Slide 1Slide 1

Chapter 33-1

Alternating Current Circuits AC Sources Resistors in AC Circuits (R, RL, RC) The RLC Series Circuit Power in an AC Circuit Resonance in a RLC Series Circuit Transformers and Power Transmission Rectifiers and Filters

Slide 2

The voltage supplied by an AC source is sinusoidal with a period T = = 1/f.

A circuit consisting of a resistor of resistanceR connected to an AC source, indicated by

Resistor in AC circuit

vvR = Vmax sint

iR = vR/R = (Vmax/R) sint = ImaxsintKirchhoff’s loop rule: v + vR= 0

amplitude~

Slide 3

Fig 33-3, p.1035

(a) Plots of the instantaneous current iR and instantaneous voltage vR across a resistor as functions of time. The current is in phase with the voltage. At time t =T, one cycle of the time-varying voltage and current has been completed. (b) Phasor diagram for the resistive circuit showing that the current is in phase with the voltage.

vR = Vmaxsint

iR = vR/R = (Vmax/R)sint = Imaxsint

Slide 4 Fig 33-4, p.1036

A voltage phasor is shown at three instants of time. In which part of the figure is the instantaneous voltage largest? Smallest?

Phasor is a vector whose magnitude is proportional to the magnitude of the variable

it represents and which rotates at the variable’s angular speed counterclockwise.

Its projection onto the vertical axis is the variable’s instantaneous value.

Slide 5

Fig 33-5, p.1037

Graph of the current (a) and of the current squared (b) in a resistor as a function of time. The average value of the current i over one cycle is zero. Notice that the gray shaded

regions above the dashed line for I 2max/2 have the same area as those below this line for

I 2max/2 . Thus, the average value of I 2 is I 2

max/2 . The root-mean-square current, Irms, is

Irms i 2 Imax

2Pav Irms

2 R Vrms Vmax

2

= I2maxsin2t

Slide 6 Fig 33-6, p.1038

A circuit consisting of an inductor of inductance L connected to an AC source.

iL Vmax

Lsint  dt

Vmax

Lcost

iL Vmax

Lsin(t / 2),  Imax

Vmax

L

XL L inductive reactance

vL -Ldi

dt= -Vmax sint ImaxXL sint

vL L Ldi / dt,  v vL 0

vLdi / dt,  di Vmax

Lsint  dt

Inductors in an AC circuit

0dt

diLv Kirchhoff’s law

Instantaneous current in the inductor

Slide 7 Fig 33-7, p.1039

(a) Plots of the instantaneous current iL and instantaneous voltage vL across

an inductor as functions of time. (b) Phasor diagram for the inductive circuit.

The current lags behind the voltage by 90°.

Slide 8 Fig 33-9, p.1041

A circuit consisting of a capacitor of ca-pacitance C connected to an AC source.

iC CVmax cost    CVmax sin(t / 2)

Imax CVmax Vmax

1 /C

Vmax

XC

q CVmax sint

capacitive reactance

vC Vmax sintC q / vC , iC dq / dt

vC ImaxXC sint

XC 1 /C

Capacitor in an AC circuit

Slide 9 Fig 33-10, p.1041

(a) Plots of the instantaneous current iC and instantaneous voltage vC across a capacitor as functions of time. The voltage lags behind the current by 90°. (b) Phasor diagram for the capacitive circuit. The current leads the voltage by 90°.

Slide 10 Fig 33-11, p.1042

At what frequencies will the bulb glow the brightest? High, low? Or is the

brightness the same for all frequencies?

Imax Vmax

XC,     P I 2

maxR,    XC 1 /C,   XL L

High freq.: low Xc, high XL

Slide 11

tVtRIv RR sinsinmax

Slide 12

)sin(

sin

max

max

tIi

tVv

Fig 33-13, p.1044

(a) A series circuit consisting of a resistor, an inductor, and a capacitor connected to an AC source. (b) Phase relationships for instantaneous voltages in the series RLC circuit.The current at all points in a series AC circuit has the same amplitude and phase.

vC ImaxXC sin(t / 2) VC costmax sin( / 2) cos L L Lv I X t V t

tVtRIv RR sinsinmax

Slide 13

Phasors for (a) a resistor, (b) an inductor, and (c) a capacitor connected in series.

(a) Phasor diagram for the series RLC circuit of Fig. 33.13a. The phasor VR is in phase with the current phasor Imax, the phasor VL leads Imax by 90°, and the phasor VC lags Imax by 90°. Vmax makes an angle with Imax.(b) Simplified version of part (a) of the figure.

Slide 14

Z

V

XXR

VI

XXRI

XIXIRI

VVVV

CL

CL

CL

CLR

max

22

maxmax

22max

2maxmax

2max

22max

)(

)(

)()(

)(

Fig 33-15, p.1045

Phasor diagrams for the series RLC circuit of Fig. 33.13a.

tan 1 XL XCR

Z is the impedance of the circuit

22 )( CL XXRZ

Slide 15

Slide 16

Fig 33-17, p.1046

Label each part of the figure as being XL > XC, XL = XC, or XL < XC.

vC VC costvL VL cost

tan 1 XL XCR

tVvR sinmax

Slide 17

Fig 33-18, p.1046

The phasor diagram for a RLC circuit with Vmax= 120 V, f = 60 Hz, R = 200 , and C = 4 F. What should be L to match the phasor diagram?

XL XC R tan L 1 /C R tan

L (XC R tan) / (XC R tan) / 2 f

L=…= 0.84 H

Slide 18

A series RLC circuit has Vmax=150 V, = 377 s-1 Hz, R= 425 , L= 1.25 H, and C = 3.5 F. Find its Z, XC, XL, Imax, the angle between current and voltage, and the maximum and instantaneous voltages across each element.

XL L ,   XC 1 /C ,   Z [R2 (XL XC )2 ]1/2    

vmax ImaxZ,   tan 1 XL XCR

XC = 471 XL = 758

Z = 513 Imax = 0.292 A

= -34o

Z

VI max

max

Slide 19

Power in an AC CircuitP ivImax sin(t )Vmax sint   ImaxVmax[sin2 t  cos   sin2t sin / 2]

Pav ImaxVmax cos / 2,   Pav IrmsVrms cos  

Pav Irms2 R 

The average power delivered by the source is converted to internal energy in the resistor

Pure resistive load:

= 0 Pav IrmsVrms

No power losses are associated with pure capacitors and pure inductors in an AC circuit

CVmax2 / 2 LImax

2 / 2

ttt

ttt

2sin2

1cossin

sincoscossin)sin(

( )

RIVVR maxmax cos

Momentary values, average is 0

Slide 20

Resonance in a Series RLC CircuitA series RLC circuit is in resonance when the current has its maximum value

The resonance frequency 0

is obtained from XL = XC, L= 1/C:

Pav Irms2 R Vrms

2 R / Z 2 ,  (XL XC )2 (L 1

C)2

L2

2 ( 2 02 )2

At resonance, when 0, the average power is maximum

and has the value

max max / I V Z /rms rmsI V Z2 2 1/ 2/[ ( ) ] rms rms L CI V R X X

0 1/ LC

2 2

2 2 2 2 2 20( )

rms

av

V RP

R L

2 / rmsV R

Slide 21

Fig 33-19, p.1050

(a) The rms current versus frequency for a series RLC circuit and three values of R. The

current reaches its maximum value at the resonance frequency 0. (b) Average powerdelivered to the circuit versus frequency for the series RLC circuit, for two values of R.

Slide 22 Fig 33-20, p.1051

Quality factor

Q0 /

Q0L / R

Average power vs frequency for series RLC circuit.The width is measured at half maximum. The power is maximum at the resonance frequency 0.

Slide 23 Fig 33-21, p.1052

An ideal transformer consists of two coils wound on the same iron core. An alternating voltage V1 is applied to the primary coil and the outputvoltage V2 is across the resistor R.

Transformers, Power Transmission

Slide 24

Fig 33-22, p.1052

Circuit diagram for a transformer

V1 N1

dB

dt

V2 N2

dB

dtN2

N1

V1

V2 N2

N1

V1

I1V1 I2V2

Req N1

N2

2

RL

**

** N2 > N1 step-up N2 < N1 step-down

transformer

Transformers, Power Transmission

Ideal transformer

*

* If the resistance is negligble

Slide 25 p.1053

Nikola Tesla (1856–1943), American Physicist

Slide 26

Fig 33-23, p.1053

The primary winding in this transformer is attached to the prongs of the

plug. The secondary winding is connected to the wire on the right, which

runs to an electronic device. (120-V to 12.5-V AC) Many of these

power-supply transformers also convert alternating current to direct current.

Slide 27 p.1053

This transformer is smaller than the one inthe opening photograph for this chapter.

In addition, it is a step-down transformer.

It drops the voltage from 4 000 V to 240 V

for delivery to a group of residences.

Slide 28Slide 28

Summary

Alternating Current Circuits AC Sources Resistors in AC Circuits (R, RL, RC) The RLC Series Circuit Power in an AC Circuit Resonance in a RLC Series Circuit Transformers and Power Transmission