In making this plot, we have made use of the fact that the propagation constant ββ can also be expressed as 2πλ 2 λ , and so for the independent variable, instead of showing ss in meters or whatever, we normalize the distance away from the load to the wavelength of the excitation signal, and hence show distance in wavelengths. What we are showing here is called a standing wave. There are places along the line where the magnitude of the voltage |Vs| V s has a maximum value. This is where V + V + and V - V - are adding up in phase with one another, and places where there is a voltage minimum, where V + V + and V - V - add up out of phase. Since | V - |=| Γ ν || V + | V - Γ ν V + , the maximum value of the standing wave pattern is 1+| Γ ν | 1 Γ ν times | V + | V + and the minimum is 1−| Γ ν | 1 Γ ν times | V + | V + . Note that anywhere on the line, the voltage is still oscillating at ⅇⅈωt ω t , and so it is not a constant, it is just that the magnitude of the oscillating signal changes as we move down the line. If we were to put an oscilloscope across the line, we would see an AC signal, oscillating at a frequency ωω.

A number of considerable interest is the ratio of the maximum voltage amplitude to the minimum voltage amplitude, called the voltage standing wave ratio, or VSWR for short. It is easy to see that:

Note that because | Γ ν |∈01 Γ ν 0 1 , VSWR∈1∞ VSWR 1 .

Although Figure 1 looks like the standing wave pattern is more or less sinusoidal, if we increase |Γ| Γ to 0.8, we see that it most definitely is not. There is also a temptation to say that the spacing between minima (or maxima) of the standing wave pattern is λλ , the wavelength of the signal, but a closer inspection of either Figure 1 or Figure 2, shows that in fact the spacing between features is only half a wavelength, or λ2 λ 2 . Why is this? Well, φs φ s goes as -2βs -2 β s and β=2πλ β 2 λ , and so every time ss increases by λ2 λ 2 , φs φ s decreases by 2π 2 and we have come one full cycle on the way |Vs| V s behaves.

Now let's go back to the Crank Diagram. At the position shown, we are at a voltage maximum, and Zs Z 0 Z s Z 0 just equals the VSWR. Note also that at this particular point, that the voltage and current phasors are in phase with one another (lined up in the same direction) and hence the impedance must be real or resistive.

We can move further down the line, and now the Vs V s phasor starts shrinking, and the Is I s phasor starts to get bigger Figure 3.

If we move even further down the line, we get to a point where the current phasor is now at a maximum value, and the voltage phasor is at a minimum value Figure 4. We are now at a voltage minimum, the impedance is again real (the voltage and current phasors are lined up with one another, so they must be in phase) and The only problem we have here is that except at a voltage minimum or maximum, finding Zs Z s from the crank diagram is not very straightforward, since the voltage and current are out of phase, and dividing the two vectors becomes somewhat tedious.

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This work is licensed by Bill Wilson under a Creative Commons Attribution License (CC-BY 1.0), and is an Open Educational Resource.

Last edited by Raymond Wagner on Aug 14, 2007 12:08 pm GMT-5.