) you get z = 1.5 + j2.0, and Gamma is about 0.644 at an angle of 37.3 degrees. The center of the chart is the matched point (Gamma = 0); the further out you go, the worse the match.

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What are the red and blue circles?

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Red ones are constant-resistance circles: all impedances with the same normalized r. Blue ones are constant-reactance circles, all sharing the same x. Move the Re(Z_L) slider and the yellow point glides along one red circle. Move Im(Z_L) and it slides along one blue circle. With practice you can read z = r + j x straight off the position of the point.

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I hear "VSWR" a lot. How is it different from Gamma?

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VSWR is a real number derived from |Gamma|: $\text{VSWR}=(1+|\Gamma|)/(1-|\Gamma|)$. Gamma is a complex number that also carries phase; VSWR is a single number that says "how mismatched are we". It is the value you usually see on a VNA or an SWR meter. With the default Gamma = 0.644 the VSWR is about 4.62. Ideally VSWR = 1, and many practical circuits aim for VSWR below 2.

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Moving the frequency slider does not change Gamma. So what is it for?

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Good catch. The terminal reflection coefficient does not depend on frequency, as long as the load Z_L itself does not. The frequency slider only updates the displayed wavelength lambda = c/f. At 2.4 GHz, lambda is about 125 mm. Stubs and matching sections are designed in units of lambda/4 or lambda/8, so it helps to keep the wavelength of the band you are working in handy.

Frequently Asked Questions

The map Gamma = (z - 1)/(z + 1) is a bilinear (Mobius) transformation, which maps lines and circles in one complex plane to lines and circles in another. Vertical lines Re(z) = r and horizontal lines Im(z) = x in the impedance plane both become circles in the Gamma plane. The whole right half-plane of impedances is compressed inside the unit disk, which makes the chart compact and bounded.

A positive Im(Z_L) is an inductive load, and the Gamma point appears in the upper half of the chart. A negative Im(Z_L) is a capacitive load, and the point appears in the lower half. Because inductors and capacitors respond oppositely to frequency, matching networks typically add a reactance of opposite sign in series or in shunt to cancel the load reactance.

|Gamma| = 1 is total reflection: all incident power bounces back. Z_L = 0 (short) is at Gamma = -1 (left edge), Z_L = infinity (open) is at Gamma = +1 (right edge), and any pure reactance (Re(Z_L) = 0) lies on the unit circle itself. Real loads always have some resistive part, so the Gamma point sits strictly inside the unit disk.

On a lossless line of length d back from the load, the apparent reflection coefficient is Gamma(d) = Gamma_L · exp(-j 2 beta d). Its magnitude is unchanged and its phase rotates by -2 beta d. On the Smith chart this is a clockwise rotation around the center; every lambda/4 of line corresponds to a 180-degree rotation. This geometric move is exactly why the Smith chart is so handy for stub and line-length matching.

Real-World Applications

Antenna and impedance matching networks: In Wi-Fi, Bluetooth and 5G front-ends, the antenna input impedance (usually not 50 ohms) must be matched to the radio. Designers still use Smith charts to step the load point toward the center with L, pi or T networks. Vector network analyzers (VNAs) draw the Smith chart live as you tune component values.

Microwave and RF circuit design: Input and output matching for LNAs and power amplifiers, plus the design of filters, couplers and mixers, all rely on the Smith chart. Stability circles, gain circles and noise-figure circles are all circles in the Gamma plane, so the chart becomes a single canvas for visual transistor design.

VSWR measurement and feedline diagnosis: VSWR meters and return-loss bridges measure |Gamma|. When debugging antennas or feedlines, the position of the Gamma point on the chart tells you not just how bad the mismatch is, but what kind of load is hiding at the other end.

Teaching electromagnetic engineering: Smith charts are standard in undergraduate RF and EM courses for showing how complex impedance relates to reflection. Sliding the point interactively makes the geometry of the bilinear transform visible in a way that paper charts cannot match.

Common Misconceptions and Pitfalls

The most common misconception is that the outer rim of the Smith chart represents infinite impedance. It does not. The rim (|Gamma| = 1) is total reflection, and different points on it correspond to different impedances: Gamma = +1 (right edge) is open (Z = infinity), Gamma = -1 (left edge) is short (Z = 0), and the upper and lower arcs are pure reactances (Re(Z) = 0). With the simulator, slide Re(Z_L) down toward 1 ohm and watch the yellow dot creep toward the rim. Remember: rim equals 100% reflection.

The second pitfall is confusing impedance with admittance (z versus 1/z). Parallel components are easier to handle in admittance y = g + j b, which has its own admittance chart (Y chart). On the Gamma plane the Y chart is the impedance chart rotated by 180 degrees (point-symmetric). When you work with shunt elements in a matching network, you must rotate the chart in your head. This simulator only shows the impedance chart, so for shunt effects you would also want an admittance overlay.

Finally, this simulator only models reflection at the load itself. On a real line the apparent Gamma rotates with position and, on a lossy line, also shrinks in magnitude. The point of this tool is to nail down the one-to-one map Z_L, Z_0 -> Gamma -> Smith-chart point. For stub matching, Q-circle design and frequency sweeps you need a dedicated RF tool (QUCS, ADS, Microwave Office). But getting the single-point intuition right is the first step everything else builds on.

Smith Chart Simulator — Reflection Coefficient and VSWR

What is the Smith Chart Simulator?

Frequently Asked Questions

Real-World Applications

Common Misconceptions and Pitfalls

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