Let’s consider the following transmission line scenario:
- Lossless;
- Characteristic Impedance Zo=1+j0Ω; and
- load impedance other than 1+j0Ω, and such that Vf=1∠0 and Vr=0.447∠-63.4° at this point.
The ratio Vr/Vf is known as the reflection coefficient, Γ. (It is also synonymous with S parameters S11, S22… Snn at the respective network ports.)
Above is a phasor diagram of the forward and reflected voltages at the load.
There are times where the normalised voltage at a point is of interest, ie the resultant Vf+Vr or 1+Γ.
Above is the construction of 1+Γ.
Above is the construction on the Smith Chart, the length of 1+Γ is scaled from the “Radially scaled parameters” scales, and the angle read by projecting the phasor to the “Angle of transmission coefficient in degrees” scale. V=1+Γ=1.26∠-18.4°.
A basic transform
Now let’s consider what the forward and reflected voltages looking into 90° of this transmission line with this load at the far end.
Again taking the forward voltage as the reference (Vf=1∠0), we can determine that the reflected voltage will be of the same magnitude (by virtue of the lossless line property), but it is delayed 90° for the path to the load, and another 90° for the return path to the source.
Above is a phasor diagram of the forward and reflected voltages at the source (using the forward voltage at this point, the source, as the reference). The red arrow shows the rotation of the Vr phasor to the new Vr’ phasor (both relative to their own Vf). We are always plotting reflected voltage relative to forward voltage at the same point.
So. these plots are of forward and reflected voltage phasors, linearly scaled in magnitude and at some angle. Because the reflected phasor adds tail to head to the forward phasor, you might think of the magnitude of the reflected voltage as radially scaled from the ‘pivot’ point or centre.
Mapping Γ to impedance and admittance
We can map Γ to impedance and vice versa.
- \(Z_l=Z_0\frac{1+\Gamma}{1-\Gamma}\)
- \(\Gamma=\frac{Z_l-Z_0}{Z_l+Z_0}\)
Note that Γ and Zo are both complex quantities, and they can each be expressed in polar form R∠θ where θ is the phase of each… but the phase of one is not the same as the phase of the other… something that seems to confuse lots of hams.
Admittance is simply the inverse of impedance, ie \(Y=\frac1{Z}\).
Whilst we might often measure Γ fairly directly, it would be convenient to map Γ to impedance and admittance using graph scales.
Phillip Smith thought so, and though it took him many years to evolve the ‘Smith Chart’ (1931-66), he eventually produced the now familiar circle chart that was used as a graphical computer for a long time, and now more often for visualisation of transmission line problems rather than the computational tool.
The Smith Chart is widely used as a display presentation on network measurement.
Impedance
Lets overlay the first phasor diagram with a set of Resistance and Reactance scales that correspond the the values of radially scaled Γ.
There are also four scales around the perimeter of the circular chart, study them carefully.
Note that our example used Zo=1 and the chart above uses Zref=1 (the impedance at the centre of the chart).
In this graphic, the phasors are plotted for explanatory purposes, but it is usual to simply plot an impedance as a point with an X at its location.
It would not be very practical to seek stationery with Zo to suit arbitrary applications as they arise, so impedances are normalised for plotting on the standard Smith Chart. The normalised impedance \(Z’=\frac{Z}{Z_0}\) and so real world impedance values are normalised to plot their points, and solution values scaled from the chart are denormalised to obtain the real world impedances.
Note that computer software and instrument displays often render denormalised or real world values for convenience.
Above is an example from Simsmith of a computer generated graphic of this transform, and in this case, the path is shown to further explain what is happening. This display uses Zo=1 because that is the problem context. The chart is rendered wrt Zo, which if it were say 50Ω, would result in denormalised display scales
Admittance
As noted, we can map Γ to impedance.
- \(Z_l=Z_0\frac{1+\Gamma}{1-\Gamma}\)
If we substitute \(Z=\frac1Y\) we can develop the admittance map relative to the impedance map.
\( Z_l=Z_0\frac{1+\Gamma}{1-\Gamma}\\\frac1{Y_l}=\frac1{Y_0}\frac{1+\Gamma}{1-\Gamma}\\
Y_l=Y_0\frac{1-\Gamma}{1+\Gamma}\\
Y_l=Y_0\frac{1+\Gamma}{1-\Gamma}\cdot 1 \angle 180°\\\)
So the to transform the impedance map \(\frac{1+\Gamma}{1-\Gamma}\) to the admittance map \(\frac{1-\Gamma}{1+\Gamma}\), we rotate it 180°. Note the peripheral scales are not rotated, just the real and imaginary scales.
Above is the admittance map of the Smith chart, note the outer scales are the same as in the impedance mapped chart. Note in the upper half of the chart, B<0, ie negative.
The foregoing explains why rotating a point 180° about the chart centre inverts Z or Y.
A simple technique for using the basic Z Smith chart for admittance was to rotate the chart so infinite conductance (short circuit) was at the left end etc. Note that B<0 (ie negative) in the upper half of the chart.
Combined Z Y chart
Smith did create a version of his chart with both Z an Y scales.
Above, Smith’s combined chart also showing here the radial scales. This combined chart can be more convenient when working with mixed series and shunt elements.
The normalised admittance \(Y’=\frac{Y}{Y_0}=Y Z_0\) and so real world admittance values are normalised to plot their points, and solution values scaled from the chart are denormalised to obtain the real world admittances.
Using forward and reflected current
The previous discussion was based on forward and reflected waves expressed as equivalent voltages. As you might expect, it is also possible to use currents. The relationships are:
- \(V_r=\Gamma V_f\)
- \(I_r=- \Gamma I_f\)
The last can be rewritten \(\Gamma=\frac{I_r}{-I_f}\), so we can construct the phasor Γ by adding If and Ir head to tail starting at the right side of the chart (Γ=1 point, when Γ=1 Il=If-Ir=0).
Having constructed the phasor Γ, we can use the impedance and admittance overlays already discussed.
So the same chart can be used, plot Γ the same way for voltage or current, and the rest of the chart applies.
There are times where the normalised current at a point is of interest, ie the resultant If+Ir or 1-Γ.
Above is the construction of 1-Γ. To read the angle of 1-Γ you could use a parallel rule to project it from the chart centre and read the angle from the “Angle of reflection coefficient in degrees” scale.
Reconciling voltage and current
Above, the voltage and current phasors combined and calculation of Z reconciles with the original statement of the scenario.
The Smith Chart really is a chart of Γ with some convenient mapping scales overlayed, with some interesting transforms conveniently available by following the constant VSWR, constant R, constant X, constant G, and constant B arcs.
Conclusions
This article is NOT about “doing Smith Charts”. “Doing Smith Charts” is the traditional way of teaching Smith Charts, teaching the mechanics of solving some simple example transmission line problems (eg a single stub tuner)… without needing or imparting an understanding of the Smith Chart.
If you didn’t witness this approach in college, Youtube abounds with contemporary examples which seem the product of “I didn’t understand <subject> at all, so I thought I would do a video on it to learn about it.” The Γ phase missionaries are an example, belief over science, and popularity determines fact.
This article has explained the essence of the Smith Chart, a foundational concept that must be understood to truly understand the Smith Chart at large, and from which more elaborate problem solving is easily developed on classic transmission line theory.