Problems

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Problems

Section 19.2 Impedance Parameters

19.1 Obtain the z parameters for the network in Fig. 19.65.

19.2 Find the impedance parameter equivalent of the network in Fig. 19.66. *

Figure 19.66

For Prob. 19.2.

* An asterisk indicates a challenging problem.

Figure 19.67

For Prob. 19.3.

Figure 19.68 For Prob. 19.4.

19.5 Obtain the z parameters for the network in Fig. 19.69 as functions of s.

Figure 19.69

For Prob. 19.5.

19.6 Compute the z parameters of the circuit in Fig. 19.70.

19.7 Calculate the z parameters of the circuit in Fig. 19.71 as functions of s.

For Prob. 19.7 and 19.80.

19.3 Find the z parameters of the circuit in Fig. 19.67. 19.8 Find the z parameters of the two-port in Fig. 19.72.

For Prob. 19.8.

19.9 The y parameters of a network are:

$$\mathbf{Y} = [\mathbf{y}] = \begin{bmatrix} 0.5 & -0.2 \\ -0.2 & 0.4 \end{bmatrix} S$$

Determine the z parameters for the network.

19.10 Construct a two-port that realizes each of the following z parameters.

(a)

$$\left[\mathbf{z}\right] = \begin{bmatrix} 25 & 20 \\ 5 & 10 \end{bmatrix} \Omega$$

\n(b) $\left[\mathbf{z}\right] = \begin{bmatrix} 1 + \frac{3}{s} & \frac{1}{s} \\ \frac{1}{s} & 2s + \frac{1}{s} \end{bmatrix} \Omega$

19.11 Determine a two-port network that is represented by the following z parameters:

$$[\mathbf{z}] = \begin{bmatrix} 6+j3 & 5-j2 \\ 5-j2 & 8-j \end{bmatrix} \Omega$$

19.12 For the circuit shown in Fig. 19.73, let

$$\begin{bmatrix} \mathbf{z} \end{bmatrix} = \begin{bmatrix} 10 & -6 \\ -4 & 12 \end{bmatrix} \Omega$$

Find

$$I_1$$

, $I_2$ , $V_1$ , and $V_2$ .

Figure 19.73 For Prob. 19.12.

19.13 Determine the average power delivered to ZL = 5 + j4 in the network of Fig. 19.74. Note: The voltage is rms.

Figure 19.74

For Prob. 19.13.

19.14 For the two-port network shown in Fig. 19.75, show that at the output terminals,

$$\mathbf{Z}_{Th} = \mathbf{z}_{22} - \frac{\mathbf{z}_{12}\mathbf{z}_{21}}{\mathbf{z}_{11} + \mathbf{Z}_s}$$

and

$$\mathbf{V}_{\mathrm{Th}} = \frac{\mathbf{z}_{21}}{\mathbf{z}_{11} + \mathbf{Z}_s} \mathbf{V}_s$$

Figure 19.75

For Probs. 19.14 and 19.41.

19.15 For the two-port circuit in Fig. 19.76,

$$\begin{bmatrix} \mathbf{z} \end{bmatrix} = \begin{bmatrix} 40 & 60 \\ 80 & 120 \end{bmatrix} \Omega$$

• (a) Find ZL for maximum power transfer to the load.
• (b) Calculate the maximum power delivered to the load.

Figure 19.76

For Prob. 19.15.

19.16 For the circuit in Fig. 19.77, at ω = 2 rad/s, z11 = 10 Ω, z12 = z21 = j6 Ω, z22 = 4 Ω. Obtain the Thevenin equivalent circuit at terminals a-b and calculate vo.

Figure 19.77 For Prob. 19.16.

Section 19.3 Admittance Parameters

19.17 Determine the z and y parameters for the circuit in Fig. 19.78. *

Figure 19.78

For Prob. 19.17.

19.18 Calculate the y parameters for the two-port in Fig. 19.79.

For Probs. 19.18 and 19.37.

19.19 Using Fig. 19.80, design a problem to help other students better understand how to find y parameters in the s-domain.

Figure 19.80 For Prob. 19.19.

19.20 Find the y parameters for the circuit in Fig. 19.81.

For Prob. 19.20.

Problems 895

19.21 Obtain the admittance parameter equivalent circuit of the two-port in Fig. 19.82.

Figure 19.82

• For Prob. 19.21.
• 19.22 Obtain the y parameters of the two-port network in Fig. 19.83.

Figure 19.83 For Prob. 19.22.

19.23 (a) Find the y parameters of the two-port in Fig. 19.84.

(b) Determine V2(s) for vs = 2u(t) V.

Figure 19.84 For Prob. 19.23.

19.24 Find the resistive circuit that represents these y parameters:

$$[\mathbf{y}] = \begin{bmatrix} \frac{1}{2} & -\frac{1}{4} \\ -\frac{1}{4} & \frac{3}{8} \end{bmatrix} S$$

19.25 Draw the two-port network that has the following y parameters:

$$[\mathbf{y}] = \begin{bmatrix} 1 & -0.5 \\ -0.5 & 1.5 \end{bmatrix} \mathbf{S}$$

19.26 Calculate [y] for the two-port in Fig. 19.85.

Figure 19.85

For Prob. 19.26.

19.27 Find the y parameters for the circuit in Fig. 19.86.

Figure 19.86

For Prob. 19.27.

• 19.28 In the circuit of Fig. 19.65, the input port is connected to a 1-A current source and the right hand side of the circuit is left open (I² = 0). Calculate the power absorbed by the circuit by using the y parameters. Confirm your result by direct circuit analysis.
• 19.29 In the bridge circuit of Fig. 19.87, I1 = 20 A and I2 = −8 A.
  • (a) Find V1 and V2 using y parameters.
  • (b) Confirm the results in part (a) by direct circuit analysis.

Figure 19.87

For Prob. 19.29.

Section 19.4 Hybrid Parameters

19.30 Find the h parameters for the networks in Fig. 19.88.

19.31 Determine the hybrid parameters for the network in Fig. 19.89.

Figure 19.89

For Prob. 19.31.

Figure 19.90 For Prob. 19.32.

19.33 Obtain the h parameters for the two-port of Fig. 19.91.

For Prob. 19.33.

19.34 Obtain the h and g parameters of the two-port in Fig. 19.92.

19.35 Determine the h parameters for the network in Fig. 19.93.

For Prob. 19.35.

19.36 For the two-port in Fig. 19.94,

$$[\mathbf{h}] = \begin{bmatrix} 16 \,\Omega & 3 \\ -2 & 0.01 \,\mathrm{S} \end{bmatrix}$$

Find:

(a)

$$V_2/V_1$$

\n(b) $I_2/I_1$
\n(c) $I_1/V_1$
\n(d) $V_2/I_1$

Figure 19.94

For Prob. 19.36.

• 19.37 The input port of the circuit in Fig. 19.79 is connected to a 10-V dc voltage source while the output port is terminated by a 5-Ω resistor. Find the voltage across the 5-Ω resistor by using h parameters of the circuit. Confirm your result by using direct circuit analysis.
• 19.38 The h parameters of the two-port of Fig. 19.95 are:

$$[\mathbf{h}] = \begin{bmatrix} 600 \,\Omega & 0.04 \\ 30 & 2 \,\text{mS} \end{bmatrix}$$

Given the Zs = 2 kΩ and ZL = 400 Ω, find Zin and Zout.

Figure 19.95 For Prob. 19.38.

19.39 Obtain the g parameters for the wye circuit of Fig. 19.96.

Problems 897

For Prob. 19.39.

19.40 Using Fig. 19.97, design a problem to help other students better understand how to find g parameters in an ac circuit.

Figure 19.97

For Prob. 19.40.

19.41 For the two-port in Fig. 19.75, show that

$$\frac{I_2}{I_1} = \frac{-g_{21}}{g_{11}Z_L + \Delta_g}$$ $$\frac{V_2}{V_s} = \frac{g_{21}Z_L}{(1 + g_{11}Z_s)(g_{22} + Z_L) - g_{21}g_{12}Z_s}$$

where ∆g is the determinant of [g] matrix.

19.42 The h parameters of a two-port device are given by

$$\mathbf{h}_{11} = 600 \ \Omega, \qquad \mathbf{h}_{12} = 10^{-3}, \qquad \mathbf{h}_{21} = 120,$$

\n

$$\mathbf{h}_{22} = 2 \times 10^{-6} \ \mathrm{S}$$

Draw a circuit model of the device including the value of each element.

Section 19.5 Transmission Parameters

19.43 Find the transmission parameters for the singleelement two-port networks in Fig. 19.98.

19.44 Using Fig. 19.99, design a problem to help other students better understand how to find the transmission parameters of an ac circuit.

Figure 19.99

• For Prob. 19.44.
• 19.45 Find the ABCD parameters for the circuit in Fig. 19.100.

Figure 19.100

For Prob. 19.45.

19.46 Find the transmission parameters for the circuit in Fig. 19.101.

Figure 19.101

For Prob. 19.46.

19.47 Obtain the ABCD parameters for the network in Fig. 19.102.

Figure 19.102

For Prob. 19.47

• 19.48 For a two-port, let A = 4, B = 30 Ω, C = 0.1 S, and D = 1.5. Calculate the input impedance Zin = V1∕I1, when:
  • (a) the output terminals are short-circuited,
  • (b) the output port is open-circuited,
  • (c) the output port is terminated by a 10-Ω load.

19.49 Using impedances in the s-domain, obtain the transmission parameters for the circuit in Fig. 19.103.

19.50 Derive the s-domain expression for the t parameters of the circuit in Fig. 19.104.

For Prob. 19.50.

19.51 Obtain the t parameters for the network in Fig. 19.105.

Figure 19.105 For Prob. 19.51.

Section 19.6 Relationships Between Parameters

19.52 (a) For the T network in Fig. 19.106, show that the h parameters are:

$$\mathbf{h}_{11} = R_1 + \frac{R_2 R_3}{R_1 + R_3}, \qquad \mathbf{h}_{12} = \frac{R_2}{R_2 + R_3}$$ $$\mathbf{h}_{21} = -\frac{R_2}{R_2 + R_3}, \qquad h_{22} = \frac{1}{R_2 + R_3}$$

Figure 19.106 For Prob. 19.52.

(b) For the same network, show that the transmission parameters are:

$$\mathbf{A} = 1 + \frac{R_1}{R_2}, \qquad \mathbf{B} = R_3 + \frac{R_1}{R_2}(R_2 + R_3)$$ $$\mathbf{C} = \frac{1}{R_2}, \qquad \mathbf{D} = 1 + \frac{R_3}{R_2}$$

• 19.53 Through derivation, express the z parameters in terms of the ABCD parameters.
• 19.54 Show that the transmission parameters of a two-port may be obtained from the y parameters as:

$$\mathbf{A} = -\frac{\mathbf{y}_{22}}{\mathbf{y}_{21}}, \qquad \mathbf{B} = -\frac{1}{\mathbf{y}_{21}}$$ $$\mathbf{C} = -\frac{\Delta_y}{\mathbf{y}_{21}}, \qquad \mathbf{D} = -\frac{\mathbf{y}_{11}}{\mathbf{y}_{21}}$$

19.55 Prove that the g parameters can be obtained from the z parameters as

$$\mathbf{g}_{11} = \frac{1}{\mathbf{z}_{11}}, \qquad \mathbf{g}_{12} = -\frac{\mathbf{z}_{12}}{\mathbf{z}_{11}},$$ $$\mathbf{g}_{21} = \frac{\mathbf{z}_{21}}{\mathbf{z}_{11}}, \qquad \mathbf{g}_{22} = \frac{\Delta_z}{\mathbf{z}_{11}}$$

19.56 For the network of Fig. 19.107, obtain Vo∕Vs.

Figure 19.107

For Prob. 19.56.

19.57 Given the transmission parameters

$$[\mathbf{T}] = \begin{bmatrix} 3 & 20 \\ 1 & 7 \end{bmatrix}$$

obtain the other five two-port parameters.

19.58 Design a problem to help other students better understand how to develop the y parameters and transmission parameters, given equations in terms of the hybrid parameters.

19.59 Given that

$$[\mathbf{g}] = \begin{bmatrix} 0.06 \text{ S} & -0.4 \\ 0.2 & 2 \Omega \end{bmatrix}$$

determine:

(a) [z] (b) [y] (c) [h] (d) [T]

Problems 899

19.60 Design a T network necessary to realize the following z parameters at ω = 10⁶ rad/s.

$$\begin{bmatrix} \mathbf{z} \end{bmatrix} = \begin{bmatrix} 4+j3 & 3 \\ 2 & 5-j \end{bmatrix} \mathbf{k} \Omega$$

• 19.61 For the bridge circuit in Fig. 19.108, obtain:
  • (a) the z parameters
  • (b) the h parameters
  • (c) the transmission parameters

Figure 19.108

For Prob. 19.61.

19.62 Find the z parameters of the op amp circuit in Fig. 19.109. Obtain the transmission parameters.

• For Prob. 19.62.
• 19.63 Determine the z parameters of the two-port in Fig. 19.110.

Figure 19.110 For Prob. 19.63.

19.64 Determine the y parameters at ω = 1,000 rad/s for the op amp circuit in Fig. 19.111. Find the corresponding h parameters.

Figure 19.111

For Prob. 19.64.

Section 19.7 Interconnection of Networks

19.65 What is the y parameter presentation of the circuit in Fig. 19.112?

Figure 19.112 For Prob. 19.65.

19.66 In the two-port of Fig. 19.113, let y12 = y21 = 0, y11 = 2 mS, and y22 = 10 mS. Find Vo∕Vs.

Figure 19.113 For Prob. 19.66.

19.67 If three copies of the circuit in Fig. 19.114 are connected in parallel, find the overall transmission

19.68 Obtain the h parameters for the network in Fig. 19.115.

For Prob. 19.68.

19.69 The circuit in Fig. 19.116 may be regarded as two two-ports connected in parallel. Obtain the y parameters as functions of s. *

19.70 For the parallel-series connection of the two two-ports in Fig. 19.117, find the g parameters. *

19.71 Determine the z parameters for the network in Fig. 19.118. *

Figure 19.118

For Prob. 19.71.

19.72 A series-parallel connection of two two-ports is shown in Fig. 19.119. Determine the z parameter representation of the network. *

Figure 19.119

For Prob. 19.72.

• 19.73 Three copies of the circuit shown in Fig. 19.70 are connected in cascade. Determine the z parameters.
• 19.74 Determine the ABCD parameters of the circuit in Fig. 19.120 as functions of s. (Hint: Partition the circuit into subcircuits and cascade them using the results of Prob. 19.43.) *

Figure 19.120

For Prob. 19.74.

19.75 For the individual two-ports shown in Fig. 19.121 where, *

$$\begin{bmatrix} \mathbf{z}_a \end{bmatrix} = \begin{bmatrix} 8 & 6 \\ 4 & 5 \end{bmatrix} \Omega \quad \begin{bmatrix} \mathbf{y}_b \end{bmatrix} = \begin{bmatrix} 8 & -4 \\ 2 & 10 \end{bmatrix} S$$

• (a) Determine the y parameters of the overall two-port.
• (b) Find the voltage ratio Vo∕Vi when ZL = 2 Ω.

Figure 19.121 For Prob. 19.75.

Section 19.8 Computing Two-Port Parameters Using PSpice

19.76 Use PSpice or MultiSim to obtain the z parameters of

Figure 19.122 For Prob. 19.76.

19.77 Using PSpice or MultiSim, find the h parameters of the network in Fig. 19.123. Take ω = 1 rad/s.

19.78 Obtain the h parameters at ω = 4 rad/s for the circuit in Fig. 19.124 using PSpice or MultiSim.

For Prob. 19.78.

19.79 Use PSpice or MultiSim to determine the z parameters of the circuit in Fig. 19.125. Take ω = 2 rad/s.

Figure 19.125

For Prob. 19.79.

• the network in Fig. 19.122. 19.80 Use PSpice or MultiSim to find the z parameters of the circuit in Fig. 19.71.
  • 19.81 Repeat Prob. 19.26 using PSpice or MultiSim.
  • 19.82 Use PSpice or MultiSim to rework Prob. 19.31.
  • 19.83 Rework Prob. 19.47 using PSpice or MultiSim.
  • 19.84 Using PSpice or MultiSim, find the transmission parameters for the network in Fig. 19.126.

Figure 19.126 For Prob. 19.84.

19.85 At ω = 1 rad/s, find the transmission parameters of the network in Fig. 19.127 using PSpice or MultiSim.

19.86 Obtain the g parameters for the network in Fig. 19.128 using PSpice or MultiSim.

Figure 19.128 For Prob. 19.86.

19.87 For the circuit shown in Fig. 19.129, use PSpice or MultiSim to obtain the t parameters. Assume ω = 1 rad/s.

Figure 19.129

For Prob. 19.87.

Section 19.9 Applications

• 19.88 Using the y parameters, derive formulas for Zin, Zout, Ai, and Av for the common-emitter transistor circuit.
• 19.89 A transistor has the following parameters in a common-emitter circuit:

$$h_{ie} = 2,640 \Omega$$

, $h_{re} = 2.6 \times 10^{-4}$

$$h_{fe} = 72$$

, $h_{oe} = 16 \,\mu\text{S}$ , $R_L = 100 \,\text{k}\Omega$

What is the voltage amplification of the transistor? How many decibels gain is this?

19.90 A transistor with

hfe = 120, hie = 2 kΩ

$$h_{re} = 10^{-4}$$

, $h_{oe} = 20 \,\mu\text{S}$

is used for a CE amplifier to provide an input resistance of 1.5 kΩ.

• (a) Determine the necessary load resistance RL.
• (b) Calculate Av, Ai, and Zout if the amplifier is driven by a 4-mV source having an internal resistance of 600 Ω.
• (c) Find the voltage across the load.
• 19.91 For the transistor network of Fig. 19.130,

$$h_{fe} = 80$$

, $h_{ie} = 1.2 \text{ k}\Omega$
$h_{re} = 1.5 \times 10^{-4}$ , $h_{oe} = 20 \mu\text{S}$

Determine the following:

• (a) voltage gain Av = Vo∕Vs,
• (b) current gain Ai = Io∕Ii,
• (c) input impedance Zin,
• (d) output impedance Zout.

19.92 Determine Av, Ai, Zin, and Zout for the amplifier shown in Fig. 19.131. Assume that *

$$h_{ie} = 4 \text{ k}\Omega, \qquad h_{re} = 10^{-4}$$ $$h_{fe} = 100, \qquad h_{oe} = 30 \text{ }\mu\text{S}$$

Figure 19.131

For Prob. 19.92.

19.93 Calculate Av, Ai, Zin, and Zout for the transistor network in Fig. 19.132. Assume that *

$$h_{ie} = 2 \text{ k}\Omega, \qquad h_{re} = 2.5 \times 10^{-4}$$ $$h_{fe} = 150, \qquad h_{oe} = 10 \text{ }\mu\text{S}$$

Figure 19.132

For Prob. 19.93.

19.94 A transistor in its common-emitter mode is specified by

$$[\mathbf{h}] = \begin{bmatrix} 200 \,\Omega & 0 \\ 100 & 10^{-6} \,\mathrm{S} \end{bmatrix}$$

Two such identical transistors are connected in cascade to form a two-stage amplifier used at audio frequencies. If the amplifier is terminated by a 4-kΩ resistor, calculate the overall Av and Zin.

19.95 Realize an LC ladder network such that

C ladder network such

$$y_{22} = \frac{s^3 + 5s}{s^4 + 10s^2 + 8}$$

19.96 Design an LC ladder network to realize a low-pass filter with transfer function

Design an LC ladder network to realize a low filter with transfer function
\n

$$H(s) = \frac{1}{s^4 + 2.613s^2 + 3.414s^2 + 2.613s + 1}$$

19.97 Synthesize the transfer function

$$H(s) = \frac{V_o}{V_s} = \frac{s^3}{s^3 + 6s + 12s + 24}$$

using the LC ladder network in Fig. 19.133.

Figure 19.133

For Prob. 19.97.

For Prob. 19.98. 19.98 A two-stage amplifier in Fig. 19.134 contains tw^o identical stages with

$$[\mathbf{h}] = \begin{bmatrix} 2 \text{k}\Omega & 0.004 \\ 200 & 500 \,\mu\text{S} \end{bmatrix}$$

If ZL = 20 kΩ, find the required value of Vs to produce Vo = 16 V.

Figure 19.134