- getting it right!

The Smith Chart was invented by Phillip H. Smith and first published in January 1939. It is a sophisticated tool designed primarily for solving transmission line problems, in which its inventor was very much involved. With all the advances in computer technology, the Smith Chart still has great relevance for radio engineers today. It is still the basic tool for determining transmission line impedances. At first glance its complexity may be somewhat of a challenge and somewhat off-putting, but if you take the trouble to find out how it works, you will find that it can be a very useful tool indeed – and one that you can use remotely - after an antenna has been installed...

As well as determining feed point impedance, the Smith Chart can be used to calculate admittances (impedance measures resistance to current flow in ohms whereas admittance measures how easily current flows in siemens), and to obtain SWR (standing wave ratio) readings and phase angles. It can be used in the design and analysis of matching networks (L and Pi networks, stub matching series, etc.), to determine how difficult matching is likely to be at different frequencies, and determine inductance and capacitance. It can also be used to plot noise figures and stability regions in microwave circuits.

For narrow band applications (typically up to about 5-10% bandwidths) plotting one frequency is usually sufficient. With wider bandwidths, it becomes necessary to use multiple plots. Multiple plot Smith Charts are an important tool in fine tuning broadband antennas and matching networks.

While most transmission lines now are 50 ohms, any characteristic impedance can be used. This is important at HF and below where feed line impedances vary greatly over broadband frequency ranges if compatibility between antenna, feedline and coupler or matching network is to be assured. When combining sensitive equipment from different manufacturers for lower frequency communications, Smith Charts often need to be consulted.

Today we rely so much on computers for complex modelling and calculations that we tend to regard older graphical/slide rule type solutions as not being accurate enough. With the Smith Chart, however, this is not so. The wavelength scale on the circumference of the chart has divisions of 1/500 of a wavelength, while the reflection coefficient scale (power reflection) can be read to 0.02. Modern day network analysers use Smith Chart graphic representations. So, even if you never need to chart one yourself, it is highly likely that you will need to understand how they work.

In techspeak the Smith Chart is a polar plot of the complex reflection coefficient symbolized by the Greek letter gamma, which examines the load where the impedance must be matched. By reflection coefficient we mean the ratio between the reflected voltage and the incident voltage. The reflection coefficient is never larger than 1 (all power is reflected back) or lower than 0 (all power is transferred – the line is perfectly matched). The greater the mismatch between the input impedance and the load impedance, the greater the amount of reflected signal from the load.

If the impedance at the input of any transmission line and the length of the line are known (or measured), you can determine the impedance at the feed point, or if the reflection coefficient is known, you can determine the characteristic impedance at the load.

Complex as this may sound, the Smith Chart is really just a specialised graph with curved coordinating lines, in the form of two sets of circles. The outer circle forms the Reactance axis. The Resistance Axis is its diameter. Values along the Resistance Axis range from 0 ( left) to infinity (right) with the Prime Centre in the middle. In fact 0, zero resistance, zero reactance is short circuit (0 +j0). Infinity, infinite resistance, is open circuit.

Touching the outer circle at infinity and centred on the Resistance Axis are a series of concentric circles representing Resistance. Where it cuts the axis, each circle is given a value. All of these values relate to the assigned value of the Prime Centre.

The use of coefficients or ratios, makes it possible to chart any impedance values for any type of uniform transmission line. Impedances (or admittances) go through a process of normalisation before being plotted, based on the value assigned to the Prime Centre. All you have to do to read a value is multiply the value on the Resistance Axis by the Prime Centre value. So, if the Prime Centre, 1.0, stands for 100 ohms, then the circle at 2.0 will have a value of 200W and the circle at 0.5 a value of 50 ohms. Similarly to plot a value you will need to divide by the Prime Centre value.

If you are just working with 50 ohm coaxial cable, it is possible to obtain charts with a 50 ohm prime centre value to make it easier.

The Reactance series of circles (actually only segments are visible) centre on infinity and fan out to the Reactance Axis. Values relating to the Prime Centre are shown where the segments cut the axis. Values above the resistance axis are positive (inductive), while those below are negative (capacitive).

When the resistance and reactance circles are superimposed, you get the Smith Chart. Using this chart you can plot complex impedances (R + jX). For clarity, the diagrams shown above only give a few values of resistance and reactance. Obviously normal Smith Charts are more detailed.

Suppose you need to plot an impedance of 50W resistance and 100W reactance (Z=50 +j100).

To maintain consistency of plotted positions, the prime centre value should represent the characteristic impedance of the line (Zo). To normalise or translate the impedance into the value to be plotted, the resistance and reactance are both divided by the prime centre value. So, if the characteristic impedance is 100W, you would get 0.5 resistance (50/100) and +1.0 reactance (100/100), which can be plotted on the chart where the 0.5 resistance circle intersects the +1.0 reactance circle.

Of course, if you are using a 50W line and a prime centre value of 50, all you need to do is just read the values off the chart. In this case, the prime centre is 50 +j0W, a pure resistance equal to the characteristic impedance of the line. Or in other words it is a perfect match with no reflected power, 1.0:1 SWR.

A Wavelength Scale, provided around the circumference of the Reactance Axis, represents the length of the transmission line over a half wavelength as it progresses from the generator (power source) to the load and vice versa. The scale starts at 0 on the centre left of the chart. Reading anticlockwise it runs from the generator/input end of the line to the load, while reading clockwise it runs from the load to the input. Because the impedances repeat every half wavelength, any length of line can be accommodated simply by subtracting the number of whole half wavelengths from a given line.

Inside the wavelength scales (see over) is the angle of Reflection Coefficient scale, which indicates both the magnitude and phase angle. The phase angle at the load shows the amount by which the reflected voltage wave leads the incident wave at the load. If the value is on the bottom half or capacitive reactance half of the chart, the value is negative, meaning that it actually lags rather than leads the incident wave.

At the bottom of the chart are further scales, which measure the SWR as a ratio or in dB, the amount of transmitted and reflected voltage in dB or as a decimal of power, and current (I) and voltage (E) phase shifts.

Scales, may be expressed slightly differently, depending on the version of chart that is used. The chart shown here is very common. In later examples, these scales vary slightly. Additional scales may also be provided both around the circumference and at the bottom of the chart.

Typical Smith Chart
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Note: The Wavelength and Reflection Coefficient Scales on the circumference and Standing Wave Transmit Loss and Reflection Scales at base.
The drawn circle represents a line with a constant 3.0 SWR.

In order to solve problems, a third set of circles, representing areas with the same SWR values, are added to the chart. These circles are centred on the Prime Centre and reach out to the reactance axis. The use of SWR circles makes it easier to see at a glance where values fall within or outside a required SWR limit. The SWR value is read from the point where the resistance circle crosses the resistance axis (in this chart to the right of Prime Centre). Prime Centre is the 1.0 VSWR circle with radius of zero.

If a circle is drawn at 3.0 and this represents a line with a constant 3.0 SWR (disregarding line losses for the time being), the impedances of this line can be determined by reading all points around the circle corresponding to the length of the line involved.

By drawing tangents to the SWR circle (as shown on back), it is possible to read the measurements of the SWR circle on the linear bottom scales. In the following example the SWR is shown as 3.0 at A on the Standing Wave Voltage Ratio/dB Scale which equates to 9.5 dB, the ratio of voltage maximum to voltage minimum.

Drawing SWR circles
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Using a Smith Chart, it is possible to solve complex mathematical problems without having to do the maths. Say you have a 50 ohm line with an electrical length of 0.3l and a terminating impedance of 25 ohms resistance 25 ohms inductive reactance (Z=25 + j25) – as the line is not terminated in 50 ohms there will be standing waves on the line. (See example on back) You can assign the value of Prime Centre to 50 ohms = 1.0 and proceed as follows:

1. Normalise the load impedance : 25/50 + 25/50 = 0.5 + j0.5 – and plot on the chart.
2. Draw an SWR circle passing through this point from Prime Centre and transfer the radius of the circle to the external scales at bottom.
3. You can now read off the SWR (2.6) and dB (8.4) values. On this chart the scale is at bottom left.
4. Draw a line from Prime Centre through the wavelength scales outside the circumference of the reactance circle and read the value off the Toward Generator scale (0.088l) as you are now plotting the input.
5. To obtain the normalised input impedance, you need to find the point on the SWR circle that is 0.3 wavelength towards the generator from the plotted load impedance. To do this add the line length, 0.3 wavelength, to the starting point, 0.088 wavelength to get 0.388 (0.3 + 0.088 = 0.388). If the result is more than 0.5 (remember the wavelength scales only cover one half wavelength) , it becomes necessary to subtract multiples of a half wavelength to achieve a value of 0.5 or less. In this case it is not necessary.
6. Locate 0.388 on the Toward Generator Scale and draw a second radial line from here to Prime Centre. Where this intersects with the SWR circle is the normalised line input impedance: 0.6 –j0.66.
7. Now convert the result by multiplying by 50 to get 30 ohm resistance -33 ohm capacitive reactance: (0.6x50 – 0.66x50).

Reading from the Angle of Reflection Coefficient Scale on the circumference, the phase angle is 116.6 degrees.

Finding the Input Impedance from the Load
Note the variations in scales compared to the chart above.
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The electrical length of a line depends on the physical length, the frequency being used, and the velocity of propagation in the line. In the above example the electrical line was given. If this is not known, it can be found by measuring input impedance measurements with short or open line terminations using an impedance measurement bridge that can produce reliable readings at high SWR values. Alternatively, it can be calculated by multiplying the line length in feet by the frequency in MHz and dividing by 984 multiplied by the velocity of propagation in the line eg:

At the date of writing, a transmission line calculator java applet is available at http://wwwfermi.la.asu.edu/w9cf/tran/ .

Working with antenna impedances is essentially the same. You need to determine the electrical length of the feed line and measure the impedance value at the input end. The antenna, whether it is to be used for transmission or reception, is the terminating or load impedance. The input (generator) end is at the measuring device. The measured input impedance is plotted on the chart. Using the wavelength scale on the circumference, the line length is plotted to find the actual impedance at the antenna (load).

So far line losses have not been taken into consideration. Frequently this does not make an appreciable difference. However, where high loss lines are involved, especially at VHF frequencies and above, such losses can have a significant impact. In accounting for line losses we take into consideration that SWR decreases from the load to the input. Cable manufacturers can provide indicative lines losses for cables at different frequencies.

Technically this gradual decrease should be represented by a spiral (falling inward and clockwise from the load to the generator/input), that takes into account the inherent attenuation in the line. However, two separate SWR circles, one for the input and one for the load, are usually drawn using the Transmission Loss or Attenuation Scale in dB at the bottom of the chart.

Say you have a have a 0.282l long 50W line with 1dB loss and the input impedance is 60 +j35 W you can account for the line loss as follows:

1. Plot the measured input impedance.
2. Draw the SWR circle as before.
3. Transfer the radius to the external scale on either side. Note the SWR and the Attenuation dB loss.
4. Move the radius towards the load (you are plotting the load) by 1dB mark (this is our given 1dB line loss).
5. Transfer the new value to the main chart by drawing a parallel line and draw the second SWR circle. In this case it will be outside the input SWR circle. (If you had measured the load and were plotting the input, it would be inside.)
6. Mark the second plot where the radius dissects the new SWR circle.

See example below

Allowing for line losses.
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Where SWR values are low, an attenuation of 1-2 dB does not have an appreciable affect, however the effect will be considerable where SWR values are high. Immediately below the Transmission loss/Attenuation Scale is the Loss Coefficient scale, which provides the factor by which the matched line loss in dB should be multiplied to account for the increased losses in the line when standing waves are present.

As mentioned earlier, network analysers are commonly in use today in the design and testing of communications equipment and these can provide information in the form a Smith Chart which can also be used for assessing compatibility between antennas and couplers, for example. Below are typical Smith Chart plots for a broadband antennas, as outputted by a network analyser.

When you are looking at Smith Charts (and VSWR plots), it can be important to know if it is measured from the antenna, at the end of a specific length of transmission line, or via another inline unit, eg. an EMP lightning protection device when these form part of the transmission line. Measurements at the antenna are valuable for calibration purposes. For operational purposes it is usually better to assess performance at the end of the cable length you will be using. Sometimes the plots at various frequencies are joined up, sometimes they are not.