Figure D.2

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• 2.27 1 A
• 2.29 3.5 Ω

2.75 8 Ω

2.77 (a) Four 20-Ω resistors in parallel

(b) One 300-Ω resistor in series with a 1.8-Ω resistor

(c) Two 24-kΩ resistors in parallel connected in series with two 56-kΩ resistors in parallel

combination of two 56-kΩ resistors

(d) A series combination of a 20-Ω resistor,

and a parallel combination of two 20-Ω resistors

300-Ω resistor, 24-kΩ resistor, and a parallel

	2.31 56 A, 8 A, 48 A, 32 A, 16 A		2.79 75 Ω
	2.33 3 V, 6 A		2.81 6.667 kΩ, 5 kΩ
	2.35 32 V, 800 mA		2.83 3.84 kΩ, ∞ Ω (best answer)
	2.37 60 Ω
	2.39 (a) 2.182 Ω, (b) 1.5 kΩ
	2.41 16 Ω	Chapter 3
	2.43 (a) 12 Ω, (b) 16 Ω	3.1	This is a design problem with several answers.
	2.45 (a) 59.8 Ω, (b) 32.5 Ω	3.3	−6 A, −3 A, −2 A, 1 A, −60 V
	2.47 24 Ω	3.5	−60 V
2.49	(a) 20 Ω, (b) Ran = 45 Ω, Rbn = 7.5 Ω, Rcn = 15 Ω	3.7	20 V
	2.51 (a) 9.231 Ω, (b) 36.25 Ω	3.9	79.34 mA
	2.53 (a) 142.32 Ω, (b) 33.33 Ω		3.11 3 V, 293.9 W, 750 mW, 121.5 W
	2.55 119.75 mA		3.13 583.3 V, 100 V
	2.57 32.44 Ω, 1.5413 A		3.15 29.45 A, 144.6 W, 129.6 W, 12 W
2.59	P40W = 102.4 W (means that this immediately burns out), P60W = 9.6 W, P100W = 16 W. The best way to
	wire the bulbs is to connect the 100-W bulb in series with a parallel combination of the 60-W bulb and the		3.17 1.73 A
	40-W bulb.		3.19 10 V, 4.933 V, 12.267 V
	2.61 Use R1 and R3 bulbs	3.21	−15 V, 0 V
	2.63 0.4 Ω, ≅ 1 W		3.23 90 V
	2.65 So, our circuit consists of the meter in series with an 18-kΩ resistor.		3.25 25.52 V, 22.05 V, 14.842 V, 15.055 V
			3.27 625 mV, 375 mV, 1.625 V
	2.67 (a) 4 V, (b) 2.857 V, (c) 28.57%, (d) 6.25%	3.29	−0.7708 V, 1.209 V, 2.309 V, 0.7076 V
	2.69 (a) 6.662 V (with), 6.786 V (without) (b) 24.61 V (with), 26.39 V (without) (c) 62.5 V (with), 75.4 V (without)		3.31 4.97 V, 4.85 V, −0.12 V
	2.71 22.5 Ω		3.33 (a) and (b) are both planar and can be redrawn as shown in Fig. D.2.
	2.73 45 Ω

3 Ω 6 Ω 1 Ω 5 Ω 2 A 2 Ω 4 Ω (a)

Figure D.2

For Prob. 3.33.

• 3.35 20 V
• 3.37 12 V
• 3.39 This is a design problem with several different answers.
• 3.41 1.188 A
• 3.43 1.7778 A, 53.33 V
• 3.45 8.561 A
• 3.47 10 V, 4.933 V, 12.267 V
• 3.49 114 V, 36 A
• 3.51 233.3 V
• 3.53 1.6196 mA, −1.0202 mA, −2.461 mA, 3 mA, −2.423 mA
• 3.55 −1 A, 0 A, 2 A
• 3.57 12 kΩ, 120 V, 80 V
• 3.59 −4.48 A, −1.0752 kV
• 3.61 −0.2813
• 3.63 −4 V, 2.105 A
• 3.65 2.17 A, 1.9912 A, 1.8119 A, 2.094 A, 2.249 A
• 3.67 −30 V

3.69 ⎡ ⎢ ⎣ 0.35 −0.1 −0.05 −0.1 0.4 −0.2 −0.05 −0.2 0.25 ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ v1 v2 v3 ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ 100 50 −10 ⎤ ⎥ ⎦

3.71 6.255 A, 1.9599 A, 3.694 A

3.73

$$\begin{bmatrix} 30 & -10 & -10 & 0 \ -10 & 40 & -10 & 0 \ -10 & -10 & 50 & -10 \ 0 & 0 & -10 & 20 \ \end{bmatrix} \begin{bmatrix} i_1 \\ i_2 \\ i_3 \\ i_4 \end{bmatrix} = \begin{bmatrix} 15 \\ 0 \\ 25 \\ -10 \end{bmatrix}$$

3.75 -3 A, 0 A, 3 A

3.77 3.111 V, 1.4444 V

3.79 −10.556 V, 20.56 V, 1.3889 V, −43.75 V

3.81 26.67 V, 6.667 V, 173.33 V, −46.67 V

3.83 See Fig. D.3; −12.5 V

Figure D.3

For Prob. 3.83.

• 3.85 9 Ω
• 3.87 −5
• 3.89 22.5 μA, 12.75 V
• 3.91 0.8078 μA, 8.345 V, 48.79 mV
• 3.93 1.333 A, 1.333 A, 2.6667 A

Chapter 4

• 4.1 600 mA, 250 V
• 4.3 (a) 0.5 V, 0.5 A, (b) 5 V, 5 A, (c) 5 V, 500 mA
• 4.5 4.5 V
• 4.7 888.9 mV
• 4.9 2 A
• 4.11 17.99 V, 1.799 A
• 4.13 8.696 V
• 4.15 1.875 A, 10.55 W
• 4.17 −8.571 V
• 4.19 −16 V

4.21	4.81
This is a design problem with multiple answers.	3.3 Ω, 10 V (Note, values obtained graphically)
4.23	4.83
1 A, 8 W	8 Ω, 72 V
4.25	4.85
−6.6 V	(a) 80 V, 30 kΩ, (b) 32 V
4.27	4.87
−48 V	(a) 10 mA, 8 kΩ, (b) 9.926 mA
4.29	4.89
3 V	(a) 99.99 μA, (b) 99.99 μA
4.31	4.91
9.13 V	(a) 150 Ω, 25 Ω, (b) 150 Ω, 250 Ω
4.33	4.93 ____________ Vs
80 V, 33 Ω, 2 A	Rs + (1 + β)Ro
4.35	4.95
−125 mV	10.667 V, 33.33 kΩ
4.37	4.97
5 kΩ, 1 mA	2 kΩ, 5 V
4.39 20 Ω, −84 V
4.41 4 Ω, −8 V, −2 A	Chapter 5
4.43	5.1
10 Ω, 0 V	60 μV
4.45	5.3
3 Ω, 15 V	10 V
4.47	5.5
20 V, 20 Ω, 1 A	0.999990
4.49	5.7
28 Ω, 3.286 V	−100 nV, −10 mV
4.51	5.9
(a) 2 Ω, 7 A, (b) 1.5 Ω, 12.667 A	2 V, 2 V
4.53	5.11
10 Ω, −3 A	This is a design problem with multiple answers.
4.55	5.13
100 kΩ, −20 mA	2.7 V, 288 μA
4.57 10 Ω, 166.67 V, 16.667 A	R____ 1R3 5.15
4.59 22.5 Ω, 40 V, 1.7778 A	(a) −(R1 + R3 + ), (b) −92 kΩ R2
4.61	5.17
1.2 Ω, 9.6 V, 8 A	(a) −2.4, (b) −16, (c) −400
4.63	5.19
−3.333 Ω, 0 A	−562.5 μA
4.65 V0 = 24 − 5I0	5.21 −3 V
4.67 25 kΩ, 49 mW	Rf ___ 5.23 − R1
4.69	5.25
∞ (theoretically)	9.375 V
4.71	5.27
8 kΩ, 1.152 W	2.7 V
4.73 20.77 W	R___2 5.29 R1
4.75	5.31
250 Ω, 12 mW	4.545 mA
4.77 (a) 3.8 Ω, 4 V, (b) 3.2 Ω, 15 V
4.79	5.33
10 Ω, 167 V	75 mW, −1 mA

5.35 If Ri = 60 k, Rf = 390 k.

5.37 −13.6 V

5.39 7 V

5.41 See Fig. D.4.

Figure D.4

For Prob. 5.41.

5.43 200 k.

5.45 This is a design problem with many correct answers. One possible design is shown in Fig. D.5.

Figure D.5

• 5.47 14.09 V
• 5.49 R1 = R3 = 20 kΩ, R2 = R4 = 80 kΩ
• 5.51 See Fig. D.6.

Figure D.6

For Prob. 5.51.

5.53 Proof.

• 5.55 7.956, 7.956, 1.989
• 5.57 6vs1 − 6vs2

5.59 −12

5.61 7.2 V

• 5.63 ________________ R2R4∕R1R⁵ − R4∕R⁶ 1 − R2R4∕R3R⁵ 5.65 2 V 5.67 −1.6 V 5.69 −25.71 mV 5.71 7.5 V 5.73 32.4 V
• 5.75 −2, 200 μA
• 5.77 −6.686 mV
• 5.79 −4.992 V
• 5.81 343.4 mV, 24.51 μA
• 5.83 The result depends on your design. Hence, let RG = 10 k ohms, R1 = 10 k ohms, R2 = 20 k ohms, R3 = 40 k ohms,
  • R4 = 80 k ohms, R5 = 160 k ohms,
  • R6 = 320 k ohms, then,

$$-v_o = (R_f/R_1)v_1 +$$

= $v_1 + 0.5v_2 + 0.25v_3 + 0.125v_4$

• $0.0625v_5 + 0.03125v_6$

• (a) |vo| = 1.1875 = 1 + 0.125 + 0.0625 = 1 + (1∕8) + (1∕16), which implies, [v¹ v² v³ v⁴ v⁵ v6] = [100110]
• (b) |vo| = 0 + (1∕2) + (1∕4) + 0 + (1∕16) + (1∕32) = (27∕32) = 843.75 mV

(c) This corresponds to [111111].
\n

$$|v_o| = 1 + (1/2) + (1/4) + (1/8) + (1/16) + (1/32)$$

\n $= 63/32 = 1.96875$ V

5.85 R = 200 kΩ, 2,000

$$5.87 \quad \left(1 + \frac{R_4}{R_3}\right) v_2 - \left[\left(\frac{R_4}{R_3}\right) + \left(\frac{R_2 R_4}{R_1 R_3}\right)\right] v_1$$

\n

$$\text{Let } R_4 = R_1 \text{ and } R_3 = R_2;$$

\n

$$\text{then } v_0 = \left(1 + \frac{R_4}{R_3}\right) (v_2 - v_1)$$

\n

$$\text{a subtractor with a gain of } \left(1 + \frac{R_4}{R_3}\right).$$

5.89 A summer with v0 = −v¹ − (5∕3)v2 where v2 = 6 V battery and an inverting amplifier with v1 = −12 vs.

$$5.91\ \ 9$$

battery and an inverting amplifier with

$$v_1 =$$

\n5.91 9
\n5.93 $A = \frac{1}{\left(1 + \frac{R_1}{R_3}\right)R_L - R_1\left(\frac{R_2 + R_L}{R_2 R_3}\right)\left(R_4 + \frac{R_2 R_L}{R_2 + R_L}\right)}$

For Prob. 5.45.

Chapter 6

6.1 15(1 − 3t)e⁻3*^t* A, 30t(1 − 3t)e⁻6*^t* W

6.3 This is a design problem with multiple answers.

6.5

$$v = \begin{cases} 20 \text{ mA}, & 0 < t < 2 \text{ ms} \\ -20 \text{ mA}, & 2 < t < 6 \text{ ms} \\ 20 \text{ mA}, & 6 < t < 8 \text{ ms} \end{cases}$$

6.7 [0.1t 2 + 10] V

6.9 13.624 V, 70.66 W

$$\textbf{6.11} \quad v(t) = \begin{cases} 10 + 3.75t \text{ V}, & 0 < t < 2s \\ 22.5 - 2.5t \text{ V}, & 2 < t < 4s \\ 12.5 \text{ V}, & 4 < t < 6s \\ 2.5t - 2.5 \text{ V}, & 6 < t < 8s \end{cases}$$

• 6.13 v1 = 42 V, v2 = 48 V
• 6.15 (a) 125 mJ, 375 mJ, (b) 70.31 mJ, 23.44 mJ
• 6.17 (a) 3 F, (b) 8 F, (c) 1 F
• 6.19 10 μF
• 6.21 2.5 μF
• 6.23 This is a design problem with multiple answers.
• 6.25 (a) For the capacitors in series,

$$Q_1 = Q_2 \rightarrow C_1 v_1 = C_2 v_2 \rightarrow \frac{v_1}{v_2} = \frac{C_2}{C_1}$$ $$v_s = v_1 + v_2 = \frac{C_2}{C_1} v_2 + v_2 = \frac{C_1 + C_2}{C_1} v_2$$ $$\rightarrow v_2 = \frac{C_1}{C_1 + C_2} v_s$$

Similarly, $v_1 = \frac{C_2}{C_1 + C_2} v_s$

(b) For capacitors in parallel,

$$v_1 = v_2 = \frac{Q_1}{C_1} = \frac{Q_2}{C_2}$$ $$Q_s = Q_1 + Q_2 = \frac{C_1 Q_2 + Q_2}{C_2} = \frac{C_1 + C_2 Q_2}{C_2}$$

or

$$Q_2 = \frac{C_2}{C_1 + C_2}$$ $$Q_1 = \frac{C_1}{C_1 + C_2} Q_s$$ $$i = \frac{dQ}{dt} \rightarrow i_1 = \frac{C_1}{C_1 + C_2} i_s,$$

\n

$$i_2 = \frac{C_2}{C_1 + C_2} i_s$$

\n6.27 2.5 µF, 40 µF
\n6.29 (a) 1.6 C, (b) 1 C
\n

$$1.5t^2 \text{ kV}, \qquad 0 < t < 1s$$

\n6.31 v(t) =

$$\begin{cases} 1.5t^2 \text{ kV}, \qquad 0 < t < 1s\\ [3t - 1.5] \text{ kV}, \qquad 1 < t < 3s\\ [0.75t^2 - 7.5t + 23.25] \text{ kV}, \qquad 3 < t < 5s \end{cases}$$

\n

$$i_1 = \begin{cases} 18t \text{ mA}, \qquad 0 < t < 1s\\ 18 \text{ mA}, \qquad 1 < t < 3s;\\ [9t - 45] \text{ mA}, \qquad 3 < t < 5s \end{cases}$$

\n

$$i_2 = \begin{cases} 12t \text{ mA}, \qquad 0 < t < 1s\\ 12 \text{ mA}, \qquad 1 < t < 3s\\ [6t - 30] \text{ mA}, \qquad 3 < t < 5s \end{cases}$$

\n6.33 15 V, 10 F
\n6.35 3.2 mH
\n6.37 4.8 cos 100t V, 96 mJ
\n6.39 [-50e^{-2t} + 50 + 20t^2 + 80t] A
\n6.41 5.977 A, 35.72 J
\n6.43 270 µJ
\n6.45 i(t) =

$$\begin{cases} 100t^2 \text{ A}, \qquad 0 < t < 1s\\ [400 - 400t + 100t^2] \text{ A}, \qquad 1 < t < 2s \end{cases}$$

\n6.47 5 $\Omega$
\n6.49 15 mH
\n6.53 20 mH
\n6.55 (a) 1.4 L, (b) 500 mL
\n6.57 6.625 H
\n6.59 Proof.

;

6.61 (a) 6.667 mH, e^−t mA, 2e^−t mA (b) −20e^−t μV (c) 1.3534 nJ

6.63 See Fig. D.7.

Figure D.7

For Prob. 6.63.

• 6.65 (a) 40 J, 40 J, (b) 80 J, (c) 5 × 10⁻⁵ (e⁻200*^t* − 1) + 4 A, 1.25 × 10⁻⁵ (e⁻200*^t* − 1) − 2 A (d) 6.25 × 10⁻⁵ (e⁻200*^t* − 1) + 2 A
• 6.67 100 cos(50t) mV
• 6.69 See Fig. D.8.

Figure D.8

For Prob. 6.69.

6.71 By combining a summer with an integrator, we get the circuit shown in Fig. D.9 where C = 5 μF, R1 = 200 kΩ, R2 = 50 kΩ, and R3 = 20 kΩ.

$$v_o = -\frac{1}{R_1C} \int v_1 dt - \frac{1}{R_2C} \int v_2 dt - \frac{1}{R_2C} \int v_2 dt$$

For the given problem, C = 2 μF : R1 = 500 kΩ, R2 = 125 kΩ, R3 = 50 kΩ.

6.73 Consider the op amp as shown in Fig. D.10.

Figure D.10 For Prob. 6.73.

Let va = vb = v. At node a,

$$\frac{0 - v}{R} = \frac{v - v_0}{R} \longrightarrow 2v - v_0 = 0$$

(1)
At node b,

$$\frac{v_i - v}{R} = \frac{v - v_0}{R} + C\frac{dv}{dt}$$ $$v_i = 2v - v_o + RC\frac{dv}{dt}$$

(2)

Combining Eqs. (1) and (2),

$$v_i = v_o - v_o + \frac{RC}{2} \frac{dv_o}{dt}$$

or $v_o = \frac{2}{RC} \int v_i dt$

showing that the circuit is a noninverting integrator.

$$6.75 -17.5 \text{ mV}$$

6.77 See Fig. D.11.

Figure D.11 For Prob. 6.77.

6.79 See Fig. D.12.

6.81 See Fig. D.13.

For Prob. 6.81.

• 6.83 Eight groups in parallel with each group made up of four capacitors in series.
• 6.85 1.25 mH inductor

Chapter 7

• 7.1 (a) 0.7143 μF, (b) 5 ms, (c) 3.466 ms

• 7.3 1.5 μs

• 7.5 This is a design problem with multiple answers.

• 7.7 15e^−t V for 0 < t < 1 sec, 5.518e⁻2(t−1) V for 1 sec < t < ∞

• 7.9 10e^−t/12 V

• 7.11 1.2e⁻3*^t* A

• 7.13 (a) 16 kΩ, 16 H, 1 ms, (b) 126.42 μJ

• 7.15 (a) 10 Ω, 500 ms, (b) 40 Ω, 250 μs

• 7.17 [−15e⁻2*^t* ] V for all t > 0.

• 7.19 5e⁻5*^t u*(t) A

• 7.21 1.618 Ω

• 7.23 10e⁻4*^t* V, t > 0, 2.5e⁻4*^t* V, t > 0

• 7.25 This is a design problem with multiple answers.

• 7.27 [5u(t + 1) + 10u(t) − 25u(t −1) + 15u(t − 2)] V

• 7.29 (c) z(t) = cos 4t δ(t − 1) = cos 4δ(t − 1) = −0.6536δ(t − 1), which is sketched below.

Figure D.14 For Prob. 7.29.

• 7.31 (a) 112 × 10 ⁻⁹ , (b) 7
• 7.33 4.5u(t − 2) A
• 7.35 (a) −e ⁻2*^t u*(t) V, (b) 2e1.5*^t u*(t) A
• 7.37 (a) 4 s, (b) 10 V, (c) (10 − 8e⁻t∕⁴ ) u(t) V
• 7.39 (a) 4 V, t < 0, 20 −16e ⁻t∕⁸ , t > 0, (b) 4 V, t < 0, 12 − 8e⁻t∕⁶ V, t > 0.
  • 7.41 This is a design problem with multiple answers.

7.43 0.8 A,

$$0.8e^{-t/480}u(t)$$

A

7.45 [20 −15e⁻14.286*^t* ] u(t) V

7.47

$$\begin{cases} 24(1 - e^{-t})V, & 0 < t < 1 \\ 30 - 14.83e^{-(t-1)}V, & t > 1 \end{cases}$$

7.49

$$\begin{cases} 8(1 - e^{-t/5}) \text{ V}, & 0 < t < 1 \\ [-16 + 31.17e^{-(t-1)}] \text{ V}, & t > 1 \end{cases}$$ $$7.51 \quad V_S = Ri + L\frac{di}{dt}$$

\n

$$\text{or } L\frac{di}{dt} = -R\left(i - \frac{V_S}{R}\right)$$

\n

$$\frac{di}{i - V_S/R} = \frac{-R}{L}dt$$

Integrating both sides,

$$\ln\left(i - \frac{V_S}{R}\right)|_{I_0}^{i(t)} = \frac{-R}{L}t$$ $$\ln\left(\frac{i - V_S/R}{I_0 - V_S/R}\right) = \frac{-t}{\tau}$$ $$\text{or } \frac{i - V_S/R}{I_0 - V_S/R} = e^{-t/\tau}$$ $$i(t) = \frac{V_S}{R} + \left(I_0 - \frac{V_S}{R}\right)e^{-t/\tau}$$

which is the same as Eq. (7.60).

• 7.53 (a) 5 A, 5e⁻t∕² u(t) A, (b) 6 A, 6e⁻2t∕³ u(t) A
• 7.55 96 V, 96e⁻4*^t u*(t) V
• 7.57 2.4e⁻2*^t u*(t) A, 600e⁻5*^t u*(t) mA
• 7.59 120e⁻4*^t u*(t) volts
• 7.61 20e⁻8*^t u*(t) V, (10 − 5e⁻8*^t* )u(t) A
• 7.63 2e⁻8*^t u*(t) A, −8e⁻8*^t u*(t) V
• 7.65 { 2(1 − e⁻2*^t* )A 1.729e⁻2(t−1)A 0 < t < 1 t > 1
• 7.67 10e–t/6u(t) V
• 7.69 48(e⁻t∕³⁰⁰⁰−1) u(t) V
• 7.71 [−5 + 5e–^t ]u(t) V
• 7.73 −9e⁻5*^t u*(t) V
• 7.75 [20 10e–^t ]u(t) V, 100μA
• 7.77 See Fig. D.15.

7.79 [1.75 – 0.75e–2*^t* ]u(t) A

7.81 See Fig. D.16.

For Prob. 7.81.

• 7.83 6.278 m/s
• 7.85 (a) 659.7 μs, (b) 16.636 s
• 7.87 441 mA
• 7.89 L < 200 mH
• 7.91 1.271 Ω

Chapter 8

• 8.1 (a) 2 A, 12 V, (b) −4 A∕s, −5 V∕s, (c) 0 A, 0 V

• 8.3 (a) 0 A, −10 V, 0 V, (b) 0 A∕s, 8 V∕s, 8 V∕s, (c) 400 mA, 6 V, 16 V

  • 8.5 (a) 0 A, 0 V, (b) 4 A∕s, 0 V∕s, (c) 2.4 A, 9.6 V

• 8.7 overdamped

• 8.9 [(10 + 50t)e⁻5*^t* ] A 8.11 [(10 + 10t)e^−t ] V

• 8.13 120 Ω

• 8.15 750 Ω, 200 μF, 25 H

• 8.17 [21.55e⁻2.679*^t* − 1.55e⁻37.32*^t* ] V

• 8.19 24 sin(0.5t) V

• 8.21 18e^−t − 2e⁻9*^t* V

• 8.23 40 mF

• 8.25 This is a design problem with multiple answers.

• 8.27 [3 − 3(cos(2t) + sin(2t))e⁻2*^t* ] volts

• 8.29 (a) 3 − 3 cos 2t + sin 2t V, (b) 2 − 4e^−t + e⁻4*^t* A,

• (c) 3 + (2 + 3t)e^−t V, (d) 2 + 2 cos 2te^−t A

  • 8.31 80 V, 40 V
  • 8.33 [30 + 0.3078e⁻4.95*^t* − 15.308e⁻0.05*^t* ] V

• 8.35 This is a design problem with multiple answers.

• 8.37 5e⁻4*^t* A

• 8.39 (−60 + [−0.2102e⁻47.83*^t* + 60.21e⁻0.167*^t* ]) V

• 8.41 [8.7 sin(4.583t)e–2*^t* ]u(t) A

• 8.43 8 Ω, 2.075 mF

• 8.45 [6 − [5 cos(1.3229t) + 1.8898 sin(1.3229t)]e⁻t∕² ] A, [7.559 sin(1.3229t)e⁻t∕² ] V

  • 8.47 (400te⁻10*^t* ) V
  • 8.49 {9 + [(3 + 6t)e–2*^t* ]} u(t) A
  • 8.51 [ ^{− i} ____0 oC sin(ot) ] V where o = 1∕ √ ___ LC
  • 8.53 (d² i∕dt² ) + 1.25(di∕dt) + 400i = 200
  • 8.55 2e–^t/2 A for t > 0

• 8.57 (a) s 2 + 10s + 9 = 0, (b) [–1.75e –t + 3.75e –9t ]u(t) A, [–21e –t + 45e –9t ]u(t) V

  • 8.59 48te–2*^t* V

• 8.61 2.4 2.667e⁻2*^t* + 0.2667e⁻5*^t* A, 9.6 – 16e⁻2*^t* + 6.4e⁻5*^t* V

$$8.63 \frac{d^2 i(t)}{dt^2} = -\frac{v_s}{RCL}$$

8.65

$$\frac{d^2v_o}{dt^2} - \frac{v_o}{R^2C^2} = 0, e^{10t} - e^{-10t} \text{ V}$$

Note, circuit is unstable.

• 8.67 −te^−t u(t) V
• 8.69 See Fig. D.17.

Figure D.17 For Prob. 8.69.

• 8.73 This is a design problem with multiple answers.
• 8.75 See Fig. D.19.

For Prob. 8.75.

8.77 See Fig. D.20.

Figure D.20 For Prob. 8.77.

• 8.79 173.61 μF
• 8.81 2.533 μH, 625 μF

8.83 ^d2 ___v dt2 + __ R L ___ dv dt + ___^R LC iD + __1 C ___ diD dt = ___ ^vs LC

Chapter 9

• 9.1 (a) 50 V, (b) 209.4 ms, (c) 4.775 Hz, (d) 44.48 V, 0.3 rad
• 9.3 (a) 10 cos(ωt − 60°), (b) 9 cos(8t + 90°), (c) 20 cos(ωt + 135°)
• 9.5 30°, v1 lags v²
• 9.7 Proof
• 9.9 (a) 50.88 ⧸−15.52°, (b) 60.02 ⧸−110.96°
• 9.11 (a) 21 ⧸−15° V, (b) 8 ⧸ 160° mA, (c) 120 ⧸−140° V, (d) 60 ⧸−170° mA

9.13 (a) −1.2749 + j0.1520, (b) −2.083, (c) 35 + j14

• 9.15 (a) −6 − j11, (b) 120.99 + j4.415, (c) −1

• 9.17 15.62 cos(50t − 9.8°) V

• 9.19 (a) 3.32 cos(20t + 114.49°), (b) 64.78 cos(50t − 70.89°), (c) 9.44 cos(400t − 44.7°)

• 9.21 (a) f(t) = 8.324 cos(30t + 34.86°), (b) g(t) = 5.565 cos(t − 62.49°), (c) h(t) = 1.2748 cos(40t − 168.69°)

• 9.23 (a) 320.1 cos(20t − 80.11°) A, (b) 36.05 cos(5t + 93.69°) A

• 9.25 (a) 0.8 cos(2t − 98.13°) A, (b) 0.745 cos(5t − 4.56°) A

• 9.27 0.289 cos(377t − 92.45°) V

• 9.29 2 sin(10⁶ t − 65°)

• 9.31 900.6 cos(2t + 51.21°) mA

• 9.33 139.64 V

• 9.35 11.015 cos(200t − 16.7°) A

• 9.37 (25 − j25) mS

• 9.39 9.135 + j27.47 Ω, 3.972 cos(10t − 71.6°) A

• 9.41 72.74 cos(t − 18.43°) V

• 9.43 1.3868 ⧸ 33.69° A

• 9.45 j5 A

• 9.47 10.598 cos(2000t + 52.63°) mA

• 9.49 22.63 sin(200t − 45°) V

• 9.51 225 cos(2t − 53.13°) A

• 9.53 23.66⧸−21.67° A

• 9.55 (2.798 − j16.403) Ω

• 9.57 0.3171 − j0.1463 S

• 9.59 (10 − j10) ohms

• 9.61 1 + j0.5 Ω

• 9.63 34.69 − j6.93 Ω

• 9.65 17.35⧸ 0.9° A, 6.83 + j1.094 Ω

• 9.67 (a) 14.8⧸−20.22° mS, (b) 19.704⧸ 74.56° mS

• 9.69 1.661 + j0.6647 S

• 9.71 1.058 − j2.235 Ω

• 9.73 0.3796 + j1.46 Ω

• 9.75 Can be achieved by the RL circuit shown in Fig. D.21.

Figure D.21

For Prob. 9.75.

• 9.77 (a) 26.57° lagging, (b) 1 MHz
• 9.79 (a) 140.2°, (b) leading, (c) 18.43 V
• 9.81 1.8 kΩ, 0.1 μF
• 9.83 104.17 mH
• 9.85 Proof
• 9.87 34.96⧸−6.54° Ω
• 9.89 25 μF
• 9.91 4 μF
• 9.93 3.592⧸−38.66° A

Chapter 10

• 10.1 1.9704 cos(10t + 5.65°) A

• 10.3 3.835 cos(4t − 35.02°) V

• 10.5 12.398 cos(4 × 10³ t + 4.06°) mA

• 10.7 124.08⧸−154° V

• 10.9 6.154 cos(10³ t + 70.26°) V 10.11 199.5⧸ 86.89° mA 10.13 29.36⧸ 62.88° A 10.15 7.906⧸ 43.49° A 10.17 9.25⧸−162.12° A 10.19 7.682⧸ 50.19° V 10.21 (a) 1, 0, − ^j __ R √ __ __L C , (b) 0, 1, ^j __ R √ __ __L C 10.23 (1 − ω² LC)Vs _________________________¹− ω² LC + jωRC(2 − ω² LC) 10.25 1.4142 cos(2t + 45°) A 10.27 7.047⧸ 95.24° A, 1.4892⧸ 37.71° A 10.29 This is a design problem with several different answers. 10.31 1.0897⧸ 61.44° A 10.33 7.906⧸ 43.49° A 10.35 1.971⧸−2.1° A 10.37 2.38⧸−96.37° A, 2.38 ⧸143.63° A, 2.38⧸23.63° A 10.39 381.4⧸109.6° mA, 344.3⧸124.4° mA, 145.5⧸60.42° mA, 100.5⧸48.5° mA 10.41 [14.142 sin (2t + 45°) + 26.83 cos(4t + 26.57°)] V 10.43 19.804 cos(2t − 129.17°) A 10.45 395.6 cos(10t + 21.47°) + 149.75 sin(4t + 176.57°) mA 10.47 [4 + 0.504 sin(t + 19.1°) + 0.3352 cos(3t − 76.43°)] A 10.49 883.9 cos(20t − 30°) mA 10.51 109.3⧸30° mA Appendix D Answers to Odd-Numbered Problems A-33

  • 10.53 27.44⧸−59.04° V
  • 10.55 (a) ZN = ZTh = 22.63 ⧸−63.43° Ω, VTh = 25⧸−150° V, IN = 1.1181⧸−86.6° A, (b) ZN = ZTh = 10 ⧸ 26° Ω, VTh = 101⧸58° V, IN = 10.176⧸32° A
  • 10.57 This is a design problem with multiple answers.

10.59	−6 + j38 Ω	11.15 90 W
	10.61 (−180 + j90) V, (−8 + j6) Ω
	10.63 11.314⧸15° A, (10 − j10) Ω
	10.65 This is a design problem with multiple answers.	11.21 19.58 Ω
	10.67 7.415⧸−84.68° V, 656.5⧸−90.16° mA, 11.243 + j1.079 Ω
	10.69 j[1/(ωRC)], Vm sin(ωt + 90°) V	11.25 8.165
	10.71 72 cos (2t + 29.52°) V	11.27 2.887 A
	10.73 21.21⧸−45° kΩ
	10.75 0.12499⧸180°	11.31 2.944 V
10.77	R2 + R3 + jωC2R2R3 ________________________ (1 + jωR1C1)(R3 + jωC2R2R3)	11.33 5.332 A
	10.79 35.78⧸−153.44° V	11.35 21.6 V
	10.81 11.27⧸128.1 V
	10.83 6.611 cos (1,000t − 159.2°) V
	10.85 This is a design problem with multiple answers.
10.87	15.91⧸169.6° V, 5.172⧸−138.6° V, 2.27⧸−152.4° V
10.89 Proof
	10.91 (a) 180 kHz, (b) 40 kΩ
10.93 Proof
10.95 Proof
	Chapter 11
	(Assume all values of currents and voltages are rms unless otherwise specified.)
11.1	[1.320 + 2.640 cos(100t + 60˚)] kW, 1.320 kW
11.3	213.4 W
11.5	P1Ω = 1.4159 W, P2Ω = 5.097 W, P3H = P0.25F = 0 W
11.7	1 kW
11.9	897 μW
	11.11 3.472 W
	11.13 28.36 W

	11.17 20 Ω, 31.25 W
	11.19 100 Ω, 6.25 W
	11.21 19.58 Ω
	11.23 This is a design problem with multiple answers.
11.25 8.165
	11.27 2.887 A
	11.29 17.321 A, 3.6 kW
	11.31 2.944 V
	11.33 5.332 A
	11.35 21.6 V
	11.37 This is a design problem with multiple answers.
	11.39 (a) 0.8575, 17.794 kW, 10.676 kVAR, (b) 585.1 μF
	11.41 (a) 0.5547 (leading), (b) 0.9304 (lagging)
	11.43 This is a design problem with multiple answers.
	11.45 (a) 46.9 V, 1.061 A, (b) 20 W
	11.47 (a) S = (339.4 + j339.4) VA, average power = 339.4 W, reactive power = 339.4 VAR (b) S = (678.8 – j678.8) VA, average power = 678.8 W, reactive power = −678.8 VAR (c) S = (7.637 + j7.637) kVA, average power = 7.637 W, reactive power = 7.637 VAR (d) S = (250 + j433) kVA, average power = 250 kW, reactive power = 433 kVAR
	11.49 (a) 4 + j2.373 kVA, (b) 1.6 – j1.2 kVA, (c) 0.4624 + j1.2705 kVA, (d) 110.77 + j166.16 VA
	11.51 (a) 0.9956 (lagging), (b) 304 W, (c) 28.64 VAR,

• (d) 305.3 VA,

• (e) [304 + j28.64] VA

• 11.53 (a) 47 ⧸29.8° A, (b) 1.0 (lagging)

• 11.55 This is a design problem with multiple answers.

• 11.57 (219 − j145.99) VA

• 11.59 j2 VAR, −j2 VAR

• 11.61 66.2⧸92.4° A, 6.62⧸−2.4° kVA

• 11.63 129.31⧸18.43° A

• 11.65 80 μW

• 11.67 (a) 12.5⧸−36.87° mVA, (b) 78.13 W

• 11.69 (a) 0.8 (lagging), (b) 6.195 kW, (c) 63.66 μF

• 11.71 (a) 50.14 + j1.7509 mΩ, (b) 0.9994 lagging, (c) 2.392⧸−2° kA

• 11.73 (a) 12.21 kVA, (b) 50.86⧸−35° A, (c) 4.083 kVAR, 188.03 μF, (d) 43.4⧸−16.26° A

• 11.75 (a) (32.14 + j7.357) kVAR, (b) 0.9748 (lagging), (c) 100.08 μF

• 11.77 157.69 W

• 11.79 50 mW

• 11.81 This is a design problem with multiple answers.

• 11.83 (a) 688.1 W, (b) 840 VA, (c) 481.8 VAR, (d) 0.8191 (lagging)

• 11.85 (a) 13 A, 21.71⧸ 166.3° A, 9.588⧸−32.43° A, (b) (4.091 + j0.617) kVA, (c) 0.9888 (lagging)

• 11.87 0.5333

• 11.89 (a) 12 kVA, 9.36 + j7.51 kVA, (b) 2.866 + j2.3 Ω

• 11.91 0.8182 (lagging), 1.398 μF

• 11.93 (a) 7.328 kW, 1.196 kVAR, (b) 0.987

• 11.95 (a) 2.814 kHz, (b) 431.8 mW

• 11.97 1.8396 kW

Chapter 12

(Assume all values of currents and voltages are rms unless otherwise specified.)

• 12.1 (a) 231⧸−30°, 231⧸−150°, 231⧸ 90° V, (b) 231⧸ 30°, 231⧸ 150°, 231⧸−90° V

• 12.3 acb sequence, 100⧸−75° V

• 12.5 207.8 cos(ωt + 62°) V, 207.8 cos(ωt − 58°) V, 207.8 cos(ωt −178°) V

• 12.7 44⧸ 53.13° A, 44⧸−66.87° A, 44⧸ 173.13° A

• 12.9 4.8⧸−36.87° A, 4.8⧸−156.87° A, 4.8⧸ 83.13° A

• 12.11 762.1 V, 366.1 A

• 12.13 20.43 A, 3.744 kW

• 12.15 13.66 A

• 12.17 4.8⧸ 53.13° A, 4.8⧸−66.87° A, 4.8⧸ 173.13° A

• 12.19 13.915⧸−18.43° A, 13.915⧸−138.43° A, 13.915⧸ 101.57° A, 24.1⧸−48.43° A, 24.1⧸−168.43° A, 24.1⧸71.57° A

• 12.21 44⧸−30° A, 76.21⧸−60° A, 0.866

• 12.23 106.61⧸ –0.65° V, 106.55⧸ 119.34° V, 106.6⧸ –120.67° V

• 12.25 17.742⧸ 4.78° A, 17.742⧸−115.22°A, 17.742⧸124.78° A

• 12.27 91.79 V

• 12.29 [5.197 + j4.586] kVA

• 12.31 (a) 6.144 + j4.608 Ω, (b) 36.08 A, (c) 207.2 μF

• 12.33 7.69 A, 360.3 V

• 12.35 (a) 14.61 − j5.953 A, (b) [10.081 + j4.108] kVA, (c) 0.9261

• 12.37 26.24 A, (5.808 − j7.744) Ω

• 12.39 432 W

• 12.41 9.021 A

• 12.43 4.373 − j1.145 kVA

• 12.45 2.109⧸ 24.83° kV

• 12.47 39.19 A (rms), 0.9982 (lagging)

• 12.49 (a) 27.65 kW, (b) 9.216 kW

• 12.51 2.078⧸ 120° A, 2.078⧸ 90° A, 2.078⧸ 150° A, 2.939⧸ 165° A, 1.0759⧸ 15° A, 2.078⧸ –150° A

• 12.53 This is a design problem with multiple answers.

• 12.55 8⧸−60° A, 28.84⧸ 133.9° A, 21.17⧸−40.89° A, (8.64 + j1.6627) kVA

• 12.57 Ia = 3.917⧸−18.1° A, Ib = 2.931⧸−130.55° A, Ic = 3.895⧸ 117.82° A

• 12.59 220.6⧸−34.56°, 214.1⧸−81.49°, 49.91⧸−50.59° V, assuming that N is grounded.

• 12.61 11.15⧸ 37° A, 230.8⧸−133.4° V, assuming that N is grounded.

• 12.63 18.67⧸ 158.9° A, 12.38⧸ 144.1° A

• 12.65 11.02⧸ 12° A, 11.02⧸−108° A, 11.02⧸ 132° A

• 12.67 (a) 97.67 kW, 88.67 kW, 82.67 kW, (b) 108.97 A

• 12.69 Ia = 94.32⧸−62.05° A, Ib = 94.32⧸ 177.95° A, Ic = 94.32⧸ 57.95° A, 28.8 + j18.03 kVA

• 12.71 (a) 2,590 W, 4,808 W, (b) 8,335 VA

• 12.73 2,360 W, −632.8 W

• 12.75 (a) 20 mA, (b) 200 mA

• 12.77 520 W

• 12.79 37.29⧸−19.65°, 37.29⧸−139.65°, 37.29⧸100.35° A, 484.7⧸ 2.97°, 484.7⧸−117.03°, 484.7⧸ 122.97° V

• 12.81 516 V

• 12.83 183.42 A

• 12.85 ZY = 2.133 Ω

• 12.87 2.77⧸−176.6° A, (4.581 + j2.604) kVA, (3.971 + j2.64) kVA

Chapter 13

(Assume all values of currents and voltages are rms unless otherwise specified.)

• 13.1 20 H
• 13.3 300 mH, 100 mH, 50 mH, 0.2887
• 13.5 (a) 247.4 mH, (b) 48.62 mH
• 13.7 1.081⧸ 144.16° V
• 13.9 2.074⧸ 21.12° V
• 13.11 461.9 cos(600t − 80.26°) mA
• 13.13 [4.308 + j4.538] Ω
• 13.15 (11.251 + j18.754) Ω, 970.1⧸−14.04° mA
• 13.17 [25.07 + j25.86] Ω
• 13.19 See Fig. D.22.

Figure D.22

For Prob. 13.19.

• 13.21 This is a design problem with multiple answers.

• 13.23 100 cos(100t − 90°) V, 5 J

• 13.25 2.2 sin(2t − 4.88°) A, 1.5085⧸ 17.9° Ω

• 13.27 191.86 W

• 13.29 0.9845, 521.6 mJ

• 13.31 This is a design problem with multiple answers.

• 13.33 12.769 + j 7.154 Ω

• 13.35 1.4754⧸−21.41° A, 77.5⧸−134.85° mA, 77⧸−110.41° mA 13.37 (a) 10, (b) 208.3 A, (c) 20.83 A 13.39 15.7⧸ 20.31° A, 78.5⧸ 20.31° A 13.41 −6 A 13.43 16.744 V, 66.98 V 13.45 36.71 mW 13.47 109.55 cos(3t + 5.48°) V 13.49 0.937 cos(2t + 51.34°) A 13.51 [8 − j1.5 Ω, 14.743⧸ 10.62° A 13.53 (a) 5, (b) 112.5 W 13.55 5 Ω 13.57 (a) 25.9⧸ 69.96°, 12.95⧸ 69.96° A (rms), (b) 21.06⧸ 147.4°, 42.12⧸ 147.4°, 42.12⧸ 147.4° V(rms), (c) 1554⧸ 20.04° VA 13.59 420.1 W, 283.6 W, 52.52 W 13.61 6 A, 0.36 A, −60 V 13.63 7.071⧸−45° A, 3.536⧸−45° A, 14.142⧸−45° A 13.65 11.05 W 13.67 (a) 352 V, (b) 14.205 A, (c) 5.682 A 13.69 200 V, (4 − j4) kΩ, (4 + j4) kΩ 13.71 0.913, 7.841 A 13.73 (a) three-phase ∆-Y transformer, (b) 8.66⧸ 156.87° A, 5⧸−83.13° A, (c) 1.8 kW 13.75 (a) 0.11547, (b) 76.98 A, 15.395 A 13.77 (a) a single-phase transformer, 1:n, n = 1∕110, (b) 7.576 mA 13.79 1.306⧸−68.01° A, 406.8⧸−77.86° mA, 1.336⧸−54.92° A

• 13.81 104.5⧸ 13.96° mA, 29.54⧸−143.8° mA, 208.824.4° mA

• 13.83 1.08⧸ 33.91° A, 15.14⧸−34.21° V

• 13.85 100 turns

• 13.87 0.5

• 13.89 0.5, 41.67 A, 83.33 A

• 13.91 (a) 1,875 kVA, (b) 7,812 A

• 13.93 (a) See Fig. D.23(a). (b) See Fig. D.23(b).

Figure D.23

For Prob. 13.93.

13.95 (a) 1∕60, (b) 139 mA

Chapter 14

$$14.1 \quad \frac{1}{1 + j\omega/\omega_o}, \omega_o = \frac{R}{L}$$

14.3 20s∕(s ² + 4s + 1)

14.5

$$\frac{(Ls + R)}{(LCs^2 + RCs + 1)}.$$

14.7 (a) 1.0116, (b) 0.5623, (c) 5.623 × 10¹⁰

.

Figure D.24

For Prob. 14.9.

14.11 See Fig. D.25.

Figure D.26 For Prob. 14.13.

Figure D.25 For Prob. 14.11.

Figure D.27 For Prob. 14.15.

Figure D.28

For Prob. 14.17.

14.19 See Fig. D.29.

14.21 See Fig. D.30.

14.21 See Fig. D.30.
14.23

$$\frac{1,000j\omega}{(1+j\omega)(10+j\omega)^2}$$

(It should be noted that this function could also have a minus sign out in front and still be correct. The magnitude plot does not contain this information. It can only be obtained from the phase plot.)

• 14.25 2 kΩ, 2 − j0.75 kΩ, 2 − j0.3 kΩ, 2 + j0.3 kΩ, 2 + j0.75 kΩ
• 14.27 R = 1 Ω, L = 0.1 H, C = 25 mF
• 14.29 4.082 krad/s, 105.55 rad/s, 38.67
• 14.31 0.5, 0.25 nF, 10 kΩ
• 14.33 125, 5 Mrad/s
• 14.35 250 μF, 40, 400 krad/s
• 14.37 2 kΩ, (1.4212 + j53.3) Ω, (8.85 + j132.74) Ω, (8.85 − j132.74) Ω, (1.4212 − j53.3) Ω
• 14.39 4.841 krad/s

Figure D.30 For Prob. 14.21.

14.41 This is a design problem with multiple answers.

14.43

$$\sqrt{\frac{1}{LC} - \frac{R^2}{L^2}}, \frac{1}{\sqrt{LC}}$$

• 14.45 447.2 rad/s, 1.067 rad/s, 419.1
• 14.47 796 kHz
• 14.49 This is a design problem with multiple answers.
• 14.51 1.256 kΩ
• 14.53 18.045 kΩ. 2.872 H, 10.5
• 14.55 1.56 kHz < f < 1.62 kHz, 25
• 14.57 (a) 1 rad/s, 3 rad/s, (b) 1 rad/s, 3 rad/s
• 14.59 2.408 krad/s, 15.811 krad/s

14.61 (a)

$$\frac{1}{1 + j\omega RC}$$

(b)

$$\frac{j\omega RC}{1 + j\omega RC}$$

• 14.63 10 MΩ, 100 kΩ

• 14.65 Proof

• 14.67 If Rf = 20 kΩ, then Ri = 80 kΩ and C = 15.915 nF.

• 14.69 Let R = 10 kΩ, then Rf = 25 kΩ, C = 7.96 nF.

• 14.71 Kf = 2 × 10⁻⁴ , Km = 5 × 10⁻³

• 14.73 9.6 MΩ, 32 μH, 0.375 pF

• 14.75 200 Ω, 400 μH, 1 μF

• 14.77 (a) 1,200 H, 0.5208 μF, (b) 2 mH, 312.5 nF, (c) 8 mH, 7.81 pF

14.79 (a)

$$8s + 5 + \frac{10}{s}$$

,
(b) $0.8s + 50 + \frac{10^4}{s}$ , 111.8 rad/s

• 14.81 (a) 0.4 Ω, 0.4 H, 1 mF, 1 mS, (b) 0.4 Ω, 0.4 mH, 1 μF, 1 mS
• 14.83 0.1 pF, 0.5 pF, 1 MΩ, 2 MΩ
• 14.85 See Fig. D.31.
• 14.87 See Fig. D.32; high-pass filter, f0 = 1.2 Hz.
• 14.89 See Fig. D.33.
• 14.91 See Fig. D.34; fo = 800 Hz.
• 14.93 _________ −RCs + 1 RCs + 1
• 14.95 (a) 0.541 MHz < fo < 1.624 MHz, (b) 67.98, 204.1
• 14.97 ^s 3 LRLC1C² __________________________________________ (sRiC1 + 1)(^s 2 LC2 + sRLC² + 1) + s 2 LC1(sRLC2 + 1)
• 14.99 8.165 MHz, 4.188 × 10⁶ rad/s
• 14.101 1.061 kΩ

14.101 1.061 kΩ
14.103

$$\frac{R_2(1+sCR_1)}{R_1+R_2+sCR_1R_2}$$

Figure D.31

For Prob. 14.85.

For Prob. 14.87.

For Prob. 14.89.

For Prob. 14.91.

15.13 (a)

$$\frac{s^2 - 1}{(s^2 + 1)^2}$$

,
\n(b) $\frac{2(s + 1)}{(s^2 + 2s + 2)^2}$ ,
\n(c) $\tan^{-1}\left(\frac{\beta}{s}\right)$
\n15.15 $5\frac{1 - e^{-s} - se^{-s}}{s^2(1 - e^{-3s})}$

15.17 This is a design problem with multiple answers.

15.19

$$\frac{1}{1 - e^{-2s}}$$

15.21

$$\frac{(2\pi s - 1 + e^{-2\pi s})}{2\pi s^2 (1 - e^{-2\pi s})}$$

15.23 (a)

$$\frac{(1 - e^{-s})^2}{s(1 - e^{-2s})}$$

(b)

$$\frac{2(1 - e^{-2s}) - 4se^{-2s}(s + s^2)}{s^3(1 - e^{-2s})}$$

15.25 (a) 18 and 0, (b) 18 and 0

• 15.27 (a) u(t) + 2e^−t u(t), (b) 3δ(t) − 11e⁻4*^t u*(t), (c) (2e^−t − 2e⁻3*^t* )u(t), (d) (3e⁻4*^t* − 3e⁻2*^t* + 6te⁻2^t )u(t)
• 15.29 [1 + 2e^−t cos (t + 90°)] u(t)
• 15.31 (a) (−5e^−t + 20e⁻2*^t* − 15e⁻3*^t* )u(t) (b) (−e^{−t +}(1 + 3*^t* ⁻ t 2 __ 2 )e⁻2*^t* ) u(t), (c) (−0.2e⁻2*^t* + 0.2e^−t cos(2t) + 0.4e^−t sin(2t))u(t)
• 15.33 (a) (3e^−t + 3 sin(t) − 3 cos(t))u(t), (b) cos(t −π)u(t − π), (c) 8 [1 − e^−t − te^−t − 0.5t 2 e−t ]u(t)

15.35 (a)

$$[2e^{-(t-6)} - e^{-2(t-6)}]u(t-6)$$

,
\n(b) $\frac{4}{3}u(t)[e^{-t} - e^{-4t}] - \frac{1}{3}u(t-2)[e^{-(t-2)} - e^{-4(t-2)}]$ ,
\n(c) $\frac{1}{13}u(t-1)[-3e^{-3(t-1)} + 3\cos 2(t-1) + 2\sin 2(t-1)]$

15.37 (a)

$$(2 - e^{-2t})u(t)$$

,
\n(b) $[0.4e^{-3t} + 0.6e^{-t} \cos t + 0.8e^{-t} \sin t]u(t)$ ,
\n(c) $e^{-2(t-4)} u(t-4)$ ,
\n(d) $\left(\frac{10}{3} \cos t - \frac{10}{3} \cos 2t\right)u(t)$

15.39 (a)

$$(-1.6e^{-t} \cos 4t - 4.05e^{-t} \sin 4t + 3.6e^{-2t} \cos 4t + (3.45e^{-2t} \sin 4t) u(t),$$

\n(b) $[0.08333 \cos 3t + 0.02778 \sin 3t + 0.0944e^{-0.551t} - 0.1778e^{-5.449t}]u(t)$

$$\mathbf{15.41} \quad z(t) = \begin{cases} 8t, & 0 < t < 2 \\ 16 - 8t, & 2 < t < 6 \\ -16, & 6 < t < 8 \\ 8t - 80, & 8 < t < 12 \\ 112 - 8t, & 12 < t < 14 \\ 0, & \text{otherwise} \end{cases}$$

15.43 (a)

$$y(t) = \begin{cases} \frac{1}{2}t^2, & 0 < t < 1 \\ -\frac{1}{2}t^2 + 2t - 1, & 1 < t < 2 \\ 1, & t > 2 \\ 0, & \text{otherwise} \end{cases}$$

(b)

$$y(t) = 2(1 - e^{-t}), t > 0,$$

(c)

$$y(t) = \begin{cases} \frac{1}{2}t^2 + t + \frac{1}{2}, & -1 < t < 0 \\ \frac{1}{2}t^2 + t + \frac{1}{2}, & 0 < t < 2 \\ \frac{1}{2}t^2 - 3t + \frac{9}{2}, & 2 < t < 3 \\ 0, & \text{otherwise} \end{cases}$$

15.45

$$(4e^{-2t} - 8te^{-2t})u(t)$$

\n15.47 (a) $[-6e^{-t} + 12e^{-2t}]u(t)$ , (b) $[6e^{-t} - 6e^{-2t}]$
\n15.49 (a) $\left(\frac{t}{a}(e^{at} - 1) - \frac{1}{a^2} - \frac{e^{at}}{a^2}(at - 1)\right)u(t)$ ,
\n(b) $[0.5 \cos(t)(t + 0.5 \sin(2t)) - 0.5 \sin(t)(\cos(t) - 1)]u(t)$

15.51 [12.5e^−t − 7.5e–3*^t* ]u(t)

15.53 cos(t) + sin(t) or 1.4142 cos(t − 45°)

$$15.55\ \left(\frac{1}{40} + \frac{1}{20}e^{-2t} - \frac{3}{104}e^{-4t} - \frac{3}{65}e^{-t}\cos(2t) - \frac{2}{65}e^{-t}\sin(2t)\right)u(t)$$

• 15.57 This is a design problem with multiple answers.
• 15.59 [−7.5e^−t + 36e⁻2*^t* − 31.5e⁻3*^t* ]u(t)
• 15.61 (a) [3 + 3.162 cos (2t − 161.12°)]u(t) volts, (b) [2 − 4e^−t + e⁻4*^t* ]u(t) amps, (c) [3 + 2e^−t + 3te^−t ]u(t) volts, (d) [2 + 2e^−t cos(2t)]u(t) amps

Chapter 16

• 16.1 [(7 + 35t)e⁻5*^t* ] u(t) A

• 16.3 [(20 + 20t)e^−t ]u(t) V

• 16.5 750 Ω, 25 H, 200 μF

• 16.7 [6 + 12e^−t cos(2t) + 2 sin(2t))]u(t) A

• 16.9 [3 + 5.924e⁻1.5505*^t* − 1.4235e⁻6.45*^t* ]u(t) mA

• 16.11 20.83 Ω, 80 μF

• 16.13 This is a design problem with multiple answers.

• 16.15 120 Ω

• 16.17 7.5 (e⁻2*^t* ⁻___2 √ __ 7 e⁻0.5*^t* sin ( √ __ ___7 2 t )) u(t) A

• 16.19 [−2.333e⁻t∕² + 2.333e⁻2*^t* ]u(t) volts

• 16.21 [10.776 e⁻2.679*^t* − 0.774e⁻37.32*^t* ]u(t) volts

• 16.23 24 cos(0.5t + 90°)u(t) volts

• 16.25 [45e^−t − 5e⁻9*^t* ]u(t) volts

• 16.27 [30 − 15.309e⁻0.05051*^t* + 0.3078e⁻4.949*^t* ]u(t) volts

• 16.29 17.5 cos(8t + 90°)u(t) amps

• 16.31 [−16 + 66.67e⁻0.8*^t* cos(0.6t − 53.13°)]u(t) volts, 13.333e⁻0.8*^t* [cos(0.6t + 90°)]u(t) amps

• 16.33 This is a design problem with multiple answers.

• 16.35 [9.091e^−t + 19.653e⁻0.0625*^t* cos(0.7044t − 117.55°] u(t) V.

• 16.37 [−60 + 60.21e⁻0.1672*^t* − 0.21e⁻47.84*^t* ]u(t) volts

• 16.39 [4.364e⁻2*^t* cos(4.583t − 90°)]u(t) amps

• 16.41 [100te⁻10*^t* ]u(t) volts

• 16.43 [9 + 9e⁻2*^t* + 6te⁻2*^t* ]u(t) amps

• 16.45 [io∕(ωC)] cos(ωt + 90°)u(t) volts

• 16.47 [60 − 40e⁻0.6*^t* cos(0.2t) − sin(0.2t))]u(t) A

• 16.49 [1.0714e⁻2*^t* − 2.572e⁻0.5*^t* cos(1.25t) + 4.791e⁻0.5*^t* sin(1.25t)]u(t) A

• 16.51 [−12 + 41.17e⁻15.125*^t* cos(4.608t − 73.06°)]u(t) amps

• 16.53 [11.547e^−t cos(1.7321t + 30°)]u(t) volts

• 16.55 [5−4e^−t − 1e⁻6*^t* ]u(t) amps, [2e^−t − 2e⁻6*^t* ]u(t) amps

• 16.57 (a) (3∕s)[1 − e^−s ], (b) [(2 − 2e⁻1.5*^t* )u(t) − (2 − 2e⁻1.5(t−1))u(t − 1)] V

• 16.59 [5e^−t − 10e⁻t∕² cos (t∕2)]u(t) V

• 16.61 [2.333 − 2.38e⁻1.2306*^t* + 2.033e⁻0.6347*^t* cos(1.4265t + 88.68°)]u(t) V

• 16.63 [7.5e⁻4*^t* cos (2t) + 345e⁻4*^t* sin (2t)]u(t) V, [6 − 9e⁻4*^t* cos (2t) − 17.062e⁻4*^t* sin (2t)]u(t) A

• 16.65 {110.1e⁻3*^t* + 192te⁻3*^t* − 10.1 cos(4t) + 34.58 sin(4t)}u(t) V

• 16.67 [e^10t − e⁻10*^t* ] u(t) volts; this is an unstable circuit!

• 16.69 240(s + 1)∕[s(s + 3)(3s ² + 8s + 1)], −120(s −1)∕ [s(s + 3)(3s ² + 8s + 1)]

16.71

$$160[2e^{-1.5t} - e^{-t}]u(t)$$

A

$$16.73 \ \ \frac{120s^2}{s^2+4}$$ $$16.75 \quad 6 + \frac{1.5s}{2(s+3)} - \frac{3s(s+2)}{s^2 + 4s + 20} - \frac{18s}{s^2 + 4s + 20}$$ $$16.77 \ \ \frac{9s}{3s^2+9s+2}$$

16.79 (a)

$$\frac{s^2 - 3}{3s^2 + 2s - 9}$$

, (b) $\frac{-3}{2s}$

• 16.81 −1∕(RLCs² ) 16.83 (a) __ R L e⁻Rt∕^L u(t), (b) (1 − e⁻Rt∕^L )u(t) 16.85 [9e^−t − 9e⁻2*^t* − 6te⁻2*^t* ]u(t)
• 16.87 This is a design problem with multiple answers.

$$16.89 \begin{bmatrix} v'_{C} \\ i'_{L} \end{bmatrix} = \begin{bmatrix} -0.25 & 1 \\ -1 & 0 \end{bmatrix} \begin{bmatrix} v'_{C} \\ i'_{L} \end{bmatrix} + \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} v_{s} \\ i_{s} \end{bmatrix};$$ $$v_{o}(t) = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \begin{bmatrix} v_{C} \\ i_{L} \end{bmatrix} + \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} v_{s} \\ i_{s} \end{bmatrix}$$ $$16.91 \begin{bmatrix} x'_{1} \\ x'_{2} \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -3 & -4 \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix} + \begin{bmatrix} 0 \\ 1 \end{bmatrix} z(t);$$ $$y(t) = \begin{bmatrix} 1 & 0 \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix} + \begin{bmatrix} 0 \end{bmatrix} z(t)$$

16.93

$$\begin{bmatrix} x'_{1} \\ x'_{2} \\ x'_{3} \end{bmatrix} = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -6 & -11 & -6 \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} z(t);$$ $$y(t) = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end{bmatrix} + \begin{bmatrix} 0 \end{bmatrix} z(t)$$

16.95

$$[-2.4 + 4.4e^{-3t}\cos(t) - 0.8e^{-3t}\sin(t)]u(t),$$ $$[-1.2 - 0.8e^{-3t}\cos(t) + 0.6e^{-3t}\sin(t)]u(t)$$

• 16.97 (a) 7(e^−t − e⁻4*^t* )u(t), (b) The system is stable.
• 16.99 500 μF, 333.3 H
• 16.101 100 μF

16.103 −100, 400, 2 × 104

16.105 If you let L = R² C then Vo/Io = sL.

Chapter 17

17.1 (a) periodic, 2, (b) not periodic, (c) periodic, 2 π, (d) periodic, π, (e) periodic, 10, (f) not periodic, (g) not periodic

17.3 See Fig. D.35.

Figure D.36 For Prob. 17.7.

• 17.9 a0 = 0.7958, a1 = 1.25, a2 = 0.5305, a3 = 0, b1 = 0 = b2 = b³
• 17.11 ^∑n=−∞ ∞ ⁷⁵ 4n² π2 [2 − 2 cos(nπ∕2) − 2j sin(nπ∕2) ⁺jnπ cos(nπ∕2) + nπ (sin(nπ∕2))]ejn π t∕²
• 17.13 This is a design problem with multiple answers.

17.15 (a)

$$10 + \sum_{n=1}^{\infty} \sqrt{\frac{16}{(n^2 + 1)^2} + \frac{1}{n^6}}$$

\n $\cos\left(10nt - \tan^{-1}\frac{n^2 + 1}{4n^3}\right)$ ,
\n(b) $10 + \sum_{n=1}^{\infty} \sqrt{\frac{16}{(n^2 + 1)} + \frac{1}{n^6}}$
\n $\sin\left(10nt + \tan^{-1}\frac{4n^3}{n^2 + 1}\right)$

17.17 (a) neither odd nor even, (b) even, (c) odd, (d) even, (e) neither odd nor even

$$17.19 \frac{5}{n^2 \omega_o^2} \sin n\pi/2 - \frac{10}{n\omega_o} (\cos \pi n - \cos n\pi/2)$$ $$- \frac{5}{n^2 \omega_o^2} (\sin \pi n - \sin n\pi/2) - \frac{2}{n\omega_o} \cos n\pi - \frac{\cos \pi n/2}{n\omega_o}$$ $$17.21 \frac{5}{2} + \sum_{n=1}^{\infty} \frac{40}{n^2} \Big[ 1 - \cos \Big( \frac{n\pi}{2} \Big) \Big] \cos \Big( \frac{n\pi t}{2} \Big)$$

17.21

$$\frac{5}{2} + \sum_{n=1}^{\infty} \frac{40}{n^2 \pi^2} \left[1 - \cos\left(\frac{n\pi}{2}\right)\right] \cos\left(\frac{n\pi}{2}\right)$$

17.23 This is a design problem with multiple

answers.

17.25
\n

$$\sum_{n=1}^{\infty} \left\{ \left[ \frac{6}{\pi^2 n^2} \left( \cos \left( \frac{2\pi n}{3} \right) - 1 \right) + \frac{4}{\pi n} \sin \left( \frac{2\pi n}{3} \right) \right] \cos \left( \frac{2\pi n}{3} \right) \right\}$$

\n

$$+ \left[ \frac{6}{\pi^2 n^2} \sin \left( \frac{2\pi n}{3} \right) - \frac{4}{n \pi} \cos \left( \frac{2\pi n}{3} \right) \right] \sin \left( \frac{2\pi n}{3} \right)$$

17.27 (a) odd, (b) −0.315, (c) 2.681

17.29

$$2\sum_{k=1}^{\infty} \left[\frac{2}{n^2 \pi} \cos(nt) - \frac{1}{n} \sin(nt)\right], n = 2k - 1$$

17.31

$$\omega'_{o} = \frac{2\pi}{T'} = \frac{2\pi}{T/\alpha} = \alpha \omega_{o}$$ $$a'_{n} = \frac{2}{T'} \int_{0}^{T'} f(\alpha t) \cos n\omega'_{o} t \, dt$$

Let $\alpha t = \lambda$ , $dt = d\lambda/\alpha$ , and $\alpha T' = T$ . Then

$$a'_{n} = \frac{2\alpha}{T} \int_{0}^{T} f(\lambda) \cos n\omega_{o} \lambda \, d\lambda/\alpha = a_{n}$$

Similarly, $b'_{n} = b_{n}$

17.33

$$v_o(t) = \sum_{n=1}^{\infty} A_n \sin(n \pi t - \theta_n) \text{ V},$$

\n

$$A_n = \frac{10(4 - 2n^2 \pi^2)}{\sqrt{(20 - 10n^2 \pi^2)^2 - 64n^2 \pi^2}},$$

\n

$$\theta_n = 90^\circ - \tan^{-1} \left(\frac{8n \pi}{20 - 10n^2 \pi^2}\right)$$

17.35

$$\frac{3}{8} + \sum_{n=1}^{\infty} A_n \cos\left(\frac{2\pi n}{3} + \theta_n\right)$$

, where

$$A_n = \frac{\frac{6}{n\pi} \sin\frac{2n\pi}{3}}{\sqrt{9\pi^2 n^2 + (2\pi^2 n^2/3 - 3)^2}},$$ $$\theta_n = \frac{\pi}{2} - \tan^{-1}\left(\frac{2n\pi}{9} - \frac{1}{n\pi}\right)$$

17.37

$$\sum_{n=1}^{\infty} \frac{2(1 - \cos \pi n)}{\sqrt{1 + n^2 \pi^2}} \cos (n \pi t - \tan^{-1} n \pi)$$ $$17.39 \frac{1}{10} + \frac{400}{\pi} \sum_{k=1}^{\infty} I_n \sin(n\pi t - \theta_n), n = 2k - 1,$$ $$\theta_n = 90^\circ + \tan^{-1} \frac{2n^2 \pi^2 - 1,200}{802n\pi},$$ $$I_n = \frac{1}{n\sqrt{(804n\pi)^2 + (2n^2 \pi^2 - 1,200)}}$$

17.41

$$\frac{200}{\pi} + \sum_{n=1}^{\infty} A_n \cos(2nt + \theta_n)$$

where

$$A_n = \frac{2,000}{\pi (4n^2 - 1)\sqrt{16n^2 - 40n + 29}}$$

and

$$\theta_n = 90^\circ - \tan^{-1}(2n - 2.5)$$

17.43 (a) 33.91 V, (b) 6.782 A, (c) 203.1 W

17.45 4.263 A, 181.7 W

17.47 10%

17.49

\n(a)

$$3.162$$

,

\n(b) $3.065$ ,

\n(c) $3.068\%$

17.51 This is a design problem with multiple answers.

17.53

$$\sum_{n=-\infty}^{\infty} \frac{0.6321e^{j2n\pi t}}{1+j2n\pi}$$

17.55

$$\sum_{n=-\infty}^{\infty} \frac{1+e^{-jn\pi}}{2\pi(1-n^2)} e^{jnt}$$ $$17.57 -3 + \sum_{n=\infty, n\neq 0}^{\infty} \frac{3}{n^3 - 2} e^{j50nt}$$

17.59

$$-\sum_{\substack{n=-\infty\\n\neq 0}}^{\infty} \frac{j4e^{-j(2n+1)\pi t}}{(2n+1)\pi}$$

17.61 (a) 6 + 2.571 cos t − 3.83 sin t + 1.638 cos 2t − 1.147 sin 2t + 0.906 cos 3t − 0.423 sin 3t + 0.47 cos 4t − 0.171 sin 4t, (b) 6.828

17.63 See Fig. D.37.

Figure D.37

17.67 DC COMPONENT = 2.000396E + 00

HARMONIC NO	FREQUENCY (HZ)	FOURIER COMPONENT	NORMALIZED COMPONENT	PHASE (DEG)	NORMALIZED PHASE (DEG)
1	1.667E-01	2.432E+00	1.000E+00	-8.996E+01	0.000E+00
2	3.334E-01	6.576E-04	2.705E-04	-8.932E+01	6.467E-01
3	5.001E-01	5.403E-01	2.222E-01	9.011E+01	1.801E+02
4	6.668E+01	3.343E-04	1.375E-04	9.134E+01	1.813E+02
5	8.335E-01	9.716E-02	3.996E-02	-8.982E+01	1.433E-01
6	1.000E+00	7.481E-06	3.076E-06	-9.000E+01	-3.581E-02
7	1.167E+00	4.968E-02	2.043E-01	-8.975E+01	2.173E-01
8	1.334E+00	1.613E-04	6.634E-05	-8.722E+01	2.748E+00
9	1.500E+00	6.002E-02	2.468E-02	-9.032E+01	1.803E+02

17.69 HARMONIC NO	FREQUENCY (HZ)	FOURIER COMPONENT	NORMALIZED COMPONENT	PHASE (DEG)	NORMALIZED PHASE (DEG)
1	5.000E-01	4.056E-01	1.000E+00	-9.090E+01	0.000E+00
2	1.000E+00	2.977E-04	7.341E-04	-8.707E+01	3.833E+00
3	1.500E+00	4.531E-02	1.117E-01	-9.266E+01	-1.761E+00
4	2.000E+00	2.969E-04	7.320E-04	-8.414E+01	6.757E+00
5	2.500E+00	1.648E-02	4.064E-02	-9.432E+01	-3.417E+00
6	3.000E+00	2.955E-04	7.285E-04	-8.124E+01	9.659E+00
7	3.500E+00	8.535E-03	2.104E-02	-9.581E+01	-4.911E+00
8	4.000E+00	2.935E-04	7.238E-04	-7.836E+01	1.254E+01
9	4.500E+00	5.258E-03	1.296E-02	-9.710E+01	-6.197E+00

TOTAL HARMONIC DISTORTION = 1.214285+01 PERCENT

17.71 See Fig. D.39.

17.73 300 mW

17.75 24.59 mF

17.77 (a) π, (b) −2 V, (c) 11.02 V

17.79 See below for the program in MATLAB and the results. % for problem 17.79 a = 10; c = 4.*a∕pi for n = 1:10 b(n) = c/(2*n-1); end diary n, b

n	bn
1	12.7307
2	4.2430
3	2.5461
4	1.8187
5	1.414
6	1.1573
7	0.9793
8	0.8487
9	0.7488
10	0.6700

diary off

17.81 (a)

$$\frac{A^2}{2}$$

, (b) $|c_1| = 2A/(3\pi)$ , $|c_2| = 2A/(15\pi)$ ,
$|c_3| = 2A/(35\pi)$ , $|c_4| = 2A/(63\pi)$ (c) 81.1%
(d) 0.72%

Chapter 18

Chapter 18
\n18.1

$$\frac{14(\cos 2\omega - \cos \omega)}{j\omega}$$

\n18.3

$$\frac{j8}{\omega^2} (2\omega \cos 2\omega - \sin 2\omega)$$

18.5 ^6j __ ω⁻ 6j ___ ω2 sin ω

18.7 (a)

$$\frac{2 - e^{-j\omega} - e^{-j2\omega}}{j\omega}$$

, (b) $\frac{5e^{-j2\omega}}{\omega^2} (1 + j\omega^2) - \frac{5}{\omega^2}$

18.9 (a)

$$\frac{10}{\omega} \sin 2\omega + \frac{20}{\omega} \sin \omega$$

,
\n(b) $\frac{10}{\omega^2} - \frac{10e^{-j\omega}}{\omega^2} (1 + j\omega)$
\n18.11 $\frac{11\pi}{\omega^2 - \pi^2} (e^{-j\omega^2} - 1)$

18.13 (a)

$$\pi e^{-j\pi/3} \delta(\omega - a) + \pi e^{j\pi/3} \delta(\omega + a)$$

,
\n(b) $\frac{e^{j\omega}}{\omega^2 - 1}$ , (c) $\pi[\delta(\omega + b) + \delta(\omega - b)]$
\n $+ \frac{j\pi A}{2} [\delta(\omega + a + b) - \delta(\omega - a + b) + \delta(\omega + a - b) - \delta(\omega - a - b)],$
\n(d) $\frac{1}{\omega^2} - \frac{e^{-j4\omega}}{j\omega} - \frac{e^{-j4\omega}}{\omega^2} (j4\omega + 1)$

18.15 (a)

$$2j \sin 3\omega$$

, (b) $\frac{2e^{-j\omega}}{j\omega}$ , (c) $\frac{1}{3} - \frac{j\omega}{2}$

18.17 (a)

$$\pi[\delta(\omega + 2) + \delta(\omega - 2)] - \frac{2j\omega}{\omega^2 - 4}$$

,
(b) $\frac{j\pi}{4}[\delta(\omega + 10) - \delta(\omega - 10)] - \frac{5}{\omega^2 - 100}$

$$18.19 \ \frac{2j\omega}{\omega^2 - 4\pi^2} (e^{-j\omega} - 1)$$

18.21 Proof

18.21 Proof
\n18.23 (a)

$$\frac{30}{(6 - j\omega)(15 - j\omega)}$$

,
\n(b) $\frac{20e^{-j\omega/2}}{(4 + j\omega)(10 + j\omega)}$ ,
\n(c) $\frac{5}{[2 + j(\omega + 2)][5 + j(\omega + 2)]}$ +
\n $\frac{5}{[2 + j(\omega - 2)][5 + j(\omega - 2)]}$
\n(d) $\frac{j\omega 10}{(2 + j\omega)(5 + j\omega)}$ ,
\n(e) $\frac{10}{j\omega(2 + j\omega)(5 + j\omega)} + \pi\delta(\omega)$

18.25 (a) 5e^2t u(t), (b) 6e⁻2*^t* , (c) (−10et u(t) + 10e^2t )u(t)

18.27 (a)

$$5 \operatorname{sgn}(t) - 10e^{-10t} u(t)$$

,
\n(b) $4e^{2t}u(-t) - 6e^{-3t}u(t)$ ,
\n(c) $2e^{-20t} \sin(30t) u(t)$ , (d) $\frac{1}{4} \pi$

18.29 (a)

$$\frac{1}{2\pi}(1 + 8 \cos 3t)$$

, (b) $\frac{4 \sin 2t}{\pi t}$ ,
(c) $3\delta(t + 2) + 3\delta(t - 2)$

18.31 (a)

$$x(t) = e^{-at}u(t)$$

,
\n(b) $x(t) = u(t+1) - u(t-1)$ ,
\n(c) $x(t) = \frac{1}{2}\delta(t) - \frac{a}{2}e^{-at}u(t)$

18.33 (a)

$$\frac{2j \sin t}{t^2 - \pi^2}$$

, (b) $u(t-1) - u(t-2)$

18.35 (a)

$$\frac{e^{-j\omega/3}}{6+j\omega}$$

, (b) $\frac{1}{2} \left[ \frac{1}{2+j(\omega+5)} + \frac{1}{2+j(\omega-5)} \right]$ ,
(c) $\frac{j\omega}{2+j\omega}$ , (d) $\frac{1}{(2+j\omega)^2}$ , (e) $\frac{1}{(2+j\omega)^2}$

$$18.37 \ \frac{j\omega}{4+j3\omega}$$

18.39 5 × 10³ ________ 106 + jω ( ___1 jω + ___1 ^{ω2 −} ___1 ^{ω2 e−}j^ω )

18.41 ²jω(4.5 + j2ω) ___________________ (2 + jω)(4 − 2ω² + jω)

18.43 1000(e⁻1*^t* − e⁻1.25*^t* )u(t) V

18.45 5(e^−t − e⁻2^t )u(t) A

• 18.47 16(e^−t − e⁻2^t )u(t) V
• 18.49 0.542 cos (t + 13.64°) V
• 18.51 16.667 J
• 18.53 π
• 18.55 682.5 J
• 18.57 2 J, 87.43%
• 18.59 (16e^−t − 20e⁻2^t + 4e⁻4^t )u(t) V
• 18.61 2X(ω) + 0.5X(ω + ω0) + 0.5X(ω − ω0)
• 18.63 106 stations
• 18.65 6.8 kHz
• 18.67 200 Hz, 5 ms

18.69 35.24%

Chapter 19
\n19.1

$$\begin{bmatrix} 30 & 10 \\ 10 & 30 \end{bmatrix}$$

Ω
\n19.3 $\begin{bmatrix} 10 & -j10 \\ -j10 & -j10 \end{bmatrix}$ Ω
\n19.5 $\begin{bmatrix} 10(s+2) & 10 \\ 10 & 10 \end{bmatrix}$
\n19.7 $\begin{bmatrix} 20(s+0.5) & -30 \\ -10 & -20 \end{bmatrix}$ Ω
\n19.9 $\begin{bmatrix} 2.5 & 1.25 \\ 1.25 & 3.125 \end{bmatrix}$ Ω

19.11 See Fig. D.40.

Figure D.40

For Prob. 19.11.

19.13 329.9 W

19.15 24 Ω, 1.536 kW

• 19.17 [ 9.6 −0.8 −0.8 8.4 ] Ω and [ 0.105 0.01 0.01 0.12] ^S
• 19.19 This is a design problem with multiple answers.
• 19.21 See Fig. D.41.

Figure D.41 For Prob. 19.21.

19.23

$$\begin{bmatrix} s+2 & -(s+1) \ -(s+1) & \frac{s^2+s+1}{s} \end{bmatrix}, \frac{0.8(s+1)}{s^2+1.8s+1.2}$$

19.25 See Fig. D.42.

Figure D.42 For Prob. 19.25.

• 19.27 [0.25 5 0.025 0.6 ]^S
• 19.29 (a) 44 V, 16 V, (b) same
• 19.31 [3.8 ^Ω −3.6 0.4 0.2 S] 19.33 [(3.077 + j1.2821) ^Ω −0.3846 + j0.2564 0.3846 − j0.2564 (76.9 + 282.1) mS]
• 19.35 [ ^2Ω −0.5 0.5 0 ]

19.37 3.571 V

19.39

$$g_{11} = \frac{1}{R_1 + R_2}, g_{12} = -\frac{R_2}{R_1 + R_2}$$

$g_{21} = \frac{R_2}{R_1 + R_2}, g_{22} = R_3 + \frac{R_1 R_2}{R_1 + R_2}$

19.41 Proof

19.43 (a)

$$\begin{bmatrix} 1 & \mathbf{Z} \\ 0 & 1 \end{bmatrix}$$

, (b) $\begin{bmatrix} 1 & 0 \\ \mathbf{Y} & 1 \end{bmatrix}$
\n19.45 $\begin{bmatrix} 1 & (20+j20) \Omega \\ j100 \mu s & 1 \end{bmatrix}$
\n19.47 $\begin{bmatrix} 0.3235 & 1.176 \Omega \\ 0.02941 S & 0.4706 \end{bmatrix}$

$$19.49 \begin{bmatrix} \frac{2s+1}{s} & \frac{1}{s} \Omega \\ \frac{(s+1)(3s+1)}{s} S & 2 + \frac{1}{s} \end{bmatrix}$$ $$19.51 \begin{bmatrix} 2 & 2+j5 \\ j & -2+j \end{bmatrix}$$

19.53

$$z_{11} = \frac{A}{C}$$

, $z_{12} = \frac{AD - BC}{C}$ , $z_{21} = \frac{1}{C}$ , $z_{22} = \frac{D}{C}$

19.55 Proof

19.57

$$\begin{bmatrix} 3 & 1 \\ 1 & 7 \end{bmatrix} \Omega$$

, $\begin{bmatrix} \frac{7}{20} & \frac{-1}{20} \\ \frac{-1}{20} & \frac{3}{20} \end{bmatrix} S$ , $\begin{bmatrix} \frac{20}{7} \Omega & \frac{1}{7} \\ \frac{-1}{7} & \frac{1}{7} S \end{bmatrix}$ , $\begin{bmatrix} \frac{1}{3} S & \frac{-1}{3} \\ \frac{1}{3} & \frac{20}{3} \Omega \end{bmatrix}$ , $\begin{bmatrix} 7 & 20 \Omega \\ 1 S & 3 \end{bmatrix}$

19.59

$$\begin{bmatrix} 16.667 & 6.667 \\ 3.333 & 3.333 \end{bmatrix} \Omega, \begin{bmatrix} 0.1 & -0.2 \\ -0.1 & 0.5 \end{bmatrix} S,$$ $$\begin{bmatrix} 10 \Omega & 2 \\ -1 & 0.3 \Omega \end{bmatrix}, \begin{bmatrix} 5 \Omega & 10 \Omega \\ 0.3 \Omega & 1 \end{bmatrix}$$

19.61 (a)

$$\begin{bmatrix} 5 & 4 \\ 3 & 3 \\ 4 & 5 \\ 3 & 3 \end{bmatrix}$$

$\Omega$ , (b) $\begin{bmatrix} 5 & 2 & 4 \\ 3 & 5 \\ -4 & 3 & 5 \\ 5 & 5 & 5 \end{bmatrix}$ , (c) $\begin{bmatrix} 5 & 3 & 0 \\ 4 & 4 & 0 \\ 3 & 5 & 5 \\ 4 & 4 & 4 \end{bmatrix}$

19.63

$$\begin{bmatrix} 0.8 & 2.4 \\ 2.4 & 7.2 \end{bmatrix} \Omega$$ $$19.65\begin{bmatrix} \frac{0.5}{3} & -\frac{1}{-0.5} \\ -\frac{0.5}{3} & \frac{2}{5/6} \end{bmatrix} S$$

19.67 [ ⁴ 0.1576 S 63.29 Ω 4.994 ]

19.69

$$\begin{bmatrix} \frac{s+1}{s+2} & \frac{-(3s+2)}{2(s+2)} \\ \frac{-(3s+2)}{2(s+2)} & \frac{5s^2+4s+4}{2s(s+2)} \end{bmatrix}$$

19.71

$$\begin{bmatrix} 2 & -3.334 \ 3.334 & 20.22 \end{bmatrix} \Omega$$

\n19.73

$$\begin{bmatrix} 14.628 & 3.141 \ 5.432 & 19.625 \end{bmatrix} \Omega$$

\n19.75 (a)

$$\begin{bmatrix} 0.3015 & -0.1765 \ 0.0588 & 19.625 \end{bmatrix} S$$

, (b) -0.0051
\n19.77

$$\begin{bmatrix} 0.9488/-161.6^{\circ} \\ 0.3163/-161.6^{\circ} \end{bmatrix} \begin{bmatrix} 0.3163/18.42^{\circ} \\ 0.9488/-161.6^{\circ} \end{bmatrix}$$

\n19.79

$$\begin{bmatrix} 4.669/-136.7^{\circ} \\ 2.53/-108.4^{\circ} \end{bmatrix} \begin{bmatrix} 2.53/-108.4^{\circ} \\ 1.789/-153.4^{\circ} \end{bmatrix} \Omega$$

\n19.81

$$\begin{bmatrix} 1.5 & -0.5 \\ 3.5 & 1.5 \end{bmatrix} S$$

\n19.83

$$\begin{bmatrix} 0.3235 & 1.1765 \\ 0.02941 S & 0.4706 \end{bmatrix}$$

\n19.85

$$\begin{bmatrix} 1.581/71.59^{\circ} \\ 1.587 \end{bmatrix} \begin{bmatrix} -\frac{1}{3} \\ 5.661 \times 10^{-4} \end{bmatrix}$$

5.661 × 10⁻⁴]

j S

$$19.87 \begin{bmatrix} -j1.765 & -j1.765 \Omega \\ j888.2 \text{ S} & j888.2 \end{bmatrix}$$

19.89 −1,613, 64.15 dB

19.91 (a) −25.64 for the transistor and −9.615 for the circuit. (b) 74.07, (c) 1.2 kΩ, (d) 51.28 kΩ

19.93 −17.74, 144.5, 31.17 Ω, −6.148 MΩ

19.95 See Fig. D.43.

Figure D.43

For Prob. 19.95.

19.97 250 mF, 333.3 mH, 500 mF

19.99 Proof