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5.7: Exercises

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    1. What is the skin depth on a copper microstrip line at \(10\text{ GHz}\)? Assume that the conductivity of the deposited copper forming the strip is half that of bulk single-crystal copper. Use the data in Example 5.2.1.
    2. What is the skin depth on a silver microstrip line at \(1\text{ GHz}\)? Assume that the conductivity of the fabricated silver conductor is \(75\%\) that of bulk single-crystal silver. Use the data in Example 5.2.1.
    3. A magnetic wall and an electric wall are \(2\text{ cm}\) apart and are separated by a lossless material having a relative permittivity of \(10\) and a relative permeability of \(23\). What is the cut-off frequency of the lowest-order mode in this system?
    4. The strip of a microstrip has a width of \(600\:\mu\text{m}\) and is fabricated on a lossless substrate that is \(1\text{ mm}\) thick and has a relative permittivity of \(10\).
      1. Draw the microstrip waveguide model of the microstrip line. Put dimensions on your drawing.
      2. Sketch the electric field distribution of the first transverse resonance mode and calculate the frequency at which the transverse resonance mode occurs.
      3. Sketch the electric field distribution of the first higher-order microstrip mode and calculate the frequency at which it occurs.
      4. Sketch the electric field distribution of the slab mode and calculate the frequency at which it occurs.
    5. A microstrip line has a width of \(352\:\mu\text{m}\) and is constructed on a substrate that is \(500\:\mu\text{m}\) thick with a relative permittivity of \(5.6\).
      1. Determine the frequency at which transverse resonance would first occur.
      2. When the dielectric is slightly less than onequarter wavelength in thickness the dielectric slab mode can be supported. Some of the fields will appear in the air region as well as in the dielectric, extending the effective thickness of the dielectric. Ignoring the fields in the air (use a one-quarter wavelength criterion), at what frequency will the dielectric slab mode first occur?
    6. The strip of a microstrip has a width of \(600\:\mu\text{m}\) and uses a lossless substrate that is \(635\:\mu\text{m}\) thick and has a relative permittivity of \(4.1\).
      1. At what frequency will the first transverse resonance occur?
      2. At what frequency will the first higher-order microstrip mode occur?
      3. At what frequency will the slab mode occur?
      4. Identify the useful operating frequency range of the microstrip.
    7. The strip of a microstrip has a width of \(500\:\mu\text{m}\) and is fabricated on a lossless substrate that is \(635\:\mu\text{m}\) thick and has a relative permittivity of \(12\). [Parallels Examples 5.4.1, 5.4.2, and 5.4.3]
      1. At what frequency does the transverse resonance first occur?
      2. At what frequency does the first higherorder microstrip mode first propagate?
      3. At what frequency does the substrate (or slab) mode first occur?
    8. The strip of a microstrip has a width of \(250\:\mu\text{m}\) and uses a lossless substrate that is \(300\:\mu\text{m}\) thick and has a relative permittivity of \(15\).
      1. At what frequency does the transverse resonance first occur?
      2. At what frequency does the first higherorder microstrip mode propagate?
      3. At what frequency does the substrate (or slab) mode first occur?
      4. What is the highest operating frequency of the microstrip?
    9. A microstrip line has a strip width of \(100\:\mu\text{m}\) and is fabricated on a substrate that is \(150\:\mu\text{m}\) thick and has a relative permittivity of \(9\).
      1. Draw the microstrip waveguide model and indicate and calculate the dimensions of the model.
      2. Based only on the microstrip waveguide model, determine the frequency at which the first transverse resonance occurs?
      3. Based on the microstrip waveguide model, determine the frequency at which the first higher-order microstrip mode occurs?
      4. At what frequency will the slab mode occur? For this you cannot use the microstrip waveguide model.
    10. A microstrip line has a strip width of \(100\:\mu\text{m}\) and is fabricated on a substrate that is \(150\:\mu\text{m}\) thick and has a relative permittivity of \(9\).
      1. Define the properties of a magnetic wall.
      2. Identify two situations where a magnetic wall can be used in the analysis of a microstrip line; that is, give two situations where a magnetic wall approximation can be used.
      3. Draw the microstrip waveguide model and indicate and calculate the dimensions of the model.
    11. The strip of a microstrip line has a width of \(0.5\text{ mm}\), and the microstrip substrate is \(1\text{ mm}\) thick and has a relative permittivity of \(9\) and relative permeability of \(1\).
      1. Draw the microstrip waveguide model and calculate the dimensions of the model. Clearly show the electric and magnetic walls in the model.
      2. Use the microstrip waveguide model to calculate the cut off frequency of the transverse resonance mode?
      3. A substrate mode can also be excited but the cut off frequency of this mode cannot be calculated using the microstrip waveguide model. Provide a brief description of the substrate mode and calculate the lowest frequency at which it can exist.
    12. A microstrip technology uses a substrate with a relative permittivity of \(10\) and thickness of \(400\:\mu\text{m}\). If the operating frequency is \(10\text{ GHz}\), what is the maximum width of the strip from higher-order mode considerations.
    13. A microstrip line operating at \(18\text{ GHz}\) has a \(200\:\mu\text{m}\) thick substrate with a relative permittivity of \(20\).
      1. Determine the maximum width of the strip from higher-order mode considerations. Consider the transverse resonance mode, the higher-order microstrip mode, and the slab mode.
      2. Thus determine the minimum achievable characteristic impedance.
    14. A microstrip line has a strip width of \(100\:\mu\text{m}\) and is fabricated on a \(150\:\mu\text{m}\)-thick lossless substrate with a relative permittivity of \(9\).
      1. Define the properties of a magnetic wall.
      2. Identify two situations where a magnetic wall can be used in determining multimoding on a microstrip line. That is, give two locations where a magnetic wall approximation can be used.
    15. Two magnetic walls are separated by \(1\text{ mm}\) in a lossless material having a relative permittivity of \(9\) and a relative permeability of \(1\).
      1. What is the wavelength of a \(10\text{ GHz}\) signal in this material?
      2. Now consider a field variation, i.e. a mode and not constant, established by the magnetic walls. Describe this lowest order field variation. That is how does the \(H\) field vary or how does the \(E\) field vary (one is sufficient)?
      3. What is the lowest frequency at which a field variation can be supported by those walls in the specified medium?
    16. A microstrip line has a strip width of \(250\:\mu\text{m}\) and a \(300\:\mu\text{m}\) thick substrate with a relative permittivity of \(15\). At what frequency can the substrate mode first occur?
    17. A microstrip line has a strip width of \(250\:\mu\text{m}\) and a \(300\:\mu\text{m}\) thick substrate with a relative permittivity of \(15\). At what frequency can a higherorder microstrip mode first propagate?
    18. A microstrip line has a strip width of \(250\:\mu\text{m}\) and a \(300\:\mu\text{m}\) thick substrate with a relative permittivity of \(15\). At what frequency can transverse resonance first occur?
    19. The strip of a microstrip line has a width of \(200\:\mu\text{m}\) and the substrate is \(400\:\mu\text{m}\) thick and has a relative permittivity of \(4\).
      1. Draw the effective waveguide model of a microstrip line with magnetic walls and an effective strip width, \(w_{\text{eff}}\).
      2. What is the effective relative permittivity of the microstrip waveguide model?
      3. What is \(w_{\text{eff}}\)?
      4. Can the lowest frequency at which the transverse resonance mode first occurs be determined from the microstrip waveguide model?

    5.7.1 Exercises by Section

    \(†\)challenging

    \(§5.2 1, 2\)

    \(§5.3 3\)

    \(§5.4 4†, 5†, 6†, 7†, 8†, 9†, 10†, 11†, 12, 13, 14, 15, 16, 17, 18, 19\)

    5.7.2 Answers to Selected Exercises

    1. \(2.315\:\mu\text{m}\)
    2. \(247\text{ MHz}\)
    3. (c) \(25\text{ GHz}\)
    4. (c) \(\text{DC}\) to \(63.4\text{ GHz}\)
    5. (d) \(\text{DC}\leq f\leq 48.6\text{ GHz}\)
    1. \(104.6\text{ GHz}\)
    1. (b) \(55.6\text{ GHz}\)

    This page titled 5.7: Exercises is shared under a CC BY-NC license and was authored, remixed, and/or curated by Michael Steer.

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