6.8: Exercises

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Consider the cross section of a coupled transmission line, as shown in Figure 6.2.1, with even and odd modes both traveling out of the page.
1. For an even mode on the coupled line, consider a phasor voltage of \(1\text{ V}\) on each of the lines above the ground plane at \(0\text{ V}\). Sketch the directed electric field in the transverse plane, i.e. show the direction of the electric field.
2. For the even mode, sketch the directed magnetic fields in the transverse plane (the plane of the cross section).
3. For an odd mode on the coupled line, consider a phasor voltage of \(+1\text{ V}\) on the left line and a phasor voltage of \(−1\text{ V}\) on the right line. Sketch the directed electric fields in the transverse plane (the plane of the cross section).
4. For the odd mode, sketch the directed magnetic fields in the transverse plane (the plane of the cross section).
An ideal directional coupler is lossless and there are no reflections at the ports. If the coupling factor is \(10\), what is the the magnitude of the transmission coefficient?
A directional coupler has the following characteristics: coupling factor \(C = 20\), transmission factor \(0.9\), and directivity factor \(25\text{ dB}\). Also, the coupler is matched so that there is no reflection at any of the ports. What is the isolation in decibels?
A lossy \(6\text{ dB}\) directional coupler is matched so that there is no reflection at any of the ports. The insertion loss (considering the through path) is \(2\text{ dB}\). If \(1\text{ mW}\) is input to the directional coupler, what is the power in microwatts dissipated in the directional coupler? Ignore power leaving the isolated port.
A matched directional coupler has a coupling factor \(C\) of \(20\), transmission factor \(0.9\), and directivity of \(25\text{ dB}\). What is the power dissipated in the directional coupler if the input power to Port 1 is \(1\text{ W}\).
Consider a pair of parallel microstrip lines separated by a spacing, \(s\), of \(100\:\mu\text{m}\).
1. What happens to the coupling factor of the lines as s reduces?
2. What happens to the system impedance as s reduces and no other dimensions change?
3. In terms of wavelengths, what is the optimum length of the coupled lines for maximum coupling?

6.8.1 Exercises by Section

\(†\)challenging

\(§6.2 1†\)

\(§6.5 2, 3, 4, 5, 6†\)

6.8.2 Answers to Selected Exercises

\(0.9950\)
(b) \(187\text{ mW}\)