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University Physics Volume 2

15.3 RLC Series Circuits with AC

University Physics Volume 215.3 RLC Series Circuits with AC
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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 the current varies in a resistor, a capacitor, and an inductor while in series with an ac power source
  • Use phasors to understand the phase angle of a resistor, capacitor, and inductor ac circuit and to understand what that phase angle means
  • Calculate the impedance of a circuit

The ac circuit shown in Figure 15.11, called an RLC series circuit, is a series combination of a resistor, capacitor, and inductor connected across an ac source. It produces an emf of

v(t)=V0sinωt.v(t)=V0sinωt.
Figure a shows a circuit with an AC voltage source connected to a resistor, a capacitor and an inductor in series. The source is labeled V0 sine omega t. Figure b shows sine waves of AC voltage and current on the same graph. Voltage has a greater amplitude than current and its maximum value is marked V0 on the y axis. The maximum value of current is marked I0. The two curves have the same wavelength but are out of phase. The voltage curve is labeled V parentheses t parentheses equal to V0 sine omega t. The current curve is labeled I parentheses t parentheses equal to I0 sine parentheses omega t minus phi parentheses.
Figure 15.11 (a) An RLC series circuit. (b) A comparison of the generator output voltage and the current. The value of the phase difference ϕϕ depends on the values of R, C, and L.

Since the elements are in series, the same current flows through each element at all points in time. The relative phase between the current and the emf is not obvious when all three elements are present. Consequently, we represent the current by the general expression

i(t)=I0sin(ωtϕ),i(t)=I0sin(ωtϕ),

where I0I0 is the current amplitude and ϕϕ is the phase angle between the current and the applied voltage. The phase angle is thus the amount by which the voltage and current are out of phase with each other in a circuit. Our task is to find I0andϕ.I0andϕ.

A phasor diagram involving i(t),vR(t),vC(t),andvL(t)i(t),vR(t),vC(t),andvL(t) is helpful for analyzing the circuit. As shown in Figure 15.12, the phasor representing vR(t)vR(t) points in the same direction as the phasor for i(t);i(t); its amplitude is VR=I0R.VR=I0R. The vC(t)vC(t) phasor lags the i(t) phasor by π/2π/2 rad and has the amplitude VC=I0XC.VC=I0XC. The phasor for vL(t)vL(t) leads the i(t) phasor by π/2π/2 rad and has the amplitude VL=I0XL.VL=I0XL.

Figure shows the coordinate axes, with four arrows starting from the origin. Arrow V subscripts R points up and right, making an angle omega t minus phi with the x axis. Its y intercept is V subscript R parentheses t parentheses. Arrow I0 is along arrow V subscript R, but shorter than it. Arrow V subscript L points up and left and is perpendicular to V subscript R. It makes a y intercept V subscript L parentheses t parentheses. Arrow V subscript C points down and right. It is perpendicular to V subscript R. It makes a y intercept V subscript C parentheses t parentheses. Three arrows labeled omega are each perpendicular to V subscript R, V subscript L and V subscript C, shown near their tips.
Figure 15.12 The phasor diagram for the RLC series circuit of Figure 15.11.

At any instant, the voltage across the RLC combination is vR(t)+vL(t)+vC(t)=v(t),vR(t)+vL(t)+vC(t)=v(t), the emf of the source. Since a component of a sum of vectors is the sum of the components of the individual vectors—for example, (A+B)y=Ay+By(A+B)y=Ay+By —the projection of the vector sum of phasors onto the vertical axis is the sum of the vertical projections of the individual phasors. Hence, if we add vectorially the phasors representing vR(t),vL(t),andvC(t)vR(t),vL(t),andvC(t) and then find the projection of the resultant onto the vertical axis, we obtain

vR(t)+vL(t)+vC(t)=v(t)=V0sinωt.vR(t)+vL(t)+vC(t)=v(t)=V0sinωt.

The vector sum of the phasors is shown in Figure 15.13. The resultant phasor has an amplitude V0V0 and is directed at an angle ϕϕ with respect to the vR(t),vR(t), or i(t), phasor. The projection of this resultant phasor onto the vertical axis is v(t)=V0sinωt.v(t)=V0sinωt. We can easily determine the unknown quantities I0I0 and ϕϕ from the geometry of the phasor diagram. For the phase angle,

ϕ=tan−1VLVCVR=tan−1I0XLI0XCI0R,ϕ=tan−1VLVCVR=tan−1I0XLI0XCI0R,

and after cancellation of I0,I0, this becomes

ϕ=tan−1XLXCR.ϕ=tan−1XLXCR.
(15.9)

Furthermore, from the Pythagorean theorem,

V0=VR2+(VLVC)2=(I0R)2+(I0XLI0XC)2=I0R2+(XLXC)2.V0=VR2+(VLVC)2=(I0R)2+(I0XLI0XC)2=I0R2+(XLXC)2.
Three arrows start from the origin on the coordinate axis. Arrow V subscript R points up and right, making an angle omega t minus phi with the x axis. Arrow V0 points up and right, making an angle omega t with the x axis. It makes an angle phi with the arrow V subscript R. It makes a y intercept labeled V0 sine omega t. The third arrow is labeled V subscript L minus V subscript C. It points up and left and is perpendicular to arrow V subscript R. Dotted lines indicate that the rectangle formed with its longer side being V subscript R and shorter side being V subscript L minus V subscript C, would have the arrow V0 as a diagonal. An arrow labeled omega is shown near the tip of V subscript R, perpendicular to it.
Figure 15.13 The resultant of the phasors for vL(t)vL(t), vC(t)vC(t), and vR(t)vR(t) is equal to the phasor for v(t)=V0sinωt.v(t)=V0sinωt. The i(t) phasor (not shown) is aligned with the vR(t)vR(t) phasor.

The current amplitude is therefore the ac version of Ohm’s law:

I0=V0R2+(XLXC)2=V0Z,I0=V0R2+(XLXC)2=V0Z,
(15.10)

where

Z=R2+(XLXC)2Z=R2+(XLXC)2
(15.11)

is known as the impedance of the circuit. Its unit is the ohm, and it is the ac analog to resistance in a dc circuit, which measures the combined effect of resistance, capacitive reactance, and inductive reactance (Figure 15.14).

Photograph of power capacitors at a power station.
Figure 15.14 Power capacitors are used to balance the impedance of the effective inductance in transmission lines.

The RLC circuit is analogous to the wheel of a car driven over a corrugated road (Figure 15.15). The regularly spaced bumps in the road drive the wheel up and down; in the same way, a voltage source increases and decreases. The shock absorber acts like the resistance of the RLC circuit, damping and limiting the amplitude of the oscillation. Energy within the wheel system goes back and forth between kinetic and potential energy stored in the car spring, analogous to the shift between a maximum current, with energy stored in an inductor, and no current, with energy stored in the electric field of a capacitor. The amplitude of the wheel’s motion is at a maximum if the bumps in the road are hit at the resonant frequency, which we describe in more detail in Resonance in an AC Circuit.

Figure shows one wheel of a car. Arrows show the up-down motion of its shock absorber spring.
Figure 15.15 On a car, the shock absorber damps motion and dissipates energy. This is much like the resistance in an RLC circuit. The mass and spring determine the resonant frequency.

Problem-Solving Strategy: AC Circuits

To analyze an ac circuit containing resistors, capacitors, and inductors, it is helpful to think of each device’s reactance and find the equivalent reactance using the rules we used for equivalent resistance in the past. Phasors are a great method to determine whether the emf of the circuit has positive or negative phase (namely, leads or lags other values). A mnemonic device of “ELI the ICE man” is sometimes used to remember that the emf (E) leads the current (I) in an inductor (L) and the current (I) leads the emf (E) in a capacitor (C).

Use the following steps to determine the emf of the circuit by phasors:

  1. Draw the phasors for voltage across each device: resistor, capacitor, and inductor, including the phase angle in the circuit.
  2. If there is both a capacitor and an inductor, find the net voltage from these two phasors, since they are antiparallel.
  3. Find the equivalent phasor from the phasor in step 2 and the resistor’s phasor using trigonometry or components of the phasors. The equivalent phasor found is the emf of the circuit.

Example 15.2

An RLC Series Circuit The output of an ac generator connected to an RLC series combination has a frequency of 200 Hz and an amplitude of 0.100 V. If R=4.00Ω,R=4.00Ω, L=3.00×10−3H,L=3.00×10−3H, and C=8.00×10−4F,C=8.00×10−4F, what are (a) the capacitive reactance, (b) the inductive reactance, (c) the impedance, (d) the current amplitude, and (e) the phase difference between the current and the emf of the generator?

Strategy The reactances and impedance in (a)–(c) are found by substitutions into Equation 15.3, Equation 15.8, and Equation 15.11, respectively. The current amplitude is calculated from the peak voltage and the impedance. The phase difference between the current and the emf is calculated by the inverse tangent of the difference between the reactances divided by the resistance.

Solution

  1. From Equation 15.3, the capacitive reactance is
    XC=1ωC=12π(200Hz)(8.00×10−4F)=0.995Ω.XC=1ωC=12π(200Hz)(8.00×10−4F)=0.995Ω.
  2. From Equation 15.8, the inductive reactance is
    XL=ωL=2π(200Hz)(3.00×10−3H)=3.77Ω.XL=ωL=2π(200Hz)(3.00×10−3H)=3.77Ω.
  3. Substituting the values of R, XCXC, and XLXL into Equation 15.11, we obtain for the impedance
    Z=(4.00Ω)2+(3.77Ω0.995Ω)2=4.87Ω.Z=(4.00Ω)2+(3.77Ω0.995Ω)2=4.87Ω.
  4. The current amplitude is
    I0=V0Z=0.100V4.87Ω=2.05×10−2A.I0=V0Z=0.100V4.87Ω=2.05×10−2A.
  5. From Equation 15.9, the phase difference between the current and the emf is
    ϕ=tan−1XLXCR=tan−12.77Ω4.00Ω=0.607rad.ϕ=tan−1XLXCR=tan−12.77Ω4.00Ω=0.607rad.

Significance The phase angle is positive because the reactance of the inductor is larger than the reactance of the capacitor.

Check Your Understanding 15.3

Find the voltages across the resistor, the capacitor, and the inductor in the circuit of Figure 15.11 using v(t)=V0sinωtv(t)=V0sinωt as the output of the ac generator.

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