Stub Matching Using the SMITH Chart
By David J. Jefferies

D.Jefferies email

Introduction:
ith a little "sleight of hand" using lengths of feeder or transmission line, one can construct a matching circuit between most antennas and most transmitters, such that the VSWR seen by the transmitter is close to unity even though the antenna itself may be grossly mismatched. This article attempts to explain how this is done.

The voltages and/or currents in the matching section may be much higher than on the feed; the waves rattle back and forth between the antenna and the stub (the geometry is a kind of resonator); the feed excites the resonance and the resonance excites the antenna.

A stub is a "side road" off the feed, which may be considered to be the "main road". The stub is a cul-de-sac, or dead end, or no-through-road. The far end from the junction is a short circuit, or an open circuit, or in general any pure reactance having no loss.

An Antenna Impedance Plot
To consider stub matching it helps to have a practical example. Here, we are going to think about an antenna which is being used away from its design frequency. In a real installation, it may be easier to add a stub match to the feed structure rather than altering the antenna geometry. The antenna may be mounted on a mast and relatively inaccessible.

Figure 1 shows a plot of the impedance of a 1 metre long dipole antenna, made from 1 cm diameter aluminum tubing, from 120MHz to 160MHz. The plot is made on the SMITH chart. We have normalised the actual impedance R + jX (the driving point impedance of the antenna at its centre) to the nominal characteristic impedance Zo = 75 ohms of a coaxial cable feed. Having simulated the antenna in NEC2 (21 segments, fed in the middle) to obtain R and X at a range of frequencies, we plot the values r and x on the SMITH chart, where r = R/Zo and x = X/Zo.

For a 1 metre long dipole, the frequency at which it is exactly a half-wavelength long is 150MHz. We see, as expected, that the closest approach to the centre of the SMITH chart (near where the red curve crosses the x=0 line) is a bit below this frequency, at 140 MHz. To get an idea of the intrinsic bandwidth of the antenna, we have plotted three circles on the SMITH chart. The orange (innermost) circle represents VSWR = 1.33, and at this VSWR only 2% of the power incident on the antenna (from the feed) is reflected back along the feed to the transmitter; the remaining 98% is radiated. For navigational understanding on the SMITH chart, the distance out from the centre of the chart to the orange circle is 0.141 of the distance to the perimeter. This is because the square root of 2% = 0.02 is 0.141, which is the modulus or size of the reflection coefficient gamma. We estimate the bandwidth for which there is more than 98% transmitted as 137-144 MHz. The green (middle) circle represents VSWR = 1.93, and at this VSWR 10% of the incident power is reflected and 90% transmitted. From the plot we estimate the bandwidth for which there is more than 90% transmitted power as 132-151 MHz. The blue (outer) circle represents VSWR = about 6, at which there is 50% reflected power, and 50% transmitted power. At this value of VSWR the antenna is grossly mismatched.

Figure 1
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Why match? Matching the impedance of an antenna to the impedance of a transmission line has two principal advantages. First, all the incident power is delivered to the antenna. Second, the transmitter is usually designed to work into an impedance close to common transmission line impedances. If it does so it is better behaved, the load impedance has no reactive part which can pull the transmitter frequency, and the VSWR on the line is unity or close to unity so the line length is immaterial and the line connecting the generator to the load is non-resonant. There are also fewer resistive losses on the feed, if the VSWR is closer to unity.

Single Stub Matching.
If you look at the SMITH chart you will find a circle of constant real normalised impedance r=1 which goes through the open circuit point and the centre of the chart. In our example in figure 2 this circle is drawn in red. If you plot any arbitrary normalised impedance on the SMITH chart, and follow round clockwise at constant radius towards the generator (along the green line in the example of figure 2), you must cross the r=1 circle somewhere. This transformation at constant radius represents motion along the transmission line towards the generator. One complete circuit of the SMITH chart represents a travel of one half wavelength towards the generator. At this intersection point your generalised arbitrary load impedance r + jx has transformed to 1 + jx', so at least the real part of the impedance equals the characteristic impedance of the line. Note x' is different from x in general. For each transformation around the SMITH chart, representing travel one half wavelength towards the transmitter, there are two intersections with the r=1 circle. Stubs may be placed at either of these points.

Figure 2
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In our example we have plotted the impedance of our 1 metre antenna at 120 MHz, which is 44.8 ohms - j 107 ohms, as the normalised impedance 0.597 - j 1.43 with respect to the 75 ohm coaxial line. We shall determine the position and length of a series stub which will match this antenna to the transmission line. For the geometry of the stub, please look at figure 4.

At the intersection point (red and green circles) you cut the line and add a pure reactance -jx'. The total impedance looking into the sum of the line impedance and -jx' is therefore 1 + jx' -jx' = 1 and the line is matched.

Why stubs? Stubs are shorted or open circuit lengths of transmission line which produce a pure reactance at the attachment point. Any value of reactance can be made, as the stub length is varied from zero to half a wavelength.

Again, look at the SMITH chart and find the outer circle where the modulus of the reflection coefficient is unity. On this circle are the SHORT and OPEN points, and all values of positive (top half of the SMITH chart) and negative (bottom half of the SMITH chart) reactance. The resistance is zero everywhere. It has to be zero, as a lossless transmission line with load infinity ohms (open) or zero ohms (short) has no mechanism for absorbing power. To generate a specified reactance, start at a short circuit (or maybe an open circuit) and follow the rim of the SMITH chart clockwise around towards the generator until the desired reactance is obtained. Cut the stub this number of wavelengths long.

In our example, the SMITH chart construction to find the stub length is shown in figure 3. From the blue arc in figure 2 we see the reactance at the r=1 intersection point is +j 1.86, so to cancel this out we must add a series stub having reactance -j 1.86. In figure 3 we plot the blue arc -j 1.86 and, starting from the short circuit (r = x = 0) we follow the green line around a distance of 0.328 wavelengths clockwise towards the generator, to generate this value of reactance. If we had started from an open circuit we would only travel a distance (0.328 - 0.250) = 0.078 wavelengths to generate this reactance. This open circuit stub is represented by the red arc on figure 3.

Figure 3
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The practical details of the series stub match are shown in figures 4 and 5. Figure 5 displays the physical lengths in centimetres, assuming a wave velocity on the coax (which you need to know to do this calculation) of 2x10^8 metres per second. This data is supplied by the cable manufacturer. The wave velocity and the frequency (120 MHz) allows us to calculate the wavelength in metres, and thus we can translate the "electrical lengths" from the SMITH chart into physical lengths of line.

 Figure 4 Figure 5

Notice that as the coaxial cable is an unbalanced transmission line, and the dipole is a balanced antenna, we need a balun at the connection to the dipole to stop return current from flowing down the outside of the coaxial braid and contributing to the radiation and therefore affecting the impedance.

Notice also that series matches in coaxial cable are difficult to make, for physical reasons. Series matches are much more appropriate to balanced feeder (possibly 300 ohm ribbon cable); our example could have been a 300 ohm balanced twin feeder supplying a 300 ohm (nominal) folded dipole.

It is important to keep the total stub length as short as possible, if wider bandwidths are required. Every time you add a half wavelength to the stub length the reactance of the stub comes back to the same value. It is good design practice to make stubs in the range 0 to 0.5 wavelengths long. However, this may require an impractically short stub, so then one can make the stub just a little over 0.5 wavelengths long.

Short or open stubs? If one is allowed to use either short or open stubs at will, one can always keep the total stub length in the range 0-0.25 wavelengths. A length of transmission line of 0.25 wavelengths takes us half way round the SMITH chart and transforms an open into a short, or vice versa. On microstrip, it is usually easier to leave stubs open circuit, for constructional reasons. On coax line or parallel wire line, a short circuit stub has less radiation from the ends: it is difficult to make a perfect non-radiating open circuit as there are always some end effects on the line. In a transmitting environment, there is a voltage maximum at an open circuit, which can spark and lead to electrical safety problems.

Series or Shunt Stubs?
The stub illustrated above is called a "series stub". In parallel wire line, it is connected in series with one of the wires of the feed, as shown in figures 4 and 5. More usual (in coax, at least) is to use a "shunt stub", which is connected across the two wires of the feed, as shown in figure 6. Since admittances in parallel add, whereas impedances in series add, we represent the transmission line impedance as an admittance y = g + js at the point of attachment, and we look for the g=1 circle. As explained last week, this is 180 degrees (a quarter-wavelength) around the SMITH chart from the r=1 impedance circle. Therefore, the points of attachment of shunt stubs are a quarter wavelength along the transmission line, either side of the points of attachment of a series stub. The stub line length needs to be different also as we need to compensate the parallel js with an equal and opposite shunt susceptance -js, and the value of susceptance s is different from the value of reactance x in the series-match example. We shall return to discuss shunt stub matches in more detail in next month's article.

Generalised Design Procedure
You are told, or find out, the load impedance ZL and the transmission line characteristic impedance Zo. Calculate the normalised impedance z=(ZL/Zo). Plot it on the SMITH chart. You are told the frequency and the velocity factor of the line. Calculate the wavelength in metres. (or cm). Follow the circle of constant radius on the SMITH chart towards the generator until the locus crosses the r=1 circle. Measure the number of wavelengths along the perimeter of the SMITH chart between the z point originally plotted, and the r=1 circle intersection. This tells you how far from the load to place your stub.

Read off from the r=1 intersection the reactance x' value. Starting from a short (or open) follow the r=0 circle around the outside of the SMITH chart until you come to a point of reactance -x'. Measure the number of wavelengths this represents from short/open end towards the generator. Cut your stub this long.

The stub is placed in series with one of the transmission line conductors. In coax this may be difficult to do technically. One therefore often resorts to shunt stub matching, where the stub and the original transmission line are connected in parallel. It is easier then to work in admittances. We notice that the SMITH chart can be used as an admittance chart merely by rotating it through 180 degrees. Normalised resistance becomes normalised conductance; normalised reactance becomes normalised susceptance. Admittances in parallel add; the short circuit point has infinite admittance and the open circuit point zero admittance. The design procedure is the same as for series stubs.

Double Stub Tuner Matching
Suppose that the load impedance changes. Or suppose that the transmitter is retuned and the frequency and wavelength change. Adjusting a single stub tuner is very difficult. One has to remove the stub, remake the line where the break was, and calculate the new stub length and point of attachment.

We can use two stubs permanently attached to the line at fixed points of attachment, and tune by altering the stub lengths. It is comparatively easy to make sliding shorts in rigid coaxial line. Two quantities have to be matched (r and x) and we have two variables; the length of each stub. (in the case of the single stub match method, the two variables are the length of the stub, and its position along the line with respect to the antenna).

As before, the generator-end stub has reactance -jx' and is attached at a point where the line impedance, including the effect of the other stub at its fixed point of attachment, is 1+jx'. Transforming the unit r=1 circle towards the load until you reach the load-end stub attachment, the circle r=1 transforms to another circle, call it "B", touching the outside of the SMITH chart, and also passing through its centre.

The load impedance, when transformed towards the generator up to the load-end stub position, will be a generalised impedance ZL' different from ZL. The effect of the load-end stub is to add reactance x" to ZL' so that the impedance value ZL'+jx" lies on the circle "B" above. We chose the length of the stub to make x" the required value for this to happen. If we write ZL'=r'+jx' then the effect of adding the stub is to move the reactance j(x'+x") along the constant r' curve depending on the size of x".

Triple Stub Tuners and E-H Tuners
It is just possible for the r' curve not to intersect the circle "B", in which case a double stub match is not possible for this value of load impedance, and stub placements. Generalised adjustable tuners are therefore designed with three stubs, which are spaced at unequal intervals. Such a device is called a "Triple Stub Tuner". Sliding shorts are easily arranged in coax or waveguide.

In waveguide only, there is a special type of tuner called an E-H tuner. This has shunt and series side arms consisting of sliding shorts, attached at the same point along the guide. There is no equivalent in 2-conductor transmission line (for geometrical reasons). An E-H tuner can always match any load impedance, providing the load has some loss.

As the frequency changes, the wavelength changes and the electrical length and point of attachment of the stub also changes. Thus, a stub match is only accurate at a single spot frequency, and as the frequency changes it becomes progressively less good. However, for a given power reflection coefficient (say less than 10% of power reflected) the match may be sufficiently good over the required range of frequency. Also, the addition of a stub match can, under certain circumstances, even increase the effective bandwidth of the system for a given reflection coefficient.

Stub matching is only desirable for relatively low fractional bandwidths. For wider bandwidth matching a multi-section quarter wave transformer can be used, or a tapered line. Impedance matching may be carried out using the SMITH chart for calculations and design, and lumped components taking the place of lengths of transmission line. It is possible to make undesirable reflections by using a "wrong" stub match, so care must be taken in applying stub matching in high power (e.g. transmitting) applications. It is always wise to measure the match before applying significant input power. In antenna matching situations significant mismatch can arise from alterations to the near-field environment of the antenna over time. Thus if a new antenna is added to an existing mast, it is always wise to check the matching of the pre-existing antennas.

There are practical difficulties at mm wavelengths, e.g. on microstrip at above 20GHz. Here, the precision of adjustment of the lengths of the stubs needs to be +/- 0.01 wavelengths for good quality matching. At 5mm wavelength this is a precision of +/- 50 microns. There are also practical difficulties at high |gamma| (reflection coefficient magnitude). Here the purpose of the stubs is to generate an equal and opposite reflection to cancel out the reflection from the nearly completely mismatched load. Clearly, to get effective cancellation, the stubs must be very precisely chosen and constructed, and the fringing-field effects become important to the point that they can dominate the design. A standard SMITH chart calculation as in this article is then unlikely to be very effective.

Notice that the characteristic impedance of the stub lines can be different from the characteristic impedance of the feed line. If this is the case, we use the "stub line Zo" to "un-normalise" its reactance, and we renormalise the reactance to the "feed line Zo". What this means is that the stub lengths vary(depending on their characteristic impedance) but this doesn't affect the point of attachment to the stubs. This is useful; the flexible coax feed may have a dielectric spacer and the rigid sliding-short double-stub tuner is probably air-spaced.

The Author is grateful to L B Cebik, W4RNL for helpful comments during the refereeing stage of the production of this article. L B Cebik refers to the section of feed between the stub and the load (antenna) as the "match line". On the match line (and also on the stub), the VSWR can be very different from unity, whereas for a correctly matched load, the VSWR is always unity on the feed line.

There is a discussion of a different method of matching, using successive sections of transmission line having various lengths and characteristic impedances, at http://www.cebik.com/ser.html   on L. B. Cebik's site.

Motorola has a page accessing PC-based software for displaying S-parameter plots and performing matching calculations. As I haven't tried to use this I cannot comment on its effectiveness. -30-