Presentation on theme: "Alternating Current Circuits"— Presentation transcript:
1 Alternating Current Circuits Chapter 28Alternating Current Circuits
2 AC Circuit Δv = ΔVmax sin ωt Δv: instantaneous voltage An AC circuit consists of a combination of circuit elements and an AC generator or sourceThe output of an AC power source is sinusoidal and varies with time according to the following equationΔv = ΔVmax sin ωtΔv: instantaneous voltageΔVmax is the maximum voltage (amplitude) of the generatorω is the angular frequency of the AC voltage
3 Resistors in an AC Circuit Consider a circuit consisting of an AC source and a resistorΔvR = ΔVmax sin ωtΔvR is the instantaneous voltage across the resistorThe instantaneous current in the resistor isThe instantaneous voltage across the resistor is also given as ΔvR = ImaxR sin ωt
4 Resistors in an AC Circuit The graph shows the current through and the voltage across the resistorThe current and the voltage reach their maximum values at the same timeThe current and the voltage are said to be in phaseThe direction of the current has no effect on the behavior of the resistor
5 Resistors in an AC Circuit The rate at which electrical energy is dissipated in the circuit is given byi: instantaneous currentThe heating effect produced by an AC current with a maximum value of Imax is not the same as that of a DC current of the same valueThe maximum current occurs for a small amount of time
6 rms Current and Voltage The rms current is the direct current that would dissipate the same amount of energy in a resistor as is actually dissipated by the AC currentAlternating voltages can also be discussed in terms of rms valuesThe average power dissipated in resistor in an AC circuit carrying a current I is
7 Ohm’s Law in an AC Circuit rms values will be used when discussing AC currents and voltagesAC ammeters and voltmeters are designed to read rms valuesMany of the equations will be in the same form as in DC circuitsOhm’s Law for a resistor, R, in an AC circuitΔVR,rms = Irms RThe same formula applies to the maximum values of v and i
8 Capacitors in an AC Circuit Consider a circuit containing a capacitor and an AC sourceKirchhoff’s loop rule gives:ΔvC: instantaneous voltage across the capacitor
9 Capacitors in an AC Circuit The voltage across the capacitor lags behind the current by 90°The impeding effect of a capacitor on the current in an AC circuit is called the capacitive reactance (measured in ohms):
10 Chapter 28 Problem 22A capacitor and a 1.8-kΩ resistor pass the same current when connected across 60-Hz power. Find the capacitance.
11 Inductors in an AC Circuit Consider an AC circuit with a source and an inductorKirchhoff’s loop rule gives:ΔvL: instantaneous voltage across the inductor
12 Inductors in an AC Circuit The voltage across the inductor always leads the current by 90°The effective resistance of a coil in an AC circuit is called its inductive reactance (measured in ohms):
13 LC Circuit A capacitor is connected to an inductor in an LC circuit Assume the capacitor is initially charged and then the switch is closedAssume no resistance and no energy losses to radiationThe current in the circuit and thecharge on the capacitor oscillatebetween maximum positive andnegative values
14 LC CircuitWith zero resistance, no energy is transformed into internal energyIdeally, the oscillations in the circuit persist indefinitely (assuming no resistance and no radiation)The capacitor is fully charged and the energy in the circuit is stored in the electric field of the capacitorQ2max / 2CNo energy is stored in the inductorThe current in the circuit is zero
15 LC Circuit The switch is then closed The current is equal to the rate at which the charge changes on the capacitorAs the capacitor discharges, the energy stored in the electric field decreasesSince there is now a current, someenergy is stored in the magneticfield of the inductorEnergy is transferred from theelectric field to the magnetic field
16 LC CircuitEventually, the capacitor becomes fully discharged and it stores no energyAll of the energy is stored in the magnetic field of the inductor and the current reaches its maximum valueThe current now decreases in magnitude, recharging the capacitor with its plates having opposite their initial polarityThe capacitor becomes fullycharged and the cycle repeatsThe energy continues to oscillatebetween the inductor and the capacitor
17 LC CircuitThe total energy stored in the LC circuit remains constant in timeSolution:
18 LC CircuitThe angular frequency, ω, of the circuit depends on the inductance and the capacitanceIt is the natural frequency of oscillation of the circuitThe current can be expressed as a function of time:
19 LC CircuitQ and I are 90° out of phase with each other, so when Q is a maximum, I is zero, etc.
20 Energy in LC CircuitsThe total energy can be expressed as a function of timeThe energy continually oscillatesbetween the energy stored in theelectric and magnetic fieldsWhen the total energy is stored inone field, the energy stored in theother field is zero
21 Energy in LC CircuitsIn actual circuits, there is always some resistanceTherefore, there is some energy transformed to internal energyRadiation is also inevitable in this type of circuitThe total energy in the circuit continuously decreases as a result of these processes
22 Chapter 28 Problem 27An LC circuit with a 20-µF capacitor oscillates with period 5.0 ms. The peak current is 25 mA. Find (a) the inductance and (b) the peak voltage.
23 The RLC Series CircuitThe resistor, inductor, and capacitor can be combined in a circuitThe current in the circuit is the same at any time and varies sinusoidally with time
24 The RLC Series CircuitThe instantaneous voltage across the resistor is in phase with the currentThe instantaneous voltage across the inductor leads the current by 90°The instantaneous voltage across the capacitor lags the current by 90°
25 Phasor Diagrams Because of the different phase relationships with the current, the voltagescannot be added directlyTo simplify the analysis of AC circuits, a graphical constructor called a phasor diagram can be usedA phasor is a vector rotating CCW; its length is proportional to the maximum value of the variable it representsThe vector rotates at an angular speed equal to the angular frequency associated with the variable, and the projection of the phasor onto the vertical axis represents the instantaneous value of the quantity
26 Phasor DiagramsThe voltage across the resistor is in phase with the currentThe voltage across the inductor leads the current by 90°The voltage across the capacitor lags behind the current by 90°
27 Phasor DiagramsThe phasors are added as vectors to account for the phase differences in the voltagesΔVL and ΔVC are on the same line and so the net y component is ΔVL - ΔVC
28 Phasor DiagramsThe voltages are not in phase, so they cannot simply be added to get the voltage across the combination of the elements or the voltage source is the phase angle between the current and the maximum voltageThe equations also apply to rms values
30 Impedance of a Circuit ΔVmax = Imax Z The impedance, Z, can also be represented in a phasor diagramφ: phase angleOhm’s Law can be applied to the impedanceΔVmax = Imax ZThis can be regarded as a generalized form of Ohm’s Law applied to a series AC circuit
31 Summary of Circuit Elements, Impedance and Phase Angles
32 Chapter 28 Problem 30Find the impedance at 10 kHz of a circuit consisting of a 1.5-kΩ resistor, 5.0-µF capacitor, and 50-mH inductor in series.
33 Power in an AC CircuitNo power losses are associated with pure capacitors and pure inductors in an AC circuitIn a capacitor, during 1/2 of a cycle energy is stored and during the other half the energy is returned to the circuitIn an inductor, the source does work against the back emf of the inductor and energy is stored in the inductor, but when the current begins to decrease in the circuit, the energy is returned to the circuit
34 Power in an AC Circuit Pav = Irms ΔVR,rms ΔVR, rms = ΔVrms cos The average power delivered by the generator is converted to internal energy in the resistorPav = Irms ΔVR,rmsΔVR, rms = ΔVrms cos Pav = Irms ΔVrms cos cos is called the power factor of the circuitPhase shifts can be used to maximize power outputs
35 Resonance in an AC Circuit Resonance occurs at the frequency, ω0, where the current has its maximum valueTo achieve maximum current, the impedance must have a minimum valueThis occurs when XL = XC and
36 Resonance in an AC Circuit Theoretically, if R = 0 the current would be infinite at resonanceReal circuits always have some resistanceTuning a radio: a varying capacitor changes the resonance frequency of the tuning circuit in your radio to match the station to be received
37 Chapter 28 Problem 29A series RLC circuit has R = 75 Ω, L = 20 mH, and resonates at 4.0 kHz. (a) What’s the capacitance? (b) Find the circuit’s impedance at resonance and (c) at 3.0 kHz.
38 Damped LC Oscillations The total energy is not constant, since there is a transformation to internal energy in the resistor at the rate of dU/dt = – i2RRadiation losses are still ignoredThe circuit’s operation can be expressed as:
39 Damped LC Oscillations Solution:Analogous to a damped harmonic oscillatorWhen R = 0, the circuit reduces to an LC circuit (no damping in an oscillator)
40 TransformersAn AC transformer consists of two coils of wire wound around a core of soft ironThe side connected to the input AC voltage source is called the primary and has N1 turnsThe other side, called the secondary, is connected to a resistor and has N2 turnsThe core is used to increase the magnetic flux and to provide a medium for the flux to pass from one coil to the other
41 TransformersThe rate of change of the flux is the same for both coils, so the voltages are related byWhen N2 > N1, the transformer is referred to as a step up transformer and when N2 < N1, the transformer is referred to as a step down transformerThe power input into the primary equals the power output at the secondary
42 Chapter 28 Problem 37You’re planning to study in Europe, and you want a transformer designed to step 230-V European power down to 120 V needed to operate your stereo. (a) If the transformer’s primary has 460 turns, how many should the secondary have? (b) You can save money with a transformer whose maximum primary current is 1.5 A. If your stereo draws a maximum of 3.3 A, will this transformer work?
43 Answers to Even Numbered Problems Chapter 28:Problem 14V = (325 V) sin[(314 s−1)t]
44 Answers to Even Numbered Problems Chapter 28:Problem 2622 to 190 pF