Smith Chart Impedance Calculator
Smith Chart Impedance Calculator
A load made of R, L and C, traced across a frequency band as a locus on the reflection-coefficient plane — with the complex impedance, Γ in magnitude and angle, VSWR, return loss and the normalised r and x at the frequency you pick, and the worst VSWR anywhere in the band.
Γ, VSWR and the locus of an R-L-C load across a band
a series load of 25 Ω, 8 nH and 3 pF in a 50 Ω system, read at 1,200 MHz and swept from 600 to 1,600 MHz
One bilinear map, written in real arithmetic
Re Γ = (R² − Z₀² + X²) ÷ [(R + Z₀)² + X²]
Im Γ = 2·X·Z₀ ÷ [(R + Z₀)² + X²]
|Γ|² = [(R − Z₀)² + X²] ÷ [(R + Z₀)² + X²]
r = R ÷ Z₀, x = X ÷ Z₀, VSWR = (1 + |Γ|) ÷ (1 − |Γ|), return loss = −20·log₁₀|Γ|
- Γ
- the reflection coefficient at the load: the complex ratio of the reflected wave to the incident one. The Smith chart is a graticule drawn on the plane it lives in, and a passive load always sits inside the unit circle
- r, x
- the normalised resistance and reactance. The Smith chart’s two families of circles are the images of constant r and constant x under the map above
- locus
- the path Γ takes as the frequency changes. A static chart cannot show it, and it is what decides whether a match is broadband or a single-point tune
Worked example
a series load of 25 Ω, 8 nH and 3 pF in a 50 Ω system, read at 1,200 MHz and swept from 600 to 1,600 MHz
At 1,200 MHz the inductor is +60.32 Ω and the capacitor is −44.21 Ω, so the load is 25.00 + j16.11 Ω — normalised, r = 0.5000 and x = 0.3222
Γ = (Z − Z₀) ÷ (Z + Z₀) = -0.2745 + j0.2738, which is 0.3877 at 135.08°
That is a VSWR of (1 + 0.3877) ÷ (1 − 0.3877) = 2.266 : 1, a return loss of 8.23 dB, 15.03% of the power reflected and 0.707 dB of mismatch loss
Across the band the story is different. The load's own series resonance is at 1.027 GHz; at the bottom of the band the VSWR is 5.016 : 1 and at the top 4.040 : 1, so the worst anywhere in the band is 5.016 : 1 — more than twice the figure at the marker. That is what the locus is for: the match is a single-point tune, not a band-wide one
The same mismatch, in the four units people quote it in
| VSWR | |Γ| | Return loss | Power reflected | Mismatch loss |
|---|---|---|---|---|
| 1.10 : 1 | 0.0476 | 26.44 dB | 0.23% | 0.010 dB |
| 1.20 : 1 | 0.0909 | 20.83 dB | 0.83% | 0.036 dB |
| 1.50 : 1 | 0.2000 | 13.98 dB | 4.00% | 0.177 dB |
| 2.00 : 1 | 0.3333 | 9.54 dB | 11.11% | 0.512 dB |
| 2.50 : 1 | 0.4286 | 7.36 dB | 18.37% | 0.881 dB |
| 3.00 : 1 | 0.5000 | 6.02 dB | 25.00% | 1.249 dB |
| 5.00 : 1 | 0.6667 | 3.52 dB | 44.44% | 2.553 dB |
| 10.00 : 1 | 0.8182 | 1.74 dB | 66.94% | 4.807 dB |
What a locus shows that a point does not
Everything on a Smith chart is one formula: Γ = (Z − Z₀) ÷ (Z + Z₀). That map takes every passive impedance — the whole right half of the complex plane — into the unit disc, and it takes the lines of constant resistance and constant reactance into the two families of circles the chart is printed with. The centre is Γ = 0, a perfect match; the right-hand edge is an open circuit, the left-hand edge a short, and the rim is total reflection. A chart is a graticule, not a calculation.
Why the locus is the point of this page. A single reading tells you the match at one frequency, and a load can be perfectly matched at one frequency and useless 10% away. The locus — where Γ goes as the frequency sweeps — shows the difference immediately. A short arc huddled near the centre is a broadband match. A long arc that swings out to the rim is a high-Q load that has been tuned at a point, and the further it swings for a given fractional bandwidth the higher that Q is. The example on this page is the second kind: 2.3:1 at the marker, more than 5:1 at the bottom of the band.
About the shape of the plot. The chart below plots Im Γ against Re Γ, and the two faint outer curves are the |Γ| = 1 boundary — every point on them is exactly on the unit circle. They will not look like a circle. The plot box is wider than it is tall, and worse, the two axes are scaled independently to fit their own data, so the horizontal scale and the vertical scale are generally different by a large factor. Read coordinates off the axes; do not read the shape. The boundary curves are there to show where the rim is and to make the stretching self-evident, not to be admired as a circle. The undistorted picture — a real circle with the load’s arrangement drawn beside it — is the diagram above the chart, and the numbers are in the figures and the table.
What this page does not do. It does not convert between VSWR, return loss, reflection coefficient and mismatch loss as a general-purpose tool — the VSWR and return loss calculator owns those conversions, takes any one of them as its input and gives the other three with the power split. It also does not design the matching network: the L-network calculator takes a source and a load and returns both two-element solutions with their inductor and capacitor values. This page’s job is the one neither of those does: to show where a real load sits, and where it goes.
Reading the chart in your head. Adding series reactance moves Γ round a circle of constant r — clockwise for inductance, anticlockwise for capacitance. Adding shunt susceptance moves it round a circle of constant conductance. Moving along a lossless transmission line towards the generator rotates Γ clockwise about the centre at constant |Γ|, a full turn every half wavelength — which is why VSWR is the same everywhere on a lossless line while the impedance is not. The coax calculator gives the line’s own Z₀ and velocity factor, which set how far round one physical length takes you.
Frequently asked questions
Why does the unit circle look like a flat line rather than a circle?
Because the plot’s two axes are scaled independently. Each is fitted to the range of its own data, so if the locus spans 0.5 horizontally and 1.3 vertically the horizontal scale is roughly three times the vertical one before the box aspect is even considered. Every point drawn on those two outer curves is exactly on |Γ| = 1; what is wrong is the aspect ratio, not the points. Read numbers off the axes and off the table.
What is the difference between this and the VSWR calculator?
The VSWR page converts between the four ways of quoting one number — VSWR, return loss, |Γ| and mismatch loss — and gives the power split. It works with the magnitude of Γ only, at one frequency. This page starts from a load made of components, so it has the complex impedance and therefore the ANGLE of Γ as well, and it sweeps frequency to give a locus. Use that page when you have a measured VSWR; use this one when you have a load.
My load is not R, L and C. Can I still use this?
If you can measure or simulate R and X at your frequency, then at that one frequency yes — set the arrangement to series, put your R in, and choose an L or a C that gives your X (X = 2πfL, or X = −1 ÷ 2πfC). The locus will not be right, because your load’s reactance does not vary with frequency the way a single component’s does, but the marker figures will be.
Is the VSWR at the load the same as the VSWR at the other end of the cable?
On a lossless line, yes — the magnitude of Γ does not change along the line, only its angle, so the VSWR is the same everywhere. On a real line with loss it is not: the reflected wave is attenuated twice, so an instrument at the far end reads a BETTER VSWR than the load actually has. A long lossy feeder can make a very poor antenna look acceptable.
Why is the worst VSWR in the band so much worse than the one at my marker?
Because the load is resonant and the marker is near its resonance. Away from resonance the reactance grows quickly and the locus swings towards the rim. The rate at which that happens is the load’s Q, and the higher the Q the narrower the usable band. Nothing about the matching network can change that; it is a property of the load.
What is mismatch loss, and is it the same as return loss?
No. Return loss is how far below the incident wave the reflected one is, in decibels. Mismatch loss is how much of the incident power fails to reach the load because it is reflected — −10·log₁₀(1 − |Γ|²). At 2:1 the return loss is 9.54 dB but the mismatch loss is only 0.51 dB, which is why a 2:1 match costs far less power than its return loss figure makes it sound.
Related calculators
References
- P. H. Smith, “Transmission Line Calculator”, Electronics, vol. 12, no. 1, January 1939, pp. 29–31, and “An Improved Transmission Line Calculator”, Electronics, vol. 17, no. 1, January 1944, pp. 130–133 with continuation pages. The original publications of the chart this page plots a locus on; the journal, volume, issue, date and pagination of both were verified. The papers themselves were not fetched, and nothing is quoted from them: the chart’s construction as the image of the impedance half-plane under Γ = (Z − Z₀)/(Z + Z₀) is verified here numerically against complex arithmetic instead.
- D. M. Pozar, Microwave Engineering, 4th edition, Wiley 2012, §2.4 (the Smith chart) and §2.3 (the terminated lossless transmission line, where VSWR and return loss are defined). The standard graduate text for this material.
- IEEE Std 1785.1 and the IEC 61169 series cover the connectors and waveguide interfaces at which these quantities are specified in practice. Cited by number only; nothing from them is reproduced here.