Smith Chart

The Smith chart provides a graphical procedure for the solution of transmission line problems, such as the determination of the impedance or impedances and their placement position along a mismatched transmission line, in order to match that line. Developed by P.H.Smith , the chart is a plot of normalized impedance , and/or normalized admittance , against the polar coordinate plot of magnitude and angle of the voltage reflection coefficient r . Mr. Smith developed the present form of the chart through an interative process in his attempts to simplify the tedious work of matching open-wire telephone and telegraph lines. Prior to its development, transmission line impedance analysis was carried out using the rigorous hyperbolic transmission line equations, presented in chapter 1 of the attached book on Smith Charts, and experimental work focussed on a trial and error approach perhaps supplemented with transmission line measurements of standing wave ratios. Thus, the ability to represent and manipulate an impedance vector on a single graphical chart was a significant step forward at that time.

The pair of coordinates used for plotting the complex values of impedance and admittance accommodates all possible values, and permit impedance to be converted to its equivalent admittance, and vice-versa. The advantages of using a Smith chart for the solution of transmission line and waveguide problems is the ease in arriving at a graphical solution over the alternative computational methods and the intuitive understanding of the effect that individual variables have on the final solution.

The Smith chart consists of a real axis with values that vary from zero to infinity, with unity at the centre. And a series of circles centred on the real axis, that represent the real parts of the normalized impedance. It also comprises a series of arcs of circles that start from the infinity point on the real axis, representing the imaginary parts of the normalized impedance. In addition to the circles and arcs, the edge of the chart is marked with scales of the angle of the reflection coefficient (in degrees), and the distance (in wavelengths) along the line from either the source or the generator. Movement around the edge of the chart in the clockwise direction corresponds to movement along the transmission line towards the generator. Movement around the edge of the chart in the anti-clockwise direction corresponds to movement along the transmission line towards the load. One complete circle around the chart represents a movement of half a wavelength along the transmission line.

In any transmission system, a source sends energy to a load, such as an antenna by means of a transmission line. To ensure that the signal sent from the source arrives with the minimum disturbance, ideally, we design the transmission network such that the characteristic impedance of the source, the transmission line and the load are all identical. In the real world this match is not ideal, and by careful design is not far from being perfect under normal conditions.

For example, if an antenna (the load) is to be used over a broad range of frequencies, its characteristic impedance will vary with over this range due to its physical design. The wider the frequency-band, the larger the variation of its characteristic impedance.

When the transmission line impedance is not the same as the load, that is, it does not match that of the load, part of the transmitted waveform is reflected back towards the source. The reflected wave, which varies in phase and magnitude, adds to the incident (transmitted or forward) wave and the result is a standing wave. This standing wave can cause considerable damage to the output of a transmitter if the load becomes an open circuit, because its voltage can be up to twice the voltage produced by the transmitter. This can cause output circuits to be damaged. The Voltage Standing Wave Ratio (VSWR) is a measure of how much of the signal is reflected back down the line from the load is. The VSWR or SWR (Standing Wave Ratio) is the ratio between the maximum and minimum peak amplitudes of the total standing wave waveform. When a mismatch occurs, this VSWR will be greater than one. When an open circuit or short circuit occurs on the line, the VSWR will go to infinity.

If there is no reflected wave, that is, if the impedance match is perfect, the reflection coefficient, that is the ratio of the reflected to forward signal amplitude, will be zero and due to its relationship with VSWR a VSWR of unity will occur. A VSWR = 1 indicates maximum power transfer to the load, whereas a VSWR ® ¥ indicates zero power transfer to the load and maximum power reflected back to the source.

The applet

If we know the reflection coefficient, we can also determine the normalized impedance of the load by using a Smith Chart. The Smith Chart has circles of constant resistance and circles of constant reactance. The relationship between reflection coefficient and normalized impedance is shown in the applet.

The Smith Chart can be used to translate the reflection coefficient into impedance and VSWR. Given a reflection coefficient (which has amplitude and phase angle), place the reflection coefficient at the desired polar value on the Smith Chart (using the mouse to point and drag to the position). The applet will display a yellow circle with a radius equal to the reflection coefficient magnitude (constant VSWR circle). If you move the reflection coefficient anywhere on this circle, you can see from the waveform at the right and the screen printout of the VSWR, that the VSWR is the same, only its phase changes. This may be difficult to do as the VSWR circle and the circles of constant resistance and reactance follow the cross-hair placement to different positions dictated by the mouse. To fix a VSWR circle, so that you can follow it, activate the "lock1" checkbox. Phase values are shown for movement of the reflection coefficient r , around the chart in this applet by observing the screen printout of r (q ° ) that represents r Ð q ° , on the top right hand side of the applet canvas. In addition to the yellow VSWR circle, a white circle that is the circle of constant resistance and the blue circle (arc) that is the circle of constant reactance will intersect at the place where the reflection coefficient has been placed. At this point, the value of normalized resistance and reactance is shown printed out on the top right hand side of the canvas.

If a value of normalized impedance (or admittance) is known, then to make it easier to place the cross-hair on the Smith chart, the text fields can be used. Enter the value of the resistance and the reactance (or if these values are conductance and susceptance) into the appropriate text fields. Then press "enter". A blue cross-hair will indicate the position on the Smith chart. In Figure 1 below, the value of normalized impedance  is shown entered into the text fields and the blue cross-hair labelled Z(Y) shown plotted on the chart.

Figure 1 Setting up impedance or admittance using the text fields rather than using the mouse

If the mouse pointer is placed exactly dead centre of the cross-hair, then the screen print out will indicate the position that the mouse was placed. It may not agree to three decimal places with your text field requirement due to the limitation of the pixel spacing on the screen, but it will be very close to the text field value. In this case the mouse was placed at . Where the mouse pointer was placed, a VSWR circle and R and X circle will be drawn through that point. This is shown in Figure 2 below.

Figure 2 Setting up the VSWR and R and X circles for the chosen impedance or admittance point.

By selecting with the mouse the opposite point on the VSWR circle along the straight line passing through your required impedance (admittance) point and the centre of the chart, the admittance (or impedance) can be determined. The screen printout will show the value in white for conductance (resistance) and blue for susceptance (reactance). Figure 3 shows this. The value of admittance (or impedance) is shown in the screen printout as . If the load was an impedance, then this construct allows the admittance to be determined. Vice-versa. For a parallel matching stub problem, the impedance must be first converted to an admittance before the stub design can continue. Throughout this and the following discussion, impedance and admittance in reality means normalized impedance and normalized admittance. This is because the Smith chart can only be used for normalized impedance or admittance values. The normalized value is the value of the actual resistance divided by the characteristic impedance of the line, it is therefore a dimensionless quantity.

Figure 3 The conversion of normalized impedance or admittance into normalized admittance or impedance

Amongst other designs, this applet allows the design of a single parallel short-circuit stub matching network as described below.

Design of a single parallel short-circuit stub matching network.

Complete the steps above to obtain a load admittance from the load impedance that is required to be matched.

By using the both the lock1 and lock2 checkboxes. The procedure for designing a single-stub matching network is as follows;

• With the mouse, select the "initialize save" check box. This will allow the data from your chosen admittance (yet to be fixed) to be stored into memory.
• Select "lock1". This will place the load admittance and the initial position of the angle in wavelengths on the outside of the chart, into memory. This is shown in Figure 4.

Figure 4 Saving the admittance data and angular position using "lock1"

• Move the cross-hair (r ) along the yellow VSWR circle, until the value of R and X (upper right hand side screen printout) indicates that R is equal or close to 1.0. If you move inside or outside of the fixed VSWR another VSWR circle will appear. This circle must adjusted to be the same as the fixed VSWR circle. When R is equal or close to 1.00 a reactance value (the blue coloured value of X= 1.375 in Figure 4) is given. This value of X is the reactance that must be cancelled by the stub. The distance in wavelengths from the load is given as "Difference in angles (l - l ₁)". This will be the distance that the short-circuit stub is to be placed from the load. Which for the example given, distance from the load = 0.013l
• Select "lock2". This will activate the second part of the circuit allowing the distance from the short circuit admittance point (the 0.25l or the ¥ point on the chart) to the point where the cross-hair will be placed (in the next step). Figure 5 shows a screen dump of this part of the solution.

• On moving the cross-hair away from the last position, the upper right-hand side of the canvas will give a screen print out indicating "Want to match out a reactance of xxx,". In this example, Figure 5 shows this to be -1.375. This is the value of reactance that the stub must be to match the load. Move the cross hair to this position by watching the blue value of reactance change until it reaches a value of "xxx". In this case a value of -1.378 was obtained. Notice that as only -1.378 is to be found, and no resistance, we must move along the R = 0 circle, or the boundary of the Smith chart to obtain this value. Once reached, the screen printout "Difference in angles (l - 0.25l ) = yyy" will indicate the length of the short circuit stub. Which in this case was found to be 0.099l .
• Thus the top right hand side screen print out will indicate the distance from the load and the length of the short-circuit stub required to match out the load impedance to give the source a characteristic impedance to look into. In the example given, to match a load impedance 

By moving the cross-hair to the extreme limits of the horizontal line through the centre of the chart (X=0) the coaxial diagram below the standing wave diagram will indicate open circuit or short circuit load. The standing wave diagram also indicates a maximum voltage (Open-circuit) or a minimum voltage (Short-circuit) at these two extremes.

Associated with this applet is a book on Smith Charts written by myself that can be downloaded in part or in total.

Tony Townsend, tonyart@ieee.org

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