Copyright R. Janow – Spring 2015 Electricity and Magnetism Lecture 13 - Physics 121 Electromagnetic Oscillations in LC & LCR Circuits, Y&F Chapter 30,

Copyright R. Janow – Spring 2015

Electricity and MagnetismLecture 13 - Physics 121 Electromagnetic Oscillations in LC & LCR Circuits, Y&F Chapter 30, Sec. 5 - 6Alternating 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 Simple

Resistive, Capacitive, Inductive Circuits. • Summary


LC and RC circuits with constant EMF - Time dependent effects

dt

diL )t( LE

constant time inductive L L/R

C

)t(Q)t(Vc

constant time capacitive C RC

R

C

E R

L

E

R

i e i)t(i inf/t

infL

E 1 ECQ eQ)t(Q inf

RC/tinf 1 Growth Phase

Ri ei)t(i 0

/t LE

0ECQ eQ)t(Q 0

RC/t 0 Decay Phase

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

C

R

L

RL C

• External AC drives circuit at frequency D=2f which may or may not be at resonance

• Resonant oscillations in LC circuit 21/

res )LC(

• Generalized resistances: reactances, impedance ]Ohms[ LC L )C/( R 1

R Z/

CL2122

• Damped oscillation in LCR circuit

Voltage across C related to integral of current

Voltage across L related to derivative of current


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

122

1

2

122

12

LiUmvK C

QU kxU LblockCel

LC/ m/k 1/Ck Lm iv Qx 120

20

Can convert mechanical to LC equations by substituting:

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

With solutions like this...

)tcos(x)t(x 00

Systems that oscillateobey equations like this...

k/m x dt

xd 0

202

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

ma kx F :force restoring Law sHooke'

no friction 0dt

dxv

dt

dxkx

dt

dx

dt

xdm

dt

dEmech 2

20

OR


Electrical Oscillations in an LC circuit, zero resistancea

L CE

+b

Charge capacitor fully to Q0=CE then switch to “b”

Kirchoff loop equation: 0 - V LC E

2

2

dt

QdL

dt

diL

C

QV L and C ESubstitute:

Peaks of current and charge are out of phase by 900

)t(QLC

dt

)t(Qd 12

2 An oscillator

equation wheresolution: ECQ )tcos(Q)t(Q 0 00

Qi )tsin(idt

)t(dQ)t(i 00000

01/LC

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

R/L ,RC

Lcwhere

Lc

12

0

)t(cosC

Q

C)t(Q

UE 02

20

2

22)t(sin

Li

)t(LiUB 0

220

2

22

00BE U U

0E

20

20

20

20

20

0B U C2

Q

LC

1

2

LQ

2

QL

2

Li U

Show: Total potential energy is constant

totU is constant

Peak ValuesAre Equal

)t(U (t)U U BEtot


Details: Energy conservation used to deduce oscillations

• The total energy:C2

)t(Q)t(Li

2

1UUU

22

EB

dt

dQ

C

Q

dt

diLi

dt

dU0• It is constant so:

dt

dQi

2

2

dt

Qd

dt

di• The definition and imply that: )

C

Q

dt

QdL(

dt

dQ 0

2

2

• Oscillator solution: )tcos(QQ 00

)tsin(Qdt

dQ 000

)tcos(Qdt

Qd 0

2002

20

120002

2

LC )tcos(Q

LC

Q

dt

Qd

LC

1

0

• To evaluate plug the first and second derivatives of the solution into the

differential equation.

• The resonant oscillation frequency is:

0dt

dQ• Either (no current ever flows) OR:

LC

Q

dt

Qd 0

2

2

oscillatorequation


13 – 1: What do you think will happen to the oscillations in an ideal LC 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?


Potential Energy alternates between all electrostatic and all magnetic – two reversals per period

C is fully charged twice each cycle (opposite

polarity)


Example: A 4 µF capacitor is charged to E = 5.0 V, and then discharged through a 0.3 Henry inductance in an LC circuit

Use preceding solutions with = 0

C L

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

mA18913 x 5 x610x4CQi

t)(sinQ - )tcos(dt

dQ

dt

dQ)t(i

000max

000000

EECQ

)tcos(Q)t(Q

0

00

a) Find the oscillation period and frequency f = ω / 2π

rad/s913

1LC

1ω)610 x 4)(3.0(0

ms9.6 0

π2f

1PeriodT

Hz145π2ωf 0

c) When does the first current maximum occur? When |sin(0t)| = 1

Maxima of Q(t): All energy is in E fieldMaxima 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: ements ms incr.43

ms7.1 4T


Examplea) Find the voltage across the capacitor in the circuit as a function of time. L = 30 mH, C = 100 F

The capacitor is charged to Q0 = 0.001 Coul. 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(V 00

C

rad/s 4.577)F10)(H103(

1

LC

1420

C volts )tcos()tcos(F

C)t(V 57710577

10

104

3

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

c) How long until the capacitor charge is exactly reversed? That occurs every ½ period, given by:

amps )t577sin(577.0)t577sin()rad/s 577)(C10()tsin(Qdt

dQi 3

000

ms 44.52

T

0


13 – 2: The expressions below could be used to represent the charge on a capacitor in an LC circuit. Which one has the greatest maximum current magnitude?

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

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

C. Q(t) = 2 cos(4t+/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


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.

LC circuit oscillation frequencies

T1/LC

20

C1

C iser1 CC ipara

I. II. III.


LCR circuits with series resistance: Oscillations but with damping Charge capacitor fully to Q0=CE then switch to “b”Stored energy decays with time due to the resistance

22

22 )t(LiC

)t(Q)t(U (t)U U BEtot

a

L CE

+b

R

t

t- ωe 0

Critically damped22

0 2)

L

R(ω 0'ω

Underdamped:

π2

tω'

Q(t)

220 2

)L

R( ω

Overdamped

t

onentialexpdcomplicate

220 2

)L

R(ω imaginary is

' ω

Solution is cosine with exponential decay (weak or under-damped case)

CQ )t'cos(eQ)t(Q 0L/Rt E 2

0

Shifted resonant frequency ’ can be real or imaginary

2/1220 L)2(R / ω ω'

Resistance dissipates storedenergy. The power is: R)t(i

dt

dUtot 2

1/LC )t(Qdt

dQ

L

R

dt

)t(Qd 0

202

2

0Oscillator equation results, but with damping (decay) term


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 higher than for the ideal one (’ > ).

B. The resonant frequency for the damped circuit is lower than 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 imaginary value of ’.

Resonant frequency with damping

2/1220 L) 2(R / ω ω'


Alternating Current (AC) EMF• Commercial electric power (home or office) is AC, not DC.• AC frequency is 60 Hz in U.S, 50 Hz in most other countries • AC transmission voltage can be changed for distribution by

using transformers (lower).• High transmission voltage Lower i2R loss than DC

• Sketch: a crude AC generator.

• Generators convert mechanical energy to electrical energy. Mechanical power can come from a water or steam turbine, windmill, diesel or turbojet engine.

• EMF appears in rotating coils in a magnetic field (Faraday’s Law)

• Slip rings and brushes take EMF from the coil without twisting the wires.

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

Outputs are sinusoidal functions: is the phase angle between current and EMF. You might (wrongly) expect = 0


External AC EMF driving a circuit

External, instantaneous EMF acts on load:

)t(ωcos(t ) Dmax EE amplitudemax E

)t ( ω cos Ii( t ) Dmax

Current in load has same frequency D ... but may be retarded or advanced

(relative to E) by “phase angle” for the whole circuit(due to inertia L and stiffness 1/C).

(t)Eload

i(t)

)t( ( t )load

load the across potential The V E ωD the driving frequency of EMF

ωD resonant frequency 0, in general

USE KIRCHHOFF LOOP & JUNCTION RULES, INSTANTANEOUS QUANTITIES

R

L

CE

In a Series LCR Circuit:• Everything oscillates at driving frequency D

• is zero at resonance – circuit acts purely resistively.

• At “resonance”:

1/LC ω ω 0D

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

• Current is the same everywhere (including phase)

Current has the same amplitude and phase everywhere in a branchVoltage drops have the same amplitude and phase across parallel branchesAt junctions, instantaneous currents in & out are conserved


Phasor Picture: Show current and potentials as vectors all rotating at frequency D

• The lengths of the vectors are the peak amplitudes. • The measured instantaneous values of i( t ) and E( t ) are projections of the phasors on the x-axis.

“phase angle” measured when peaks pass Φ and are independentΦ t D

• Current is the same (phase included) everywhere in a single (series) branch of any circuit • Use rotating current vector as reference.• EMF E(t) applied to the circuit can lead or lag the current by a phase angle in the range [-/2, +/2]

maxI

maxEDt

)t (ωcos(t ) Dmax EE )t ( ω osc Ii( t ) Dmax

x

y

In SERIES LCR circuit: Relate voltage peaks to phase of the current

VR

VC

VL

• 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>XCXC>XL


The meaningful quantities in AC circuits may be instantaneous, peak, or average

Instantaneous voltages and currents:

• oscillatory, depend on time through argument “t”• possibly advanced or retarded relative to each other by phase angles• represented by rotating “phasors” – see slides below

Peak voltage and current amplitudes are just the coefficients out front i max maxE

Simple time averages of periodic quantities are zero (and useless).• Example: Integrate over a whole number of periods – one is enough (=2)

0 )dtt(ωcos1

(t )dt1

0 D max0av

EEE

0 t)dt(ωcos1

(t )dt1

0 D max0av

iii

0

1 0dt ) tcos(

Integrand is odd on a full cycle

)t ( ω cos ii( t ) Dmax )t(ωcos(t ) Dmax EE

So how should “average” be defined?


Averaging Definitions for AC circuits

“RMS” averages are used the way instantaneous quantities were in DC circuits:

• “RMS” means “root, mean, squared”.

2

(t )dt1

max

2/1

0

22/1

av

2rms )t( EEEE

i

(t )dti (t )i i max//

avrms

2

1 21

02

212

0

21 2/1dt ) t(cos

Integrand is positive on a whole cycle after squaring

So-called “rectified average values” are seldom used in AC circuits:

•Integrate |cos(t)| over one full cycle or cos(wt) over a positive half cycle

max4/4/2/

1maxrav i

2 |)dttcos(| i i

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

NOTE:


For a Resistor only: AC current i(t) and voltage vR(t) are in phase

vR( t )εi(t)

Peak current and peak voltage are in phase in a purely resistive part of a circuit, rotating at the driving frequency D

Voltage drop across R: RwhereDR VIR )(ωcosR I tR )t(i)t(v

I

V R RThe ratio of the AMPLITUDES

(peaks) VR to I is the resistance:

0 Φ The phases of vR(t) and i(t) coincide

)t (ωcos V )t( ε(t ) v DRR

0(t )vR - )t(ε Kirchoff loop rule:

)(ωcos Ii(t ) tDCurrent has time dependence: Applied EMF: )t (ωcos(t ) Dmax EE


For an Inductor only: 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

dt

i dL (t ) v DDDL

vL(t)Lεi(t)

) 2/t(ωcost) (ωsin DD Note:

I

V X L

L The RATIO of the peaks (AMPLITUDES) VL to I is the inductive reactance XL:

)DL (OhmsL ω Definition: inductive reactance

so...phase angle for inductor

LIV )2/tcos(LI(t )v DLpeak whereDDL

Voltage phasor leads current by +/2 in inductive part of a circuit ( positive)

Inductive Reactance limiting cases• 0: Zero reactance.

Inductor acts like a wire.• infinity: Infinite reactance. Inductor acts like a broken wire.

)t (ωcos V )t( ε(t ) v DLL

0(t )vL - )t(ε Kirchoff loop rule:


Sine from derivative


For a Capacitor only: AC current i(t) leads the voltage vC(t) by 900

) 2/t(ωcost) (ωsin DD Note:

I

V C

C The RATIO of the AMPLITUDES VC to I is the capacitive reactance XC:

) (Ohms Cω

1

DC

Definition: capacitive reactance

so...phase angle for capacitor

CDwhereDDC V)C/(I )2/tcos()C/(I (t )v

Voltage phasor lags the current by /2 in a pure capacitive circuit branch ( negative)

εi

vC ( t )

C

Capacitive Reactance limiting cases• 0: Infinite reactance. DC blocked. C acts like broken wire.• infinity: Reactance is zero. Capacitor acts like a simple wire

)t (ωcos V )t( ε(t ) v DCC

0(t )vC - )t(ε Kirchoff loop rule:


Voltage drop across Cis proportional to q(t):

)tsin(C

I dt)tcos(

C

I dt)t(i

C

1

C

q(t) (t ) v D

DDC

Sine from

integral


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

• Apply sinusoidal current i (t) = Imcos(Dt)• For pure R, L, or C loads, phase angles for voltage drops are 0, /2, -/2• “Reactance” means ratio of peak voltage to peak current (generalized resistances).

VR& Im in phase Resistance

RI/V mR C1I/V

DCmC LI/V DLmL

VL leads Im by /2Inductive Reactance

VC lags Im by /2Capacitive Reactance

Phases of voltages in series components are referenced to the current phasor

currrent

SamePhase


Same frequency dependance as E (t)• Same phase for the current in E, R, L, & C, but...... • Current leads or lags E (t) by a constant phase angle

Phasor Model for a Series LCR circuit, AC voltage

RL

C

E

vR

vC

vL

Apply EMF: )tcos()t( Dmax EE

Im

Em

Dt

VL

VC

VR

Show component voltage phasors rotating at D with phases relative to the current phasor Im and magnitudes below :

• VR has same phase as Im

• VC lags Im by /2• VL leads Im by /2

RIV mR

CmC XIV

LmL XIV

0byLlagsCCLR m 180 V V )VV( V

Ealong Im perpendicular to Im

ImEm

Dt+Dt

Phasors all rotate CCW at frequency D

• Lengths of phasors are the peak values (amplitudes) • The “x” components are instantaneous values measured

0)t(v)t(v)t(v)t( CLR EKirchoff Loop rule for series LRC:

)tcos(I)t(i Dmax The current i(t) is the same everywhere in the singlebranch in the circuit:

Documents

Copyright R. Janow – Spring 2015 Electricity and Magnetism Lecture 13 - Physics 121 Electromagnetic Oscillations in LC & LCR Circuits, Y&F Chapter 30,