Physics 218, Lecture V1 Physics 218 Lecture 5 Dr. David Toback.
Electricity and Magnetism Lecture 13 - Physics 121tyson/P122-ECE_Lecture12_Extra_Pt1.pdf · Lecture...
Transcript of 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
Copyright R. Janow – Fall 2013
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
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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
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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
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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
Copyright R. Janow – Fall 2013
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?
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Potential Energy alternates between all electrostatic and
all magnetic – two reversals per period
C is fully charged twice each cycle (opposite polarity)
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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
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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
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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
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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.
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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
Copyright R. Janow – Fall 2013
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'
Copyright R. Janow – Fall 2013
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:
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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
Copyright R. Janow – Fall 2013
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
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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
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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
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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
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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
Copyright R. Janow – Fall 2013
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
Copyright R. Janow – Fall 2013
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)
Copyright R. Janow – Fall 2013
Summary: Lecture 13/14 Chapter 31 - LCR & AC Circuits, Oscillations
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Summary: RC and RL circuit results