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AMATEUR RADIO BY GARRY CRATT, VK2YBX The Smith Chart – what it is & how you use it Possibly one of the most useful graphic tools available today to the RF engineer is the Smith Chart. This chart, named after its inventor, Mr Phillip Smith, an engineer at Bell Labora­tories during the 1930s, first appeared in “Electronics” magazine in the USA in January 1939. problems, the precise reason for its creation. In fact, the Smith Chart is really a special type of graph, having curved co-ordinate lines, instead of the rectangular lines encountered on standard graph paper. Quite complex mathematical reasons exist behind the construction of the chart but these do not need to be understood by the user. It is easy to be put off by the Smith Chart. At first glance, it looks like a nightmare, with all those apparently spiralling curves, but after you’ve read this article you should be quite at ease with it. It really is a very useful chart Different curves for the amateur radio operator. Essentially, the Smith Chart is used to graphically repre­ sent the reflection characteristics and impedance of an RF cir­cuit. It is ideally suited for the solution of transmission line To make the Smith Chart easy to understand, we’ll show its different curves separately and then bring them all together as a simplified composite chart. Fig.1 is the first set of curves 0 REACTANCE AXIS -0.2 +0.2 0 RESISTANCE AXIS 0.2 -0. 5 0.5 PRIME CENTRE .5 +0 RESISTANCE CIRCLES 1 -1 +1 2 h Fig.1: the first components of a Smith Chart are the resistance circles. The values are “normalised” to a value of 1.00, the prime centre of this plot. +5 50 +2 -2 -5 5 h Fig.2: the second component of the Smith Chart is the reactance plot. Again the reactance values are normalised. June 1993  53 tance lines plotted on the one graph. Note that this is a greatly simplified Smith Chart, as those normally pub­ lished have the circles at much closer intervals which is why they look so complicated at first glance. Now let’s see how you might plot a particular impedance on the chart. Consider an example where we have an impedance con­sisting of 50Ω resistive and 100Ω inductive reactance (50 + j100), and a prime centre value of 50Ω. This particular impedance can be plotted at the intersection of 1.0 on the resistive scale and 2.0 on the positive reactance circle. AMATEUR RADIO – CTD 0 + j0 (SHORT CIRCUIT) -0.2 +0.2 0 0.2 -0. 5 .5 +0 0.5 1 -1 +1 SWR circles 2 0.5 + j1 +2 -2 5 1 + j2 50 +5 -5 1 - j2 OPEN CIRCUIT Fig.3: this simplified Smith Chart shows the resistance and reactance circles plotted together. Note that the resistance axis coincides with the zero reactance line. which are resistance circles. Each resistance circle is assigned a particular value, shown where the circle cuts the vertical resistance axis. That value remains the same for all points along that circle. In fact, the values range from zero at the top of the axis to infinity at the bottom and represent a ratio with respect to the centre point of the chart which is mark­ ed “1”. By assigning the centre point or “prime centre” of the chart a particular value, each circle represents a value of resistance scaled in accordance with the ratio for that circle. For example, if you allocate a value of 100Ω to the centre point, any point lying on the 0.5 circle has a value of 100 x 0.5 = 50Ω. Similarly, any point on the 2.0 circle has a value of 200Ω. This also means that the resistance value of any point on the chart can be calculated by multiplying the ratio of the particular line with the value assigned to the prime centre. The value you would normally assign to the centre point is the same as the value of the characteristic impedance of the line being matched, typically 54  Silicon Chip 50Ω. In fact, special printed charts are available having a prime centre of 50Ω. All resistance and reac­ tance values can then be plotted directly, without having to “normalise” impedances. Reactance circles Fig.2 shows curves which are reactance circles. Note that these circles originate from the left and right hand sides of the vertical zero reactance line. The circumference of the circle is the reactance axis. Just as each resistance circle was assigned a particular value, so are the reactance lines. Any point along a reactance circle has the same value and these values can be multiplied (or normalised). Points located to the right of the zero reactance axis are positive (inductive) and values to the left are negative (capacitive). Note also that the vertical zero reactance line on Fig.2 coincides with the vertical resistance axis on Fig.1. That makes sense because any “pure” resistance will have zero reactance. Fig.3 is the composite Smith Chart, with the resistance circles and reac- Now we come to the nub of the matter, as far as most ama­teur radio operators will be concerned. A useful addition to the Smith chart is the standing wave circle. A series of these can be drawn on the chart using a draughting compass, centred on the prime centre. The point at which a circle for a given SWR crosses the resistance axis is the value of SWR. So the circle represent­ing an SWR of 2:1 has its centre at the prime centre and the radius crossing 2.0 on the resistance axis. Fig.4 shows a simplified Smith Chart with SWR circles added. If we wish to match a 50Ω transmission line, having a length of 2¼ wavelengths to a terminating im­pedance of 25Ω resistive and 25Ω inductive reactance (25 + j25), the following procedure should be used. First, we “normalise” the terminating impedance by dividing both components by 50. This equates to 0.5 +j0.5. We then plot this impedance at the inter­section of the 0.5Ω resistance line and the 0.5Ω reactance cir­cle. We know the reactance is positive (inductive), so it must be located on the right hand side of the resistance axis. By drawing a circle, whose centre is at the prime centre and whose radius is the distance from the prime centre to the impedance point, we have plotted (0.5 + j0.5). By noting where the circle intersects the resistance axis, it can be seen that a voltage ratio of 2.6:1 exists at that point. Wavelength scale A comprehensive Smith Chart, as distinct from the simpli­fied examples used here, also bears a wavelength 5.0 SWR CIRCLE -0.2 +0.2 0 2.0 SWR CIRCLE 0.2 -0. 5 .5 +0 0.5 1 -1 +1 2 GW QUALITY SCOPES 100MHz +2 -2 50 PLUS FREE DMM +5 -5 5 Fig.4: plotting SWR circles on a Smith Chart is a useful step in the process of matching a transmitter to an antenna, while avoiding the need for tedious mathematical calculations. scale around the perimeter of the chart. The scale is marked in fractions of a wavelength of a transmission line. One scale runs anticlockwise, starting at the “generator” end, which is normally the input end, and running towards the load. Another scale runs in the opposite direction from load to generator. The complete circumference equals one half wavelength. Using our matching example above, we could further progress towards a solution by drawing a line from the prime centre, through the plotted 0.5 + j0.5 point, and to the wavelength scale. As our plotted impedance point is looking from the load end of the network, we use the “towards generator” scale to read 0.088 wavelength at the point of intersection. We know that our 50-ohm cable has a length of 2.25 wave­lengths, and as the complete scale on the chart represents a half wavelength and any impedance reflections will be repeated every half wavelength, we need only use 0.25 as our transmission line length for this calculation. By adding 0.25 to the 0.088 indicated on the wavelength scale, we can locate the resultant 0.338 on the wavelength scale and draw a line from that point to the prime centre. The point where this line intersects our 2.6:1 SWR circle is the line input impedance, in this example 1.0 - j1.0. To find the line im­pedance, we simply multiply by 50, and this gives 50Ω resistive and 50Ω capacitive. This is the impedance that the transmitter must match. Line loss & multi-element matching The Smith Chart can be used to calculate line loss and also to facilitate the design of multi-element matching networks. A comprehensive guide to the use of Smith Charts can be found in the Sams publication “RF Circuit Design” by Chris Bo­wick. Good background material can also be SC found in the ARRL Antenna Handbook. 40MHz ESCORT EDM-1133 20MHz • • • • • • 3¾ Digits Autoranging 8 Functions DC V, AC V DC A, AC A Ohms Valued at $127! 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