Smith chart: try out impedance, SWR and matching

Chart Admittance SWR circle

Display ▾

Angles on the rim Target circles r = 1, g = 1 Label arcs Grid in Ω

Explain Zoom

Move the pointer over the chart and this tells you what is there.

← → show steps · ↑ ↓ examples · V other route · + − zoom · Esc ends

Load

Example

Load type

R Ω

X Ω

Z0 Ω

Band

f MHz

Or drag the point in the chart.

Values Time domain ↗

File ▾

Open network file

	Load	after the chain

Circuit & matching Z at the nodes

No components yet. Click ＋ above or let the L-network below make a suggestion. You can drag the end point of each arc in the chart.

Steps

L-network

Two components that bring the load to Z0 at the set frequency. There are usually two or four solutions.

Measurement (.s1p)

Drop a Touchstone file (.s1p) here or choose one.

Frequency sweep show

Calculates the whole chain across the band. The point becomes a curve, and you see how wide the match is.

Inductors, capacitors and lines are ideal until you give them a Q or a loss (column “Q, line” or for all of them under Settings). A lossy line slowly pulls the point towards the centre as it turns. You can measure your antenna under Measure antenna; this page shows a measurement file (.s1p) from your analyser as a curve.

Chart Target circles Show route Frequency curve

Task

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How to play

The red dot is the load. The goal is the green circle in the centre: there the SWR is low enough.
Drag L, C or a line into the schematic. Dropped on the rail it goes in series, below it in parallel (to ground).
Dragging a placed component sideways changes its value, a click removes it, Ctrl+Z brings it back. In the chart you can also drag the end point, which changes the last component.
In series the point moves on circles of equal r, in parallel on circles of equal g. The two target circles r = 1 and g = 1 pass through the centre: land on one of them and you only need one more component.
Without a mouse: the buttons below the schematic add a component, Tab selects a component, ← → changes its value (with Shift in fine steps), Del removes it.

Circuit

L C Transmission line

No component yet. Drag one here from the parts box at the top right.

Why no resistors? A resistor can match too, but it burns power. A matching network of L and C (and line) gets the power to the antenna without loss.

Status

SWR –

Components: 0
Moves: 0
Time: 0:00
Solved: 0/0
Streak: 0

Across frequency

Two ropes

Weight right to left 4 : 1

Pulse heights

In the Smith chart

Part of it comes back

Send a pulse along a light rope tied to a heavy one. Two things happen at the knot: part of it carries on into the heavy rope, slower and smaller. Another part comes back, upside down. The other way round, from heavy to light rope, the reflected pulse comes back upright, and the transmitted pulse is even taller than the incident one.

How much comes back depends only on how different the ropes are. Ropes of equal weight let everything through. The bigger the difference, the more is reflected. This ratio of back to forward, with its sign, is called the reflection coefficient Γ.

Fixed and free end

If the rope is tied to a wall, the knot cannot move. Everything comes back, fully upside down: Γ = −1. If the end slides freely on a pole with a ring, the ring swings twice as far, and everything comes back upright: Γ = +1.

A wave on a cable behaves exactly the same. The heavy rope is the low impedance. A short at the end is the wall (the voltage must be zero there, Γ = −1), an open end the free ring (Γ = +1). A load of Z0 is the rope of equal weight, nothing comes back. In the Smith chart Γ lies on the horizontal axis: short on the left, open on the right, match in the centre.

The maths

The characteristic impedance of a rope grows with the square root of its weight per metre, Z ∝ √μ. For the displacement Γ = (Z1 − Z2)/(Z1 + Z2) and for the transmitted part τ = 1 + Γ. For a cable it is the same with the load: Γ = (Z − Z0)/(Z + Z0). How the wave runs back and forth on a real line is shown by the transmission line in the time domain.

From rectangle to Smith chart

side by side

Bend angle 0°

← → bend angle · Space play · L scale · Esc ends

Load

R Ω

jX Ω

Reference Z0 Ω

The same load as in the simulator. Or drag the point.

Values

What goes where

Left, the vertical line R = 0, becomes the outer circle. Only pure inductors and capacitors live there.

Right (R = ∞), top (+j∞) and bottom (−j∞) shrink into one single point on the right edge: open circuit.

Top stays inductive, bottom capacitive. The horizontal axis stays the axis of pure resistances.

Z0 (r = 1, x = 0) ends up in the centre. The vertical R lines become full circles, the horizontal X lines become arcs, all through the point ∞.

On the way every stage is a Smith chart itself: the outline is a circle, and at its centre sits a different reference that moves from very large to Z0. At first this circle is huge, so the view zooms out over the first degree until it is fully visible. In non-linear mode the compression is undone first, then the bending happens.

Why ∞ ends up on the edge

In the rectangular plane R runs from 0 to infinity and X from minus to plus infinity. No sheet is big enough for that: with a linear scale only a section ever fits, and an antenna with 2000 Ω simply falls off.

The non-linear scale compresses more and more towards the outside: 10, 25, 50, 100, 250 Ω … ∞. Now every impedance has a place, and Z0 sits in the centre. But infinity is not a place, it is a whole edge: right, top and bottom at once.

The bend angle turns it into a point. It bends the grid step by step until all lines running off to infinity meet on the right edge. At the end you get exactly Γ = (z − 1)/(z + 1): the reflection coefficient, and that is the Smith chart. The circles and arcs are not arbitrary, the mapping turns every straight line into a circle.

Every intermediate stage is already such a circle mapping (a Möbius transformation): infinity is a point on the right, the outline R = 0 a closed circle, and at its centre lies a reference that moves from ∞ to Z0. In other words: along the way you see Smith charts for smaller and smaller references. In non-linear mode the compression is undone first (straight lines stay straight), then the same bending follows.

Probe on the cable Open in /leitung

Drag the probe or move it with ← and →. Transmitter on the left, load on the right.

In the Smith chart

Cable and load

Load

Cable

Length

with loss

What the probe measures

Two waves run on the cable: one towards the load, one back. Where they are in phase the probe sees a high voltage (V max), where they oppose each other a low one (V min). There is always a quarter wave in between. For the current it is exactly the other way round.

Move the probe from the load towards the transmitter and the point turns clockwise on its SWR circle in the Smith chart; half a wavelength of cable is one full turn. On the right of the axis is V max (the impedance there is real and high, Z0 times SWR), on the left V min (Z0 divided by SWR). The ratio of V max to V min is the SWR.

With loss the circle becomes an inward spiral: the longer the cable, the less comes back from the load, and at the transmitter the SWR looks better than it is at the antenna.

How the waves run back and forth is shown by the transmission line in the time domain with the envelope of the standing wave.

Filters in the Smith chart

As the frequency sweeps, the input impedance of a filter traces a curve. In the passband it stays near the centre, in the stopband it runs out to the edge.

Γ across frequency

Sphere

Drag or use the arrow keys to rotate.

Curve

Source

Outlook: the sphere

With “Sphere” the chart is laid onto a sphere: the whole familiar Smith chart (|Γ| ≤ 1) is the lower half, the outer circle is the equator. On top lies what a passive load cannot have: |Γ| > 1, a negative resistance, as an amplifier or oscillator can show.

The two markers: 25 Ω is at the bottom (Γ = −1/3), −25 Ω at the top (Γ = −3).

Why three dimensions

The Smith chart shows Γ but does not tell you at which frequency the curve is where. Add frequency as a third axis and the curve becomes a helix in a cylinder. Seen from the side (Re Γ over f or Im Γ over f) you can read off when the curve passes through the centre. Seen from above it is the Smith chart again.

The shadows on the walls are the same curve seen from the side: Re Γ at the bottom, Im Γ at the back.

What the chart shows

Every impedance that can hang on your cable has exactly one place in the circle. The distance from the centre tells you how much power comes back, the angle tells you how the wave comes back. The chart is nothing more than the reflection coefficient Γ in a plane, with a grid of resistance and reactance circles laid over it so you can read the impedance directly.

All values in the grid are normalised: divided by the reference impedance Z0, usually 50 Ω. So the circle “1” means 50 Ω, “2” means 100 Ω, “0.5” means 25 Ω. Set Z0 to 75 Ω and the picture stays the same, only the ohms behind it change.

The centre

Here the load is exactly Z0. Nothing comes back, the SWR is 1. Every match has one goal: get the point to the centre.

The edge

On the outer circle everything comes back (|Γ| = 1). Only pure reactances live there, lossless inductors and capacitors, plus short and open circuit. The closer the point is to the edge, the higher the SWR.

Left and right

The horizontal line through the centre is the axis of pure resistances. Far left is the short circuit (0 Ω), far right the open circuit (infinity). 25 Ω lies left of the centre, 100 Ω right, both the same distance away, both SWR 2.

Top and bottom

Above the axis the load is inductive, it has a positive reactance (+jX) like an inductor. Below the axis it is capacitive (−jX) like a capacitor. An antenna that is too long sits at the top, one that is too short at the bottom.

The circles

The full circles that all pass through the same point on the right are circles of constant resistance. The important one is the circle r = 1 through the centre: if you are on it, you only need one more component in series. The arcs from the right edge going up and down are circles of constant reactance.

The SWR circle

All points at the same distance from the centre have the same SWR. A cable with Z0 changes the impedance but not the SWR: the point moves along this circle. That is why a different cable cannot improve the SWR at the rig, only a matching network can (a real line has loss and flatters the reading a little).

Admittance

Parallel components are easier to calculate with the admittance Y = 1/Z. The admittance grid is the same grid mirrored about the centre. A parallel component moves the point on a circle of this grid, a series component on a circle of the normal grid.

How to match

Every component moves the point along a fixed path:

Series inductor: clockwise on the resistance circle, upwards.
Series capacitor: anticlockwise on the resistance circle, downwards.
Parallel capacitor: clockwise on the conductance circle, downwards.
Parallel inductor: anticlockwise on the conductance circle, upwards.
Line: in a circle around the centre, clockwise. Half a wavelength is once all the way round, a quarter is half a turn.
Stub (parallel): acts like a parallel inductor or capacitor, depending on its length and whether its end is open or shorted.

An L-network takes two paths: first onto the circle r = 1 (or g = 1), then along it into the centre. If the load lies outside the circle r = 1, you start at the load with a parallel component; if it lies inside the circle g = 1, with a series one. In between either works. A series inductor and a parallel capacitor make a low-pass that also attenuates harmonics. That is often the better choice.

Converting with the chart

Every point in the chart is an impedance, and at a given frequency each reactance corresponds to exactly one component. So under the values each impedance appears twice: as a series circuit (resistor with inductor or capacitor in series) and as a parallel circuit. Both behave the same at this frequency, but not at others. +j20 Ω at 100 MHz is an inductor of 31.8 nH, −j20 Ω a capacitor of 79.6 pF. The line under the chart works this out for every point under the pointer.

You can also type values with a prefix: 22p, 4,7 µH, 1,5k or 3.3e-6. You enter a line in λ, degrees, m or mm, the electrical length is shown next to it.

Losses

Real inductors have a Q: their loss resistance is ωL / Q and sits in series with the inductor. For capacitors Q is usually so high that you can leave it out, so they are lossless by default. You set the Q for all components under Settings or per component in the table. With losses an arc no longer hits the centre exactly, and the SWR can deceive you: an inductor with Q 50 and 700 Ω has 14 Ω of loss, the SWR drops, but the power turns into heat instead of being radiated.

Cables have a loss per 100 m, and it grows roughly with the square root of frequency. With loss the point on a line no longer moves in a circle but in a spiral towards the centre. An “R in series” or “R parallel” models resistors directly, such as an attenuator or a termination.

Loss across frequency

Besides the SWR the sweep shows a second curve: how many dB are lost from the transmitter to the load. It contains both the reflection and the losses in components and cables. The transmitter has the source impedance Rg (Z0 by default). This lets you see a low-pass as a filter: 0 dB at the frequency, above it the curve falls. With a logarithmic frequency axis it becomes a Bode plot.

An example to check: 4.5 dB per 100 m at 10 MHz becomes 4.5 · √2.8 = 7.5 dB per 100 m at 28 MHz, so 30 m of cable loses 2.3 dB. Markers in the sweep (major and minor, under Settings) show in the chart which point belongs to which frequency.

The numbers on the right

Γ (reflection coefficient) is the ratio of reflected to forward voltage, with magnitude and angle. SWR = (1 + |Γ|) / (1 − |Γ|). The return loss tells you in dB how much weaker the returning wave is, the bigger the better. The mismatch loss is the power that does not reach the load because of the reflection, and reflected is the same share in percent: |Γ|² times 100.

At SWR 2, 11 % of the power comes back and the loss is 0.5 dB. Nobody at the other end will hear that. A high SWR is mainly a problem for the power amplifier, and on a long, lossy cable.

One click loads the example into the simulator. In full screen you page through the examples with ↑ and ↓, through the steps with ← and →, and switch to the other route with V.