Problems1

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Problems1

Section 11.2 Instantaneous and Average Power

• 11.1 If v(t) = 160 cos 50t V and i(t) = −33 sin (50t − 30°)A, calculate the instantaneous power and the average power.

• 11.2 Given the circuit in Fig. 11.35, find the average power supplied or absorbed by each element.

• 11.3 A load consists of a 60-Ω resistor in parallel with a 90-μF capacitor. If the load is connected to a voltage source vs(t) = 160 cos 2000t, find the average power delivered to the load.

• 11.4 Using Fig. 11.36, design a problem to help other students better understand instantaneous and average power.

¹Starting with problem 11.22, unless otherwise specified, assume that all values of currents and voltages are rms.

Problems 489

11.5 ssuming that vs = 8 cos(2t − 40°) V in the circuit of Fig. 11.37, find the average power delivered to each of the passive elements.

Figure 11.37 For Prob. 11.5.

11.6 For the circuit in Fig. 11.38, is = 6 cos 103 t A. Find the average power absorbed by the 50-Ω resistor.

Figure 11.38 For Prob. 11.6.

11.7 Given the circuit of Fig. 11.39, find the average power absorbed by the 10-Ω resistor.

For Prob. 11.7.

11.8 In the circuit of Fig. 11.40, determine the average power absorbed by the 40-Ω resistor.

11.9 For the op amp circuit in Fig. 11.41, Vs = 2⧸30*° V*. Find the average power absorbed by the 20-kΩ resistor.

• For Prob. 11.9.
• 11.10 In the op amp circuit in Fig. 11.42, find the total average power absorbed by the resistors.

Figure 11.42 For Prob. 11.10.

11.11 For the network in Fig. 11.43, assume that the port impedance is

$$\mathbf{Z}_{ab} = \frac{R}{\sqrt{1 + \omega^2 R^2 C^2}} \sqrt{-\tan^{-1} \omega RC}$$

Find the average power consumed by the network when R = 10 kΩ, C = 200 nF, and i = 33 sin(377t + 22°) mA.

Figure 11.43 For Prob. 11.11.

Section 11.3 Maximum Average Power Transfer

11.12 For the circuit shown in Fig. 11.44, determine the load impedance ZL for maximum power transfer (to ZL). Calculate the maximum power absorbed by the load.

Figure 11.44

For Prob. 11.12.

• 11.13 The Thevenin impedance of a source is ZTh = 120 + j60 Ω, while the peak Thevenin voltage is VTh = 165 + j0 V. Determine the maximum available average power from the source.
• 11.14 Using Fig. 11.45, design a problem to help other students better understand maximum average power transfer to a load Z.

Figure 11.45 For Prob. 11.14.

11.15 In the circuit of Fig. 11.46, find the value of ZL that will absorb the maximum power and the value of the maximum power.

For Prob. 11.15.

11.16 For the circuit in Fig. 11.47, find the value of ZL that will receive the maximum power from the circuit. Then calculate the power delivered to the load ZL.

11.17 Calculate the value of ZL in the circuit of Fig. 11.48 in order for ZL to receive maximum average power. ‒j3 ^Ω What is the maximum average power received by ZL? ^4Ω

Figure 11.48

For Prob. 11.17.

11.18 Find the value of ZL in the circuit of Fig. 11.49 for maximum power transfer.

Figure 11.49

For Prob. 11.18.

11.19 The variable resistor R in the circuit of Fig. 11.50 is adjusted until it absorbs the maximum average power. Find R and the maximum average power absorbed.

Figure 11.50 For Prob. 11.19.

11.20 The load resistance RL in Fig. 11.51 is adjusted until it absorbs the maximum average power. Calculate the value of RL and the maximum average power.

For Prob. 11.20.

11.21 Assuming that the load impedance is to be purely resistive, what load should be connected to terminals a-b of the circuits in Fig. 11.52 so that the maximum power is transferred to the load?

Figure 11.52 For Prob. 11.21.

Section 11.4 Effective or RMS Value

11.22 Find the rms value of the offset sine wave shown in Fig. 11.53.

For Prob. 11.22.

11.23 Using Fig. 11.54, design a problem to help other students better understand how to find the rms value of a waveshape.

• For Prob. 11.23.
• 11.24 Determine the rms value of the waveform in Fig. 11.55.

Figure 11.55 For Prob. 11.24.

Figure 11.56 For Prob. 11.25.

11.26 Find the effective value of the voltage waveform in Fig. 11.57.

For Prob. 11.26.

11.27 Calculate the rms value of the current waveform of Fig. 11.58.

11.28 Find the rms value of the voltage waveform of Fig. 11.59 as well as the average power absorbed by a 2-Ω resistor when the voltage is applied across the resistor.

11.29 Calculate the effective value of the current waveform in Fig. 11.60 and the average power delivered to a 12-Ω resistor when the current runs through the resistor.

Figure 11.60 For Prob. 11.29.

11.30 Compute the rms value of the waveform depicted in Fig. 11.61.

Figure 11.61

For Prob. 11.30.

11.31 Find the rms value of the signal shown in Fig. 11.62.

Figure 11.62 For Prob. 11.31.

11.32 Obtain the rms value of the current waveform shown in Fig. 11.63.

Figure 11.63 For Prob. 11.32.

11.33 Determine the rms value for the waveform in Fig. 11.64.

11.34 Find the effective value of f(t) defined in Fig. 11.65.

Figure 11.65 For Prob. 11.34.

11.35 One cycle of a periodic voltage waveform is depicted in Fig. 11.66. Find the effective value of the voltage. Note that the cycle starts at t = 0 and ends at t = 6 s.

Figure 11.66 For Prob. 11.35.

11.36 Calculate the rms value for each of the following functions:

(a) i(t) = 10 A (b) v(t) = 4 + 3 cos 5t V (c) i(t) = 8 − 6 sin 2t A (d) v(t) = 5 sin t + 4 cost V

11.37 Design a problem to help other students better understand how to determine the rms value of the sum of multiple currents.

Section 11.5 Apparent Power and Power Factor

11.38 For the power system in Fig. 11.67, find: (a) the average power, (b) the reactive power, (c) the power factor. Note that 440 V is an rms value.

11.39 An ac motor with impedance ZL = 2 + j1.2 Ω is supplied by a 220-V, 60-Hz source. (a) Find pf, P, and Q. (b) Determine the capacitor required to be connected in parallel with the motor so that the power factor is corrected to unity.

11.40 Design a problem to help other students better understand apparent power and power factor.

11.41 Obtain the power factor for each of the circuits in Fig. 11.68. Specify each power factor as leading or lagging.

Figure 11.68

For Prob. 11.41.

Section 11.6 Complex Power

• 11.42 A 110-V rms, 60-Hz source is applied to a load impedance Z. The apparent power entering the load is 120 VA at a power factor of 0.707 lagging.
  • (a) Calculate the complex power.
  • (b) Find the rms current supplied to the load.
  • (c) Determine Z. (d) Assuming that Z = R + jωL, find the values of R and L.

11.43 Design a problem to help other students understand complex power.

11.44 Find the complex power delivered by vs to the network in Fig. 11.69. Let vs = 100 cos 2000t V.

Figure 11.69 For Prob. 11.44.

11.45 The voltage across a load and the current through it are given by

$$v(t) = 20 + 60 \cos 100t$$ $$i(t) = 1 - 0.5 \sin 100t$$

A

Find:

(a) the rms values of the voltage and of the current (b) the average power dissipated in the load

11.46 For the following voltage and current phasors, calculate the complex power, apparent power, real power, and reactive power. Specify whether the pf is leading or lagging.

(a)

$$V = 220/30^{\circ}$$

V rms, $I = 0.5/60^{\circ}$ A rms

(b)

$$V = 250 \div 10^{\circ} \text{ V rms}$$

,

$$I = 6.2 \angle -25^{\circ}$$

A rms

(c)

$$V = 120/0$$

° V rms, $I = 2.4/15$ ° A rms

• (d) V = 160⧸45*°* V rms, I = 8.5⧸90*°* A rms
• 11.47 For each of the following cases, find the complex power, the average power, and the reactive power:

(a)

$$v(t) = 169.7 \sin(377t + 45^\circ)$$

V,
$i(t) = 5.657 \sin(377t)$ A

(b)

$$v(t) = 339.4 \sin (377t + 90^\circ)$$

V,

$$i(t) = 5.657 \sin (377t + 45^{\circ}) \text{ A}$$

(c) V =

$$900/90^{\circ}$$

V rms, Z = $75/45^{\circ}$ $\Omega$

(d)

$$I = 100/60^{\circ}$$

A rms, $Z = 50/60^{\circ}$ $\Omega$

• 11.48 Determine the complex power for the following cases:
  • (a) P = 269 W, Q = 150 VAR (capacitive)
  • (b) Q = 2000 VAR, pf = 0.9 (leading)

(c)

$$S = 600$$

VA, $Q = 450$ VAR (inductive)

(d) Vrms = 220 V, P = 1 kW,

∣Z∣ = 40 Ω (inductive)

11.49 Find the complex power for the following cases:

• (a) P = 4 kW, pf = 0.86 (lagging) (b) S = 2 kVA, P = 1.6 kW (capacitive) (c) Vrms = 208⧸20*°* V, Irms = 6.5⧸−50*°* A (d) Vrms = 120⧸30*°* V, Z = 40 + j60 Ω
• 11.50 Obtain the overall impedance for the following cases:
  • (a) P = 1000 W, pf = 0.8 (leading), Vrms = 220 V
  • (b) P = 1500 W, Q = 2000 VAR (inductive), Irms = 12 A

(c)

$$S = 4500/60^{\circ}$$

VA, $V = 120/45^{\circ}$ V

• 11.51 For the entire circuit in Fig. 11.70, calculate:
  • (a) the power factor
  • (b) the average power delivered by the source
  • (c) the reactive power
  • (d) the apparent power
  • (e) the complex power

Figure 11.70

For Prob. 11.51.

• 11.52 In the circuit of Fig. 11.71, device A receives 2 kW at 0.8 pf lagging, device B receives 3 kVA at 0.4 pf leading, while device C is inductive and consumes 1 kW and receives 500 VAR.
  • (a) Determine the power factor of the entire system.
  • (b) Find I given that Vs = 120⧸45*°* V rms.

Figure 11.71 For Prob. 11.52.

• 11.53 In the circuit of Fig. 11.72, load A receives 4 kVA at 0.8 pf leading. Load B receives 2.4 kVA at 0.6 pf lagging. Box C is an inductive load that consumes 1 kW and receives 500 VAR.
  • (a) Determine I.
  • (b) Calculate the power factor of the combination.

Figure 11.72 For Prob. 11.53.

Section 11.7 Conservation of AC Power

11.54 For the network in Fig. 11.73, find the complex power absorbed by each element.

Figure 11.73

For Prob. 11.54.

Figure 11.74

For Prob. 11.55.

11.56 Obtain the complex power delivered by the source in the circuit of Fig. 11.75.

Figure 11.75

For Prob. 11.56.

For Prob. 11.57.

11.58 Obtain the complex power delivered to the 10-kΩ resistor in Fig. 11.77 below.

• 11.59 Calculate the reactive power in the inductor and capacitor in the circuit of Fig. 11.78.
• Figure 11.78 100 Ω 100 Ω j100 Ω 100 0° mA ‒j200 Ω 20 0°^{V +} ‒

For Prob. 11.59.

11.60 For the circuit in Fig. 11.79, find Vo and the input power factor.

Figure 11.79 For Prob. 11.60.

11.61 Given the circuit in Fig. 11.80, find Io and the overall

Figure 11.80 For Prob. 11.61.

11.62 For the circuit in Fig. 11.81, find Vs.

complex power supplied.

11.63 Find Io in the circuit of Fig. 11.82.

11.64 Determine Is in the circuit of Fig. 11.83, if the voltage source supplies 6 kW and 1.2 kVAR (leading).

Figure 11.83

For Prob. 11.64.

11.65 In the op amp circuit of Fig. 11.84, vs = 4 cos 104 t V. Find the average power delivered to the 50-kΩ resistor.

For Prob. 11.65.

11.66 Obtain the average power absorbed by the 10-Ω resistor in the op amp circuit in Fig. 11.85.

Figure 11.85

For Prob. 11.66.

11.67 For the op amp circuit in Fig. 11.86, calculate:

• (a) the complex power delivered by the voltage source
• (b) the average power dissipated in the 10-Ω resistor

11.68 Compute the complex power supplied by the current source in the series RLC circuit in Fig. 11.87.

Figure 11.87

For Prob. 11.68.

Section 11.8 Power Factor Correction

11.69 Refer to the circuit shown in Fig. 11.88.

• (a) What is the power factor?
• (b) What is the average power dissipated?
• (c) What is the value of the capacitance that will give a unity power factor when connected to the load?

Figure 11.88

For Prob. 11.69.

• 11.70 Design a problem to help other students better understand power factor correction.
  • 11.71 Three loads are connected in parallel to a 120⧸0*°* V rms source. Load 1 absorbs 60 kVAR at pf = 0.85 lagging, load 2 absorbs 90 kW and 50 kVAR leading, and load 3 absorbs 100 kW at pf = 1. (a) Find the equivalent impedance. (b) Calculate the power factor of the parallel combination. (c) Determine the current supplied by the source.
  • 11.72 Two loads connected in parallel draw a total of 2.4 kW at 0.8 pf lagging from a 120-V rms, 60-Hz line. One load absorbs 1.5 kW at a 0.707 pf lagging. Determine: (a) the pf of the second load, (b) the parallel element required to correct the pf to 0.9 lagging for the two loads.
  • 11.73 A 240-V rms 60-Hz supply serves a load that is 10 kW (resistive), 15 kVAR (capacitive), and 22 kVAR (inductive). Find:
    • (a) the apparent power
    • (b) the current drawn from the supply
    • (c) the kVAR rating and capacitance required to improve the power factor to 0.96 lagging
    • (d) the current drawn from the supply under the new power-factor conditions

For Prob. 11.67.

• 11.74 A 120-V rms 60-Hz source supplies two loads connected in parallel, as shown in Fig. 11.89.
  • (a) Find the power factor of the parallel combination.
  • (b) Calculate the value of the capacitance connected in parallel that will raise the power factor to unity.

Figure 11.89 For Prob. 11.74.

• 11.75 Consider the power system shown in Fig. 11.90. Calculate:
  • (a) the total complex power
  • (b) the power factor
  • (c) the parallel capacitance necessary to establish a unity power factor

For Prob. 11.77.

11.78 Find the wattmeter reading of the circuit shown in Fig. 11.93.

11.79 Determine the wattmeter reading of the circuit in

Figure 11.94

For Prob. 11.79.

For Prob. 11.80.

• 11.80 The circuit of Fig. 11.95 portrays a wattmeter connected into an ac network.
  • (a) Find the magnitude of the load current.
  • (b) Calculate the wattmeter reading.

For Prob. 11.78.

Fig. 11.94.

Section 11.9 Applications

11.76 Obtain the wattmeter reading of the circuit in Fig. 11.91.

For Prob. 11.76.

• 11.81 Design a problem to help other students better understand how to correct power factor to values other than unity.
• 11.82 A 240-V rms 60-Hz source supplies a parallel combination of a 5-kW heater and a 30-kVA induction motor whose power factor is 0.82. Determine:
  • (a) the system apparent power
  • (b) the system reactive power
  • (c) the kVA rating of a capacitor required to adjust the system power factor to 0.9 lagging
  • (d) the value of the capacitor required
• 11.83 Oscilloscope measurements indicate that the peak voltage across a load and the peak current through it are, respectively, 210⧸ 60*°* V and 8⧸25*°* A. Determine:
  • (a) the real power
  • (b) the apparent power
  • (c) the reactive power
  • (d) the power factor

11.84 A consumer has an annual consumption of 1200 MWh with a maximum demand of 2.4 MVA. The maximum demand charge is $30 per kVA per annum, and the energy charge per kWh is 4 cents.

(a) Determine the annual cost of energy.