1. Electricity and Magnetism Lecture 13 - Physics 121 (for ECE and non-ECE) Electromagnetic Oscillations in LC & LCR Circuits, Y&F Chapter 30, Sec Alternating Current Circuits, Y&F Chapter 31, Sec
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.

2. Time dependent effects - LC and RC circuits with constant EMF
Voltage across C related to Q, integral of current
Voltage across L related to derivative of current
Growth Phase
Decay Phase
Now LCR in same circuit + periodic EMF –> New effects C R L
Circuit driven by external AC frequency wD=2pf
which may or may not be at resonance
Resonant oscillations in LC circuit – ignore damping
Generalized resistances: reactances, impedance
Damped oscillation in LCR circuit - not driven

3. Model: Resonant mechanical oscillations (SHM)
Definition: oscillating system
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, undamped)
no friction OR
Energy oscillates between 100% kinetic and 100% potential: Kmax = Umax
With solutions like this...
Systems that oscillate
obey equations like this...
What oscillates for a spring in SHM? position & velocity, spring force,
Analogous roles:
Mechanical
Electrical

4. Electrical Oscillations in an LC circuit, zero resistance+ b
Charge capacitor fully to Q0=CE then switch to “b”
Kirchoff loop equation:
Substitute:
An oscillator
equation results
solution:
derivative:
What oscillates? Charge, current, B & E fields, UB, UE
Peaks of current and charge are out of phase by 900
Claim: Total energy is constant:
is constant
Peak Values
Are Equal

5. Details: Derive Oscillator Equation from Energy Conservation
The total energy:
It is constant so:
The definition and imply that:
Either (no current ever flows) OR:
oscillator
equation
Oscillator solution:
To evaluate w0: plug the first and second derivatives of the solution into the differential equation.
The resonant oscillation frequency w0 is:

6. 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?
They will stop after one complete cycle.
They will continue forever.
They will continue for awhile, and then suddenly stop.
They will continue for awhile, but eventually die away.
There is not enough information to tell what will happen.

7. Potential Energy alternates between all electrostatic and all magnetic – two reversals per period t
C is fully charged twice
each cycle (opposite polarity)
So is L
Peaks are 90o out of phase

8. Use preceding solutions with f = 0
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 L
Use preceding solutions with f = 0
a) Find the oscillation period and frequency f = ω / 2π
b) Find the maximum (peak) current (differentiate Q(t) )
c) When does current i(t) have its first maximum? 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 at:
Others at:

9. Example
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 = 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 f = 0 to match initial condition.
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:

10. 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 peak current magnitude?
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)

11. 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.
I, II, III.
II, I, III.
III, I, II.
III, II, I.
II, III, I.
I II III.

12. “damped radian frequency”
LCR Circuits with Series Resistance: “Natural” Oscillations but with Damping
Charge capacitor fully to Q0=CE then switch to “b” a L C E + b R
Modified oscillator equation has damping (decay) term:
The resistor dissipates stored energy. The power is:
Find transient solution: oscillations with exponential decay (under- damped case)
Shifted resonant frequency w’ can be real or imaginary
“damped radian frequency”
Critically damped
Underdamped:
Q(t)
Overdamped

13. 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 is not negligible?
The resonant frequency for the non-ideal, damped circuit is higher than for the ideal one (w’ > w0).
The resonant frequency for the damped circuit is lower than for the ideal one (w’ < w0).
The resistance in the circuit does not affect the resonant frequency—they are the same (w’ = w0).
The damped circuit has an imaginary value of w’.

14. Alternating Voltage (AC) Source
Commercial electric power (home or office) is AC, not DC.
AC frequency = 60 Hz in U.S, 50 Hz in most other countries
High transmission voltage  Lower i2R loss than DC
AC transmission voltage is reduced for distribution to customers by using transformers.
Generators convert mechanical 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 AC voltage and current.
External circuits are driven by sinusoidal EMF
Current in external circuit may lead or lag EMF due
to character of the load connected.
“Steady State” outputs - sinusoids with 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
Real, instantaneous

15. Sinusoidal AC Source driving a circuit
USE KIRCHHOFF LOOP & JUNCTION RULES, INSTANTANEOUS QUANTITIES
load
the driving frequency of EMF
resonant frequency w0, in general
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).
Application of KIRCHHOFF RULES:
. 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 net currents in & out are conserved R L C
In a Series LCR Circuit:
Everything oscillates at driving frequency wD
F is zero at resonance  circuit acts purely resistively.
At “resonance”:
Otherwise F can be positive (current lags applied EMF)
or negative (current leads applied EMF)
Current is the same everywhere (including phase)

16. represent currents and potentials as vectors rotating at frequency wD in complex plane
Phasors
Imaginary Y
wDt F
Real X
Peak amplitudes  lengths of the phasors.
M measurable, real, instantaneous values of i( t ) and
E( t ) are projections of the phasors onto the x-axis.
“phase angle”, the angle from current
phasor to EMF phasor in the driven circuit.
and are independent
Current is the same (phase included) everywhere
in a single essential branch of any circuit
Current vector is reference for series LCR circuit.
EMF Emax(t) applied leads or lags the current by a
phase angle F in the range [-p/2, +p/2]
Phases for individual basic circuit elements in series (to be shown): 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>XC
XC>XL

17. So how should “average” be defined?
The meaningful types of quantities in AC circuits are instantaneous, peak, & average (RMS)
Instantaneous voltages and currents refer to a specific time:
oscillatory, depend on time through argument “wt”
possibly advanced or retarded relative to each other by phase angle
represented by rotating “phasors” – see slides below
Peak voltage and current amplitudes are just the coefficients out front
Simplistic time averaging of periodic quantities gives zero (and useless) results).
Example: Integrate over a whole number of periods – one t is enough (wt=2p)
Integrand is odd over a full cycle
So how should “average” be defined?

18. Definitions of Average AC Quantities
“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
“RMS averages” are used the way instantaneous quantities were in DC circuits:
“RMS” means “root, mean, squared”.
Integrand is positive on a whole cycle after squaring
Prescription: RMS Value = Peak value / Sqrt(2)
NOTE:

19. Current/Voltage Phases in pure R, C, and L circuit elements
Sinusoidal current i (t) = Imcos(wDt). Peak is Im
Peak voltage drops across R, L, or C lead/lag current by 0, p/2, -p/2 radians
Reactances (generalized resistances) are ratios of peak voltages to peak currents
Phases of voltages in a series branch are referenced to the current phasor
VR& Im in phase
Resistance
currrent
Same
Phase
VC lags Im by p/2
Capacitive Reactance
VL leads Im by p/2
Inductive Reactance

20. AC current i(t) and voltage vR(t) in a resistor are in phase
From Kirchhoff
loop rule:
Time dependent current:
Applied EMF:
vR( t )
i(t)
Voltage drop across R:
The phases of vR(t) and i(t) coincide
Peak current and peak voltage are in phase in a purely resistive part of a circuit, rotating at the driving frequency wD
The ratio of the AMPLITUDES (peaks) VR to Im is the resistance:

21. AC current i(t) lags the voltage vL(t) by 900 in an inductor
From Kirchhoff
loop rule:
Time dependent current:
Applied EMF:
vL(t) L
i(t)
Sine from derivative
Voltage drop across L
is due to Faraday Law:
Note:
so...phase angle F = + p/2 for inductor
Voltage phasor leads current by +p/2 in inductive part of a circuit (F positive)
The RATIO of the peaks (AMPLITUDES) VL to Im is the inductive reactance XL:
Definition: inductive reactance
Inductive Reactance
limiting cases
w  0: Zero reactance.
Inductor acts like a wire.
w  infinity: Infinite reactance.
Inductor acts like a broken wire.

22. AC current i(t) leads the voltage vC(t) by 900 in a capacitor
From Kirchhoff
loop rule:
Time dependent current:
Applied EMF: i
vC ( t ) C
Voltage drop across C
is proportional to Q(t):
Sine from integral
Note:
so...phase angle F = - p/2 for capacitor
Voltage phasor lags the current by p/2 in a pure capacitive circuit branch (F negative)
The RATIO of the AMPLITUDES VC to Im is the capacitive reactance XC:
Definition: capacitive reactance
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

23. Phasors applied to a Series LCR circuit
Applied EMF: R L C E vR vC vL
Current:
Recall:
Phasors rotate CCW at frequency wD
Lengths of phasors are the peak values (amplitudes)
Instantaneous values are the “x” components
Reference voltages to current phasor
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 F
Same current everywhere in the single branch Im Em F
wDt+F
wDt
Apply Loop Rule to instantaneous voltages: Im Em F
wDt VL VC VR
Voltage phasors for VR, VL, & VC all rotate at wD :
VC lags Im by p/2
VR has same phase as Im
VL leads Im by p/2
along Im
perpendicular to Im
Voltage phasors add like vectors