lecture notes 13

8/12/2019 Lecture Notes 13

1/24

Copyright R. JanowFall 2013

Electricity and Magnetism

Lecture 13 - Physics 121Electromagnetic Oscillations in LC & LCR Circuits,

Y&F Chapter 30, Sec. 5 - 6

Alternating Current Circuits,Y&F Chapter 31, Sec. 1 - 2

Summary: RC and LC circuits

Mechanical Harmonic Oscillator

LC Circuit Oscillations

Damped Oscillations in an LCR Circuit AC Circuits, Phasors, Forced Oscillations

Phase Relations for Current and Voltage in SimpleResistive, Capacitive, Inductive Circuits.

Summary


2/24


Recap: LC and RC circuits with constant EMF -Time dependent effects

Riei)t(i 0/t L

E

0

R

iei)t(i inf/t

infL

E1

dt

diL)t( LE

constanttimeinductiveL L/R

ECQeQ)t(Q infRC/tinf 1C

)t(Q)t(Vc

constanttimecapacitiveC RC

ECQeQ)t(Q 0RC/t 0

R

C

ER

L

E

Grow th Phase

Decay Phase

Now LCR in same circuit, time varying EMF > New effects

C

R

L

R

L C

External AC can drive circuit, frequency wD=2pf

New behavior: resonant oscillation in LC circuit21/

res )LC(

Generalized resistances: reactances, impedance

]Ohms[LC L)C/(R w1 RZ /CL 2122

New behavior: damped oscillation in LCR circuit


3/24


Recall: Resonant mechanical oscillations

Definition of

an oscillating

system:

Periodic, repetitive behavior System state ( t ) = state( t + T ) = = state( t + NT ) T = period = time to complete one complete cycle State can mean: position and velocity, electric and magnetic fields,

22

12

2

1

2

12

2

12

LiUmvKC

QUkxU LblockCel LC/m/k1/CkLmivQx 120

2

0 Substitutions can convert mechanical to LC equations:

Energy oscillates between 100% kinetic and 100% potential: Kmax= Umax

With solutions like this... )tcos(x)t(x 00Systems that oscillateobey equations like this... k/mx

dt

xd 0

2

02

2

What oscillates for a spring in SHM? position & velocity, spring force,

Mechanical example: Spring oscillator (simple harmonic motion)

22

2

1

2

1kxmvUKE spblockmech constant

makxF:forcerestoringLawsHooke'

no friction 0

dt

dxv

dt

dxkx

dt

dx

dt

xdm

dt

dEmech

2

2

0

OR


4/24


Electrical Oscillations in an LC circuit, zero resistancea

L CE

+b

Charge capacitor fully to Q0=CE then switch to bKirchoff loop equation: 0-V LC E

2

2

dt

QdL

dt

diL

C

QV

Land

CESubstitute:

Peaks of current and charge are out of phase by 900

)t(QLC

dt

)t(Qd 12

2 An oscillatorequation where

solution: ECQ)tcos(Q)t(Q 000Qi)tsin(idt

)t(dQ

)t(i 00 000 w

01/LC

What oscillates? Charge, current, B & E fields, UB, UE

R/L,RC Lcwhere

Lc 12

0

)t(cosC

QC

)t(QUE 0220222

)t(sinLi)t(LiUB

0

2

2

02

22

00

BE UU 00

EBU

C

Q

LC

1

LQ

QL

LiU

2222

2

0

2

0

2

0

2

0

2

0

Show: Total potential energy is constant

tot

U is constant

Peak ValuesAre Equal

)t(U(t)UU

BEtot


5/24


Details: use energy conservation to deduce oscillations

The total energy:C2

)t(Q)t(Li

2

1UUU

22

EB dtdQ

CQ

dtdiLi

dtdU 0 It is constant so:

dt

dQi

2

2

dt

Qd

dt

di The definitions and imply that: )C

Q

dt

QdL(

dt

dQ 0

2

2

Oscillator solution: )tcos(QQ 00

)tsin(Qdt

dQ 000)tcos(Q

dt

Qd 020022

012

0002

2

LC

)tcos(QLC

Q

dt

Qd

LC

1

0

To evaluate w0:plug the first and second derivatives of the solution into thedifferential equation.

The resonant oscillation frequency w0is:

0

dt

dQ Either (no current ever flows) or:

LC

Q

dt

Qd 0

2

2

oscillatorequation


6/24


13 1: What do you think will happen to the oscillations in a trueLC circuit (versus a real circuit) over a long time?

A.They will stop after one complete cycle.

B.They will continue forever.

C.They will continue for awhile, and then suddenly stop.D.They will continue for awhile, but eventually die away.

E.There is not enough information to tell what will happen.

Oscillations Forever?


7/24


Potential Energy alternates between all electrostatic and

all magnetic two reversals per period

C is fully charged twice eachcycle (opposite polarity)


8/24


Example: A 4 F capacitor is charged to = 5.0 V, andthen discharged through a 0.3 Henry

inductance in an LC circuit

Use preceding solutions with = 0C L

b) Find the maximum (peak) current (differentiate Q(t) )

mA18913x5x6

10x4CQi

t)(sinQ-)tcos(dt

dQ

dt

dQ)t(i

000max

000000

w

EECQ

)tcos(Q)t(Q

0

00

w

a) Findthe oscillation period and frequency f = / 2

rad/s913

1LC

1)610x4)(3.0(

0

ms9.60

2f

1PeriodT wHz1452f 0

c) Whendoes the first current maximum occur? When |sin(w0t)| = 1Maxima of Q(t): All energy is in E field

Maxima of i(t): All energy is in B field

Current maxima at T/4, 3T/4, (2n+1)T/4

First One at:

Others at: ementms incr.43

ms7.14T


9/24


Examplea) Find the voltage across the capacitor in the circuit as a function of time.

L = 30 mH, C = 100 mFThe capacitor is charged to Q0= 0.001Coul. at time t = 0.

The resonant frequency is:

The voltage across the capacitor has the same

time dependence as the charge:

At time t = 0, Q = Q0, so choose phase angle = 0.C

)tcos(Q

C

)t(Q

)t(V00

C

rad/s4.577)F10)(H103(

1

LC

1420

Cvolts)tcos()tcos(

F

C)t(V 57710577

10

10

4

3 b) What is the expression for the current in the circuit? The current is:

c) How long until the capacitor charge is reversed? That happens every

period, given by:

amps)t577sin(577.0)t577sin()rad/s577)(C10()tsin(Qdt

dQi 3000

ms44.52

T

0w

p


10/24


13

2: The expressions below could be used to represent thecharge on a capacitor in an LC circuit. Which one has the greatest

maximum currentmagnitude?

A. Q(t) = 2 sin(5t)

B. Q(t) = 2 cos(4t)C. Q(t) = 2 cos(4t+p/2)D. Q(t) = 2 sin(2t)

E. Q(t) = 4 cos(2t)

Which Current is Greatest?

dt

)t(dQi )tcos(Q)t(Q 00


11/24


13 3: The three LC circuits below have identical inductors and

capacitors. Order the circuits according to their oscillation

frequency in ascending order.

A. I, II, III.

B. II, I, III.

C. III, I, II.

D. III, II, I.

E. II, III, I.

Time needed to discharge the capacitor in LC circuit

T1/LC

p 20 C1C iser1CC ipara

I. II. III.


12/24


LCR circuits: Add series resistanceCircuits still oscillate but oscillation isdamped

Charge capacitor fully to Q0=CE then switch to bStored energy decays with time due to resistance

22

22 )t(Li

C

)t(Q)t(U(t)UU

BEtot

a

L CE

+b

R

t- e 0

Critically damped22

02

)L

R(0'

Underdamped:

2

t'

Q(t)

22

02

)L

R(

Overdamped

onentialexpdcomplicate

22

02

)L

R(imaginaryis'

Solution is cosine withexponential decay (weak

or under-damped case)CQ)t'cos(eQ)t(Q 0

L/RtE

20

Shifted resonant frequency wcan be real or imaginary

21220 2

/L)(R /'

Resistance dissipates stored

energy. The power is: R)t(idt

dUtot 21/LC)t(Q

dt

dQ

L

R

dt

)t(Qd 02022

0Oscillator equation results, but

with damping (decay) term


13/24


13 4: How does the resonant frequency for an ideal LC circuit(no resistance) compare with for an under-damped circuit whose

resistance cannot be ignored?

A. The resonant frequency for the non-ideal, damped circuit is higherthan

for the ideal one (> ).B. The resonant frequency for the damped circuit is lowerthan for the ideal

one (< ).C. The resistance in the circuit does not affect the resonant frequency

they are the same (

= ).

D. The damped circuit has an imaginaryvalue of w.

Resonant frequency with damping

2/1220 L)2(R /'


14/24


Alternating Current (AC) - EMF

AC is easier to transmit than DC

AC transmission voltage can be changed by using a transformer. Commercial electric power (home or office) is AC, not DC.

The U.S. AC frequency is 60 Hz. Most other countries use 50 Hz.

Sketch: a crude AC generator.

EMF appears in a rotating a coil of wire in a

magnetic field (Faradays Law) Slip rings and brushes allow the EMF to be

taken off the coil without twisting the wires.

Generators convert mechanical energy to

electrical energy. Power to rotate the coil

can come from a water or steam turbine,

windmill, or turbojet engine.

)tcos()t( dmaxind w)tcos(Ii dmax

Outputs are sinusoidalfunctions:is the phase angleYou might (wrongly) expect = 0


15/24


External AC EMF driving a circuit

Apply external, sinusoidal, instantaneous EMF to load:

t)(cos(t ) DmaxEE amplitudemax E

)t(cosIi( t ) Dmax

Currentin load has same frequency wD ... but may be retarded or advanced(relative to ) by phase angle (due to inertia L and stiffness 1/C).Current has the sameamplitude and phase everywhere in a branch

(t)E load

i(t)

)t(( t )load

loadtheacrosspotentialThe V E

R

L

CE

In a Series LCR Circuit:

Everything oscillates at driving frequency wD

is zero at resonance circuit acts purely resistively.At resonance: 1/LC 0D

Otherwise is positive (current lags applied EMF)or negative (current leads applied EMF)

D the driving frequencyD resonant frequency w0, in general


16/24


Instantaneous, peak, and average quantities for AC circuits

Instantaneous voltages and currents:

oscillatory, depend on time through argument: wtpossibly advanced or retarded relative to each other by phase anglesrepresented by rotating phasors see below

Peak voltage and current amplitudes are just the coefficients out front

i maxmaxE

Simple time averages of periodic quantities are zero (and useless).

Example: Integrate over a wholenumber of periods one is enough (w=2p)0t )dt(cos

1(t )dt

1

0Dmax

0av EEE

0)dtt(cos1

(t )dt1

0

Dmax0

av iii 0 0dt)tcos(Integrand is odd on a full cycle

)t(cosii( t ) Dmax t )(cos(t ) DmaxEE


17/24


RMS Quantities for AC circuits

RMS averages are used the way instantaneousquantities were in DC circuits:

RMS means root, mean, squared.

2

(t )dt1

max2/1

0

22/1

av

2rms )t( EEEE

i

(t )dti(t )ii max//

avrms

2

1 21

0

221

2

0 21 2/1dt)t(cosIntegrand is even on a whole cycle

So-called rectified average values areseldom used in AC circuits:

Integrate |cos(wt)| over one full cycleor cos(wt) over a positive half cycle

max

4/

4/2/

1

maxrav I2

|)dttcos(|II p

Prescription: RMS Value = Peak value / Sqrt(2)


18/24


Phasor Picture: Show current and potentials asvectors all rotating at frequency wD

The lengths of the vectors are the peakamplitudes.The measured instantaneous values of i( t ) and ( t )

are projections of the phasors on the x-axis.

phase angle measures when peaks pass and are independent tD

Current is the same (phase included) everywhere

in a single (series) branchof any circuit EMF (t) applied to the circuit can lead or lag the

current by a phase angle in the range [-p/2, +p/2] Use rotating current vector as reference.

maxI

maxE wDt

)t(cos(t ) Dmax EE)t( oscIi( t ) Dmax

x

y

Series LCR circuit: Relate internal voltage drops to phase of the current

VR

VCVL

Voltage across R is in phase with the current.Voltage across C lags the current by 900.

Voltage across L leads the current by 900.

XL XC XC XL


19/24


Driven Resistance: AC current i(t) and voltage vR(t) in phase

vR( t )

i(t)

Voltage drop across R:

t )(cosV)t((t )v DRR )(cosIi(t ) tD

0(t )vR-)t( Kirchoff loop rule:

Current flows through R, causing the voltage drop, withsame time dependence governed by cos(wt).

I

VR R

(definition ofresistance)

Peak current and peak voltage are in

phase in a purely resistive part of a

circuit, rotating at the driving

frequency wD

R)t(i)t(vR

The ratio of the AMPLITUDESVRto I is the resistance:

0 Peaks, zeros, and valleys of v and I coincideChoose: Then:


20/24


Driven Inductance: AC current i(t) lags the voltage vL(t) by 900

Voltage drop across Lis due to Faraday Law:

)tsin(LI)]t[cos(dt

dLI(t )v DDDL w

0(t )vL-)t( Kirchoff loop rule:

I

VX LL The RATIO of theAMPLITUDES VLto I is the

inductive reactance XL:

)(cosIi(t ) tDChoose:

Then:

vL(t )L

i(t)

dt

idL(t )vL

)2/t(cost)(sin DD pNote:

so...phase angle p/2 for inductor)2/tcos(V(t )v DLL p

)DL (OhmsL Definition:inductivereactance

Voltage phasor leads the

current phasor by +p/2 in apure inductive part of a

circuit (positive)Inductive Reactance

limiting cases

w 0: Zero reactance.Inductance acts like a wire.

w infinity: Infinite reactance.Inductance acts like a broken wire.


21/24


Driven Capacitance: AC current i(t) leads the voltage vC(t) by 900

Voltage drop across Cis proportional to q(t):

)tsin(C

Idt)tcos(

C

Idt)t(i

C

1(t )v D

D

DC ww

0(t )vc-)t( Kirchoff loop rule:

I

VX CC The RATIO of theAMPLITUDES VCto I is the

capacitive reactance XC:

)(cosIi(t ) tDChoose:

Then:

C

q(t)(t )vC

)2/t(cost)(sin DD pNote:

so...phase angle p/2 for capacitor)2/tcos(V(t )v DCC p

)(OhmsC

1

D

C Definition:capacitivereactance

Voltage phasor lags the

current phasor by -p/2 in apure capacitive part of a

circuit (negative)

i

vC ( t )

C

dti(t)qd

Capacitive Reactance

limiting cases

w 0: Infinite reactance. DCblocked. C acts like broken wire.

w infinity: Reactance is zero.Capacitor acts like a simple wire


22/24


Current & voltage phases in pure R, C, and L circuits

Apply sinusoidal voltage E (t) = Emcos(wDt)For pure R, L, or C loads, phase angles are 0, p/2, -p/2Reactance means ratio of peakvoltage to peakcurrent (generalized resistances).

VR& iRin phaseResistance

Ri/V Rmax C1i/V DCCmax w Li/V DLLmax wVLleads iLby p/2

Inductive ReactanceVClags iCby p/2

Capacitive Reactance

Current is the same everywhere in a single branch (including phase).Phases of voltages in series components are referenced to the current phasor

currrent

SamePhase


23/24


Series LCR circuit driven by an external AC voltage

ImEm

wDt+ wDt

R

LC

E

VR

VC

VL

Apply EMF:

)tcos()t( Dmax EE wDis the driving frequencyThe current i(t) is the same everywhere in the circuit

Same frequency dependance as (t)but...

Current leads or lags E (t) by a constant phase angle Same phase for the current in , R, L, & C

)tcos(I)t(i Dmax w

Phasors all rotate CCW at frequency wD

Lengths of phasors are the peak values (amplitudes)The x components are the measured values.

ImEm

wDt

VL

VC

VR

Plot voltages phasors rotating at wD with phasesrelative to the current phasor Im and magnitudes below :VRhas samephase as ImVClagsImby p/2VLleadsImby p/2

RIV mRCmC XIVLmL XIV

0byLlagsCCLRm 180VV)VV(V

Ealong im perpendicular to im

0)t(v)t(v)t(v)t( CLR EKirchoff Loop rule for potentials (measured along x)


24/24

lecture notes 13

Documents