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University Physics Volume 2

15.4 Power in an AC Circuit

University Physics Volume 215.4 Power in an AC Circuit
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  1. Preface
  2. Unit 1. Thermodynamics
    1. 1 Temperature and Heat
      1. Introduction
      2. 1.1 Temperature and Thermal Equilibrium
      3. 1.2 Thermometers and Temperature Scales
      4. 1.3 Thermal Expansion
      5. 1.4 Heat Transfer, Specific Heat, and Calorimetry
      6. 1.5 Phase Changes
      7. 1.6 Mechanisms of Heat Transfer
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    2. 2 The Kinetic Theory of Gases
      1. Introduction
      2. 2.1 Molecular Model of an Ideal Gas
      3. 2.2 Pressure, Temperature, and RMS Speed
      4. 2.3 Heat Capacity and Equipartition of Energy
      5. 2.4 Distribution of Molecular Speeds
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    3. 3 The First Law of Thermodynamics
      1. Introduction
      2. 3.1 Thermodynamic Systems
      3. 3.2 Work, Heat, and Internal Energy
      4. 3.3 First Law of Thermodynamics
      5. 3.4 Thermodynamic Processes
      6. 3.5 Heat Capacities of an Ideal Gas
      7. 3.6 Adiabatic Processes for an Ideal Gas
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 4 The Second Law of Thermodynamics
      1. Introduction
      2. 4.1 Reversible and Irreversible Processes
      3. 4.2 Heat Engines
      4. 4.3 Refrigerators and Heat Pumps
      5. 4.4 Statements of the Second Law of Thermodynamics
      6. 4.5 The Carnot Cycle
      7. 4.6 Entropy
      8. 4.7 Entropy on a Microscopic Scale
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  3. Unit 2. Electricity and Magnetism
    1. 5 Electric Charges and Fields
      1. Introduction
      2. 5.1 Electric Charge
      3. 5.2 Conductors, Insulators, and Charging by Induction
      4. 5.3 Coulomb's Law
      5. 5.4 Electric Field
      6. 5.5 Calculating Electric Fields of Charge Distributions
      7. 5.6 Electric Field Lines
      8. 5.7 Electric Dipoles
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    2. 6 Gauss's Law
      1. Introduction
      2. 6.1 Electric Flux
      3. 6.2 Explaining Gauss’s Law
      4. 6.3 Applying Gauss’s Law
      5. 6.4 Conductors in Electrostatic Equilibrium
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    3. 7 Electric Potential
      1. Introduction
      2. 7.1 Electric Potential Energy
      3. 7.2 Electric Potential and Potential Difference
      4. 7.3 Calculations of Electric Potential
      5. 7.4 Determining Field from Potential
      6. 7.5 Equipotential Surfaces and Conductors
      7. 7.6 Applications of Electrostatics
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 8 Capacitance
      1. Introduction
      2. 8.1 Capacitors and Capacitance
      3. 8.2 Capacitors in Series and in Parallel
      4. 8.3 Energy Stored in a Capacitor
      5. 8.4 Capacitor with a Dielectric
      6. 8.5 Molecular Model of a Dielectric
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    5. 9 Current and Resistance
      1. Introduction
      2. 9.1 Electrical Current
      3. 9.2 Model of Conduction in Metals
      4. 9.3 Resistivity and Resistance
      5. 9.4 Ohm's Law
      6. 9.5 Electrical Energy and Power
      7. 9.6 Superconductors
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    6. 10 Direct-Current Circuits
      1. Introduction
      2. 10.1 Electromotive Force
      3. 10.2 Resistors in Series and Parallel
      4. 10.3 Kirchhoff's Rules
      5. 10.4 Electrical Measuring Instruments
      6. 10.5 RC Circuits
      7. 10.6 Household Wiring and Electrical Safety
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    7. 11 Magnetic Forces and Fields
      1. Introduction
      2. 11.1 Magnetism and Its Historical Discoveries
      3. 11.2 Magnetic Fields and Lines
      4. 11.3 Motion of a Charged Particle in a Magnetic Field
      5. 11.4 Magnetic Force on a Current-Carrying Conductor
      6. 11.5 Force and Torque on a Current Loop
      7. 11.6 The Hall Effect
      8. 11.7 Applications of Magnetic Forces and Fields
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    8. 12 Sources of Magnetic Fields
      1. Introduction
      2. 12.1 The Biot-Savart Law
      3. 12.2 Magnetic Field Due to a Thin Straight Wire
      4. 12.3 Magnetic Force between Two Parallel Currents
      5. 12.4 Magnetic Field of a Current Loop
      6. 12.5 Ampère’s Law
      7. 12.6 Solenoids and Toroids
      8. 12.7 Magnetism in Matter
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    9. 13 Electromagnetic Induction
      1. Introduction
      2. 13.1 Faraday’s Law
      3. 13.2 Lenz's Law
      4. 13.3 Motional Emf
      5. 13.4 Induced Electric Fields
      6. 13.5 Eddy Currents
      7. 13.6 Electric Generators and Back Emf
      8. 13.7 Applications of Electromagnetic Induction
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    10. 14 Inductance
      1. Introduction
      2. 14.1 Mutual Inductance
      3. 14.2 Self-Inductance and Inductors
      4. 14.3 Energy in a Magnetic Field
      5. 14.4 RL Circuits
      6. 14.5 Oscillations in an LC Circuit
      7. 14.6 RLC Series Circuits
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    11. 15 Alternating-Current Circuits
      1. Introduction
      2. 15.1 AC Sources
      3. 15.2 Simple AC Circuits
      4. 15.3 RLC Series Circuits with AC
      5. 15.4 Power in an AC Circuit
      6. 15.5 Resonance in an AC Circuit
      7. 15.6 Transformers
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    12. 16 Electromagnetic Waves
      1. Introduction
      2. 16.1 Maxwell’s Equations and Electromagnetic Waves
      3. 16.2 Plane Electromagnetic Waves
      4. 16.3 Energy Carried by Electromagnetic Waves
      5. 16.4 Momentum and Radiation Pressure
      6. 16.5 The Electromagnetic Spectrum
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  4. A | Units
  5. B | Conversion Factors
  6. C | Fundamental Constants
  7. D | Astronomical Data
  8. E | Mathematical Formulas
  9. F | Chemistry
  10. G | The Greek Alphabet
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
    14. Chapter 14
    15. Chapter 15
    16. Chapter 16
  12. Index

Learning Objectives

By the end of the section, you will be able to:
  • Describe how average power from an ac circuit can be written in terms of peak current and voltage and of rms current and voltage
  • Determine the relationship between the phase angle of the current and voltage and the average power, known as the power factor

A circuit element dissipates or produces power according to P=IV,P=IV, where I is the current through the element and V is the voltage across it. Since the current and the voltage both depend on time in an ac circuit, the instantaneous power p(t)=i(t)v(t)p(t)=i(t)v(t) is also time dependent. A plot of p(t) for various circuit elements is shown in Figure 15.16. For a resistor, i(t) and v(t) are in phase and therefore always have the same sign (see Figure 15.5). For a capacitor or inductor, the relative signs of i(t) and v(t) vary over a cycle due to their phase differences (see Figure 15.7 and Figure 15.9). Consequently, p(t) is positive at some times and negative at others, indicating that capacitive and inductive elements produce power at some instants and absorb it at others.

Figures a through d show sine waves on graphs of P versus t. All have the same amplitude and frequency. Figure a is labeled resistor. P bar is equal to half I0 V0. The sine wave is above the x axis, with the minimum y value being 0. It starts from a trough. Figure b is labeled capacitor. P bar is equal to 0. The equilibrium position of the sine wave is along the x axis. It starts at equilibrium with a positive slope. Figure c is labeled inductor. P bar is equal to 0. The equilibrium position of the sine wave is along the x axis. It starts at equilibrium with a negative slope. Figure d is labeled AC source. P bar is equal to half I0 V0 cos phi. The equilibrium position of the sine wave is above the x axis, with the minimum y-value of the wave being negative.
Figure 15.16 Graph of instantaneous power for various circuit elements. (a) For the resistor, Pave=I0V0/2,Pave=I0V0/2, whereas for (b) the capacitor and (c) the inductor, Pave=0.Pave=0. (d) For the source, Pave=I0V0(cosϕ)/2,Pave=I0V0(cosϕ)/2, which may be positive, negative, or zero, depending on ϕ.ϕ.

Because instantaneous power varies in both magnitude and sign over a cycle, it seldom has any practical importance. What we’re almost always concerned with is the power averaged over time, which we refer to as the average power. It is defined by the time average of the instantaneous power over one cycle:

Pave=1T0Tp(t)dt,Pave=1T0Tp(t)dt,

where T=2π/ωT=2π/ω is the period of the oscillations. With the substitutions v(t)=V0sinωtv(t)=V0sinωt and i(t)=I0sin(ωtϕ),i(t)=I0sin(ωtϕ), this integral becomes

Pave=I0V0T0Tsin(ωtϕ)sinωtdt.Pave=I0V0T0Tsin(ωtϕ)sinωtdt.

Using the trigonometric relation sin(AB)=sinAcosBsinBcosA,sin(AB)=sinAcosBsinBcosA, we obtain

Pave=I0V0cosϕT0Tsin2ωtdtI0V0sinϕT0Tsinωtcosωtdt.Pave=I0V0cosϕT0Tsin2ωtdtI0V0sinϕT0Tsinωtcosωtdt.

Evaluation of these two integrals yields

1T0Tsin2ωtdt=121T0Tsin2ωtdt=12

and

1T0Tsinωtcosωtdt=0.1T0Tsinωtcosωtdt=0.

Hence, the average power associated with a circuit element is given by

Pave=12I0V0cosϕ.Pave=12I0V0cosϕ.
(15.12)

In engineering applications, cosϕcosϕ is known as the power factor, which is the amount by which the power delivered in the circuit is less than the theoretical maximum of the circuit due to voltage and current being out of phase. For a resistor, ϕ=0,ϕ=0, so the average power dissipated is

Pave=12I0V0.Pave=12I0V0.

A comparison of p(t) and PavePave is shown in Figure 15.16(d). To make Pave=(1/2)I0V0Pave=(1/2)I0V0 look like its dc counterpart, we use the rms values IrmsandVrmsIrmsandVrms of the current and the voltage. By definition, these are

Irms=iave2andVrms=vave2,Irms=iave2andVrms=vave2,

where

iave2=1T0Ti2(t)dtand vave2=1T0Tv2(t)dt.iave2=1T0Ti2(t)dtand vave2=1T0Tv2(t)dt.

With i(t)=I0sin(ωtϕ)andv(t)=V0sinωt,i(t)=I0sin(ωtϕ)andv(t)=V0sinωt, we obtain

Irms=12I0andVrms=12V0.Irms=12I0andVrms=12V0.

We may then write for the average power dissipated by a resistor,

Pave=12I0V0=IrmsVrms=Irms2R.Pave=12I0V0=IrmsVrms=Irms2R.
(15.13)

This equation further emphasizes why the rms value is chosen in discussion rather than peak values. Both equations for average power are correct for Equation 15.13, but the rms values in the formula give a cleaner representation, so the extra factor of 1/2 is not necessary.

Alternating voltages and currents are usually described in terms of their rms values. For example, the 110 V from a household outlet is an rms value. The amplitude of this source is 1102V=156 V.1102V=156 V. Because most ac meters are calibrated in terms of rms values, a typical ac voltmeter placed across a household outlet will read 110 V.

For a capacitor and an inductor, ϕ=π/2andπ/2rad,ϕ=π/2andπ/2rad, respectively. Since cosπ/2=cos(π/2)=0,cosπ/2=cos(π/2)=0, we find from Equation 15.12 that the average power dissipated by either of these elements is Pave=0.Pave=0. Capacitors and inductors absorb energy from the circuit during one half-cycle and then discharge it back to the circuit during the other half-cycle. This behavior is illustrated in the plots of Figure 15.16, (b) and (c), which show p(t) oscillating sinusoidally about zero.

The phase angle for an ac generator may have any value. If cosϕ>0,cosϕ>0, the generator produces power; if cosϕ<0,cosϕ<0, it absorbs power. In terms of rms values, the average power of an ac generator is written as

Pave=IrmsVrmscosϕ.Pave=IrmsVrmscosϕ.

For the generator in an RLC circuit,

tanϕ=XLXCRtanϕ=XLXCR

and

cosϕ=RR2+(XLXC)2=RZ.cosϕ=RR2+(XLXC)2=RZ.

Hence the average power of the generator is

Pave=IrmsVrmscosϕ=VrmsZVrmsRZ=Vrms2RZ2.Pave=IrmsVrmscosϕ=VrmsZVrmsRZ=Vrms2RZ2.
(15.14)

This can also be written as

Pave=Irms2R,Pave=Irms2R,

which designates that the power produced by the generator is dissipated in the resistor. As we can see, Ohm’s law for the rms ac is found by dividing the rms voltage by the impedance.

Example 15.3

Power Output of a Generator An ac generator whose emf is given by

v(t)=(4.00V)sin[(1.00×104rad/s)t]v(t)=(4.00V)sin[(1.00×104rad/s)t]

is connected to an RLC circuit for which L=2.00×10−3HL=2.00×10−3H, C=4.00×10−6FC=4.00×10−6F, and R=5.00ΩR=5.00Ω. (a) What is the rms voltage across the generator? (b) What is the impedance of the circuit? (c) What is the average power output of the generator?

Strategy The rms voltage is the amplitude of the voltage times 1/21/2. The impedance of the circuit involves the resistance and the reactances of the capacitor and the inductor. The average power is calculated by Equation 15.14, or more specifically, the last part of the equation, because we have the impedance of the circuit Z, the rms voltage VrmsVrms, and the resistance R.

Solution

  1. Since V0=4.00V,V0=4.00V, the rms voltage across the generator is
    Vrms=12(4.00V)=2.83V.Vrms=12(4.00V)=2.83V.
  2. The impedance of the circuit is
    Z=R2+(XLXC)2={(5.00Ω)2+[(1.00×104rad/s)(2.00×10−3H)1(1.00×104rad/s)(4.00×10−6F)]2}1/2=7.07Ω.Z=R2+(XLXC)2={(5.00Ω)2+[(1.00×104rad/s)(2.00×10−3H)1(1.00×104rad/s)(4.00×10−6F)]2}1/2=7.07Ω.
  3. From Equation 15.14, the average power transferred to the circuit is
    Pave=Vrms2RZ2=(2.83V)2(5.00Ω)(7.07Ω)2=0.801W.Pave=Vrms2RZ2=(2.83V)2(5.00Ω)(7.07Ω)2=0.801W.

Significance If the resistance is much larger than the reactance of the capacitor or inductor, the average power is a dc circuit equation of P=V2/R,P=V2/R, where V replaces the rms voltage.

Check Your Understanding 15.4

An ac voltmeter attached across the terminals of a 45-Hz ac generator reads 7.07 V. Write an expression for the emf of the generator.

Check Your Understanding 15.5

Show that the rms voltages across a resistor, a capacitor, and an inductor in an ac circuit where the rms current is IrmsIrms are given by IrmsR,IrmsXC,andIrmsXL,IrmsR,IrmsXC,andIrmsXL, respectively. Determine these values for the components of the RLC circuit of Equation 15.12.

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