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
Copyright R. Janow – Spring 2015
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
Copyright R. Janow – Spring 2015
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
Copyright R. Janow – Spring 2015
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
Copyright R. Janow – Spring 2015
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
Copyright R. Janow – Spring 2015
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?
Copyright R. Janow – Spring 2015
Potential Energy alternates between all electrostatic and all magnetic – two reversals per period
C is fully charged twice each cycle (opposite
polarity)
Copyright R. Janow – Spring 2015
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
Copyright R. Janow – Spring 2015
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
Copyright R. Janow – Spring 2015
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
Copyright R. Janow – Spring 2015
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.
Copyright R. Janow – Spring 2015
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
Copyright R. Janow – Spring 2015
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'
Copyright R. Janow – Spring 2015
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
Copyright R. Janow – Spring 2015
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
Copyright R. Janow – Spring 2015
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
Copyright R. Janow – Spring 2015
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?
Copyright R. Janow – Spring 2015
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:
Copyright R. Janow – Spring 2015
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
Copyright R. Janow – Spring 2015
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:
)(ωcos Ii(t ) tDCurrent has time dependence: Applied EMF: )t (ωcos(t ) Dmax EE
Sine from derivative
Copyright R. Janow – Spring 2015
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:
)(ωcos Ii(t ) tDCurrent has time dependence: Applied EMF: )t (ωcos(t ) Dmax EE
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
Copyright R. Janow – Spring 2015
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
Copyright R. Janow – Spring 2015
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: