Section 9.3 Phasors

← Back to Fundamentals of Electric Circuits Overview (a) 10 sin(ωt + 30°) (b) −9 sin (8t) (c) −20 sin(ωt + 45°)

• 9.4 Design a problem to help other students better understand sinusoids.
• 9.5 Given v1 = 45 sin(ωt + 30°) V and v2 = 50 cos(ωt − 30°) V, determine the phase angle between the two sinusoids and which one lags the other.
• 9.6 For the following pairs of sinusoids, determine which one leads and by how much.

(a)

$$v(t) = 10 \cos(4t - 60^{\circ})$$

and
$i(t) = 4 \sin(4t + 50^{\circ})$

• (b) v1(t) = 4 cos(377t + 10°) and v2(t) = −20 cos 377t
• (c) x(t) = 13 cos 2t + 5 sin 2t and y(t) = 15 cos(2t − 11.8°)

Section 9.3 Phasors

• 9.7 If f(ϕ) = cos ϕ + j sin ϕ, show that f(ϕ) = e j^ϕ .
• 9.8 Calculate these complex numbers and express your results in rectangular form:

(a)

$$\frac{60/45^{\circ}}{7.5 - j10} + j2$$

\n(b)

$$\frac{32/20^{\circ}}{(6 - j8)(4 + j2)} + \frac{20}{-10 + j24}$$

\n(c)

$$20 + (16/-50^{\circ})(5 + j12)$$

9.9 Evaluate the following complex numbers and leave your results in polar form:

(a)

$$5/30^{\circ}$$

$\left(6 - j8 + \frac{3/60^{\circ}}{2 + j}\right)$
(b) $\frac{(10/60^{\circ})(35/ -50^{\circ})}{(2 + j6) - (5 + j)}$

9.10 Design a problem to help other students better understand phasors.

9.11 Find the phasors corresponding to the following signals:

(a)

$$v(t) = 21 \cos(4t - 15^\circ) \text{V}$$

(b)

$$i(t) = -8 \sin(10t + 70^{\circ})$$

mA

(c)

$$v(t) = 120 \sin(10t - 50^{\circ})$$

V

(d)

$$i(t) = -60 \cos(30t + 10^{\circ})$$

mA

9.12 Let X = 4⧸ 40° and Y = 20⧸−30°. Evaluate the following quantities and express your results in polar form:

$$\text{(a)}\,(X+Y)X^*$$ $$(b) (X - Y)^*$$

(c) (X + Y)∕X

9.13 Evaluate the following complex numbers:

(a)

$$\frac{2+j3}{1-j6} + \frac{7-j8}{-5+j11}$$

\n(b)

$$\frac{(5/10°)(10/-40°)}{(4/-80°)(-6/50°)}$$

\n(c)

$$\begin{vmatrix} 2+j3 & -j2 \\ -j2 & 8-j5 \end{vmatrix}$$

9.14 Simplify the following expressions:

Simplify the following expressions:
\n(a)

$$\frac{(5 - j6) - (2 + j8)}{(-3 + j4)(5 - j) + (4 - j6)}$$

\n(b)

$$\frac{(240/75^\circ + 160/–30^\circ)(60 - j80)}{(67 + j84)(20/32^\circ)}$$

\n(c)

$$\left(\frac{10 + j20}{3 + j4}\right)^2 \sqrt{(10 + j5)(16 - j20)}$$

9.15 Evaluate these determinants:

(a)

$$\begin{vmatrix} 10 + j6 & 2 - j3 \ -5 & -1 + j \end{vmatrix}$$

\n(b)

$$\begin{vmatrix} 20 \underline{/-30^{\circ}} & -4 \underline{/-10^{\circ}} \\ 16 \underline{/-0^{\circ}} & 3 \underline{/-40^{\circ}} \end{vmatrix}$$

\n(c)

$$\begin{vmatrix} 1 - j & -j & 0 \\ j & 1 & -j \\ 1 & j & 1 + j \end{vmatrix}$$

9.16 Transform the following sinusoids to phasors:

(a) −20 cos(4t + 135°) (b) 8 sin(20t + 30°) (c) 20 cos (2t) + 15 sin (2t)

• 9.17 Two voltages v1 and v2 appear in series so that their sum is v = v1 + v2. If v1 = 10 cos(50t − π∕3) V and v2 = 12 cos(50t + 30°) V, find v.
• 9.18 Obtain the sinusoids corresponding to each of the following phasors:

(a)

$$V_1 = 60/15^{\circ}$$

V, $\omega = 1$
\n(b) $V_2 = 6 + j8$ V, $\omega = 40$
\n(c) $I_1 = 2.8e^{-j\pi/3}$ A, $\omega = 377$
\n(d) $I_2 = -0.5 - j1.2$ A, $\omega = 10^3$

9.19 Using phasors, find:

(a) 3 cos(20t + 10°) − 5 cos(20t − 30°)

• (b) 40 sin 50t + 30 cos(50t − 45°)
• (c) 20 sin 400t + 10 cos(400t + 60°)

$$-5\sin(400t-20^\circ)$$

9.20 A linear network has a current input 7.5 cos(10t + 30°) A and a voltage output 120 cos(10t + 75°) V. Determine the associated impedance.

9.21 Simplify the following:

(a)

$$f(t) = 5 \cos(2t + 15^\circ) - 4 \sin(2t - 30^\circ)$$

(b) $g(t) = 8 \sin t + 4 \cos(t + 50^\circ)$

• (c) h(t) = ∫ 0 (10 cos 40t + 50 sin 40t) dt
• 9.22 An alternating voltage is given by v(t) = 55 cos(5t + 45°) V. Use phasors to find

$$10v(t) + 4\frac{dv}{dt} - 2\int_{-\infty}^{t} v(t) dt$$

Assume that the v alue of the inte gral is zero at t = −∞.

9.23 Apply phasor analysis to evaluate the following:

(a)

$$v = [110 \sin(20t + 30^\circ) + 220 \cos(20t - 90^\circ)]$$

V
(b) $i = [30 \cos(5t + 60^\circ) - 20 \sin(5t + 60^\circ)]$ A

9.24 Find v(t) in the following integrodifferential equations using the phasor approach:

(a)

$$v(t) + \int v dt = 10 \cos t$$

\n(b) $\frac{dv}{dt} + 5v(t) + 4 \int v dt = 20 \sin(4t + 10^{\circ})$

9.25 Using phasors, determine i(t) in the following equations:

(a)

$$2\frac{di}{dt} + 3i(t) = 4\cos(2t - 45^{\circ})$$

\n(b) $10 \int i \, dt + \frac{di}{dt} + 6i(t) = 5\cos(5t + 22^{\circ})$ A

9.26 The loop equation for a series RLC circuit gives

$$\frac{di}{dt} + 2i + \int_{-\infty}^{t} i \, dt = \cos 2t \, A$$

Assuming that the v alue of the inte gral at t = −∞ is zero, find i(t) using the phasor method.

9.27 A parallel RLC circuit has the node equation

$$\frac{dv}{dt} + 50v + 100 \int v \, dt = 110 \cos(377t - 10^{\circ}) \, \text{V}$$

Determine v(t) using the phasor method. You may assume that the value of the integral at t = −∞ is zero.

Section 9.4 Phasor Relationships for Circuit Elements

• 9.28 Determine the current that flows through an 20-Ω resistor connected to a voltage source vs = 120 cos (377t + 37°) V.

• 9.29 Given that vc(0) = 2 cos(155°) V, what is the instantaneous voltage across a 2-μF capacitor when the current through it is i = 4 sin(106 t + 25°) A?

• 9.30 A voltage v(t) = 100 cos(60t + 20°) V is applied to a parallel combination of a 40-kΩ resistor and a 50-μF capacitor. Find the steady-state currents through the resistor and the capacitor.

• 9.31 A series RLC circuit has R = 80 Ω, L = 240 mH, and C = 5 mF. If the input voltage is v(t) = 115 cos 2t, find the current flowing through the circuit.

• 9.32 Using Fig. 9.40, design a problem to help other students better understand phasor relationships for circuit elements.

• 9.33 A series RL circuit is connected to a 220-V ac source. If the voltage across the resistor is 170 V, find the voltage across the inductor.

• 9.34 What value of ω will cause the forced response, vo, in Fig. 9.41 to be zero?

Figure 9.41 For Prob. 9.34.

Section 9.5 Impedance and Admittance

9.35 Find the steady-state current i in the circuit of Fig. 9.42, when vs(t) = 115 cos 200t V.

Figure 9.42 For Prob. 9.35.

9.36 Using Fig. 9.43, design a problem to help other students better understand impedance.

9.37 Determine the admittance Y for the circuit in Fig. 9.44.

Figure 9.45

For Prob. 9.38.

9.39 For the circuit shown in Fig. 9.46, find Zeq and use that to find current I. Let ω = 10 rad/s.

Figure 9.46 For Prob. 9.39.

9.40 In the circuit of Fig. 9.47, find io when:

(a)

$$\omega = 1$$

rad/s (b) $\omega = 5$ rad/s
(c) $\omega = 10$ rad/s

Figure 9.47 For Prob. 9.40.

9.41 Find v(t) in the RLC circuit of Fig. 9.48.

Figure 9.48

For Prob. 9.41.

9.42 Calculate vo(t) in the circuit of Fig. 9.49.

Figure 9.49

For Prob. 9.42.

9.43 Find current Io in the circuit shown in Fig. 9.50.

Figure 9.50 For Prob. 9.43.

9.44 Calculate i(t) in the circuit of Fig. 9.51.

Figure 9.51 For prob. 9.44.

Figure 9.52 For Prob. 9.45.

200 mH 100 mF 2 Ω ^{2Ω i} o ^{vs +} ‒

Figure 9.53

For Prob. 9.46.

9.47 In the circuit of Fig. 9.54, determine the value of is(t).

Figure 9.54

For Prob. 9.47.

9.48 Given that vs(t) = 20 sin(100t − 40°) in Fig. 9.55, determine ix(t).

Figure 9.55

For Prob. 9.48.

9.49 Find v ^s (t) in the circuit of Fig. 9.56 if the current ix through the 1-Ω resistor is 8 sin 200t A.

Figure 9.56 For Prob. 9.49.

9.50 Determine vx in the circuit of Fig. 9.57. Let is(t) = 5 cos(100t + 40°) A.

Figure 9.57

For Prob. 9.50.

9.51 If the voltage vo across the 2-Ω resistor in the circuit of Fig. 9.58 is 90 cos 2t V, obtain is.

Figure 9.58

For Prob. 9.51.

9.52 If Vo = 8⧸ 30° V in the circuit of Fig. 9.59, find Is.

Figure 9.59

For Prob. 9.52.

Figure 9.60

For Prob. 9.53.

9.54 In the circuit of Fig. 9.61, find Vs if Io = 30⧸ 0° A.

Figure 9.61 For Prob. 9.54.

Figure 9.62

For Prob. 9.55.

Section 9.7 Impedance Combinations

9.56 At ω = 377 rad/s, find the input impedance of the circuit shown in Fig. 9.63.

Figure 9.63 For Prob. 9.56.

9.57 At ω = 1 rad/s, obtain the input admittance in the circuit of Fig. 9.64.

Figure 9.64

For Prob. 9.57.

9.58 Using Fig. 9.65, design a problem to help other students better understand impedance combinations.

* An asterisk indicates a challenging problem.

* 9.59 For the network in Fig. 9.66, find Zin. Let ω = 100 rad/s.

Figure 9.66

For Prob. 9.59.

9.60 Obtain Zin for the circuit in Fig. 9.67.

Figure 9.67

For Prob. 9.60.

9.61 Find Zeq in the circuit of Fig. 9.68.

Figure 9.68

For Prob. 9.61.

9.62 For the circuit in Fig. 9.69, find the input impedance Zin at 10 krad/s.

Figure 9.69 For Prob. 9.62.

9.63 For the circuit in Fig. 9.70, find the value of ZT.

Figure 9.70 For Prob. 9.63.

9.64 Find ZT and Vo in the circuit in Fig. 9.71. Let the value of the inductance equal j20 Ω.

9.65 Determine ZT and I for the circuit in Fig. 9.72.

Figure 9.72

For Prob. 9.65.

9.66 For the circuit in Fig. 9.73, calculate ZT and Vab.

Figure 9.73 For Prob. 9.66.

9.67 At ω = 103 rad/s, find the input admittance of each of the circuits in Fig. 9.74.

Figure 9.74

For Prob. 9.67.

9.68 Determine Yeq for the circuit in Fig. 9.75.

Figure 9.75

For Prob. 9.68.

9.69 Find the equivalent admittance Yeq of the circuit in Fig. 9.76.

Figure 9.76

For Prob. 9.69.

9.70 Find the equivalent impedance of the circuit in Fig. 9.77.

Figure 9.77 For Prob. 9.70.

9.71 Obtain the equivalent impedance of the circuit in Fig. 9.78.

Figure 9.78

For Prob. 9.71.

Figure 9.79

For Prob. 9.72.

9.73 Determine the equivalent impedance of the circuit in Fig. 9.80.

Section 9.8 Applications

• 9.74 Design an RL circuit to provide a 90° leading phase shift.
• 9.75 Design a circuit that will transform a sinusoidal
• voltage input to a cosinusoidal voltage output.
• 9.76 For the following pairs of signals, determine if v¹ leads or lags v2 and by how much.

(a) v1 = 10 cos(5t − 20°), v2 = 8 sin 5t

(b)

$$v_1 = 19 \cos(2t + 90^\circ)$$

, $v_2 = 6 \sin 2t$

(c)

$$v_1 = -4 \cos 10t$$

, $v_2 = 15 \sin 10t$

• 9.77 Refer to the RC circuit in Fig. 9.81.
  • (a) Calculate the phase shift at 2 MHz.
  • (b) Find the frequency where the phase shift is 45°.

Figure 9.81

For Prob. 9.77.

• 9.78 A coil with impedance 8 + j6 Ω is connected in series with a capacitive reactance X. The series combination is connected in parallel with a resistor R. Given that the equivalent impedance of the resulting circuit is 5⧸ 0° Ω, find the value of R and X.
• 9.79 (a) Calculate the phase shift of the circuit in Fig. 9.82. (b) State whether the phase shift is leading or lagging (output with respect to input).
  • (c) Determine the magnitude of the output when the input is 120 V.

Figure 9.82

For Prob. 9.79.

• 9.80 Consider the phase-shifting circuit in Fig. 9.83. Let Vi = 120 V operating at 60 Hz. Find:
  • (a) Vo when R is maximum
  • (b) Vo when R is minimum
  • (c) the value of R that will produce a phase shift of 45°

Figure 9.83

For Prob. 9.80.

• 9.81 The ac bridge in Fig. 9.37 is balanced when R1 = 400 Ω, R2 = 600 Ω, R3 = 1.2 kΩ, and C2 = 0.3 μF. Find Rx and Cx. Assume R2 and C2 are in series.
• 9.82 A capacitance bridge balances when R1 = 100 Ω, R2 = 2 kΩ, and Cs = 40 μF. What is Cx, the capacitance of the capacitor under test?
• 9.83 An inductive bridge balances when R1 = 1.2 kΩ, R2 = 500 Ω, and Ls = 250 mH. What is the value of Lx, the inductance of the inductor under test?

9.84 The ac bridge shown in Fig. 9.84 is known as a Maxwell bridge and is used for accurate measurement of inductance and resistance of a coil in terms of a standard capacitance Cs. Show that when the bridge is balanced,

$$L_x = R_2 R_3 C_s \qquad \text{and} \qquad R_x = \frac{R_2}{R_1} R_3$$

Find Lx and Rx for R1 = 40 k Ω, R2 = 1.6 k Ω, R3 = 4 kΩ, and Cs = 0.45 μF.

Figure 9.84 Maxwell bridge; For Prob. 9.84.

^f = ____________ 1 2π^√

9.86 The circuit shown in Fig. 9.86 is used in a television receiver. What is the total impedance of this circuit?

Comprehensive Problems

For Prob. 9.86.

9.87 The network in Fig. 9.87 is part of the schematic describing an industrial electronic sensing device. What is the total impedance of the circuit at 4 kHz?

Figure 9.87 For Prob. 9.87.

• (a) What is the impedance of the circuit?
• (b) If the frequency were halved, what would be the impedance of the circuit?

Figure 9.88

For Prob. 9.88.

9.89 An industrial load is modeled as a series combination of an inductor and a resistance as shown in Fig. 9.89. Calculate the value of a capacitor C across the series combination so that the net impedance is resistive at a frequency of 2 kHz.

For Prob. 9.89.

9.90 An industrial coil is modeled as a series combination of an inductance L and resistance R, as shown in Fig. 9.90. Since an ac voltmeter measures only the magnitude of a sinusoid, the following