Electricity and Magnetism
Lecture 13 - Physics 121 (for ECE and non-ECE)
Electromagnetic Oscillations in LC & LCR Circuits,
Y&F Chapter 30, Sec. 5 - 6
Alternating Current Circuits,
Y&F Chapter 31, Sec. 1 - 2
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Summary: RC and LC circuits
Mechanical Harmonic Oscillator – Prototypical Oscillator
LC Circuit - Natural Oscillations
LCR Circuit - Damped Oscillations
AC Circuits, Phasors, Forced Oscillations
Phase Relations for Current and Voltage in Simple
Resistive, Capacitive, Inductive Circuits.
Copyright R. Janow – Fall 2015
LC and RC circuits with constant EMF - Time dependent effects
R
E
Vc ( t ) 
C
Q(t )
C
L  L/R
Voltage across C related
to integral of current

Q(t)  Q 0e  t / RC

Qinf  CE
Q0  CE
EL (t )   L
L
C  RC  capacitive time constant
Q(t)  Qinf 1  e t / RC
R
E
 inductive time constant
Voltage across L related
to derivative of current
Growth Phase
Decay Phase
di
dt

i(t )  iinf 1  e  t / L
i(t )  i0e  t / L
i0 

E
iinf 
E
R
R
Now LCR in same circuit, periodic EMF –> New effects
R
• Resonant oscillations in LC circuit – ignore damping
wres  (LC)1/ 2
L
C
R
L
C
• Damped oscillation in LCR circuit - not driven
• External AC drives circuit at frequency wD=2pf
which may or may not be at resonance
• Generalized resistances: reactances, impedance
R
C  1 /(w C)
L  wL
[ Ohms ]




1/ 2
| Z | R 2  L   C 2
Copyright R. Janow – Fall 2015
Recall: Resonant mechanical oscillations
Definition:
oscillating
system
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Periodic, repetitive behavior
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,…
Mechanical example: Spring oscillator (simple harmonic motion)
F  kx  ma OR
1
1
 Usp  mv 2  kx 2
2
2
Hooke' s Law restoring force :
Emech  constant  Kblock
no friction
dEmech
dt
Systems that oscillate
obey equations like this...
With solutions like this...
d2x dx
dx
0 m 2
 kx
dt
dt dt
v
dx
0
dt
d2x
2


w
x w0  k/m
0
2
dt
x(t )  x 0 cos(w0 t  )
What oscillates for a spring in SHM? position & velocity, spring force,
Energy oscillates between 100% kinetic and 100% potential: Kmax = Umax
Can convert mechanical to LC circuit equations by substituting:
x  Q v  i m  L k  1/C
w02  k / m  w02  1 / LC
2
1
1Q
1
1 2
2
Uel  2 kx  UC  2
Kblock  2 mv 2  UCopyright
L  2 Li R. Janow – Fall 2015
C
Electrical Oscillations in an LC circuit, zero resistance
Charge capacitor fully to Q0=CE then switch to “b”
Kirchoff loop equation: VC - EL  0
Q
di
d2Q
Substitute:
VC 
and EL  L
 L 2
C
dt
dt
d2Q(t)
a
+
b
E
L
1
An oscillator
1/LC  w0
Q
(
t
)
2
LC
equation
where
dt
solution: Q(t)  Q0 cos(w0t  )
Q0  CE
dQ(t)
 i0 sin(w0 t  )
i0  w0Q 0
derivative: i(t) 
dt
 
What oscillates? Charge, current, B & E fields, UB, UE
Peaks of current and charge are out of phase by 900
w02 
1
 c L
where  c  RC, L  L / R
Claim: Total potential energy is constant Utot  UE (t)  UB (t )
Q 2 (t )
UE 
2C

2
Li2 (t) Li0
UB 

sin2 (w0 t  )
2
2
Q 02
cos2 (w0 t  )
2C
Peak Values
Are Equal
UB
0
Li20
Lw20Q20
LQ20 1
Q20




 UE
0
2
2
2 LC
2C
UE  UB
0
0
Utot is constant
Copyright R. Janow – Fall 2015
C
Details: Oscillator Equation follows from Energy Conservation
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•
•
1
Q( t ) 2
2
The total energy: U  UB  UE  Li(t ) 
2
2C
dU
di
Q dQ
It is constant so:
 0  Li

dt
dt
C dt
dQ
di d2Q
i
The definition
and
imply that:

2
dt
dt dt
d2Q
dQ d2Q Q
(L 2  )  0
dt
C
dt
oscillator
equation
•
dQ
Either
 0 (no current ever flows) OR:
dt
•
Oscillator solution: Q  Q 0 cos(w0 t  )
•
To evaluate w0: plug the first and second derivatives of the solution into the
differential equation.
dQ
 Q 0w0 sin(w0 t  )
dt
d 2Q
 Q 0w02 cos(w0 t  )
2
dt
•
dt 2

Q
0
LC
d2Q
Q

 Q0 cos(w0 t  )
dt 2 LC
The resonant oscillation frequency w0 is:
w0 
1 

2

w

 0 LC   0
1
Copyright R. Janow – Fall 2015
LC
Oscillations Forever?
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.
Copyright R. Janow – Fall 2015
Potential Energy alternates between all electrostatic and
all magnetic – two reversals per period 
• C is fully charged twice
each cycle (opposite polarity)
• So is L
• Peaks are 90o out of phase
Copyright R. Janow – Fall 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
C
Use preceding solutions with  = 0
a) Find the oscillation period and frequency f = ω / 2π
ω0  1
LC
 1
f  ω0 2π  145 Hz
( 0.3 )( 4 x 10  6 )
T  Period  1f  2 π
w0  6.9 ms
 913 rad/s
b) Find the maximum (peak) current (differentiate Q(t) )
Q( t )  Q 0 cos( w0 t )
Q 0  CE
i( t ) 
dQ
d
 Q0
cos( w 0 t )  - Q 0 w 0 sin 0 (w 0 t)
dt
dt
imax  Q 0 w 0  CEw 0  4 x10
c) When does the first current maximum occur?
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
6
x 5 x 913  18 mA
When |sin(w0t)| = 1
First One at:
Others at:
T
4
 1.7 ms
3.4 ms incr ements
Copyright R. Janow – Fall 2015
L
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:
1
1
w0 

 577.4 rad/s
2
4
LC
(3  10 H)(10 F)
The voltage across the capacitor has the same
time dependence as the charge:
Q(t ) Q 0 cos(w0 t  )
VC (t ) 

C
C
At time t = 0, Q = Q0, so choose phase angle  = 0.
VC (t) 
10 3 C
10
4
cos(577 t)  10 cos(577 t) volts
F
b) What is the expression for the current in the circuit? The current is:
dQ
 Q 0 w0 sin(w0 t )  (10  3 C)(577 rad/s ) sin(577t )  0.577 sin(577t ) amps
dt
c) How long until the capacitor charge is exactly reversed? That occurs every
½ period, given by:
i
T
p

 5.44 ms
2 w0
Copyright R. Janow – Fall 2015
Which Current is Greatest?
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.
B.
C.
D.
E.
Q(t) = 2 sin(5t)
Q(t) = 2 cos(4t)
Q(t) = 2 cos(4t+p/2)
Q(t) = 2 sin(2t)
Q(t) = 4 cos(2t)
Q(t )  Q 0 cos(w0 t  )
i
dQ(t )
dt
Copyright R. Janow – Fall 2015
LC circuit oscillation frequencies
13 – 3: The three LC circuits below have identical inductors and
capacitors. Order the circuits according to their oscillation
frequency in ascending order.
A.
B.
C.
D.
E.
I, II, III.
II, I, III.
III, I, II.
III, II, I.
II, III, I.
I.
w0  1/LC 
II.
2p
T
III.
Cpara   Ci
1
Cser
  1C
i
Copyright R. Janow – Fall 2015
LCR Circuits with Series Resistance:
“Natural” Oscillations but with Damping
Charge capacitor fully to Q0=CE then switch to “b”
2
a
2
Q (t )
Li (t )
Utot  UE (t)  UB (t ) 

2C
2
Resistance dissipates stored dUtot
 i2 (t )R
energy. The power is:
dt
+
R
b
E
C
L
Modified oscillator equation has damping (decay) term:
Find transient solution:
oscillations with exponential
decay (under- damped case)
Shifted resonant frequency w’
can be real or imaginary
Underdamped:
ω 02  (
R 2
)
2L
d2Q(t )
dt

2
R dQ
 w02Q(t )  0
L dt
Q(t )  Q 0e Rt / 2L cos(w' t  )
ω' 
ω
2
0

 (R / 2L)2 1/2
ω 02  (
Q(t)
e
R 2
)
2L
Q0  CE
“damped radian
frequency”
Overdamped
Critically damped
ω'  0
'
w0  1/LC
ω'
is imaginary
ω 02  (
R 2
)
2L
complicate d
exp onential
- ω0 t
ω t
2π
t
t
Copyright R. Janow – Fall 2015
Resonant frequency with damping
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’.
ω' 
ω
2
0
 (R / 2 L)2

1/ 2
Copyright R. Janow – Fall 2015
Alternating Voltage (AC) Source
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Commercial electric power (home or office) is AC, not DC.
AC frequency = 60 Hz in U.S, 50 Hz in most other countries
AC transmission voltage can be lowered for distribution by using
transformers.
High transmission voltage  Lower i2R loss than DC
Generators convert mechanical energy to electrical
energy.
Sinusoidal EMF appears in rotating coils in a
magnetic field (Faraday’s Law)
Slip rings and brushes take EMF from the coil and
produce sinusoidal output voltage and current.
E(t )  Emax cos (ωDt  F )
i( t )  I max cos ( ωDt )
Real, instantaneous
“Steady State” outputs are sinusoids with same frequency wD:
 “Phase angle” F is between current and EMF in the driven circuit.
 You might (wrongly) expect F = 0
 F is positive when Emax leads Imax and negative otherwise
Copyright R. Janow – Fall 2015
Sinusoidal AC Source
driving a circuit
USE KIRCHHOFF LOOP & JUNCTION RULES, INSTANTANEOUS QUANTITIES
E(t )  Emax cos(ωDt  F)
i(t)
Emax  amplitude
The potential across the load 
E (t)
ωD  the driving frequency of EMF
ωD  resonant frequency w0, in general
E(t)
load
Consider STEADY STATE RESPONSE (after transients die away):
. Current in load is sinusoidal, has same frequency wD as source ... but
. Current may be retarded or advanced relative to E by “phase angle” F for the
whole circuit (due to inertia L and stiffness 1/C).
i( t )  I max cos ( ωDt )
KIRCHHOFF LAW APPLICATION
. In a series branch, current has the same amplitude and phase everywhere.
. Across parallel branches, voltage drops have the same amplitudes and phases
. At nodes (junctions), instantaneous currents in & out are conserved
In a Series LCR Circuit:
R
E
C
L
• Everything oscillates at driving frequency wD
• Current is the same everywhere (including phase)
• At “resonance”: ωD  ω0  1/LC
• F is zero at resonance  circuit acts purely resistively.
• Otherwise F is positive (current lags applied EMF)
or negative (current leads applied
EMF)
Copyright R. Janow – Fall 2015
Phasors: Represent currents and potentials as vectors
Imaginary Y rotating at frequency wD in complex plane
Emax
• The lengths of the vectors are the peak amplitudes.
• The measured instantaneous values of i( t ) and E( t )
are projections of the phasors onto the x-axis.
Imax
F
Φ  “phase angle”, the angle from current
wDt
E(t )  Emax cos(ωDt
Φ
Real X
phasor to EMF phasor in the driven circuit.
and wDt are independent
• Current is the same (phase included) everywhere
in a single essential (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 F in the range [-p/2, +p/2]
 F)
i( t )  I max cos ( ωD t )
In individual basic circuit elements, in series:
VR
VC
• 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.
VL
XL>XC
XC>XL
Copyright R. Janow – Fall 2015
The meaningful quantities in AC circuits are
instantaneous, peak, & average (RMS)
Instantaneous voltages and currents:
E(t )  Emax cos(ωDt  F )
i( t )  I max cos ( ωDt )
• oscillatory, depend on time through argument “wt”
• 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
Emax
I max
Simple time averages of periodic quantities are zero (and useless).
• Example: Integrate over a whole number of periods – one

1
 0

cos(wt  F) dt  0
Integrand is odd on a full cycle
Eav

is enough (w=2p)
1 
1 
E
(t
)dt

E
cos (ωD t  F )dt  0
max
 0
 0
1 
1 
iav   i(t )dt  Imax  cos (ωD t)dt  0
 0
 0

So how should “average” be defined?
Copyright R. Janow – Fall 2015
Averaging Definitions for AC circuits
So-called “rectified average values” are
useful for DC but seldom used in AC circuits:
•Integrate |cos(wt)| over one full cycle
or cos(wt) over a positive half cycle
1  / 4
 / 2  / 4
irav  Imax

| cos(wt)dt | 
2
imax
p
“RMS” averages are used the way instantaneous
quantities were in DC circuits:
• “RMS” means “root, mean, squared”.
1 
 0
 cos (wt
NOTE:
2
 F) dt  1/ 2
Integrand is positive on a whole cycle after squaring
Erms

 E (t) 
1/ 2
2
av

irms  i (t )

1/ 2
2
av
1 

   E 2(t )dt 
 0

1 

   i2(t )dt 
 0

1/ 2

1/ 2

Emax
2
Imax
2
Prescription: RMS Value = Peak value / Sqrt(2)
Copyright R. Janow – Fall 2015
Relative Current/Voltage Phases in pure R, C, and L elements
• Apply sinusoidal current i (t) = Imcos(wDt)
• For pure R, L, or C loads, phase angles for voltage drops are 0, p/2, -p/2
• “Reactance” means ratio of peak voltage to peak current (generalized resistances).
VR& Im in phase
Resistance
VR / Im  R
VC lags Im by p/2
Capacitive Reactance
VC / Im   C  1
wDC
VL leads Im by p/2
Inductive Reactance
VL / Im  L  wDL
Phases of voltages in series components are referenced to the current phasor
Same
Phase
currrent
Copyright R. Janow – Fall 2015
AC current i(t) and voltage vR(t) in a resistor are in phase
Current has time dependence: i(t )  I cos(ωDt)
Applied EMF: E(t )  Emax cos(ωDt  F)
Kirchoff loop rule:
ε
ε(t) - vR(t )  0
i(t)
vR( t )
vR(t )  ε(t)  vR(t )  VR cos(ωDt  F)
Voltage drop across R:
vR (t)  i(t)R  I R cos(ωDt)
The phases of vR(t) and i(t) coincide
The ratio of the AMPLITUDES
(peaks) VR to I is the resistance:
where
IR  VR
Φ  0
R
VR
I
Peak current and peak voltage are in
phase in a purely resistive part of a
circuit, rotating at the driving
frequency wD
Copyright R. Janow – Fall 2015
AC current i(t) lags the voltage vL(t) by 900 in an inductor
Current has time dependence: i(t )  I cos(ωDt)
Applied EMF:
Kirchoff loop
E(t )  Emax cos(ωDt  F)
rule: ε(t) - vL(t )  0
ε
i(t)
L
vL(t)
vL(t )  ε(t)  vL(t )  VL cos(ωDt  F)
Sine from
Voltage drop across L v (t )  L d i  LI d [cos(w t)]   Iw L sin(w t)
derivative
D
D
D
is due to Faraday Law: L
dt
dt
Note:  sin(ωDt)  cos(ωDt  p / 2 )  vL(t )  IwDL cos(wDt  p / 2) where peak VL  IwDL
so...phase angle F   p/2 for inductor
Voltage phasor leads current by +p/2 in inductive part of a circuit (F positive)
Definition:
inductive
reactance
L  ωDL
(Ohms )
V
XL  L
I
The RATIO of the peaks
(AMPLITUDES) VL to I is
the inductive reactance XL:
Inductive Reactance
limiting cases
• w  0: Zero reactance.
Inductor acts like a wire.
• w  infinity: Infinite reactance.
Inductor acts like a broken wire.
Copyright R. Janow – Fall 2015
AC current i(t) leads the voltage vC(t) by 900 in a capacitor
Current has time dependence: i(t )  I cos(ωDt)
Applied EMF:
Kirchoff loop
E(t )  Emax cos(ωDt  F)
rule: ε(t) - vC(t )  0
ε
vC(t )  ε(t)  vC(t )  VC cos(ωDt  F)
i
vC ( t )
C
Voltage drop across C v (t )  q(t)  1 i(t)dt  I cos(w t)dt  I sin(w t) Sine from
C
D
D
C
C
C
wDC
integral
is proportional to q(t):
Note:
sin(ωDt)  cos(ωDt  p / 2 )  vC(t )  I /(wDC) cos(wDt  p / 2) where I /(wDC)  VC
so...phase angle F   p/2 for capacitor
Voltage phasor lags the current by p/2 in a pure capacitive circuit branch (F negative)
Definition:
capacitive
reactance
C 
1
ωDC
(Ohms )
V
C  C
I
The RATIO of the
AMPLITUDES VC to I is the
capacitive reactance XC:
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
Copyright R. Janow – Fall 2015
Phasors applied to a Series LCR circuit
E
vR
Applied EMF: E (t)  Emax cos(wD t  F)
R
Current: i(t)  Imax cos(wDt)
L
Same everywhere in the single essential branch
vL
• Same frequency dependance as E (t)
C
• Same phase for the current in E, R, L, & C, but......
• Current leads or lags E (t) by a constant phase angle F
vC
Phasors all rotate CCW at frequency wD
• Lengths of phasors are the peak values (amplitudes)
F
Em
Im
wDt+F
• The “x” components are instantaneous values measured
wDt
VL
Em
Im
F
wDt
VR
VC
Kirchoff Loop rule for series LRC:
E(t)  vR (t)  vL (t)  vC (t)  0
Component voltage phasors all rotate at wD with phases
relative to the current phasor Im and magnitudes below :
VR  ImR
• VR has same phase as Im
• VC lags Im by p/2
VC  ImXC
• VL leads Im by p/2
VL  ImXL






Em  VR  (VL  VC )
VC lags VL by 1800
along Im
perpendicular
to I–mFall 2015
Copyright R. Janow
Voltage addition rule for series LRC circuit
2
Magnitude of Em Em
 VR2  (VL  VC )2
Em
in series circuit:
1


Reactances: L  wDL
C
VL-VC
wDC
IL L  VL
ICC  VC
IRR  VR
XL-XC
Same current amplitude in each component:
Im
F
Z
VR
R
Im  IR  IL  IC
Impedance magnitude
E
is the ratio of peak EMF | Z |  m
Im
to peak current:
peak applied voltage
peak current
wDt
[ Z ]  ohms
For series LRC circuit, divide each voltage in |Em| by (same) peak current
2
2 1/2
Magnitude of Z: | Z |  [ R  (L  C ) ]
Phase angle F:
tan(F ) 
VL  VC
  C
 L
VR
R
F measures the power absorbed by the circuit:
Applies to a single series
branch with L, C, R
See phasor diagram
 
P  Em  Im  EmIm cos(F)
• R ~ 0  tiny losses, no power absorbed  Im normal to Em  F ~ +/- p/2
Copyright R.
Janow – Fall 2015
• XL=XC  Im parallel to Em  F  0  Z=R  maximum current
(resonance)