chapter 8 alternating current circuits. ac circuit an ac circuit consists of a combination of...
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Chapter 8Chapter 8
Alternating Current Alternating Current Circuits Circuits
AC CircuitAC Circuit
An AC circuit consists of a combination An AC circuit consists of a combination of circuit elements and an AC generator of circuit elements and an AC generator or sourceor source
The output of an AC generator is The output of an AC generator is sinusoidal and varies with time sinusoidal and varies with time according to the following equationaccording to the following equation Δv = ΔVΔv = ΔVmaxmax sin 2 sin 2ƒtƒt
Δv is the instantaneous voltageΔv is the instantaneous voltage ΔVΔVmaxmax is the maximum voltage of the generator is the maximum voltage of the generator ƒ is the frequency at which the voltage changes, ƒ is the frequency at which the voltage changes,
in Hzin Hz
Resistor in an AC CircuitResistor in an AC Circuit Consider a circuit Consider a circuit
consisting of an AC consisting of an AC source and a resistorsource and a resistor
The graph shows the The graph shows the current through and the current through and the voltage across the voltage across the resistorresistor
The current and the The current and the voltage reach their voltage reach their maximum values at the maximum values at the same timesame time
The current and the The current and the voltage are said to be voltage are said to be in in phasephase
More About Resistors in an More About Resistors in an AC CircuitAC Circuit
The direction of the current has no effect The direction of the current has no effect on the behavior of the resistoron the behavior of the resistor
The rate at which electrical energy is The rate at which electrical energy is dissipated in the circuit is given bydissipated in the circuit is given by P = iP = i22 R R
where i is the where i is the instantaneous currentinstantaneous current the heating effect produced by an AC current with a the heating effect produced by an AC current with a
maximum value of Imaximum value of Imaxmax is not the same as that of a is not the same as that of a DC current of the same valueDC current of the same value
The maximum current occurs for a small amount of The maximum current occurs for a small amount of timetime
rms Current and Voltagerms Current and Voltage
The The rms currentrms current is the direct is the direct current that would dissipate the current that would dissipate the same amount of energy in a same amount of energy in a resistor as is actually dissipated by resistor as is actually dissipated by the AC currentthe AC current
Alternating voltages can also be Alternating voltages can also be discussed in terms of rms valuesdiscussed in terms of rms values
maxmax
rms I707.02
II
maxmax
rms V707.02
VV
Ohm’s Law in an AC Ohm’s Law in an AC CircuitCircuit
rms values will be used when rms values will be used when discussing AC currents and voltagesdiscussing AC currents and voltages AC ammeters and voltmeters are AC ammeters and voltmeters are
designed to read rms valuesdesigned to read rms values Many of the equations will be in the Many of the equations will be in the
same form as in DC circuitssame form as in DC circuits Ohm’s Law for a resistor, R, in an Ohm’s Law for a resistor, R, in an
AC circuitAC circuit ΔVΔVrmsrms = I = Irmsrms R R
Also applies to the maximum values of v Also applies to the maximum values of v and iand i
Capacitors in an AC CircuitCapacitors in an AC Circuit
Consider a circuit containing a capacitor Consider a circuit containing a capacitor and an AC sourceand an AC source
The current starts out at a large value and The current starts out at a large value and charges the plates of the capacitorcharges the plates of the capacitor There is initially no resistance to hinder the flow There is initially no resistance to hinder the flow
of the current while the plates are not chargedof the current while the plates are not charged As the charge on the plates increases, the As the charge on the plates increases, the
voltage across the plates increases and the voltage across the plates increases and the current flowing in the circuit decreasescurrent flowing in the circuit decreases
More About Capacitors in More About Capacitors in an AC Circuitan AC Circuit
The current The current reverses directionreverses direction
The voltage across The voltage across the plates the plates decreases as the decreases as the plates lose the plates lose the charge they had charge they had accumulatedaccumulated
The voltage across The voltage across the capacitor lags the capacitor lags behind the current behind the current by 90°by 90°
Capacitive Reactance and Capacitive Reactance and Ohm’s LawOhm’s Law
The impeding effect of a capacitor on the The impeding effect of a capacitor on the current in an AC circuit is called the current in an AC circuit is called the capacitive reactancecapacitive reactance and is given by and is given by
When ƒ is in Hz and C is in F, XWhen ƒ is in Hz and C is in F, XCC will be in will be in ohmsohms
Ohm’s Law for a capacitor in an AC circuitOhm’s Law for a capacitor in an AC circuit ΔVΔVrmsrms = I = Irmsrms X XCC
Cƒ2
1XC
Inductors in an AC CircuitInductors in an AC Circuit
Consider an AC Consider an AC circuit with a source circuit with a source and an inductorand an inductor
The current in the The current in the circuit is impeded by circuit is impeded by the back emf of the the back emf of the inductorinductor
The voltage across The voltage across the inductor always the inductor always leads the current by leads the current by 90°90°
Inductive Reactance and Inductive Reactance and Ohm’s LawOhm’s Law
The effective resistance of a coil in The effective resistance of a coil in an AC circuit is called its an AC circuit is called its inductive inductive reactancereactance and is given by and is given by XXLL = 2 = 2ƒLƒL
When ƒ is in Hz and L is in H, XWhen ƒ is in Hz and L is in H, XLL will be in will be in ohmsohms
Ohm’s Law for the inductorOhm’s Law for the inductor ΔVΔVrmsrms = I = Irmsrms X XLL
The RLC Series CircuitThe RLC Series Circuit
The resistor, The resistor, inductor, and inductor, and capacitor can be capacitor can be combined in a combined in a circuitcircuit
The current in the The current in the circuit is the same circuit is the same at any time and at any time and varies sinusoidally varies sinusoidally with timewith time
Current and Voltage Current and Voltage Relationships in an RLC Relationships in an RLC CircuitCircuit
The instantaneous The instantaneous voltage across the voltage across the resistor is in phase resistor is in phase with the currentwith the current
The instantaneous The instantaneous voltage across the voltage across the inductor leads the inductor leads the current by 90°current by 90°
The instantaneous The instantaneous voltage across the voltage across the capacitor lags the capacitor lags the current by 90°current by 90°
Phasor DiagramsPhasor Diagrams To account for the To account for the
different phases of the different phases of the voltage drops, vector voltage drops, vector techniques are usedtechniques are used
Represent the voltage Represent the voltage across each element across each element as a rotating vector, as a rotating vector, called a called a phasorphasor
The diagram is called The diagram is called a a phasor diagramphasor diagram
Phasor Diagram for RLC Phasor Diagram for RLC Series CircuitSeries Circuit
The voltage across the The voltage across the resistor is on the +x resistor is on the +x axis since it is in phase axis since it is in phase with the currentwith the current
The voltage across the The voltage across the inductor is on the +y inductor is on the +y since it leads the since it leads the current by 90°current by 90°
The voltage across the The voltage across the capacitor is on the –y capacitor is on the –y axis since it lags axis since it lags behind the current by behind the current by 90°90°
Phasor Diagram, contPhasor Diagram, cont
The phasors are The phasors are added as vectors to added as vectors to account for the account for the phase differences in phase differences in the voltagesthe voltages
ΔVΔVLL and ΔV and ΔVCC are on are on the same line and the same line and so the net y so the net y component is ΔVcomponent is ΔVL L - - ΔVΔVCC
ΔVΔVmaxmax From the Phasor From the Phasor DiagramDiagram
The voltages are not in phase, so they The voltages are not in phase, so they cannot simply be added to get the cannot simply be added to get the voltage across the combination of the voltage across the combination of the elements or the voltage sourceelements or the voltage source
is the is the phase anglephase angle between the between the current and the maximum voltagecurrent and the maximum voltage
R
CL
2CL
2Rmax
V
VVtan
)VV(VV
Impedance of a CircuitImpedance of a Circuit
The impedance, The impedance, Z, can also be Z, can also be represented in a represented in a phasor diagramphasor diagram
R
XXtan
)XX(RZ
CL
2CL
2
Impedance and Ohm’s Impedance and Ohm’s LawLaw
Ohm’s Law can be applied to the Ohm’s Law can be applied to the impedanceimpedance ΔVΔVmaxmax = I = Imaxmax Z Z
Summary of Circuit Summary of Circuit Elements, Impedance and Elements, Impedance and Phase AnglesPhase Angles
Problem Solving for AC Problem Solving for AC CircuitsCircuits
Calculate as many unknown Calculate as many unknown quantities as possiblequantities as possible For example, find XFor example, find XLL and X and XCC
Be careful of units -- use F, H, Be careful of units -- use F, H, ΩΩ Apply Ohm’s Law to the portion of Apply Ohm’s Law to the portion of
the circuit that is of interestthe circuit that is of interest Determine all the unknowns asked Determine all the unknowns asked
for in the problemfor in the problem
Power in an AC CircuitPower in an AC Circuit
No power losses are associated with No power losses are associated with capacitors and pure inductors in an AC capacitors and pure inductors in an AC circuitcircuit In a capacitor, during one-half of a cycle In a capacitor, during one-half of a cycle
energy is stored and during the other half the energy is stored and during the other half the energy is returned to the circuitenergy is returned to the circuit
In an inductor, the source does work against In an inductor, the source does work against the back emf of the inductor and energy is the back emf of the inductor and energy is stored in the inductor, but when the current stored in the inductor, but when the current begins to decrease in the circuit, the energy is begins to decrease in the circuit, the energy is returned to the circuitreturned to the circuit
Power in an AC Circuit, Power in an AC Circuit, contcont
The average power delivered by The average power delivered by the generator is converted to the generator is converted to internal energy in the resistorinternal energy in the resistor PPavav = I = IrmsrmsΔVΔVRR = = IIrmsrmsΔVΔVrmsrms cos cos cos cos is called the is called the power factorpower factor of the of the
circuitcircuit Phase shifts can be used to Phase shifts can be used to
maximize power outputsmaximize power outputs
Resonance in an AC CircuitResonance in an AC Circuit
ResonanceResonance occurs at occurs at the frequency, ƒthe frequency, ƒoo, , where the current has where the current has its maximum valueits maximum value To achieve maximum To achieve maximum
current, the impedance current, the impedance must have a minimum must have a minimum valuevalue
This occurs when XThis occurs when XLL = = XXCC
LC2
1ƒo
Resonance, contResonance, cont Theoretically, if R = 0 the current would be Theoretically, if R = 0 the current would be
infinite at resonanceinfinite at resonance Real circuits always have some resistanceReal circuits always have some resistance
Tuning a radioTuning a radio A varying capacitor changes the resonance frequency A varying capacitor changes the resonance frequency
of the tuning circuit in your radio to match the station of the tuning circuit in your radio to match the station to be receivedto be received
Metal DetectorMetal Detector The portal is an inductor, and the frequency is set to a The portal is an inductor, and the frequency is set to a
condition with no metal presentcondition with no metal present When metal is present, it changes the effective When metal is present, it changes the effective
inductance, which changes the current which is inductance, which changes the current which is detected and an alarm soundsdetected and an alarm sounds
TransformersTransformers An AC transformer An AC transformer
consists of two consists of two coils of wire wound coils of wire wound around a core of around a core of soft ironsoft iron
The side The side connected to the connected to the input AC voltage input AC voltage source is called the source is called the primaryprimary and has N and has N11 turnsturns
Transformers, 2Transformers, 2
The other side, called the The other side, called the secondarysecondary, is connected to a , is connected to a resistor and has Nresistor and has N22
turnsturns The core is used to increase the The core is used to increase the
magnetic flux and to provide a magnetic flux and to provide a medium for the flux to pass from medium for the flux to pass from one coil to the otherone coil to the other
The rate of change of the flux is the The rate of change of the flux is the same for both coilssame for both coils
Transformers, 3Transformers, 3
The voltages are related byThe voltages are related by
When NWhen N22 > N > N11, the transformer is , the transformer is referred to as a referred to as a step upstep up transformer transformer
When NWhen N22 < N < N11, the transformer is , the transformer is referred to as a referred to as a step down step down transformertransformer
11
22 V
N
NV
Transformer, finalTransformer, final
The power input into the primary The power input into the primary equals the power output at the equals the power output at the secondarysecondary II11ΔVΔV11 = I = I22ΔVΔV22
You don’t get something for nothingYou don’t get something for nothing This assumes an ideal transformerThis assumes an ideal transformer
In real transformers, power efficiencies In real transformers, power efficiencies typically range from 90% to 99%typically range from 90% to 99%