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6.11 Finding zl

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How to find the load impedence using the Smith Chart and VSWR circle.

Let's move on to some other Smith Chart applications. Suppose, somehow, we can obtain a plot of V s on a line with some unknown load on it. The data might look like . What can we tell from this plot? Well, V max 1.7 and V min 0.3 which means

VSWR 1.7 0.3 5.667
and hence
Γ VSWR 1 VSWR 1 4.667 6.667 0.7

A standing wave pattern

Since r s Γ , we can plot r s on the Smith Chart, as shown here . We do this by setting the compass at a radius of 0.7 and drawing a circle! Now, Z L Z 0 is somewhere on this circle. We just do not know where yet! There is more information to begleaned from the VSWR plot however.

The vswr circle

Firstly, we note that the plot has a periodicity of about 10 cm. This means thatλthe wavelength of the signal on the line is 20 cm. Why? According to this equation, V s goes as φ s and φ s θ Γ 2 β s and β 2 λ , thus V s goes as 4 s λ . Thus each λ 2 , we are back to where we started.

Secondly, we note that there is a voltage minima at about 2.5 cm away from the load. Where on would we expect to find a voltage minima? It would be where r s has a phase angle of 180 ° or point "A" shown in here . The voltage minima is always where the VSWR circle passes through the real axis on the left hand side. (Conversely a voltagemaxima is where the circle goes through the real axis on the right hand side.) We don't really care about Z s Z 0 at a voltage minima, what we want is Z s 0 Z 0 , the normalized load impedance. This should be easy! If we start at "A" and go 2.5 20 0.125 λ towards the load we should end up at the point corresponding to Z L Z 0 . The arrow on the mini-Smith Chart says "Wavelengths towards generator" If we start at A, and want to go towards the load , we had better go around the opposite direction from the arrow. (Actually, as you can see on a real Smith Chart, there are arrows pointing in both directions, and they are appropriately marked for yourconvenience.)

Location of a vmin

So we start at "A" go 0.125 λ in a counter-clockwise direction, and mark a new point "B" which represents our Z L Z 0 which appears to be about 0.35 -0.95 or so . Thus, the load in this case (assuming a 50 Ω line impedance) is a resistor, again by co-incidence of about 50 Ω , in series with a capacitor with a negative reactance of about 47.5 Ω . Note that we could have started at the minima at 12.5 cm or even 22.5 cm, and then have rotated 12.5 20 0.625 λ or 22.5 20 1.125 λ towards the load. Since λ 2 0.5 λ means one complete rotation around the Smith Chart, we would have ended up at the same spot, with the same Z L Z 0 that we already have! We could also have started at a maxima, at say 7.5 cm, marked our starting point on the right handside of the Smith chart, and then we would go 0.375 λ counterclockwise and again, we'd end up at "B".

Moving from vmin to the load

Now, here is another example. In this case the VSWR 1.5 0.5 3 , which means Γ 0.5 and we get a circle as shown in . The wavelength λ 2 25 10 30 cm . The first minima is thus a distance of 10 30 0.333 λ from the load. So we again start at the minima, "A" and now rotate as distance 0.333 λ towards the load .

Another standing wave pattern

The vswr circle
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Source:  OpenStax, Introduction to physical electronics. OpenStax CNX. Sep 17, 2007 Download for free at http://cnx.org/content/col10114/1.4
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