# 2.12: Exercises

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1. A coaxial line is short-circuited at one end and is filled with a dielectric with a relative permittivity of $$64$$. [Parallels Example 2.1.1]
1. What is the free-space wavelength at $$18\text{ GHz}$$?
2. What is the wavelength in the dielectric-filled coaxial line at $$18\text{ GHz}$$?
3. The first resonance of the coaxial resonator is at $$18\text{ GHz}$$. What is the physical length of the resonator?
2. A transmission line has the following $$RLGC$$ parameters: $$R = 100\:\Omega\text{/m},\: L = 85\text{ nH/m},\: G = 1\text{ S/m},$$ and $$C = 150\text{ pF/m}$$. Consider a traveling wave on the transmission line with a frequency of $$1\text{ GHz}$$. [Parallels Example 2.2.2]
1. What is the attenuation constant?
2. What is the phase constant?
3. What is the phase velocity?
4. What is the characteristic impedance of the line?
5. What is the group velocity?
3. A transmission line has the per-unit length parameters $$L = 85\text{ nH/m},\: G = 1\text{ S/m},$$ and $$C = 150\text{ pF/m}$$. Use a frequency of $$1\text{ GHz}$$. [Parallels Example 2.2.2]
1. What is the phase velocity if $$R = 0\:\Omega\text{/m}$$?
2. What is the group velocity if $$R = 0\:\Omega\text{/m}$$?
3. If $$R = 10\text{ k}\Omega\text{/m}$$ what is the phase velocity?
4. If $$R = 10\text{ k}\Omega\text{/m}$$ what is the group velocity?
4. A line is $$10\text{ cm}$$ long and at the operating frequency the phase constant $$\beta$$ is $$40\text{ rad/m}$$. What is the electrical length of the line? [Parallels Example 2.1.2]
5. A dielectric-filled lossless transmission line carrying a $$1\text{ GHz}$$ signal has the parameters $$L = 80\text{ nH/m}$$ and $$C = 200\text{ pF/m}$$. When the dielectric is replaced by air the line’s capacitance is $$C_{\text{air}} = 50\text{ pF/m}$$. What is the relative permittivity of the dielectric?
6. A coaxial transmission line is filled with lossy dielectric material with a relative permittivity of $$5 −\jmath 0.2$$. If the line is air-filled it would have a characteristic impedance of $$100\:\Omega$$. What is the input impedance of the line if it is $$1\text{ km}$$ long? Use reasonable approximations. [Hint: Does the termination matter?]
7. A transmission line has the per unit length parameters $$R = 2\Omega\text{/cm},\: L=100\text{ nH/m},\: G = 1\text{ mS/m},\: C = 200\text{ pF/m}$$.
1. What is the propagation constant of the line at $$5\text{ GHz}$$?
2. What is the characteristic impedance of the line at $$5\text{ GHz}$$?
3. Plot the magnitude of the characteristic impedance versus frequency from $$100\text{ MHz}$$ to $$10\text{ GHz}$$.
8. A line is $$20\text{ cm}$$ long and at $$1\text{ GHz}$$ the phase constant $$\beta$$ is $$20\text{ rad/m}$$. What is the electrical length of the line in degrees?
9. What is the electrical length of a line that is a quarter of a wavelength long,
1. in degrees?
10. A lossless transmission line has an inductance of $$8\text{ nH/cm}$$ and a capacitance of $$40\text{ pF/cm}$$.
1. What is the characteristic impedance of the line?
2. What is the phase velocity on the line at $$1\text{ GHz}$$?
11. A $$50\:\Omega$$ coaxial airline is a coaxial line without a dielectric (i.e., it is air-filled) and with thin dielectric discs supporting the inner conductor have negligible effect. If the air is replaced by a dielectric having a relative permittivity of $$20$$, what is the characteristic impedance of the dielectric-filled line?
12. A transmission line has an attenuation of $$2\text{ dB/m}$$ and a phase constant of $$25\text{ radians/m}$$ at $$2\text{ GHz}$$. [Parallels Example 2.2.3]
1. What is the complex propagation constant of the transmission line?
2. If the capacitance of the line is $$50\text{ pF}\cdot\text{m}^{−1}$$ and $$G = 0$$, what is the characteristic impedance of the line?
13. A very low-loss microstrip transmission line has the following per unit length parameters: $$R = 2\:\Omega\text{/m},\: L = 80\text{ nH/m},\: C = 200\text{ pF/m},$$ and $$G = 1\:\mu\text{S/m}$$.
1. What is the characteristic impedance of the line if loss is ignored?
2. What is the attenuation constant due to conductor loss?
3. What is the attenuation constant due to dielectric loss?
14. A lossless transmission line carrying a $$1\text{ GHz}$$ signal has the following per unit length parameters: $$L = 80\text{ nH/m},\: C = 200\text{ pF/m}$$.
1. What is the attenuation constant?
2. What is the phase constant?
3. What is the phase velocity?
4. What is its characteristic impedance?
15. A transmission line has a characteristic impedance $$Z_{0}$$ and is terminated in a load with a reflection coefficient of $$0.8\angle 45^{\circ}$$. A forward-traveling voltage wave on the line has a power of $$1\text{ dBm}$$.
1. How much power is reflected by the load?
2. What is the power delivered to the load?
16. A transmission line has an attenuation of $$0.2\text{ dB/cm}$$ and a phase constant of $$50\text{ radians/m}$$ at $$1\text{ GHz}$$.
1. What is the complex propagation constant of the transmission line?
2. If the capacitance of the line is $$100\text{ pF/m}$$ and $$G = 0$$, what is the complex characteristic impedance of the line?
3. If the line is driven by a source modeled as an ideal voltage and a series impedance, what is the impedance of the source for maximum transfer of power to the transmission line?
4. If $$1\text{ W}$$ is delivered (i.e. in the forward-traveling wave) to the transmission line by the generator, what is the power in the forward-traveling wave on the line at $$2\text{ m}$$ from the generator?
17. The transmission line shown in Figure 2.5.1 consists of a source with Thevenin impedance $$Z_{1} = 40\:\Omega$$ and source $$E = 5\text{ V}$$ (peak) connected to a $$\lambda/4$$ long line of characteristic impedance $$Z_{01} = 50\:\Omega$$, which in turn is connected to an infinitely long line of characteristic impedance $$Z_{02} = 100\:\Omega$$. The transmission lines are lossless. Two reference planes are shown in Figure 2.5.1. At reference plane $$\mathsf{1}$$ the incident power is $$P_{I1}$$, the reflected power is $$P_{R1}$$, and the transmitted power is $$P_{T1}$$. $$P_{I2},\: P_{R2},$$ and $$(P_{T2})$$ are similar quantities at reference plane $$\mathsf{2}$$. [Parallels Examples 2.6.4 and 2.6.6]
1. What is $$P_{I1}$$?
2. What is $$P_{T2}$$?
18. A lossless, $$10\text{ cm}$$-long, $$75\:\Omega$$ transmission line is driven by a $$1\text{ GHz}$$ generator with a Thevenin equivalent impedance of $$50\:\Omega$$. The maximum power that can be delivered to a load attached to the generator is $$2\text{ W}$$. The line is terminated in a load that has a complex reflection coefficient (referred to $$50\:\Omega$$) of $$0.65 +\jmath 0.65$$. The effective relative permittivity, $$\varepsilon_{e}$$, of the non-magnetic transmission line is $$2.0$$.
1. Calculate the forward-traveling voltage wave (at the generator end of the transmission line). Ignore reflections from the load at the end of the $$75\:\Omega$$ line.
2. What is the load impedance?
3. What is the wavelength of the forward-traveling voltage wave?
4. What is the VSWR on the line?
5. What is line’s propagation constant?
6. What is the input reflection coefficient (at the generator end) of the line?
7. What is the power delivered to the load?
19. The first resonance of a lossless non-magnetic open-circuited transmission line is at $$30\text{ GHz}$$. The effective relative permittivity of the line is $$12$$.
1. What is the resonator’s input impedance?
2. Draw its LC equivalent circuit.
3. What is its electrical length?
4. What is its physical length?
20. An open-circuited transmission line is used as a resonator. What is the electrical length of the line at its first resonance?
21. The second resonance of an open-circuited transmission line is used as a resonator.
1. What is its input impedance?
2. What is its electrical length?
22. A lossless transmission line is driven by a $$1\text{ GHz}$$ generator having a Thevenin equivalent impedance of $$50\:\Omega$$. The transmission line is lossless, has a characteristic impedance of $$75\:\Omega$$, and is infinitely long. The maximum power that can be delivered to a load attached to the generator is $$2\text{ W}$$.
1. What is the total (phasor) voltage at the input to the transmission line?
2. What is the magnitude of the forward-traveling voltage wave at the generator side of the line?
3. What is the magnitude of the forward-traveling current wave at the generator side of the line?
23. A transmission line is terminated in a short circuit. What is the ratio of the forward- and backward-traveling voltage waves at the termination? [Parallels Example 2.3.1]
24. A $$50\:\Omega$$ transmission line is terminated in a $$40\:\Omega$$ load. What is the ratio of the forward- to the backward-traveling voltage waves at the termination? [Parallels Example 2.3.1]
25. A $$50\:\Omega$$ transmission line is terminated in an open circuit. What is the ratio of the forward-to the backward-traveling voltage waves at the termination? [Parallels Example 2.3.1]
26. The resonator below is constructed from a $$3.0\text{ cm}$$ length of $$100\:\Omega$$ air-filled coaxial line, shorted at one end and terminated with a capacitor at the other end.

Figure $$\PageIndex{1}$$

1. What is the lowest resonant frequency of this circuit without the capacitor (ignore the $$10\text{ k}\Omega$$ resistor)?
2. What is the capacitor value to achieve the lowest-order resonance at $$6.0\text{ GHz}$$ (ignore the $$10\text{ k}\Omega$$ resistor)?
3. Assume that loss is introduced by placing a $$10\text{ k}\Omega$$ resistor in parallel with the capacitor. What is the $$Q$$ of the circuit?
4. Approximately what is the bandwidth of the circuit?
1. A $$50\:\Omega$$ transmission line is terminated in a load that results in a reflection coefficient of $$0.5+\jmath 0.5$$.
1. What is the load impedance?
2. What is the $$\text{VSWR}$$ on the line?
3. What is the input impedance if the line is one-half wavelength long?
2. Communication filters are often constructed using several shorted transmission line resonators that are coupled by passive elements such as capacitors. Consider a coaxial line that is short-circuited at one end. The dielectric filling the coaxial line has a relative permittivity of $$64$$ and the resonator is to be designed to resonate at a center frequency, $$f_{0}$$, of $$800\text{ MHz}$$. [Parallels Example 2.7.2]
1. What is the wavelength in the dielectricfilled coaxial line?
2. What is the form of the equivalent circuit (in terms of inductors and capacitors) of the one-quarter wavelength long resonator if the coaxial line is lossless?
3. What is the length of the resonator?
4. If the diameter of the inner conductor of the coaxial line is $$2\text{ mm}$$ and the inside diameter of the outer conductor is $$5\text{ mm}$$, what is the characteristic impedance of the coaxial line?
5. Calculate the input admittance of the dielectric-filled coaxial line at $$0.99f_{0},\: f_{0},$$ and $$1.01f_{0}$$. Determine the numerical derivative of the line admittance at $$f_{0}$$.
6. Derive the values of the equivalent circuit of the resonator at the resonant frequency and derive the equivalent circuit of the resonator. Hint: Match the derivative expression derived in (e) with the actual derivative derived in Example 2.7.2.
3. Develop an analytic formula relating a reflection coefficient $$(\Gamma_{1})$$ in one reference system $$(Z_{01})$$ to a reflection coefficient $$(\Gamma_{2})$$ in another reference system $$(Z_{02})$$.
4. A line has a characteristic impedance $$Z_{0}$$ and is terminated in a load with a reflection coefficient of $$0.8$$. A forward-traveling voltage wave on the line has a power of $$1\text{ W}$$.
1. How much power is reflected by the load?
2. What is the power delivered to the load?
5. A load consists of a shunt connection of a capacitor of $$10\text{ pF}$$ and a resistor of $$25\:\Omega$$. The load terminates a lossless $$50\:\Omega$$ transmission line. The operating frequency is $$1\text{ GHz}$$. [Parallels Example 2.3.2]
1. What is the impedance of the load?
2. What is the normalized impedance of the load (normalized to the characteristic impedance of the line)?
3. What is the reflection coefficient of the load?
4. What is the current reflection coefficient of the load?
5. What is the standing wave ratio (SWR)?
6. What is the current standing wave ratio (ISWR)?
6. A $$50\:\Omega$$ air-filled transmission line is connected between a $$40\text{ GHz}$$ source with a Thevenin equivalent impedance of $$50\:\Omega$$ and a load. The SWR on the line is $$3.5$$.
1. What is the magnitude of the reflection coefficient, $$\Gamma_{L}$$, at the load?
2. What is the phase constant, $$\beta$$, of the line?
3. If the first minimum of the standing wave voltage on the transmission line is $$2\text{ mm}$$ from the load, determine the electrical distance (in degrees) of the SWR minimum from the load.
4. Determine the angle of $$\Gamma_{L}$$ at the load.
5. What is $$\Gamma_{L}$$ in magnitude-phase form?
6. What is $$\Gamma_{L}$$ in rectangular form?
7. Determine the load impedance, $$Z_{L}$$.
7. A load consists of a resistor of $$100\:\Omega$$ in parallel with a $$5\text{ pF}$$ capacitor with an electrical signal at $$2\text{ GHz}$$.
1. What is the load impedance?
2. What is the reflection coefficient in a $$50\:\Omega$$ reference system?
3. What is the SWR on a $$50\:\Omega$$ transmission line connected to the load?
8. An amplifier is connected to a load by a transmission line matched to the amplifier. If the SWR on the line is $$1.5$$, what percentage of the available amplifier power is absorbed by the load?
9. An output amplifier can tolerate a mismatch with a maximum SWR of $$2.0$$. The amplifier is characterized by a Thevenin equivalent circuit with an impedance of $$50\:\Omega$$ and is connected directly to an antenna characterized by a load resistance $$R_{L}$$. Determine the tolerance limits on $$R_{L}$$ so that the amplifier does not self-destruct.
10. A load has a reflection coefficient of $$0.5$$ when referred to $$50\:\Omega$$. The load is placed at the end of a $$100\:\Omega$$-transmission line.
1. What is the complex ratio of the forward-traveling wave to the backward-traveling wave on the $$100\:\Omega$$ line at the load end of the line?
2. What is the VSWR on the $$100\:\Omega$$ line?
11. A load has a reflection coefficient of $$0.5$$ when referred to $$50\:\Omega$$. The load is at the end of a line with a $$50\:\Omega$$ characteristic impedance.
1. If the line has an electrical length of $$45^{\circ}$$, what is the reflection coefficient calculated at the input of the line?
2. What is the VSWR on the $$50\:\Omega$$ line?
12. A $$100\:\Omega$$ resistor in parallel with a $$5\text{ pF}$$ capacitor terminates a $$100\:\Omega$$ transmission line. Calculate the SWR on the line at $$2\text{ GHz}$$.
13. A lossless $$50\:\Omega$$ transmission line has a $$50\:\Omega$$ generator at one end and is terminated in $$100\:\Omega$$. What is the VSWR on the line?
14. A lossless $$75\:\Omega$$ line is driven by a $$75\:\Omega$$ generator. The line is terminated in a load that with a reflection coefficient (referred to $$50\:\Omega$$) of $$0.5 + \jmath 0.5$$. What is the VSWR on the line?
15. A load with a $$20\text{ pF}$$ capacitor in parallel with a $$50\:\Omega$$ resistor terminates a $$25\:\Omega$$ line. The operating frequency is $$5\text{ GHz}$$. [Parallels Example 2.3.3]
1. What is the VSWR?
2. What is ISWR?
16. A load $$Z_{L} = 55−\jmath 55\:\Omega$$ and the system reference impedance, $$Z_{0}$$, is $$50\:\Omega$$. [Parallels Example 2.3.4]
1. What is the load reflection coefficient $$\Gamma_{L}$$?
2. What is the current reflection coefficient?
3. What is the VSWR on the line?
4. What is the ISWR on the line?
5. Now consider a source connected directly to the load. The source has a Thevenin equivalent impedance $$Z_{G} = 60\:\Omega$$ and an available power of $$1\text{ W}$$. Use $$\Gamma_{L}$$ to find the power delivered to $$Z_{L}$$.
6. What is the total power absorbed by $$Z_{G}$$?
17. A slotted line, as shown in Figure 2.3.7(c), is used to characterize a $$50\:\Omega$$ line terminated in a load $$Z_{L}$$. $$V_{\text{max}} = 1\text{ V}$$ and $$V_{\text{min}} = 0.1\text{ V}$$, and the first minimum is $$5\text{ cm}$$ from the load. The guide wavelength is $$30\text{ cm}$$. What is $$Z_{L}$$? [Parallels Example 2.3.5]
18. A shorted coaxial line is used as a resonator. The first resonance is determined to be a parallel resonance and is at $$1\text{ GHz}$$.
1. Draw the lumped-element equivalent circuit of the resonator.
2. What is the electrical length of the resonator?
3. What is the impedance looking into the line at resonance?
4. If the resonator is $$\lambda/4$$ longer, what is the impedance of the resonator now?
19. A load of $$100\:\Omega$$ is to be matched to a transmission line with a characteristic impedance of $$50\:\Omega$$. Use a quarter-wave transformer. What is the characteristic impedance of the quarter-wave transformer?
20. Determine the characteristic impedance of a quarter-wave transformer used to match a load of $$50\:\Omega$$ to a generator with a Thevenin equivalent impedance of $$75\:\Omega$$.
21. A transmission line is to be inserted between a $$5\:\Omega$$ line and a $$50\:\Omega$$ load so that there is maximum power transfer to the $$50\:\Omega$$ load at $$20\text{ GHz}$$.
1. How long is the inserted line in terms of wavelengths at $$20\text{ GHz}$$?
2. What is the characteristic impedance of the line at $$20\text{ GHz}$$?
22. The resonator below is constructed from a $$3.0\text{ cm}$$ length of $$100\:\Omega$$ air-filled coaxial line shorted at one end and terminated with a capacitor at the other end:

Figure $$\PageIndex{2}$$

1. What is the lowest resonant frequency of this circuit without the capacitor (ignore the resistor)?
2. What is the capacitor value required to achieve resonance at $$6.0\text{ GHz}$$?
3. Assume that loss is introduced by placing a $$10\text{ k}\Omega$$ resistor in parallel with the capacitor. What is the $$Q$$ of the circuit?
4. What is the bandwidth of the circuit?
1. A coaxial transmission line is filled with lossy material with a relative permittivity of $$5 −\jmath 0.2$$. If the line is air-filled it would have a characteristic impedance of $$100\:\Omega$$.
1. What is the characteristic impedance of the dielectric-filled line?
2. What is the propagation constant at $$500\text{ MHz}$$?
3. What is the input impedance of the line if it has an electrical length of $$280^{\circ}$$ and is terminated in a $$35\:\Omega$$ resistor?
2. A coaxial line is filled with a very slightly lossy material with a relative permittivity of $$5$$. The line would have a characteristic impedance of $$100\:\Omega$$ if it was air-filled.
1. What is the characteristic impedance of the dielectric-filled line?
2. What is the propagation constant at $$500\text{ MHz}$$? Use the fact that the velocity of an EM wave in a lossless air-filled line is the same as that of free-space propagation in air.
3. What is the input impedance of the line if it has an electrical length of $$90^{\circ}$$ and it is terminated in a $$35\:\Omega$$ resistor?
4. What is the input impedance of the line if it has an electrical length of $$180^{\circ}$$ and is terminated in an impedance $$\jmath 35\:\Omega$$?
5. What is the input impedance of the line if it is $$1\text{ km}$$ long? Use reasonable approximations.
3. A lossy transmission line with a characteristic impedance of $$60 −\jmath 2\:\Omega$$ is driven by a generator with a Thevenin equivalent impedance $$Z_{g}$$. If the line is infinitely long, what is $$Z_{g}$$ for maximum power transfer from the generator to the line?
4. A $$25\:\Omega$$-transmission line is driven by a generator with an available power of $$23\text{ dBm}$$ and a Thevenin equivalent impedance of $$60\:\Omega$$. [Parallels Example 2.6.3]
1. What is the Thevenin equivalent generator voltage?
2. What is the magnitude of the forward-traveling voltage wave on the line? Assume the line is infinitely long.
3. What is the power of the forward-traveling voltage wave?
5. A open-circuited coaxial line is used as a resonator. The first resonance is a series resonance at $$2\text{ GHz}$$. [Parallels Example 2.5.1]
1. Draw the lumped-element equivalent circuit of the resonator.
2. What is the resonator’s electrical length?
3. What is the impedance looking into the line at resonance?
4. If the resonator is $$3\lambda_{g}/4$$ longer, what is the input impedance of the resonator?
6. The forward-traveling wave on a $$60\:\Omega$$ line has a power of $$2\text{ mW}$$. The line is terminated in a resistance of $$50\:\Omega$$. How much power is delivered to the $$50\:\Omega$$ load.
7. The forward-traveling wave on a $$40\:\Omega$$ line has a power of $$2\text{ mW}$$. The line is terminated in a resistance of $$60\:\Omega$$. How much power is in the backward traveling wave?
8. The forward-traveling wave on a $$60\:\Omega$$ line has a power of $$2\text{ mW}$$. The line is terminated in a resistance of $$50\:\Omega$$. Draw the lumped-element equivalent circuit at the interface of the line and the load. [Parallels Example 2.6.1]
9. A source is connected to a load by a one wavelength long transmission line having a loss of $$1.5\text{ dB}$$. The source reflection coefficient (referred to the transmission line) is $$0.2$$ and the load reflection coefficient is $$0.5$$.
1. What is the transmission coefficient?
2. Draw the bounce diagram using the transmission and reflection coefficients. Determine the overall effective transmission coefficient from the source to the load. Calculate the power delivered to the load from a source with an available power of $$600\text{ mW}$$.
10. Consider a coaxial line that is short-circuited at one end. The dielectric filling the line has $$\varepsilon_{r} = 20$$ and the line has its lowest frequency resonance at $$2.4\text{ GHz}$$. [Parallels Example 2.7.1]
1. What is the guide wavelength?
2. Draw the resonator’s equivalent circuit.
3. What is the resonator’s physical length?
11. Consider a lossless coaxial line that is open-circuited at one end and is used as a resonator that is resonant at $$f_{0} = 2.4\text{ GHz}$$. The line’s dielectric has $$\varepsilon_{r} = 81$$. [Parallels Example 2.7.2]
1. What is the wavelength in the line?
2. Draw the lumped-element equivalent circuit of a $$\lambda_{g}/4$$ long resonator?
3. What is the resonator’s physical length?
4. What is the derivative with respect to frequency of the admittance of the $$LC$$ equivalent circuit developed in (b).
5. If the diameter of the inner conductor of the line is $$1\text{ mm}$$ and the inside diameter of the outer conductor is $$3\text{ mm}$$, what is the characteristic impedance of the line?
6. Determine the numerical frequency derivative of the line admittance at $$f_{0}$$.
7. Derive the values of the equivalent circuit of the resonator at resonance.
12. Develop the lumped-element model of a half wavelength long line having characteristic impedance $$Z_{0}$$. [Parallels Example 2.7.3]
13. The diameter of the inner conductor of a coaxial line is $$2\text{ mm}$$ and the interior diameter of the outer conductor $$8\text{ mm}$$. The coaxial line is filled with polyimide which has a relative permittivity of $$3.2$$.
1. What is the characteristic impedance of the line?
2. Describe the conditions by which a non-TEM mode can be supported. Refer to two different families of higher-order modes.
3. For the coaxial line here, at what frequency will a second propagating mode be first supported?

## 2.12.1 Exercises By Section

$$†$$challenging, $$‡$$very challenging

$$§2.1\: 1$$

$$§2.2\: 2†, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14†, 15$$

$$§2.3\: 16†, 17, 18‡, 19‡, 20, 21, 22, 23, 24, 25, 26†, 27‡, 28‡, 29, 30†, 31†, 32‡, 33‡, 34†, 35†, 36†, 37, 38, 39, 40, 41, 42, 43$$

$$§2.4\: 44†, 45, 46, 47$$

$$§2.5\: 48‡, 49†, 50†, 51, 52‡, 53$$

$$§2.6\: 54, 55, 56, 57†$$

$$§2.7\: 58, 59†, 60$$

$$§2.9\: 61$$

## 2.12.2 Answers to Selected Exercises

1. $$11.2\:\Omega$$
2. $$0.23+\jmath 25\text{ m}^{-1}$$
1. (b) $$74.0\text{ mW}$$
1. (c) $$1.17\text{ cm}$$
1. (e) $$\jmath 4.55\cdot 10^{-11}\text{ S}\cdot\text{ s}$$
1. (f) $$0.544+\jmath 0.116$$
2. (f) $$41.45$$
1. $$25\:\Omega\leq Z_{L}\leq 100\:\Omega$$
1. $$61.2\:\Omega$$
1. $$354.2\text{ mW}$$

This page titled 2.12: Exercises is shared under a not declared license and was authored, remixed, and/or curated by Michael Steer.