# 3.11: Exercises

- Page ID
- 41032

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- 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 \(1\).
- 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?
- 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?
- 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?
- 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?
- 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?
- 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]
- What is the effective relative permittivity of the line?
- What is the characteristic impedance of the line?
- What is the propagation constant at \(3\text{ GHz}\) ignoring any losses?
- 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 \(3\text{ GHz}\)?

- 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.
- What is line’s effective permittivity.
- What is its characteristic impedance?

- A \(600\:\mu\text{m}\)-wide microstrip line on a \(500\:\mu\text{m}\) thick alumina substrate. Use Table 3.5.2.
- What is line’s effective permittivity.
- What is its characteristic impedance?

- A microstrip line on a \(1\text{ mm}\)-thick FR4 substrate has a width of \(0.497\text{ mm}\). Use Table 3.5.2.
- What is line’s effective permittivity.
- What is its characteristic impedance?

- Consider a microstrip line on a substrate with a relative permittivity of \(12\) and thickness of \(1\text{ mm}\).
- What is the minimum effective permittivity of the microstrip line if there is no limit on the minimum or maximum width of the strip?
- What is the maximum effective permittivity of the microstrip line if there is no limit on the minimum or maximum width of the strip?

- 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?
- The substrate of a microstrip line has a relative permittivity of \(16\) but the calculated effective permittivity is \(12\). What is the filling factor?
- 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?
- 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?
- 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 \(8\).
- What is the filling factor, \(q\), of the line?
- What is the line’s effective relative permittivity?
- What is the characteristic impedance of the line?

- 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 \(q\)?
- 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?
- 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.
- What is the total resistance of the line in \(\Omega\text{/m}\)?
- What is the attenuation constant in \(\text{Np/m}\)?
- What is the attenuation constant in \(\text{dB/cm}\)?

- A microstrip line has a characteristic impedance of \(50\:\Omega\), a strip resistance of \(10\:\Omega\text{/m}\), and a ground plane resistance of \(3\:\Omega\text{/m}\).
- What is the total resistance of the line in \(\Omega\text{/m}\)?
- What is the attenuation constant in \(\text{Np/m}\)?
- What is the attenuation constant in \(\text{dB/cm}\)?

- 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.
- What is the low frequency resistance (in \(\Omega\text{/m}\)) of the strip?
- What is the low frequency resistance of the ground plane?
- What is the total low frequency resistance of the microstrip line?

- 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.
- What is the low frequency resistance (in \(\Omega\text{/m}\)) of the strip?
- What is the low frequency resistance of the ground plane?
- What is the total low frequency resistance of the microstrip line?
- What is the attenuation in \(\text{dB/m}\) of the line at low frequencies?

- 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]
- 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 \(7\).
- What is the conductance of the line in \(\text{S/m}\)?
- What is the attenuation constant in \(\text{Np/m}\)?
- What is the attenuation constant in \(\text{dB/cm}\)?

- 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.
- What is \(G\) at \(1\text{ GHz}\)?
- What is the magnitude of the characteristic impedance at \(1\text{ GHz}\)?
- What is the dielectric attenuation constant of the line at \(1\text{ GHz}\) in \(\text{dB/m}\)?

- 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 \(0.1\:\Omega\text{/m}\).
- What is the per unit length resistance of the microstrip line at \(1\text{ GHz}\)?
- What is the magnitude of the characteristic impedance at \(1\text{ GHz}\)?
- What is the conductive attenuation constant in \(\text{Np/m}\)?
- What is the dielectric attenuation constant of the line at \(1\text{ GHz}\) in \(\text{dB/m}\)?

- 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 \(0.001\).
- What is the filling factor, \(q\), of the line?
- What is the line’s effective relative permittivity?
- What is the line’s attenuation in \(\text{Np/m}\)?
- What is the line’s attenuation in \(\text{dB/m}\)?

- 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 \(100\text{ pF/m}\).
- What is the line conductance in \(\text{S/m}\)?
- What is the line’s attenuation in \(\text{Np/m}\)?
- What is the line’s attenuation in \(\text{dB/m}\)?

- 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.
- A load has an impedance \(Z = 75 +\jmath 15\:\Omega\).
- What is the load reflection coefficient, \(\Gamma_{L}\), with reference impedance of \(75\:\Omega\)?
- 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.)
- 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.)
- 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.

- 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]
- What is the width of the line?
- What is the effective permittivity of the line?

- 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 \(\jmath 60\:\Omega\).

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

- 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]
- What is the width of the line? Ignore the thickness of the strip and frequency-dependent effects.
- What is the effective permittivity of the line?

- A load has an impedance \(Z = 75 +\jmath 15\:\Omega\).
- What is the load reflection coefficient, \(\Gamma_{L}\), if the system reference impedance is \(75\:\Omega\)?
- 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.)
- 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.)
- 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.

- 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.
- What is the effective permittivity of the stripline?
- What is the characteristic impedance of the stripline at \(1\text{ GHz}\)?

- 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 \(0.1\text{ mm}\).
- What is the effective permittivity of the stripline?
- What is the characteristic impedance of the stripline?
- What is the total fringing capacitance in \(\text{pF/m}\)?

- 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 \(0.1\text{ mm}\).
- What is the effective permittivity of the stripline?
- What is the characteristic impedance of the stripline?
- What is the total fringing capacitance in \(\text{pF/m}\)?

- 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?
- 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?
- 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 \(200\:\mu\text{m}\).
- Draw the effective waveguide model of a stripline with magnetic walls and an effective strip width, \(w_{\text{eff}}\).
- What is the effective relative permittivity of the stripline waveguide model?
- What is \(w_{\text{eff}}\)?

- 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]
- 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 \(\text{dB/m}\)?
- 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]
- What is the line’s effective permittivity?
- What is its characteristic impedance?
- What is the attenuation constant of the line in \(\text{dB/m}\) at \(2\text{ GHz}\)?

- 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.
- 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]
- What is the line’s effective permittivity?
- 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

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

- (c) \(0.579\text{ dB/m}\)
- (a) \(13\:\Omega\text{/m}\)

- (b) \(44.72\:\Omega\)

- (a) \(340\:\mu\text{m}\)

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

- \(245.5\:\mu\text{m}\)
- (c) \(969\:\mu\text{m}\)