The Smith Chart - Load Impedance Measured Down the Transmission Line

7/30/2019 The Smith Chart - Load Impedance Measured Down the Transmission Line

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[2]

We can calculate the input impedances using equation [1]:

[3]

The impedances in equation [3] are plotted in Figure 2:

Figure 2. Input Impedances of Equation [3] Plotted on the Smith Chart.

From Figure 2, something interesting emerges. We could go through a ton of math equations to prove

this, but that's not real fruitful. Each of the points in Figure 2 are the same distance away from the

center of the Smith Chart. That is, the complicated input impedance equation ([2] above) translates

into a simple circular motion on the Smith Chart. Hence, you can find the impedance of a load a

distanceL down a transmission line simply by moving in a circular fashion around the Smith Chart.

Let's take an example. Let zL = 2.0 again as above. If we draw a circle centered at the center of the

Smith Chart and travelling through zL, then we get the curve given in Figure 3 by the black x's:


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Figure 3. Constant VSWR circle.

A few points are plotted along this circle. If you travel lambda/8 (one eighth of a wavelength) down

the transmission line in Figure 1, the resulting input impedance can be found by rotating 90 degrees in

the clockwise direction on the Smith Chart. Similarly, if you want the input impedance lambda/4 (one

quarter of a wavelength) from the load impedance, the resulting input impedance can be found by

rotatin 180 degrees in the clockwise direction around the Smith Chart. Hence, the input impedance

(from equation [1] or the Smith Chart) repeats itself every half-wavelength. That is, a half-wavelength along the transmission line corresponds to a complete rotation on the Smith Chart (back

to where you started).

The circle in Figure 3 that runs through zL is known as a constant VSWR or constant SWR circle.

Since this circle is centered at the center of the Smith Chart, the magnitude of is constant along

this curve. Hence, the VSWRis constant everywhere along this curve.

Example

An example will help cement the above idea. Suppose we know that the input impedance, z1=0.1 (so

Z1=0.1*50=5), at location 1 in Figure 4. What is the input impedance of the load, ZL, and Z2 (the

impedance at the generator), assuming Z0=50 Ohms?

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Figure 4. Diagram of Transmission Line Problem.

Solution. This problem is very simple thanks to the Smith Chart. First, plot the known value of z1,

along with the constant VSWR curve (the circle centered in the Smith Chart going through z1):

Fi ure 5. The First Ste : Plottin z1 and the Constant VSWR Circle.


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To determine ZL, we want to move on the Smith Chart towards the load. Remember:

Moving towards the load impedance (the antenna) corresponds to a counter-clockwise movement

on the Smith Chart

Moving towards the generator or transmitter/receiver corresponds to a clockwise rotation on the

Smith Chart

Hence, to find ZL, we want to move in the counter-clockwise direction. Note thatL1 = 5*lambda/8.

Recall that a distance of lambda/2 on a transmission line corresponds to a complete rotation on the

Smith Chart. Hence, this is equivalent to moving 5*lambda/8 - lambda/2 = lambda/8. Hence, we can

simply rotate in the counter-clockwise direction by 90 degrees (one quarter turn on the Smith Chart).

We can then read off the value for this impedance, and the result is:

zL = 0.198 - i*0.9802 ==> ZL = 9.901 - i*49.0099

Similarly, to find z2, the impedance at the generator, we simply move lambda/8 in the clockwise

direction (since we are traveling away from the antenna/load). The result is a 90 degree rotation,plotted in Figure 6. The result is:

z2 = 0.198 + i*0.9802 ==> Z2 = 9.901 + i*49.0099

The correspond Smith Chart is shown, with only the needed constant resistance circles and constant

reactance curves:

Figure 6. The Second Step: Using the Constant SWR Circle to Get zL and z2.

In Figure 6, the circle corresponds to the magnitude of being 0.8182. This can be found from

using the equation for (equation [1] here). In Figure 6, I only plotted the reactance curves for

= =-
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. . . .

On a real smith chart, you would simply interpolate between the closest reactance curves. Similarly, I

only plotted the resistance circles given by Re[Z]=0.1 (for the z1 impedance) and Re[Z]=0.198 (for

zL and z2). All other curves are irrelevant on the Smith Chart for this problem.

Finally, on complicated, detailed Smith Charts, you will see a scale along the outer perimeter as

shown below:

I zoomed in on a Smith Chart in the above picture. The numbers correspond to wavelengths, so that

you can more easily figure out the proper rotation when determining the resultant impedance due to a

length of transmission line.

In conclusion, this might seem all kind of stupid. But actually it will be very useful for understanding

antenna responses at an intuitive level, and to help visualize how impedances change due to

transmission lines. More importantly, it is very useful in impedance matching, as we will see.


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Next: Introduction to Impedance Matching - Series L and C

Previous: Constant Reactance Curves

Smith Chart Table of Contents

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The Smith Chart - Load Impedance Measured Down the Transmission Line

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