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%