Question

A load Zₗ = 100+j25 is to be matched to a lossless 50Ω line using a shorted stub. The wavelength on the transmission line is 20 cm. Use the Smith Chart to determine

a) the distance between the load and the stub, d.
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Answer 1

To determine the distance between the load and the stub using the Smith Chart, follow these steps: convert the load impedance to normalized coordinates, plot the normalized load impedance on the Smith Chart, draw a constant VSWR circle, find the intersection point, and use the Smith Chart scale to read the distance.

Step-by-step explanation:

To determine the distance between the load and the stub using the Smith Chart, you need to follow the steps below:

1. Convert the load impedance to normalized coordinates by dividing the real part by the characteristic impedance and the imaginary part by the characteristic impedance. In this case, Zl = 100+j25 and the characteristic impedance is 50Ω, so the normalized load impedance is Zl_norm = (100+j25)/50 = 2+j0.5.
2. Plot the normalized load impedance on the Smith Chart.
3. Draw a constant standing wave ratio (VSWR) circle on the Smith Chart that passes through the origin and the normalized load impedance point.
4. Find the intersection point of the constant VSWR circle and the radial line passing through the center of the Smith Chart.
5. The distance between the load and the stub is given by d = (theta - 180°) × (lambda/360°), where theta is the angle between the radial line and the horizontal axis of the Smith Chart, and lambda is the wavelength on the transmission line.

In this case, since the load impedance is inductive (positive imaginary part), the intersection point will be in the inductive region of the Smith Chart. You can use the Smith Chart scale to read the distance between the load and the stub.