Chapter 22 Alternating-Current Circuits and Machines

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

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

DC vs. AC Sources

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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