3.11: Exercises

1. At low frequencies a microstrip line has a capacitance of \(1\text{ nF/m}\) and when the dielectric is replaced by air its capacitance is \(0.5\text{ nF/m}\). What is the phase velocity of signals on the line with the dielectric substrate in place? Consider that the relative magnetic permeability is \(1\).
2. A non-magnetic microstrip line has a capacitance of \(100\text{ pF/m}\) and when the dielectric is replaced by air it has a capacitance of \(25\text{ pF/m}\). What is the phase velocity of signals on the line with the dielectric?
3. At low frequencies a non-magnetic microstrip transmission line has a capacitance of \(10\text{ nF/m}\) and when the dielectric is replaced by air it has a capacitance of \(2.55\text{ nF/m}\). What is the effective permittivity of the microstrip line of signals on the line with the dielectric substrate in place?
4. A microstrip line on \(250\:\mu\text{m}\) thick GaAs has a minimum and maximum strip widths of \(50\:\mu\text{m}\) and \(250\:\mu\text{m}\) respectively. What is the range of characteristic impedances that can be used in design?
5. A microstrip line with a substrate having a relative permittivity of \(10\) has an effective permittivity of \(8\). What is the wavelength of a \(10\text{ GHz}\) signal propagating on the microstrip?
6. A microstrip line has a width of \(500\:\mu\text{m}\) and a substrate that is \(635\:\mu\text{m}\) thick with a relative permittivity of \(20\). What is the effective permittivity of the line?
7. The strip of a microstrip has a width of \(250\:\mu\text{m}\) and is fabricated on a lossless substrate that is \(500\:\mu\text{m}\) thick and has a relative permittivity of \(2.3\). [Parallels Example 3.5.2]
   1. What is the effective relative permittivity of the line?
   2. What is the characteristic impedance of the line?
   3. What is the propagation constant at \(3\text{ GHz}\) ignoring any losses?
   4. If the strip has a resistance of \(0.5\:\Omega\text{/cm}\) and the ground plane resistance can be ignored, what is the attenuation constant of the line at \(3\text{ GHz}\)?
8. A microstrip line on a \(250\:\mu\text{m}\)-thick silicon substrate has a width of \(200\:\mu\text{m}\). Use Table 3.5.2.
   1. What is line’s effective permittivity.
   2. What is its characteristic impedance?
9. A \(600\:\mu\text{m}\)-wide microstrip line on a \(500\:\mu\text{m}\) thick alumina substrate. Use Table 3.5.2.
   1. What is line’s effective permittivity.
   2. What is its characteristic impedance?
10. A microstrip line on a \(1\text{ mm}\)-thick FR4 substrate has a width of \(0.497\text{ mm}\). Use Table 3.5.2.
    1. What is line’s effective permittivity.
    2. What is its characteristic impedance?
11. Consider a microstrip line on a substrate with a relative permittivity of \(12\) and thickness of \(1\text{ mm}\).
    1. What is the minimum effective permittivity of the microstrip line if there is no limit on the minimum or maximum width of the strip?
    2. What is the maximum effective permittivity of the microstrip line if there is no limit on the minimum or maximum width of the strip?
12. A microstrip line has a width of \(1\text{ mm}\) and a substrate that is \(1\text{ mm}\) thick with a relative permittivity of \(20\). What is the geometric filling factor of the line?
13. The substrate of a microstrip line has a relative permittivity of \(16\) but the calculated effective permittivity is \(12\). What is the filling factor?
14. A microstrip line has a strip width of \(250\:\mu\text{m}\) and a substrate with a relative permittivity of \(10\) and a thickness of \(125\:\mu\text{m}\). What is the filling factor?
15. A microstrip line has a strip width of \(250\:\mu\text{m}\) and a substrate with a relative permittivity of \(4\) and thickness of \(250\:\mu\text{m}\). Determine the filling factor and thus the effective relative permittivity of the line?
16. A microstrip line has a strip with a width of \(100\:\mu\text{m}\) and the substrate which is \(250\:\mu\text{m}\) thick and a relative permittivity of \(8\).
    1. What is the filling factor, \(q\), of the line?
    2. What is the line’s effective relative permittivity?
    3. What is the characteristic impedance of the line?
17. An inhomogeneous transmission line is fabricated using a medium with a relative permittivity of \(10\) and has an effective permittivity of \(7\). What is the fill factor \(q\)?
18. A microstrip technology uses a substrate with a relative permittivity of \(10\) and thickness of \(400\:\mu\text{m}\). The minimum strip width is \(20\:\mu\text{m}\). What is the highest characteristic impedance that can be achieved?
19. A microstrip transmission line has a characteristic impedance of \(75\:\Omega\), a strip resistance of \(5\:\Omega\text{/m}\), and a ground plane resistance of \(5\:\Omega\text{/m}\). The dielectric of the line is lossless.
    1. What is the total resistance of the line in \(\Omega\text{/m}\)?
    2. What is the attenuation constant in \(\text{Np/m}\)?
    3. What is the attenuation constant in \(\text{dB/cm}\)?
20. A microstrip line has a characteristic impedance of \(50\:\Omega\), a strip resistance of \(10\:\Omega\text{/m}\), and a ground plane resistance of \(3\:\Omega\text{/m}\).
    1. What is the total resistance of the line in \(\Omega\text{/m}\)?
    2. What is the attenuation constant in \(\text{Np/m}\)?
    3. What is the attenuation constant in \(\text{dB/cm}\)?
21. A microstrip line has \(10\:\mu\text{m}\)-thick gold metallization for both the strip and ground plane. The strip has a width of \(125\:\mu\text{m}\) and the substrate is \(125\:\mu\text{m}\) thick.
    1. What is the low frequency resistance (in \(\Omega\text{/m}\)) of the strip?
    2. What is the low frequency resistance of the ground plane?
    3. What is the total low frequency resistance of the microstrip line?
22. A \(50\:\Omega\) microstrip line has \(10\:\mu\text{m}\)-thick gold metallization for both the strip and ground plane. The strip has a width of \(250\:\mu\text{m}\) and the lossless substrate is \(250\:\mu\text{m}\) thick.
    1. What is the low frequency resistance (in \(\Omega\text{/m}\)) of the strip?
    2. What is the low frequency resistance of the ground plane?
    3. What is the total low frequency resistance of the microstrip line?
    4. What is the attenuation in \(\text{dB/m}\) of the line at low frequencies?
23. A \(50\:\Omega\) microstrip line with a lossless substrate has a \(0.5\text{ mm}\)-wide strip with a sheet resistance of \(1.5\text{ m}\Omega\) and the ground plane resistance can be ignored. What is the attenuation constant at \(1\text{ GHz}\)? [Parallels Example 3.5.3]
24. A microstrip line operating at \(10\text{ GHz}\) has a substrate with a relative permittivity of \(10\) and a loss tangent of \(0.005\). It has a characteristic impedance of \(50\:\Omega\) and an effective permittivity of \(7\).
    1. What is the conductance of the line in \(\text{S/m}\)?
    2. What is the attenuation constant in \(\text{Np/m}\)?
    3. What is the attenuation constant in \(\text{dB/cm}\)?
25. A microstrip line has the per unit length parameters \(L = 2\text{ nH/m}\) and \(C = 1\text{ pF/m}\), also at \(10\text{ GHz}\) the substrate has a conductance \(G\) of \(0.001\text{ S/m}\). The substrate loss is solely due to dielectric relaxation loss and there is no substrate conductive loss. The resistances of the ground and strip are zero.
    1. What is \(G\) at \(1\text{ GHz}\)?
    2. What is the magnitude of the characteristic impedance at \(1\text{ GHz}\)?
    3. What is the dielectric attenuation constant of the line at \(1\text{ GHz}\) in \(\text{dB/m}\)?
26. A microstrip line has the per unit length parameters \(L = 1\text{ nH/m}\) and \(C = 1\text{ pF/m}\), also at \(1\text{ GHz}\) the substrate has a conductance \(G\) of \(0.001\text{ S/m}\). The substrate loss is solely due to dielectric relaxation loss and there is no substrate conductive loss. The resistance of the strip is \(0.5\:\Omega\text{/m}\) and the resistance of the ground plane is \(0.1\:\Omega\text{/m}\).
    1. What is the per unit length resistance of the microstrip line at \(1\text{ GHz}\)?
    2. What is the magnitude of the characteristic impedance at \(1\text{ GHz}\)?
    3. What is the conductive attenuation constant in \(\text{Np/m}\)?
    4. What is the dielectric attenuation constant of the line at \(1\text{ GHz}\) in \(\text{dB/m}\)?
27. A microstrip line operating at \(2\text{ GHz}\) has perfect metallization for both the strip and ground plane. The strip has a width of \(250\:\mu\text{m}\) and the substrate is \(250\:\mu\text{m}\) thick with a relative permittivity of \(10\) and a loss tangent of \(0.001\).
    1. What is the filling factor, \(q\), of the line?
    2. What is the line’s effective relative permittivity?
    3. What is the line’s attenuation in \(\text{Np/m}\)?
    4. What is the line’s attenuation in \(\text{dB/m}\)?
28. A \(50\:\Omega\) microstrip line operating at \(1\text{ GHz}\) has perfect metallization for both the strip and ground plane. The substrate has a relative permittivity of \(10\) and a loss tangent of \(0.001\). Without the dielectric the line has a capacitance of \(100\text{ pF/m}\).
    1. What is the line conductance in \(\text{S/m}\)?
    2. What is the line’s attenuation in \(\text{Np/m}\)?
    3. What is the line’s attenuation in \(\text{dB/m}\)?
29. Design a microstrip line having a \(50\:\Omega\) characteristic impedance. The substrate has a permittivity of \(2.3\) and is \(250\:\mu\text{m}\) thick. The operating frequency is \(18\text{ GHz}\). You need to determine the width of the microstrip line.
30. A load has an impedance \(Z = 75 +\jmath 15\:\Omega\).
    1. What is the load reflection coefficient, \(\Gamma_{L}\), with reference impedance of \(75\:\Omega\)?
    2. Design an open-circuited stub at the load that will make the impedance of the load plus the stub, call this \(Z_{1}\), be purely real. Choose a stub characteristic impedance of \(75\:\Omega\). At this stage do an electrical design only. (This requires complete electrical information such as the electrical length of the stub.)
    3. Following on from (b), now design a quarter-wave transformer between the source and the stub that will present \(50\:\Omega\) at the input. (The design must include the characteristic impedance of the transmission line and its electrical length. Thus the structure is a \(\lambda /4\) transformer, a stub, and the load.)
    4. Now convert the electrical specifications of the design into a physical design at \(1\text{ GHz}\) using microstrip technology with substrate thickness \(h = 0.5\text{ mm}\) and relative permittivity \(\varepsilon_{r} = 10\). You must design the widths and lengths of the stub and the quarterwave transformer.
31. Design a microstrip line to have a characteristic impedance of \(65\:\Omega\) at \(5\text{ GHz}\). The substrate is \(635\:\mu\text{m}\) thick with a relative permittivity of \(9.8\). Ignore the thickness of the strip. [Parallels Example 3.6.1]
    1. What is the width of the line?
    2. What is the effective permittivity of the line?
32. Design a microstrip shorted stub at \(10\text{ GHz}\) with the following characteristics:
    • Characteristic impedance of \(60\:\Omega\).
    • A substrate with a relative permittivity of \(9.6\) and thickness of \(500\:\mu\text{m}\).
    • Input impedance of \(\jmath 60\:\Omega\).

1. What is the width of the microstrip line?
2. What is the length of the line in centimeters?
3. What is the effective permittivity of the line?
4. If the line is one-quarter wavelength longer than that calculated in (b), what will the input reactance be?
5. Regardless of your calculations above, what is the input admittance of a one-quarter wavelength long shorted stub?

33. Design a microstrip line to have a characteristic impedance of \(20\:\Omega\). The microstrip is to be constructed on a substrate that is \(1\text{ mm}\) thick with a relative permittivity of \(12\). [Parallels Example 3.6.1]
    1. What is the width of the line? Ignore the thickness of the strip and frequency-dependent effects.
    2. What is the effective permittivity of the line?
34. A load has an impedance \(Z = 75 +\jmath 15\:\Omega\).
    1. What is the load reflection coefficient, \(\Gamma_{L}\), if the system reference impedance is \(75\:\Omega\)?
    2. Design a shorted stub at the load that will make the impedance of the load plus the stub, call this \(Z_{1}\), be purely real; that is, the reflection coefficient of the effective load, \(\Gamma_{1}\), has zero phase. Choose a stub characteristic impedance of \(75\:\Omega\). At this stage do an electrical design only. (This require complete electrical information, e.g. the electrical length of the stub.)
    3. Following on from (b), now design a quarter-wave transformer between the source and the stub that will present \(50\:\Omega\) at the input. (The design must include the characteristic impedance of the transmission line and its electrical length. Thus the structure is a \(\lambda /4\) transformer, a stub, and the load.)
    4. Now convert the electrical design into a physical design at \(1\text{ GHz}\) using microstrip technology with substrate thickness \(h = 0.5\text{ mm}\) and relative permittivity \(\varepsilon_{r} = 10\). You must design the widths and lengths of the stub and the quarter-wave transformer.
35. The strip of a symmetrical stripline has a width of \(1\text{ mm}\) and the ground planes of the stripline are separated by \(2\text{ mm}\). The dielectric has a relative permittivity of \(4.2\). The strip has negligible thickness.
    1. What is the effective permittivity of the stripline?
    2. What is the characteristic impedance of the stripline at \(1\text{ GHz}\)?
36. The strip of a symmetrical stripline has a width of \(500\:\mu\text{m}\) and the ground planes of the stripline are separated by \(1\text{ mm}\). The dielectric has a relative permittivity of \(10\). The strip has a thickness of \(0.1\text{ mm}\).
    1. What is the effective permittivity of the stripline?
    2. What is the characteristic impedance of the stripline?
    3. What is the total fringing capacitance in \(\text{pF/m}\)?
37. The strip of a symmetrical stripline has a width of \(200\:\mu\text{m}\) and the ground planes of the stripline are separated by \(1\text{ mm}\). The dielectric has a relative permittivity of \(4\). The strip has a thickness of \(0.1\text{ mm}\).
    1. What is the effective permittivity of the stripline?
    2. What is the characteristic impedance of the stripline?
    3. What is the total fringing capacitance in \(\text{pF/m}\)?
38. The strip of a symmetrical stripline has a width of \(50\:\mu\text{m}\) and the ground planes of the stripline are separated by \(300\:\mu\text{m}\). The dielectric has a relative permittivity of \(10\). The strip has a thickness of \(10\:\mu\text{m}\). What is the characteristic impedance of the stripline?
39. The strip of a symmetrical stripline has a width of \(0.25\text{ mm}\) and the ground planes of the stripline are separated by \(1\text{ mm}\). The dielectric has a relative permittivity of \(80\). What is the effective width of the strip?
40. The strip of a symmetrical stripline has a width of \(100\:\mu\text{m}\) and is embedded in a lossless medium that is \(400\:\mu\text{m}\) thick and has a relative permittivity of \(13\), thus the separation, \(h\), from the strip to each of the ground planes is \(200\:\mu\text{m}\).
    1. Draw the effective waveguide model of a stripline with magnetic walls and an effective strip width, \(w_{\text{eff}}\).
    2. What is the effective relative permittivity of the stripline waveguide model?
    3. What is \(w_{\text{eff}}\)?
41. A symmetrical stripline has a thin strip with a width of \(200\:\mu\text{m}\), is embedded in a dielectric of relative permittivity \(12\), and is between ground planes separated by \(500\:\mu\text{m}\). What is \(Z_{0}\) of the line? [Parallels Example 3.7.1]
42. At \(1\text{ GHz}\) a \(60\:\Omega\) stripline has the per unit parameters \(R =2\:\Omega\text{/m}\) and \(G = 1\text{ mS/m}\). What is the attenuation of the line in \(\text{dB/m}\)?
43. A \(50\:\Omega\) symmetrical stripline has a \(0.5\text{ mm}\) wide strip and the ground planes are separated by \(1.2\text{ mm}\). The strip has a sheet resistance of \(1.5\text{ m}\Omega\) and each ground plane has a sheet resistance of \(1\text{ m}\Omega\). (Ignore high frequency effects on resistance.) The substrate has a loss tangent of \(0.005\) and a relative permittivity of \(6\). [Parallels Example 3.7.2]
    1. What is the line’s effective permittivity?
    2. What is its characteristic impedance?
    3. What is the attenuation constant of the line in \(\text{dB/m}\) at \(2\text{ GHz}\)?
44. The strip of a CPW line has a width \(w = 400\:\mu\text{m}\) and separations from the in-plane grounds of \(s = 250\:\mu\text{m}\). The substrate is \(h = 1000\:\mu\text{m}\) thick and the thickness of the metal is \(t = 5\:\mu\text{m}\). What is the effective permittivity and characteristic impedance of the CPW line.
45. A CPW line with a \(250\:\mu\text{m}\) thick GaAs substrate, has a width of \(125\:\mu\text{m}\) and thickness of \(3\:\mu\text{m}\), and a gap of \(125\:\mu\text{m}\) between the strip and ground planes. [Parallels Example 3.8.1]
    1. What is the line’s effective permittivity?
    2. What is the \(Z_{0}\) of the line?

3.11.1 Exercises by Section

\(†\)challenging, \(‡\)very challenging

\(§3.2\: 1, 2, 3\)

\(§3.5\: 4, 5, 6†, 7†, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25†, 26†, 27, 28\)

\(§3.6\: 29†, 30†, 31†, 32‡, 33†\)

\(§3.7\: 34‡, 35†, 36†, 37†, 38†, 39†, 40‡, 41†, 42, 43\)

\(§3.8\: 44, 45†\)

3.11.2 Answers to Selected Exercises

6. \(12.75\)
7. (c) \(\jmath 84.1\text{ m}^{-1}\)

19. (c) \(0.579\text{ dB/m}\)
20. (a) \(13\:\Omega\text{/m}\)

25. (b) \(44.72\:\Omega\)

31. (a) \(340\:\mu\text{m}\)

33. (b) \(9.17\)
34. (b) \(100.9^{\circ}\) for open stub, \(10.89^{\circ}\) for shorted stub

39. \(245.5\:\mu\text{m}\)
40. (c) \(969\:\mu\text{m}\)