1. 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

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 circuits have numerous advantages over DC circuits

4. DC vs. AC Sources

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
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
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
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. Devices come with a power rating
A single number that tells you about the power usage of the device
The instantaneous power varies between Vmax Imax and 0
The average power is ½ the maximum power
Pavg = ½ (Vmax Imax ) = Vrms Irms
Ohm’s Law can again be used to express the power in different ways
Section 22.2

14. 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

15. 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 again 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

16. 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

17. 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

18. 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

19. 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

20. RL Circuit Example
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 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

21. RC Circuit Example
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

22. 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

23. Transformers, cont.
The wires are covered with a nonconducting 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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
The input coil induces a magnetic field that circulates through the horseshoe magnet
Section 22.10

31. 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

32. 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

33. 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 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

34. Advantages, final
The power lines have typical maximum voltages of 500,000 V
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