RF and Microwave Engineering: Fundamentals of Wireless Communications

6.14 Problems

6.1. A transmission line is short-circuited at both ends. Demonstrate that the line is at resonance if it is one half-wavelength (or multiples of λ/2) in length. A transmission line is short-circuited at one end and open-circuited at the other end. Demonstrate that the line's shortest resonant length is one quarter of a wavelength.

6.2. The following real load impedances

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and complex load impedances

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shall be matched to an impedance of Z₀ = 50 Ω. Use a printed Smith chart or solve the matching problem with a computerized Smith chart tool (e.g. [4]).

6.3. A load impedance of R_A = 1100 Ω shall be matched to a source with an internal resistance of R_I = 50 Ω at a frequency of f = 1 GHz.

1. Design a simple (single-stage) LC matching network.

2. Increase the bandwidth by designing a two-stage LC matching network.

3. Increase the bandwidth further by designing a three-stage LC matching network. Use the same impedance ratio R_A/Z_in for each stage.

Use a Smith chart to visualize the transformation path.

6.4. Figure 6.14 shows a simple power divider with microstrip lines on an alumina substrate (height h = 635 μm; relative permittivity ε_r = 9.8). The power divider shall be matched to Z₀ = 50 Ω at a frequency of f = 5 GHz.

1. Design the microstrip quarter-wave ...