# 5.16: Exercises

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1. The Thevenin equivalent output impedance of each of the amplifiers in Figure 5.12.8(a) is $$5\:\Omega$$, and the system impedance, $$R_{0}$$, is $$75\:\Omega$$. Choose the transformer windings for maximum power transfer. [Parallels Example 5.12.4]
2. A spiral inductor is modeled as an ideal inductor of $$10\text{ nH}$$ in series with a $$5\:\Omega$$ resistor. What is the $$Q$$ of the spiral inductor at $$1\text{ GHz}$$?
3. Consider the design of a $$50\text{ dB}$$ resistive $$\mathsf{T}$$ attenuator in a $$75\:\Omega$$ system. [Parallels Example 5.5.1]
1. Draw the topology of the attenuator.
2. Write down the design equations.
3. Complete the design of the attenuator.
4. Consider the design of a $$50\text{ dB}$$ resistive Pi attenuator in a $$75\:\Omega$$ system. [Parallels Example 5.5.1]
1. Draw the topology of the attenuator.
2. Write down the design equations.
3. Complete the design of the attenuator.
5. A $$20\text{ dB}$$ attenuator in a $$17\:\Omega$$ system is ideally matched at both the input and output. Thus there are no reflections and the power delivered to the load is reduced by $$20\text{ dB}$$ from the applied power. If a $$5\text{ W}$$ signal is applied to the attenuator, how much power is dissipated in the attenuator?
6. A resistive Pi attenuator has shunt resistors of $$R_{1} = R_{2} = 294\:\Omega$$ and a series resistor $$R_{3} = 17.4\:\Omega$$. What is the attenuation (in decibels) and the characteristic impedance of the attenuator?
7. A resistive Pi attenuator in a system with characteristic impedance $$Z_{0}$$ has equal shunt resistors of $$R_{1} = R_{2}$$ and a series resistor $$R_{3}$$. Show that $$Z_{0} = \sqrt{(R_{1}^{2}R_{3})/(2R_{1} + R_{3})}$$ and the attenuation factor $$K = \sqrt{(R_{1} + Z_{0})/(R_{1} − Z_{0})}$$. [Start with Equation (5.5.2).]
8. Design a resistive Pi attenuator with an attenuation of $$10\text{ dB}$$ in a $$100\:\Omega$$ system.
9. Design a $$3\text{ dB}$$ resistive Pi attenuator in a $$50\:\Omega$$ system.
10. A resistive Pi attenuator has shunt resistors $$R_{1} = R_{2} = 86.4\:\Omega$$ and a series resistor $$R_{3} = 350\:\Omega$$. What is the attenuation (in decibels) and the system impedance of the attenuator?
11. Derive the $$50\:\Omega$$ scattering parameters of the ideal transformer shown below where the number of windings on the secondary side (Port $$\mathsf{2}$$) is twice the number of windings on the primary side (Port $$\mathsf{1}$$).

Figure $$\PageIndex{1}$$

1. What is $$S_{11}$$? [Hint: Terminate Port $$\mathsf{2}$$ in $$50\:\Omega$$ and determine the input reflection coefficient.]
2. What is $$S_{21}$$?
3. What is $$S_{22}$$?
4. What is $$S_{12}$$?
1. Derive the two-port $$50\:\Omega$$ scattering parameters of the magnetic transformer below. The primary (Port $$\mathsf{1}$$) has $$10$$ turns, the secondary (Port $$\mathsf{2}$$) has $$25$$ turns.

Figure $$\PageIndex{2}$$

1. What is $$S_{11}$$?
2. What is $$S_{21}$$?
3. What is $$S_{22}$$?
4. What is $$S_{12}$$?
1. An ideal quadrature hybrid has the scattering parameters
$S_{90^{\circ}}=\frac{1}{\sqrt{2}}\left[\begin{array}{cccc}{0}&{-\jmath}&{1}&{0}\\{-\jmath}&{0}&{0}&{1}\\{1}&{0}&{0}&{-\jmath}\\{0}&{1}&{-\jmath}&{0}\end{array}\right]\nonumber$
Draw the signal flow graph of the hybrid, labeling each of the edges and assigning $$a_{1},\: b_{1},$$ etc., to the nodes. (Do not start with the SFG of a generic 4-port network.)
2. An ideal $$180^{\circ}$$ hybrid has the scattering parameters
$S_{180^{\circ}}=\frac{1}{\sqrt{2}}\left[\begin{array}{cccc}{0}&{1}&{-1}&{0}\\{1}&{0}&{0}&{1}\\{-1}&{0}&{0}&{1}\\{0}&{1}&{1}&{0}\end{array}\right]\nonumber$
Draw the signal flow graph of the hybrid, labeling each of the edges and assigning $$a_{1},\: b_{1},$$ etc., to the nodes. (Note, do not start with the SFG of a generic 4-port network.)
3. A signal is applied to Ports $$\mathsf{2}$$ and $$\mathsf{3}$$ of a $$180^{\circ}$$ hybrid, as shown in Figure 5.8.2(b). If the signal consists of a differential component of $$0\text{ dBm}$$ and a common mode component of $$10\text{ dBm}$$:
1. Determine the power delivered at Port $$\mathsf{1}$$.
2. Determine the power delivered at Port $$\mathsf{4}$$. Assume that the hybrid is lossless.
4. Silicon RFICs use differential signal paths to minimize the introduction of substrate noise. As well, differential amplifiers are an optimum topology in current-biased circuits. Off-chip signals are often on microstrip lines and so the source and load, being off-chip, are not differential. The off-chip circuits are then called single-ended. Using $$180^{\circ}$$ hybrids, diagrams, and explanations, outline a system architecture accommodating this mixed differential and single-ended environment.
5. Consider the hybrid shown in the figure below. If the number of windings of Coils $$\mathsf{2}$$ and $$\mathsf{3}$$ are twice the number of windings of Coil $$\mathsf{1}$$, show that for matched hybrid operation $$2Z_{2} = Z_{3} = 8Z_{0}$$.

Figure $$\PageIndex{3}$$

1. The balun of Figure 5.9.3 transforms an unbalanced system with a system impedance of $$Z_{0}$$ to a balanced system with an impedance of $$4Z_{0}$$. The actual impedance transformation is determined by the number of windings of the coils. Design a balun of the type shown in Figure 5.9.3 that transforms an unbalanced $$50\:\Omega$$ system to a balanced $$377\:\Omega$$ system. [Hint: Find the ratio of the windings of the coils.]
2. A balun can be realized using a wire-wound transformer, and by changing the number of windings on the transformer it is possible to achieve impedance transformation as well as balanced-to-unbalanced functionality. A $$500\text{ MHz}$$ balun based on a magnetic transformer is required to achieve impedance transformation from an unbalanced impedance of $$50\:\Omega$$ to a balanced impedance of $$200\:\Omega$$. If there are $$20$$ windings on the balanced port of the balun transformer, how many windings are there on the unbalanced port of the balun?
3. Design a lumped-element two-way power splitter in a $$75\:\Omega$$ system at $$1\text{ GHz}$$. Base your design on a Wilkinson power divider.
4. Design a three-way power splitter in a $$75\:\Omega$$ system. Base your design on a Wilkinson power divider using transmission lines and indicate lengths in terms of wavelengths.
5. Design a lumped-element three-way power splitter in a $$75\:\Omega$$ system at $$1\text{ GHz}$$. Base your design on a Wilkinson power divider.
6. A resistive power splitter is a three-port device that takes power input at Port $$\mathsf{1}$$ and delivers power at Ports $$\mathsf{2}$$ and $$\mathsf{3}$$ that are equal; that is, $$S_{21} = S_{31}$$. However, the sum of the power at Ports $$\mathsf{2}$$ and $$\mathsf{3}$$ will not be equal to the input power due to loss. Design a $$75\:\Omega$$ resistive three-port power splitter with matched inputs, $$S_{11} =0= S_{22} = S_{33}$$. That is, draw the resistive circuit and calculate its element values.
7. Design a balun based on a magnetic transformer if the balanced load is $$300\:\Omega$$ and the unbalanced impedance is $$50\:\Omega$$.
1. Draw the schematic of the balun with the load and indicate the ratio of windings.
2. If the number of windings on the unbalanced side of the transformer is $$20$$, how many windings are on the unbalanced side?
8. Develop the electrical design of a rat-race hybrid at $$30\text{ GHz}$$ in a $$50\:\Omega$$ system.
9. Develop the electrical design of a rat-race hybrid at $$30\text{ GHz}$$ in a $$100\:\Omega$$ system.
10. Design a lumped-element $$180^{\circ}$$ hybrid at $$1900\text{ MHz}$$ using $$1\text{ nH}$$ inductors.
11. Design a $$90^{\circ}$$ lumped-element hybrid at $$1900\text{ MHz}$$ using $$1\text{ nH}$$ inductors.
12. Design a $$90^{\circ}$$ lumped-element hybrid at $$500\text{ MHz}$$ for a $$75\:\Omega$$ system.
13. Design a lumped-element $$180^{\circ}$$ hybrid at $$1900\text{ MHz}$$ matched to $$50\:\Omega$$ source and load impedances.

## 5.16.1 Exercises by Section

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

$$§5.1\: 1$$

$$§5.2\: 2, 3, 4, 5$$

$$§5.5\: 6†, 7†, 8, 9, 10†$$

$$§5.8\: 11†, 12†$$

$$§5.9\: 13†, 14†, 15†, 16†, 17‡$$

$$§5.10\: 18†, 19†$$

$$§5.11\: 20†, 21†, 22†$$

$$§5.12\: 23†$$

$$§5.13\: 24‡, 25†, 26†, 27, 28, 29, 30$$

## 5.16.2 Answers to Selected Exercises

1. $$12.57$$
2. $$R_{1}=R_{2}=74.5\:\Omega$$
3. $$75.48\:\Omega$$
4. $$4.95\text{ W}$$
1. (c) $$0.6$$
2. $$-0.6897$$
1. $$10\text{ dBm}$$

Figure $$\PageIndex{4}$$

$$L=20.7\text{ nH}$$, $$C_{1}=3.68\text{ pF}$$

$$C_{2}=1.23\text{ pF}$$, $$R=75\:\Omega$$

1. ratio of windings is $$2.45$$
1. Fig 5.13.3(a)
$$L=5.92\text{ nH}$$, $$C=1.19\text{ pF}$$

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