# 6.7: Exercises

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1. A four-layer circuit board has four levels of metalization with the two inner layers being supply and ground planes are RF grounds. The top layer has a very dense concentration of signal lines and for the purposes of understanding the fields in the dielectric between the top two layers of metalization both layers of metalization can be considered to be solid. The situation now reduces to a parallel plate waveguide which allows signals to travel in the dielectric from one region of the circuit board to another. The dielectric has a relative permittivity of $$4$$ and the thickness of the dielectric between the metal layers is $$1\text{ mm}$$.
1. What is the mode that propagates at the lowest frequency and what is the cut-off wavenumber and cut-off frequency of this mode?
2. What is the wave impedance of this mode at $$1\text{ GHz}$$?
3. What is the phase velocity of this mode at $$1\text{ GHz}$$?
2. A multi-layer circuit board forms a parallel plate waveguide between the metal planes (even if they are broken up a little). Thus EM energy propagates around the circuit board and not all of the energy is confined by the circuit board lines.
1. What is the EM mode that is mostly to propagates?
2. Draw the $$\text{E}$$ and $$\text{H}$$ fields of the mode in cross section.
3. If the distance between the plates is $$1\text{ mm}$$, what is the lowest frequency at which this lowest order mode can propagate.
3. Two adjacent layers of a circuit board are continuous ground planes separated by a $$0.5\text{ mm}$$-thick dielectric having a relative permittivity of $$4$$.
1. What is the lowest frequency at which the TEM mode propagates.
2. What is the wave impedance of the TEM mode at $$10\text{ GHz}$$.
4. Two adjacent layers of a circuit board are continuous ground planes separated by a $$0.5\text{ mm}$$-thick dielectric having a relative permittivity of $$4$$. What is the lowest frequency at which the $$\text{TM}_{1}$$ mode propagates.
5. Two adjacent layers of a circuit board are continuous ground planes separated by a $$0.5\text{ mm}$$-thick dielectric having a relative permittivity of $$4$$. What is the lowest frequency at which the $$\text{TE}_{1}$$ mode propagates.
6. An air-filled $$\text{W}$$-band waveguide has a $$75– 100\text{ GHz}$$ operating frequency, extends in the $$z$$ direction and has internal dimensions of $$\Delta_{x} = 0.100\text{ in.}$$ by $$\Delta_{y} = 0.050\text{ in.}$$ $$f_{c}$$ is the cut-off frequency, and $$k_{c}$$ is the cutoff wavenumber.
1. Does the waveguide support a transverse electromagnetic (TEM) mode provide your reasons?
2. What is the distinguishing feature of the TE waveguide modes?
3. What is the distinguishing feature of the TM waveguide modes?
4. What modes other than TEM, TE, and TM modes propagate in rectangular waveguide?
5. Are $$\text{TE}_{00}$$ and $$\text{TM}_{00}$$ modes supported in the waveguide and why?
6. What is $$k_{c}$$ of the $$\text{TE}_{10}$$ mode?
7. What is $$k_{c}$$ of the $$\text{TE}_{01}$$ mode?
8. What is $$k_{c}$$ of the $$\text{TE}_{11}$$ mode?
9. What is $$f_{c}$$ of the $$\text{TE}_{10}$$ mode?
10. What is $$f_{c}$$ of the $$\text{TE}_{01}$$ mode?
11. What is the wavenumber of the $$\text{TE}_{01}$$ mode at $$90\text{ GHz}$$, discuss if your answer is not real?
12. What is the wavenumber of the $$\text{TE}_{10}$$ mode at $$90\text{ GHz}$$, discuss if your answer is not real?
13. What is $$f_{c}$$ of the $$\text{TE}_{11}$$ mode?
14. What is $$k_{c}$$ of the $$\text{TM}_{10}$$ mode?
15. What is $$k_{c}$$ of the $$\text{TM}_{01}$$ mode?
16. What is $$k_{c}$$ of the $$\text{TM}_{11}$$ mode?
17. What is $$k_{c}$$ of the $$\text{TM}_{10}$$ mode?
18. What is $$f_{c}$$ of the $$\text{TM}_{01}$$ mode?
19. What is $$f_{c}$$ of the $$\text{TM}_{11}$$ mode?
20. What can you say about the relative cut-off frequency of the $$\text{TE}_{mn}$$ and $$\text{TM}_{mn}$$, $$m + n\geq 2$$, modes relative to the cutoff frequencies of the $$\text{TE}_{mn}$$ and $$\text{TM}_{mn}$$, $$m + n = 1$$, modes?
21. Sketch and label the dispersion curves for the $$\text{TE}_{mn}$$ and $$\text{TM}_{mn}$$, $$m, n = 0, 1,$$ modes, i.e. $$\beta$$ versus frequency over the frequency range $$0$$ to $$200\text{ GHz}$$.
22. What is the frequency range over which only one mode can propagate?
23. Why is it desirable that only one mode propagate?
24. Why is the operating frequency range specified for this waveguide less than the frequency range over which only one mode can exist.
7. An air-filled $$\text{E}$$-band waveguide has a $$60–90\text{ GHz}$$ operating frequency and has internal dimensions of $$0.122\text{ in}$$ by $$0.061\text{ in}$$.
1. What is the phase velocity of the $$\text{TE}_{10}$$ mode at $$60\text{ GHz}$$.
2. What is the phase velocity of the $$\text{TE}_{10}$$ mode at $$90\text{ GHz}$$.
3. What is the wave impedance of the $$\text{TE}_{10}$$ mode at $$60\text{ GHz}$$.
4. What is the wave impedance of the $$\text{TE}_{10}$$ mode at $$90\text{ GHz}$$.
5. What is the difference between characteristic impedance and wave impedance for the waveguide?
8. An $$\text{X}$$-band waveguide has the dimensions $$0.9\text{ in}\times 0.4\text{ in}$$. Calculate the cut-off frequencies for the $$\text{TE}_{10},\text{ TE}_{01},$$ and $$\text{TM}_{11}$$, modes.
9. An $$\text{X}$$-band waveguide has the dimensions $$0.9\text{ in}\times 0.4\text{ in}$$. Calculate the cut-off frequencies for the $$\text{TE}_{10},\text{ TE}_{01},$$ and $$\text{TM}_{11}$$, modes.
10. A $$\text{W}$$-band waveguide has the dimensions $$0.1\text{ in}\times 0.05\text{ in}$$ and the cut-off frequency of the $$\text{TE}_{10}$$ mode is $$59.0\text{ GHz}$$. Verify this.
1. What is the cut-off frequencies of the $$\text{TE}_{01}$$ mode?
2. What is the cut-off frequencies of the $$\text{TM}_{11}$$ mode?
3. What is the the operating frequency range over which only one mode can exist?
11. An $$\text{E}$$-band waveguide has the dimensions $$0.122\text{ in}\times 0.061\text{ in}$$ and the cut-off frequency of the $$\text{TE}_{10}$$ mode is $$48.4\text{ GHz}$$. Verify this.
1. What is the cut-off frequency of the $$\text{TE}_{01}$$ mode?
2. What is the cut-off frequency of the $$\text{TM}_{11}$$ mode?
3. What is the the frequency range over which only one mode can exist?
12. A $$\text{W}$$-band waveguide has the dimensions $$0.1\text{ in}\times 0.05\text{ in}$$. At $$100\text{ GHz}$$ only the $$\text{TE}_{10}$$ mode can propagate. At $$100\text{ GHz}$$ and for the $$\text{TE}_{10}$$ mode
1. what is the wave impedance?
2. what is the phase velocity?
3. what is the group velocity?
13. An $$\text{E}$$-band waveguide has the dimensions $$0.1\text{ in}\times 0.05\text{ in}$$. At $$38\text{ GHz}$$ only the $$\text{TE}_{10}$$ mode can propagate. At $$38\text{ GHz}$$ and for the $$\text{TE}_{10}$$ mode
1. what is the wave impedance?
2. what is the phase velocity?
3. what is the group velocity?
14. A communication system at $$60\text{ GHz}$$ has a bandwidth of $$4\text{ GHz}$$ so that the operating frequency range is $$58\text{ GHz}$$ to $$62\text{ GHz}$$. Waveguide is to be used to route the millimeter-wave signal to an antenna. What waveguide should be used to justify your answer.
15. A communication system $$15\text{ GHz}$$ has a bandwidth of $$1\text{ GHz}$$ so that the operating frequency range is $$14.5\text{ GHz}$$ to $$15.5\text{ GHz}$$. Waveguide is to be used to route the millimeter-wave signal to an antenna. What waveguide should be used to justify your answer.

## 6.7.1 Exercises by Section

$$§6.2\: 1, 2, 3, 4, 5$$

$$§6.4\: 6, 7, 8, 9, 10, 11, 12, 13, 14, 15$$

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