1. Alternating-Current Circuits and Machines
Chapter 22
Alternating-Current Circuits and Machines

2. DC Circuit Summary DC circuits DC stands for direct current
Source of electrical energy is generally a battery
If only resistors are in the circuit, the current is independent of time
If the circuit contains capacitors and resistors, the current can vary with time but always approaches a constant value a long time after closing the switch
Introduction

3. AC Circuit Introduction
AC stands for alternating current
The power source is a device that produces an electric potential that varies with time
There will be a frequency and peak voltage associated with the potential
Household electrical energy is supplied by an AC source
Standard frequency is 60 Hz
AC power has numerous advantages over DC power
Introduction

4. DC vs. AC Sources
Introduction

5. Generating AC Voltages
Most sources of AC voltage employ a generator based on magnetic induction
A shaft holds a coil with many loops of wire
The coil is positioned between the poles of a permanent magnet
The magnetic flux through the coil varies with time as the shaft turns
This changing flux induces a voltage in the coil
This induced voltage is the generator’s output
Section 22.1

6. Generators
Generators of electrical energy convert the mechanical energy of the rotating shaft into electrical energy
The principle of conservation of energy still applies
The source of electrical energy in a circuit enables the transfer of electrical energy from a generator to an attached circuit
Section 22.1

7. AC Circuits and Simple Harmonic Motion
The voltage variation of an AC circuit is reminiscent of a simple harmonic oscillator
There is also a close connection between circuits with capacitors and inductors and simple harmonic motion
Section 22.1

8. Resistors in AC Circuits
Assume a circuit consisting of an AC generator and a resistor
The voltage across the output of the AC source varies with time according to
V = Vmax sin (2 π ƒ t)
V is the instantaneous potential difference
Vmax is the amplitude of the AC voltage
Section 22.2

9. Resistors, cont. Applying Ohm’s Law:
Since the voltage varies sinusoidally, so does the current
Section 22.2

10. RMS Voltage
To specify current and voltage values when they vary with time, rms values were adopted
RMS stands for Root Mean Square
For the voltage
Section 22.2

11. RMS Current
The root-mean-square value can be defined for any quantity that varies with time
For the current
The root-mean-square values of the voltage and current are typically used to specify the properties of an AC circuit
Section 22.2

12. Power
The instantaneous power is the product of the instantaneous voltage and instantaneous current
P = I V
Since both I and V vary with time, the power also varies with time:
P = Vmax Imax sin2 (2πƒt)
Section 22.2

13. Power, cont. The instantaneous power varies between Vmax Imax and 0
The average power is ½ the maximum power
Pavg = ½ (Vmax Imax ) = Vrms Irms
This has the same mathematical form as the power in a DC circuit
Ohm’s Law can again be used to express the power in different ways
Section 22.2

14. AC Circuit Notation
It is important to distinguish between instantaneous and average values of voltage, current and power
The amplitudes of the AC variations and the rms values are also important
Section 22.2

15. Phasors AC circuits can be analyzed graphically
An arrow has a length Vmax
The arrow’s tail is tied to the origin
Its tip moves along a circle
The arrow makes an angle of θ with the horizontal
The angle varies with time according to θ = 2πƒt
Section 22.2

16. Phasors, cont.
The rotating arrow represents the voltage in an AC circuit
The arrow is called a phasor
A phasor is not a vector
A phasor diagram provides a convenient way to illustrate and think about the time dependence in an AC circuit
Section 22.2

17. Phasors, final
The current in an AC circuit can also be represented by a phasor
The two phasors always make the same angle with the horizontal axis as time passes
The current and voltage are in phase
For a circuit with only resistors
Section 22.2

18. AC Circuits with Capacitors
Assume an AC circuit containing a single capacitor
The instantaneous charge is q = C V = C Vmax sin (2 πƒ t)
The capacitor’s voltage and charge are in phase with each other
Section 22.3

19. Current in Capacitors
The instantaneous current is the rate at which charge flows onto the capacitor plates in a short time interval
The current is the slope of the q-t plot
A plot of the current as a function of time can be obtained from these slopes
Section 22.3

20. Current in Capacitors, cont.
The current is a cosine function
I = Imax cos (2πƒt)
Equivalently, due to the relationship between sine and cosine functions
I = Imax sin (2πƒt + ϕ) where ϕ = π/2
Section 22.3

21. Capacitor Phasor Diagram
The current is out of phase with the voltage
The angle π/2 is called the phase angle, ϕ, between V and I
For this circuit, the current and voltage are out of phase by 90°
Section 22.3

22. Current Value for a Capacitor
The peak value of the current is
The factor Xc is called the reactance of the capacitor
SI unit of reactance is Ohms
Reactance and resistance are different because the reactance of a capacitor depends on the frequency
If the frequency is increased, the charge oscillates more rapidly and Δt is smaller, giving a larger current
At high frequencies, the peak current is larger and the reactance is smaller
Section 22.3

23. Power In A Capacitor
For an AC circuit with a capacitor, P = VI = Vmax Imax sin (2πƒt) cos (2πƒt)
The average value of the power over many oscillations is 0
Energy is transferred from the generator during part of the cycle and from the capacitor in other parts
Energy is stored in the capacitor as electric potential energy and not dissipated by the circuit
Section 22.3

24. AC Circuits with Inductors
Assume an AC circuit containing an AC generator and a single inductor
The voltage drop is
V = L (ΔI / Δt)
= Vmax sin (2 πƒt)
The inductor’s voltage is proportional to the slope of the current-time relationship
Section 22.4

25. Current in Inductors
The instantaneous current oscillates in time according to a cosine function
I = -Imax cos (2πƒt)
A plot of the current-time relationship is shown
Section 22.4

26. Current in Inductors, cont.
The current equation can be rewritten as
I = Imax sin (2πƒt – π/2)
Equivalently,
I = Imax sin (2πƒt + Φ) where Φ = -π/2
Section 22.4

27. Inductor Phasor Diagram
The current is out of phase with the voltage
For this circuit, the current and voltage are out of phase by -90°
Remember, for a capacitor, the phase difference was +90°
Section 22.4

28. Current Value for an Inductor
The peak value of the current is
The factor XL is called the reactance of the inductor
SI unit of inductive reactance is Ohms
As with the capacitor, inductive reactance depends on the frequency
As the frequency is increased, the inductive reactance increases
Section 22.4

29. Power in an Inductor
For an AC circuit with an inductor, P = VI = -Vmax Imax sin (2πƒt) cos (2πƒt)
The average value of the power over many oscillations is 0
Energy is transferred from the generator during part of the cycle and from the inductor in other parts of the cycle
Energy is stored in the inductor as magnetic potential energy
Section 22.4

30. Properties of AC Circuits
Section 22.4

31. LC Circuit Most useful circuits contain multiple circuit elements
Will start with an LC circuit, containing just an inductor and a capacitor
No AC generator is included, but some excess charge is placed on the capacitor at t = 0
Section 22.5

32. LC Circuit, cont.
After t = 0, the charge moves from one capacitor plate to the other and current passes through the inductor
Eventually, the charge on each capacitor plate falls to zero
The inductor opposes change in the current, so the induced emf now acts to maintain the current at a nonzero value
This current continues to transport charge from one capacitor plate to the other, causing the capacitor’s charge and voltage to reverse sign
Eventually the charge on the capacitor returns to its original value
Section 22.5

33. LC Circuit, final
The voltage and current in the circuit oscillate between positive and negative values
The circuit behaves as a simple harmonic oscillator
The charge is q = qmax cos (2πƒt)
The current is I = Imax sin (2πƒt)
Section 22.5

34. Energy in an LC Circuit Capacitors and inductors store energy
A capacitor stores energy in its electric field and depends on the charge
An inductor stores energy in its magnetic field and depends on the current
As the charge and current oscillate, the energies stored also oscillate
Section 22.5

35. Energy Calculations For the capacitor, For the inductor,
The energy oscillates back and forth between the capacitor and its electric field and the inductor and its magnetic field
The total energy must remain constant
Section 22.5

36. Energy, final
The maximum energy in the capacitor must equal the maximum energy in the inductor
From energy considerations, the maximum value of the current can be calculated
This shows how the amplitudes of the current and charge oscillations in the LC circuits are related
Section 22.5

37. Frequency Oscillations – LC Circuit
In an LC circuit, the instantaneous voltage across the capacitor and inductor are always equal
Therefore, |VC| = |I XC| = |VL| = |I XL|
Simplifying, XC = XL
This assumed the current in the LC circuit is oscillating and hence applies only at the oscillation frequency
This frequency is the resonant frequency
Section 22.5

38. LRC Circuits
Let the circuit contain a generator, resistor, inductor and capacitor in series
LRC circuit
From Kirchhoff’s Loop Rule,
VAC = VL + VC + VR
But the voltages are not all in phase, so the phase angles must also be taken into account
Section 22.6

39. LRC Circuit – Phasor Diagram
All the elements are in series, so the current is the same through each one
All the current phasors have the same orientation
Resistor: current and voltage are in phase
Capacitor and inductor: current and voltage are 90° out of phase, in opposite directions
Section 22.6

40. Resonance The VC and VR values are the same at the resonance frequency
Only the resistor is left to “resist” the flow of the current
This cancellation between the voltages occurs only at the resonance frequency
The resonance frequency corresponds to the highest current
Section 22.6

41. Applications of Resonance
Resonance is used in radios, cell phones and other similar applications
Tuning a radio
Changes the value of the capacitance in the LCR circuit so the resonance frequency matches the frequency of the station you want to listen to
LCR circuits are used to construct devices that are frequency dependent
Section 22.6

42. Real Inductors in AC Circuits
A typical inductor includes a nonzero resistance
Due to the wire itself
The inductor can be modeled as an ideal inductor in series with a resistor
The current can be calculated using phasors
Section 22.7

43. Real Inductor, cont.
The elements are in series, so the current is the same through both elements
Voltages are VR = I R and VL = I XL
The voltages must be added as phasors
The phase differences must be included
The total voltage has an amplitude of
Section 22.7

44. Impedance
The impedance, Z, is a measure of how strongly a circuit “impedes” current in a circuit
The impedance is defined as Vtotal = I Z where
This is the impedance for an RL circuit only
The impedance for a circuit containing other elements can also be calculated using phasors
The angle between the current and the impedance can also be calculated
Section 22.7

45. Impedance, LCR Circuit The current phasor is on the horizontal axis
The total voltage is
The impedance is
Section 22.7

46. Elements and Frequencies in AC Circuits
Resistor
Resistors in an AC circuit behave very much like resistors in a DC circuit
The current is always in phase with the voltage
Capacitor or inductor
Both are frequency dependent
Due to the frequency dependence of the reactants
XC is largest at low frequencies, so the current through a capacitor is smallest at low frequencies
XL is largest at high frequencies, so the current through an inductor is smallest at high frequencies
Section 22.8

47. Elements at Various Frequencies – Summary
Section 22.8

48. High-Pass Filter (LR Circuit)
When the input frequency is very low, the reactance of the inductor is small
The inductor acts as a wire
Voltage drop will be 0
At very high frequencies, the inductor acts as an open circuit
No current is passed
The output voltage is equal to the input voltage
This circuit acts as a high-pass filter
Section 22.8

49. Low-Pass Filter (RC Circuit)
When the input frequency is very low, the reactance of the capacitor is large
The current is very small
The capacitor acts as an open circuit
The output voltage is equal to the input voltage
At high frequencies, the capacitor acts as a short circuit
The inductor acts as a wire
The output voltage is 0
This circuit acts as a low-pass filter
Section 22.8

50. Application of a Low-Pass Filter
A low-pass filter is used in radios and MP3 players
A music signal often contains static
Static comes from unwanted high-frequency components in the music
These high frequencies can be filtered out by using a low-pass filter
Section 22.8

51. Frequency Limits, RL Circuit
For an RL circuit, the input frequency is compared to the RL time constant
The time constant is τRL = L / R
Define a corresponding frequency as ƒRL = 1/ τRL = R / L
The high-frequency limit applies when the input frequency is much greater than ƒRL
A frequency higher than ~10 x ƒRL falls into the high-frequency limit
The low-frequency limit applies when the input frequency is much less than ƒRL
A frequency lower than ~ƒRL / 10 falls into the low-frequency limit
Section 22.8

52. Frequency Limits, RC Circuit
For an RC circuit, the input frequency is compared to the RC time constant
The time constant is τRC = R C
Define a corresponding frequency as ƒRC = 1/ τRC =
1 / RC
The high-frequency limit applies when the input frequency is much greater than ƒRC
A frequency higher than ~10 x ƒRC falls into the high-frequency limit
The low-frequency limit applies when the input frequency is much less than ƒRC
A frequency lower than ~ƒRC / 10 falls into the low-frequency limit
Section 22.8

53. Frequency Limits, LC Circuit
The resonant frequency determines the boundary between high- and low-frequency limits
Remember,
Section 22.8

54. Filter Application – Stereo Speakers
Many stereo speakers actually contain two separate speakers
A tweeter is designed to perform well at high frequencies
A woofer is designed to perform well at low frequencies
The AC signal passes through a crossover network
A combination of low-pass and high-pass filters
The outputs of the filter are sent to the speaker which is most efficient at that frequency
Section 22.8

55. Transformers
Transformers are devices that can increase or decrease the amplitude of an applied AC voltage
A simple transformer consists of two solenoid coils with the loops arranged so that all or most of the magnetic field lines and flux generated by one coil pass through the other coil
Section 22.9

56. Transformers, cont.
The wires are covered with a non-conducting layer so that current cannot flow directly from one coil to the other
An AC current in one coil will induce an AC voltage across the other coil
An AC voltage source is typically attached to one of the coils called the input coil
The other coil is called the output coil

57. Transformers, Equations
Faraday’s Law applies to both coils
If the input coil has Nin coils and the output coil has Nout turns, the flux in the coils is related by
The voltages are related by
Section 22.9

58. Transformers, final
The ratio of the turns can be greater than or less than one
Therefore, the input voltage can be transformed to a different value
Transformers cannot change DC voltages
Since they are based on Faraday’s Law
Section 22.9

59. Practical Transformers
Most practical transformers have central regions filled with a magnetic material
This produces a larger flux, resulting in a larger voltage at both the input and output coils
The ratio Vout / Vin is not affected by the presence of the magnetic material
Section 22.9

60. Applications of Transformers
Transformers are used in the transmission of electric power over long distances
Many household appliances use transformers to convert the AC voltage at a wall socket to the smaller voltages needed in many devices
Two steps are needed – converting 120 V to 9 V then AC to DC
Section 22.9

61. Transformers and Power
The output voltage of a transformer can be made much larger by arranging the number of coils
According to the principle of conservation of energy, the energy delivered through the input coil must either be stored in the transformer’s magnetic field or transferred to the output circuit
Over many cycles, the stored energy is constant
The power delivered to the input coil must equal the output power
Section 22.9

62. Power, cont.
Since P = V I, if Vout is greater than Vin, then Iout must be smaller than Iin
Pin = Pout only in an ideal transformer
In real transformers, the coils always have a small electrical resistance
This causes some power dissipation
For a real transformer, the output power is always less than the input power
Usually by only a small amount
Section 22.9

63. Motors An AC voltage source can be use to power a motor
The AC source is connected to a coil wound around a horseshoe magnet
Called the input coil
The input coil induces a magnetic field that circulates through the horseshoe magnet
Section 22.10

64. Motors, cont.
A second coil is mounted between the poles of the horseshoe magnet and attached to a rotating shaft
The forces acting on the second coil produce a torque on the coil
This causes the shaft to rotate
As the AC current in the input coil changes direction, so do the forces
The torques continue to produce a rotation that is always in the same direction
The oscillations of the AC current and field make the shaft rotate
Section 22.10

65. Advantages of AC vs. DC
Biggest advantage is in the systems that distribute electric power across long distances
The power generated at a power plant must be distributed to distance places
The power plant acts as an AC generator
Section 22.11

66. Advantages, cont. There is power dissipated in the power lines
Pave = (Irms )2 Rline
The power company wants to minimize these power losses, so they want to make Irms as small as possible
The voltage is increased by using a transformer
The increase in voltage is done in order to decrease the current
A transformer is used to drop the high voltages in the power lines to the lower voltages at the house
Section 22.11

67. Advantages, final
The power lines have typical voltages of 500,000 V or higher
The transformer reduces the voltage to a maximum voltage of 170 V
Typically 5% to 10% of the energy that leaves the power plant is dissipated in the resistance of the power lines
Section 22.11