Alternating Current Circuits and Electromagnetic Waves.ppt

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Alternating Current Alternating Current Circuits Circuits and Electromagnetic and Electromagnetic Waves Waves

Transcript of Alternating Current Circuits and Electromagnetic Waves.ppt

Page 1: Alternating Current Circuits and Electromagnetic Waves.ppt

Alternating Current Alternating Current Circuits Circuits

and Electromagnetic and Electromagnetic WavesWaves

Page 2: Alternating Current Circuits and Electromagnetic Waves.ppt

21.1 Resistor in an AC 21.1 Resistor in an AC CircuitCircuit

An AC circuit consists of a combination of circuit An AC circuit consists of a combination of circuit elements and an AC generator or sourceelements and an AC generator or source

The output of an AC generator is sinusoidal and The output of an AC generator is sinusoidal and varies with time according to the following equationsvaries with time according to the following equations vv = = VVmaxmax sin 2 sin 2ƒt, iƒt, i = = IImaxmax sin 2 sin 2ƒtƒt

vv and and ii is the instantaneous voltage and is the instantaneous voltage and current, respectivelycurrent, respectively

VVmaxmax is the maximum voltage of the generator is the maximum voltage of the generator IImaxmax is the maximum current is the maximum current ƒƒ is the frequency in Hz ( is the frequency in Hz (=2=2ƒƒ))

Page 3: Alternating Current Circuits and Electromagnetic Waves.ppt

Resistor in an AC Circuit, Resistor in an AC Circuit, cont.cont. Consider a circuit Consider a circuit

consisting of an AC source consisting of an AC source and a resistorand a resistor

The graph shows the The graph shows the current through and the current through and the voltage across the resistorvoltage across the resistor

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

Note: The average value of the current over one cycle is zero!

i, v

i

v

Page 4: Alternating Current Circuits and Electromagnetic Waves.ppt

Resistor in an AC Circuit, Resistor in an AC Circuit, cont.cont.

The average value of a sinusoidal current is zero: I = 0

However, we have to consider I2R:P=(Imax)

2R sin2t

sin2t = ½(1-cos2t) Average: ½(1-cos2t)= ½

P=(Imax)2R sin2t =(½)(Imax)

2R 

The people want the same “shape” of formula as for DC power: I2R This step requires the rms (root-mean-square) current

The “bar” indicates average value

Zero!

Page 5: Alternating Current Circuits and Electromagnetic Waves.ppt

Resistor in an AC Circuit, Resistor in an AC Circuit, cont.cont.

I2=(Imax)2sin2t =(½)(Imax)

2

Irms=(I2)½=[(½)(Imax)2]½ (Irms)2=(½)(Imax)2

P=(½)(Imax)2 R= (Irms)2R

maxmaxrms 707.02

2III

Page 6: Alternating Current Circuits and Electromagnetic Waves.ppt

AC Power delivered to a AC Power delivered to a resistorresistor

P

Pmax

Voltage across a resistor: Vrms=IrmsR (Ohm’s law)

Average power delivered: P=Pmax/2=VrmsIrms=(Irms)2R

Page 7: Alternating Current Circuits and Electromagnetic Waves.ppt

rms Current and Voltagerms Current and Voltage

Rms Current:Rms Current:

Rms Voltage:Rms Voltage:

maxmaxrms 707.02

2III

maxmaxrms 707.02

2VVV

Page 8: Alternating Current Circuits and Electromagnetic Waves.ppt

rms Current and Voltage, rms Current and Voltage, cont.cont.

The direction of the current has The direction of the current has no effect on the behavior of the no effect on the behavior of the resistorresistor

The The rms currentrms current is the DC current is the DC current that would dissipate the same that would dissipate the same amount of energy in a resistor as amount of energy in a resistor as is dissipated by the actual AC is dissipated by the actual AC currentcurrent

Page 9: Alternating Current Circuits and Electromagnetic Waves.ppt

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 resistorOhm’s Law for a resistor, , RR, in an , in an

AC circuitAC circuit VVrmsrms = = IIrmsrms RR

Also applies to the maximum values of Also applies to the maximum values of vv and and ii

Page 10: Alternating Current Circuits and Electromagnetic Waves.ppt

Example: An AC power supply with Vmax=48 V is

connected to a resistor with 12 . Calculate (a) the rms current, (b) P and (c) Pmax.

(a) Irms=(0.70748 V)/12

Irms=2.83 A

(b) P=(0.70748 V)(2.83 A)

P=96 W

(c) Pmax=2P=192 W

Page 11: Alternating Current Circuits and Electromagnetic Waves.ppt

21.2 Capacitors in an AC 21.2 Capacitors in an AC CircuitCircuit

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

Page 12: Alternating Current Circuits and Electromagnetic Waves.ppt

More About Capacitors in More About Capacitors in an AC Circuitan AC Circuit

The current reverses The current reverses directiondirection

The voltage across the The voltage across the plates decreases as plates decreases as the plates lose the the plates lose the charge they had charge they had accumulatedaccumulated

The voltage across the The voltage across the capacitor lags behind capacitor lags behind the current by 90° the current by 90° (current leads)(current leads)

Page 13: Alternating Current Circuits and Electromagnetic Waves.ppt

Reason for the phase shiftReason for the phase shift

Charging Discharging Charging Discharging Charging Discharging Charging Discharging

Current Voltage

Current leads

Page 14: Alternating Current Circuits and Electromagnetic Waves.ppt

Capacitive Reactance Capacitive Reactance XXcc

f=0 Hz, XC= (remember

the DC case)

f= Hz, Xc=0

XC=1/(C)=1/(2fC) SI unit:

Page 15: Alternating Current Circuits and Electromagnetic Waves.ppt

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 When ƒƒ is in Hz and is in Hz and CC is in is in FF, X, XCC will be in ohms will be in ohms Ohm’s Law for a capacitor in an AC circuitOhm’s Law for a capacitor in an AC circuit

VVrmsrms = = IIrmsrms XXCC

πfCX

2

1C

Page 16: Alternating Current Circuits and Electromagnetic Waves.ppt

21.3 Inductors in an AC 21.3 Inductors in an AC CircuitCircuit

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°

Page 17: Alternating Current Circuits and Electromagnetic Waves.ppt

Proof of the phase shiftProof of the phase shift

(Faraday’s law) VLd/dt

 vL=L(dI/dt)

vL=Vmaxcos t  

L(dI/dt)=Vmaxcost

dI=[Vmaxcost/L]dt

I=(Vmax/L) costdt=(Vmax/L)sint+K

 

Imax

Imax=Vmax/L XL=L inductive

reactance [unit: ]

Page 18: Alternating Current Circuits and Electromagnetic Waves.ppt

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 When ƒƒ is in Hz and is in Hz and LL is in H, is in H, XXLL will be in will be in ohmsohms

Ohm’s Law for the inductorOhm’s Law for the inductor VVrmsrms = = IIrmsrms XXLL

Page 19: Alternating Current Circuits and Electromagnetic Waves.ppt

21.4 The RLC Series 21.4 The RLC Series CircuitCircuit

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

vR vL vC

Page 20: Alternating Current Circuits and Electromagnetic Waves.ppt

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°

Page 21: Alternating Current Circuits and Electromagnetic Waves.ppt

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 as across each element as a rotating vector, called a rotating vector, called a a phasorphasor

The diagram is called a The diagram is called a phasor diagramphasor diagram

Page 22: Alternating Current Circuits and Electromagnetic Waves.ppt

Phasor Diagram for RLC Phasor Diagram for RLC Series CircuitSeries Circuit

The voltage across the The voltage across the resistor is on the +resistor is on the +xx 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 +inductor is on the +yy 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°

Current

Page 23: Alternating Current Circuits and Electromagnetic Waves.ppt

Phasor Diagram, contPhasor Diagram, cont

The phasors are The phasors are added as vectors added as vectors to account for the to account for the phase differences phase differences in the voltagesin the voltages

VVLL and and VVCC are on are on the same line and the same line and so the net y so the net y component is component is VVL L - - VVCC

VVL L - - VV

CC

VVRR

VVLL

VVCC

VVmaxmax

Page 24: Alternating Current Circuits and Electromagnetic Waves.ppt

VVmaxmax 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

tan

)(

V

VV

VVVV

Page 25: Alternating Current Circuits and Electromagnetic Waves.ppt

Impedance of a CircuitImpedance of a Circuit

The impedance, The impedance, ZZ, can also be , can also be represented in a represented in a phasor diagramphasor diagram

R

XX

XXRZ

CL

2CL

2

tan

)(

Page 26: Alternating Current Circuits and Electromagnetic Waves.ppt

Impedance and Ohm’s Impedance and Ohm’s LawLaw

Ohm’s Law can be applied Ohm’s Law can be applied to the impedanceto the impedanceVVmaxmax = = IImaxmax ZZ

Ohm’s law of the AC circuit

Page 27: Alternating Current Circuits and Electromagnetic Waves.ppt

Summary of Circuit Summary of Circuit Elements, Impedance and Elements, Impedance and Phase AnglesPhase Angles

Page 28: Alternating Current Circuits and Electromagnetic Waves.ppt

21.5 Power in an AC 21.5 Power in an AC CircuitCircuit

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

Page 29: Alternating Current Circuits and Electromagnetic Waves.ppt

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 P P = = IIrmsrmsVVRR = = IIrmsrmsVVrmsrms 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

Page 30: Alternating Current Circuits and Electromagnetic Waves.ppt

21.6 Resonance in an AC 21.6 Resonance in an AC CircuitCircuit

ResonanceResonance occurs at occurs at the frequency, the frequency, ƒƒ00, , 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 This occurs when XXLL = = XXCC

LCƒ

2

1o

Z

VI rms

rms

Page 31: Alternating Current Circuits and Electromagnetic Waves.ppt

Resonance, cont.Resonance, cont. Theoretically, if Theoretically, if RR = 0 the current would be = 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