Using Smith Chart to Find Reflection Coefficients & Input Impedance of Transmission Lines | Schemes and Mind Maps Linear Algebra

Reflection Coefficient and Transmission Lines Using the Smith Chart

As we discussed in class, the Smith Chart represents the complex plane of the reflection

coefficient. You will recall from class that the input reflection coefficient to a transmission line of

physical length l, Г, is given in terms of the load reflection coefficient Г by the expression

Г Г

󰇛1󰇜

This indicates that on the complex reflection coefficient plane (the Smith Chart), the point representing

Г can be found by a constant-radius rotation from the point representing Г. This rotation represents a

change in phase of the complex number, and is a rotation at constant radius because the magnitude of the

reflection coefficient remains constant in (1) when finding Гfrom Г (please note this by careful

examination of (1)). The phase changes by 2. This represents a rotation in the clockwise direction in

the complex Г plane (the Smith Chart) by 2 radians.

To graphically find Г from Г on the Smith Chart, locate Г (or ) on the Smith Chart. Then,

using your compass, draw a constant-radius circle centered at the center of the Smith Chart and going

through Г. Phase change of the reflection coefficient due to a transmission line will cause the value of

reflection coefficient (and impedance) to rotate along this circle in the clockwise direction. The distance

of rotation has been computed for you in the creation of the Smith Chart and is tabulated on the scale

labeled “Wavelengths toward Generator” around the outside circumference of the Chart.

We now work through an example in the Pozar book for practice and demonstration of how these

techniques work. Please pull out a Smith Chart, pencil, ruler, and compass, and work through this

problem along with this tutorial.

Pozar Example 2.2, p. 66:

“A load impedance of 40  70Ω terminates a 100 Ω transmission line that is 0.3λ long. Find the

reflection coefficient at the load, the reflection coefficient at the input to the line, the input impedance, the

standing wave ratio on the line, and the return loss.” We will leave it to Pozar to explain standing wave

ratio and return loss for now. This tutorial will focus on finding the load reflection coefficient, input

reflection coefficient and input impedance.

Solution:

The first step is to normalize the impedance:





40  70

100 0.40.7

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Reflection Coefficient and Transmission Lines Using the Smith Chart

As we discussed in class, the Smith Chart represents the complex plane of the reflection coefficient. You will recall from class that the input reflection coefficient to a transmission line of physical length l , Г (^) ௜௡, is given in terms of the load reflection coefficient Г (^) ௅ by the expression

Г (^) ௜௡ ൌ Г (^) ି݁௅ ௝ଶఉ௟^ ሺ1ሻ This indicates that on the complex reflection coefficient plane (the Smith Chart), the point representing Г (^) ௜௡ can be found by a constant-radius rotation from the point representing Г (^) ௅. This rotation represents a change in phase of the complex number, and is a rotation at constant radius because the magnitude of the reflection coefficient remains constant in (1) when finding Г (^) ௜௡from Г (^) ௅ (please note this by careful examination of (1)). The phase changes by ݈ߚെ2. This represents a rotation in the clockwise direction in the complex Г plane (the Smith Chart) by ݈ߚെ2 radians. To graphically find Г (^) ௜௡ from Г (^) ௅ on the Smith Chart, locate Г (^) ௅ (or ݖ௅) on the Smith Chart. Then, using your compass, draw a constant-radius circle centered at the center of the Smith Chart and going through Г (^) ௅. Phase change of the reflection coefficient due to a transmission line will cause the value of reflection coefficient (and impedance) to rotate along this circle in the clockwise direction. The distance of rotation has been computed for you in the creation of the Smith Chart and is tabulated on the scale labeled “Wavelengths toward Generator” around the outside circumference of the Chart. We now work through an example in the Pozar book for practice and demonstration of how these techniques work. Please pull out a Smith Chart, pencil, ruler, and compass, and work through this problem along with this tutorial.

Pozar Example 2.2, p. 66: “A load impedance of 40 ൅70 ݆ Ω terminates a 100 Ω transmission line that is 0.3λ long. Find the reflection coefficient at the load, the reflection coefficient at the input to the line, the input impedance, the standing wave ratio on the line, and the return loss.” We will leave it to Pozar to explain standing wave ratio and return loss for now. This tutorial will focus on finding the load reflection coefficient, input reflection coefficient and input impedance.

Solution: The first step is to normalize the impedance: ݖ௅ ܼൌ ܼ௅଴ൌ 40 ൅70 ݆ 100 ൌ 0.4 ൅0.7 ݆

This number can be plotted on the Smith Chart. See the location marked “1” on the Chart on the attached scanned pages of my solution. To measure Г (^) ௅, stretch the compass to place the pin on the center of the chart and the lead of the other arm on the point representing ݖ௅ (and of course this is also Г (^) ௅). Then go to the bottom of the page and use the scale labeled “RFL COEFF, E or I”. Place the pin on the location marked “CENTER” of this scale, and make a mark with your compass’ lead. Read the value associated with the mark. This is the magnitude of the reflection coefficient. This scale is nicely laid out for you to use for this purpose. This is labeled “2” on my Smith Chart attached to this document and is near the bottom of the Smith Chart page. It can be seen from the scale that the magnitude of the reflection coefficient, |Г (^) ௅ |^ is approximately 0.58. To find Г (^) ௜௡ based on this, draw a line from the center of the Smith Chart (using your ruler) through the edge of the chart. Read the reading on the outermost circumferential scale (this is the “WAVELENGTHS TOWARD GENERATOR” scale. You should read approximately 0.105. Write this on your chart. This is your starting point on this scale. I have labeled this step as “3”. Now, add the length of the line, in wavelengths, to this number 0.105. This gives the number on the “WAVELENGTHS TOWARD GENERATOR” scale that should be read when you draw a line through Г (^) ௜௡. Since in this example the line is 0.3 wavelengths long, the addition to be performed is 0.105λ + 0.3λ = 0.405λ. Now draw a line from the center of the Smith Chart through 0.405 on the “WAVELENGTHS TOWARD GENERATOR” scale. This is labeled as “4” on my Smith Chart. The intersection of this line and the circle you have drawn on the chart is the point representing Г (^) ௜௡ and ݖ௜௡. The value of Г (^) ௜௡ is now easily determined. The magnitude is the same as |Г (^) ௅ | because the point is on a constant radius circle in the reflection-coefficient plane (on the same radius as Г (^) ௅). The angle has changed, but can be easily read by the intersection of the line through Г (^) ௜௡ with the “ANGLE OF REFLECTION COEFFICIENT IN DEGREES” scale. This is the third scale from the outside on the outer circumference of the Smith Chart. It appears that the angle of the input reflection coefficient is -112° by reading this scale. This reading is shown and is labeled as “5” on the Smith Chart. The input impedance ݖ௜௡ is represented by the same point as Г (^) ௜௡, because the Smith Chart is a superimposition of the resistance and reactance circles on the complex reflection-coefficient plane. Thus we can read ݖ௜௡ by finding the resistance and reactance circles that go through this point. The resistance circle appears to be 0.37 and the reactance circle appears to be -0.61 (recall that reactance circles on the bottom half of the Smith Chart represent negative reactances). Thus, the value of input impedance is ݖ௜௡ ൌ 0.37 െ0.61 ݆ The input impedance in Ohms, ܼ ௜௡, is found by un-normalizing ݖ௜௡(multiplying it by the reference impedance ܼ ଴:

Using Smith Chart to Find Reflection Coefficients & Input Impedance of Transmission Lines, Schemes and Mind Maps of Linear Algebra

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