electricity and magnetism lecture 13 - physics 121tyson/p122-ece_lecture12_extra_pt1.pdf · lecture...

Copyright R. Janow – Fall 2013

Electricity and Magnetism

Lecture 13 - Physics 121 Electromagnetic 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 Simple Resistive, Capacitive, Inductive Circuits.

• Summary


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

Ri ei)t(i 0

/t LE

0

R

i e i)t(i inf/t

infL

E

1

dt

diL )t( LE

constant time inductive L L/R

ECQ eQ)t(Q infRC/t

inf 1

C

)t(Q)t(Vc

constant time capacitive C RC

ECQ eQ)t(Q 0RC/t

0

R

C

E R

L

E

Growth 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 Circuit 21/

res )LC( w

• Generalized Resistances: Reactances, Impedance

]Ohms[ LC L )C/( R ww 1

R Z/

CL

2122

• New behavior: Damped Oscillation in LCR Circuit


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,…

2

2

12

2

1

2

12

2

12

LiUmvK C

QU kxU LblockCel

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

20 ww

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 w 00

Systems that oscillate obey equations like this... k/m x

dt

xdww

0202

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

2

0

OR


Electrical Oscillations in an LC circuit, zero resistance a

L C E

+ 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 1

2

2

An oscillator equation where

solution: ECQ )tcos(Q)t(Q 0 w 00

Qi )tsin(idt

)t(dQ)t(i 00 000 ww

0w1/LC

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

R/L ,RC

Lcwhere

Lc

w

120

)t(cosC

Q

C

)t(QUE w 0

220

2

22)t(sin

Li

)t(LiU

Bw

02

20

2

22

00BE U U

00EB

U C

Q

LC

1

LQ

QL

Li U

w

2222

20

20

20

20

20

Show: Total potential energy is constant

totU is constant

Peak Values Are Equal

)t(U (t)U U BEtot


Details: use energy conservation to deduce oscillations

• The total energy: C2

)t(Q)t(Li

2

1UUU

22

EB

dt

dQ

C

Q

dt

diLi

dt

dU 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 w 00

)tsin(Qdt

dQww 000

)tcos(Qdt

Qdww 0

2002

20

120002

2

ww

LC )tcos(Q

LC

Q

dt

Qd

LC

1

0 w

• To evaluate w0: plug the first and second derivatives of the solution into the

differential equation.

• The resonant oscillation frequency w0 is:

0dt

dQ• Either (no current ever flows) or:

LC

Q

dt

Qd 0

2

2

oscillator equation


13 – 1: What do you think will happen to the oscillations in a true

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

c) When does the first current maximum occur? When |sin(w0t)| = 1

Maxima 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:

ms. ms . T Tf 7196 4

1

Others at:

ements ms incr.43

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

mA18913 x 5 x6

10x4CQi

t)(sinQ - )tcos(dt

dQ

dt

dQ)t(i

000max

000000

ww

www

EECQ

)tcos(Q)t(Q

0

00

w

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 w

Hz145π2ωf 0


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

L = 30 mH, C = 100 mF

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

w

rad/s 4.577)F10)(H103(

1

LC

1420

w

C

volts )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 ½

period, given by:

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

dQi 3

000 ww

ms 44.52

T

0

w

p


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+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 w


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

pw

20

C

1C iser

1 CC ipara

I. II. III.


LCR circuits: Add series resistance Circuits still oscillate but oscillation is damped Charge capacitor fully to Q0=CE then switch to “b”

Stored energy decays with time due to resistance

22

22 )t(Li

C

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

a

L C E

+ b

R

t

t- ωe 0

Critically damped 22

02

)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 0

L/Rt Ew 20

Shifted resonant frequency w’

can be real or imaginary 2122

0 2/ L)(R / ω ω'

Resistance dissipates stored

energy. The power is: R)t(idt

dUtot 2

1/LC )t(Qdt

dQ

L

R

dt

)t(Qdww 0

202

2

0Oscillator equation results, but

with damping (decay) term


13 – 4: How does the resonant frequency w0 for an ideal LC circuit

(no resistance) compare with w’ 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 (w’ > w0).

B. The resonant frequency for the damped circuit is lower than for the ideal

one (w’ < w0).

C. The resistance in the circuit does not affect the resonant frequency—they are the same (w’ = w0).

D. The damped circuit has an imaginary value of w’.

Resonant frequency with damping

2/122

0 L) 2(R / ω ω'


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., the 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 (Faraday’s 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.

tsin)t(dmind

w

)tsin(ii d w 0

Represent outputs as sinusoidal functions:


External AC EMF driving a circuit

External, sinusoidal, instantaneous EMF applied to load:

t )(ωsin(t ) Dmax EE amplitudemax E

) Φ t ( ω sin II( t ) Dmax

Current in load flows with same frequency wD ...but may be retarded or

advanced (relative to E) by “phase angle” F (due to inertia L and stiffness 1/C)

Current has the same amplitude and phase everywhere in a branch

(t)Eload

I(t)

)t( ( t )load

load the across potential The V E

R

L

C E

Example: Series LCR Circuit

Everything oscillates at driving frequency wD

F is zero at resonance – circuit acts purely resistively.

At “resonance”: 1/LC ω ω 0D

Otherwise F is + or – (current leads or lags applied EMF)

ωD the driving frequency

ωD resonant frequency w0, in general


Instantaneous, peak, average, and RMS quantities for AC circuits

) Φ t ( ω sin ii( t ) Dmax

t )(ωsin(t ) Dmax EE

Instantaneous voltages and currents:

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

Peak voltage and current amplitudes are just the coefficients out front

i max maxESimple time averages of periodic quantities are zero (and useless).

• Example: Integrate over a whole number of periods – one is enough (wt=2p)

t )dt(ωsin(t )dt D maxav

00

011

EEE

Fw

00dt ) tsin(

)dtt(ωsin (t )dt D

maxav

F

00

011

iiiIntegrand is odd

“RMS” averages are used the way instantaneous quantities were in DC circuits • “RMS” means “root, mean, squared”.

(t )dt max//

avrms

2

121

0

221

2 EEEE

Fw

0

221 /dt ) t(sin

1

i (t )dti (t )i i max

//

avrms

2

121

0

221

2

Integrand is even


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

• The measured instantaneous values of i( t ) and E( t )

are projections of the phasors on the y-axis.

• The lengths of the vectors are the peak amplitudes.

“phase angle” measures when peaks pass Φ

and are independent Φ t Dw• Current is the same (phase included) everywhere

in a single branch of any circuit.

• EMF E(t) applied to the circuit can lead or lag the

current by a phase angle F in the range [-p/2, +p/2].

Series LCR circuit: Relate internal voltage drops 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.

maxI

maxEwDt- F

F

t )(ωsin(t ) Dmax EE

)Φ t ( ω sin iI( t ) Dmax

x

y


AC circuit, resistance only current and voltage in phase

VR ( t ) εi

t ωD

VR

IR

(t )VR

I (t )

Voltage drop across R:

t )(ωsin (t ) V DMR ε)t(ε

0(t )V -R

ε

)t - Φ(ωsin i(t ) i DmaxRR

Kirchoff loop rule:

Current:

R )t(i)t(V RR

Φ 0 RRim

E

Example s 240

1 t at )t i(current the Find Resistor. Ω 20 a to applied Hz 60V, f10

Mε

t

εM

ε

60

11

f τ

240

1t

A 2

1

2

π sin

20

10 ) i(t

1/240) . 120sin(R

t )(ωsin it) (im

DRmaxp

E

wD = 2pf = 120p = 1/60 s. t = /4

(definition of resistance)

Peak current and peak voltage are in phase in a purely

resistive part of a circuit, rotating at the driving frequency wD


Capacitance-only AC circuit: current leads voltage by p/2

0 )t( VC

)t(ε

εi

VC ( t )

C ;t )(ωsinV C(t )C VQ(t ) DmaxCC

)tsin(i t )(ωcosV C ωdt

dQ(t ) (t ) i DCmaxDmaxCDC Fw

)π

t(ωsin t ) ( ωcos DD2

so...phase angle F p/2 for capacitor

)2

πt (ωsin

V(t ) i D

C

maxCC

Current leads the voltage by p/2 in a pure

capacitive part of a circuit (F negative)

t ωD

maxCV

maxCi(t )iC

(t )VC

090

Phasor Picture Capacitive Reactance

limiting cases

• w 0: Infinite reactance.

DC blocked. C acts like

broken wire.

• w infinity: Reactance is

zero. Capacitor acts like a

simple wire

)R i V ( i V RR recallCCmaxCmax

Reactances are ratios of peak values

Kirchoff loop rule:

Charge:

Current:

mmaxCDmaxCc V ; t )(ωsin V(t ) V E

Note:

(Ohms )

D

Cω

1C

Definition: capacitive reactance


Inductance-only AC circuit: current lags voltage by p/2

)tsin(i t)(ωcosLω

Vt) dtωsin(

L

Vdi(t )i DLD

D

LD

maxLLL max

max Fw

L

)t(V

dt

i d

dt

i dL(t )V LLL

L

0(t )V)t(L

E

) 2

πt(ωsint ) (ωcos DD

VL L εi

so...phase angle F p/2 for inductor

Kirchoff loop rule:

maxLmDmL V ; t )(ωsin (t ) V EE

Faraday Law:

Current:

Note:

LmaxLmaxL i V

Reactances are ratios of peak values

)2

π t(ωsin max

V (t )i D

L

LmaxL

Current lags the voltage by p/2 in a pure

inductive part of a circuit (F positive)

Inductive Reactance

limiting cases

• w 0: Zero reactance.

Inductance acts like a wire.

• w infinity: Infinite reactance.

Inductance acts like a broken

wire.

maxLV

maxLi

090

ω t

)(t VL

(t )iL

Phasor Picture

)DL (OhmsL ω Definition: inductive reactance


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

• Apply sinusoidal voltage E (t) = Emsin(wDt)

• For pure R, L, or C loads, phase angles are 0, p/2, -p/2

• Reactance” means ratio of peak voltage to peak current (generalized resistances).

VR& iR in phase

Resistance

Ri/V Rmax C

1i/VD

CCmax w Li/V DLLmax w

VL leads iL by p/2

Inductive Reactance

VC lags iC by p/2

Capacitive Reactance

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


Series LCR circuit driven by an external AC voltage

im Em F

wDt wDt-F

R

L

C

E

VR

VC

VL

Apply EMF:

)tsin()t( Dmax w EE wD is the driving frequency

The current i(t) is the same everywhere in the circuit

• Same frequency dependance as E (t) but...

• Current leads or lags E (t) by a constant phase angle F

• Same phase for the current in E, R, L, & C

)tsin(i)t(i Dmax Fw

Phasors all rotate CCW at frequency wD

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

im

Em

F

wDt-F

VL

VC

VR

Plot voltages in components with phases

relative to current phasor im :

• VR has same phase as im

• VC lags im by p/2

• VL leads im by p/2

RiV mR

CmC XiV

LmL XiV

0 )t(V)t(V)t(V)t( CLR

E

0byLlagsCCLR m 180 V V )VV( V

E

along im perpendicular to im

Kirchoff Loop rule for potentials (measured along y)


Summary: Lecture 13/14 Chapter 31 - LCR & AC Circuits, Oscillations


Summary: RC and RL circuit results

electricity and magnetism lecture 13 - physics 121tyson/p122-ece_lecture12_extra_pt1.pdf · lecture...

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