1. Summary: RC and LC circuits Mechanical Harmonic Oscillator – prototype
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 – prototype
LC circuits oscillate forever
LCR circuits oscillate with damping
AC circuits oscillate at forced frequency
AC circuits look simpler using “phasors”
Currents and voltages have fixed relative phasing that differs for simple resistive, capacitive, inductive components in circuits.

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

3. Model: Resonant mechanical oscillations (SHM)
Definition: oscillating system
Periodic behavior. T is the period. f = 1/T is the frequency.
Time to complete one complete cycle period T
State ( t ) = state( t + T ) = ……. = state( t + NT ).
State can mean: position and velocity, electric and magnetic fields,…
Mechanical Example: Simple harmonic oscillator (undamped)
no friction
Mechanical energy is conserved`
Systems that oscillate
obey equations like this...
With solutions like this...
Energy oscillates between 100% kinetic and 100% potential: Kmax = Umax
What oscillates? position & velocity, acceleration, spring force,

4. Electrical Oscillations in an LC circuit, zero resistance
Charge capacitor fully to Q0=CE then switch to “b” a L C E + b
Kirchhoff loop equation:
Substitute:
An oscillator
equation results
solution:
derivative:
Peaks of current and charge are out of phase by 900
What oscillates? Charge, current, B & E fields, UB, UE
Note: could have chosen sin rather than cos, with phase constant f adjusted

5. Electrical Oscillations in an LC circuit, zero resistance
Total electrical energy is constant:
The peak values UE0 and UB0 are equal:

6. 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:

7. 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.

8. 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

9. Use preceding solutions with f = 0
Example:
A 4 µF capacitor is charged to E = 5.0 V, and then connected to 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
Maxima of current at T/4, 3T/4, … (2n+1)T/4
First One at:
Others at:

10. 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 chose phase angle f = 0 to match initial condition.
Find the expression for the current in the circuit:
c) How long until the capacitor charge is exactly reversed? That occurs every ½ period, given by:

11. Which Current is Greatest?
13 – 2: The expressions below could represent the charge on a capacitor in an LC circuit as a function of time. 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)

12. 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.

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

14. 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 LCR 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’.

15. Alternating External Voltage (AC) Source
Apply external voltage
Real, instantaneous current flows
“Steady State” sinusoidal signals flow 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 (convention)
Commercial electric power (home or office) is AC, with frequency = 60 Hz in U.S, 50 Hz in most other countries.
High transmission voltage  Lower i2R loss than DC, and is reduced for distribution to customers by using transformers.
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 according to character of the load connected.

16. Apply Sinusoidal AC Source to circuit
load
the driving frequency
resonant frequency
STEADY STATE RESPONSE (after transients die away):
. Current in load is sinusoidal, has same frequency 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)

17. 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.
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.
Applied EMF Emax(t) leads or lags the current by a
phase angle F in the range [-p/2, +p/2]
Relative phases for individual series circuit elements (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

18. So how should “average” be defined?
The meaningful types of quantities in AC circuits can be instantaneous, peak, or 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 x-component of rotating “phasors” – see slides below
Peak voltage and current amplitudes are just the coefficients out front
represented by lengths of rotating phasors
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?

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

20. Current/Voltage Phases in Resistors, Capacitors, and Inductors
Sinusoidal current i (t) = Imcos(wDt). Peak is Im
vR(t) ~ i(t) vL(t) ~ derivative of i(t) Vc(t) ~ integral of i(t)
Peak voltages across R, L, or C lead/lag current by 0, p/2, -p/2 radians
Ratios of peak voltages to peak currents are called Reactances in C’s and L’s
Phases of voltages in a series branch are relative to the current phasor.
VR& Im in phase
Resistance
currrent
Same
Phase
VL leads Im by p/2
Inductive Reactance
VC lags Im by p/2
Capacitive Reactance

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

22. 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
From voltage drop across L (Faraday Law):
Note:
so...phase angle F = + p/2 for inductor
Voltage phasor across an inductor leads current by +p/2 (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.
Extra Discussion

23. 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
From voltage drop across C
(proportional to Q(t)):
Sine from integral
Note:
so...phase angle F = - p/2 for capacitor
Voltage phasor across capacitor lags the current by p/2 (F negative)
Extra Discussion
The RATIO of the AMPLITUDES VC to Im 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
Definition: capacitive reactance

24. Phasors applied to a Series LCR circuit
Applied EMF:
Current: R L C E vR vC vL
Refer voltage phasors to current phasor
Everything varies with driving frequency
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
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
Instantaneous voltages add normally (Loop Rule) Im Em F
wDt VL VC VR
along Im
perpendicular to Im
Phasor magnitudes add like vectors

25. Applies to a single series branch with L, C, R
Impedance Definition applied to the Series LCR Circuit
Voltage peak addition
rule in series circuit: Z Im Em F
wDt
VL-VC VR
XL-XC R
Reactances:
Same peak current flows in each component:
peak applied voltage
peak current
Define: Impedance is the ratio of peak EMF to peak current.
Divide each voltage in |Em| by (same) peak current:
Magnitude of Z:
Applies to a single series branch with L, C, R
Phase angle F:
See phasor diagram
R ~ 0  Im normal to Em  F ~ +/- p/2  tiny losses, no power absorbed
XL=XC  Im parallel to Em  F = 0  Z = R  maximum current (resonance)
F measures the power absorbed by the circuit:

26. Summary: AC Series LCR CircuitZ Im Em F
wDt
VL-VC VR
XL-XC R
sketch shows
XL > XC L C E vR vC vL
VL = ImXL
+90º (p/2)
Lags VL by 90º
XL=wdL L
Inductor
VC = ImXC
-90º (-p/2)
Leads VC by 90º
XC=1/wdC C
Capacitor
VR = ImR
0º (0 rad)
In phase with VR R
Resistor
Amplitude Relation
Phase Angle
Phase of Current
Resistance or Reactance
Symbol
Circuit Element