EM Oscillations
Chapter 31
EM Oscillations
What are EM
Oscillations?
Passive Circuits
Alternating Current
(AC)
Power
“I’ve got an oscillating fan at my house. The fan goes back and forth. It looks like
the fan is saying ‘No.’ So I like to ask it questions that a fan would say ‘no’ to. Do
you keep my hair in place? Do you keep my documents in order? Do you have 3
settings? Liar! My fan... lied to me. Now I will pull the pin up. Now you ain’t
sayin’ [nothin’].”
-Mitch Hedberg
David J. Starling
Penn State Hazleton
PHYS 212
EM Oscillations
Chapter 31
EM Oscillations
What are EM
Oscillations?
Passive Circuits
Objectives
(a) Describe the behavior of LC and RLC circuits and
characterize them in terms of simple and damped
harmonic oscillators.
(b) Characterize AC circuits and determine the reactance
of resistors, capacitors, and inductors, and describe the
impedance of an AC circuit graphically as phasors or
numerically as complex impedances.
Alternating Current
(AC)
Power
What are EM Oscillations?
We have seen circuits that grow/decay exponentially:
I
RC circuit: q(t) = CV(1 − e−t/τC ) or q(t) = CVe−t/τC
I
RL circuit: i(t) = (V/R)(1 − e−t/τL ) or
i(t) = (V/R)e−t/τL
I
But what if we put a charged capacitor together with an
inductor?
I
Here, there is no energy dissipation!
Chapter 31
EM Oscillations
What are EM
Oscillations?
Passive Circuits
Alternating Current
(AC)
Power
Chapter 31
EM Oscillations
What are EM Oscillations?
What are EM
Oscillations?
Capacitors and inductors store energy:
I
UE =
q2
2C
I
UB =
Li2
2
Passive Circuits
Z
P = iV = i(Ldi/dt) → UB =
I
Pdt = Li2 /2
Initially, all the energy is stored in the capacitor and no
current flows
I
Eventually, the capacitor discharges and current flows
I
The energy is transfered from the capacitor to the
inductor... and then back!
Alternating Current
(AC)
Power
What are EM Oscillations?
Chapter 31
EM Oscillations
What are EM
Oscillations?
Passive Circuits
Alternating Current
(AC)
Power
What are EM Oscillations?
We can measure
Chapter 31
EM Oscillations
What are EM
Oscillations?
Passive Circuits
I
the voltage across the capacitor V = q/C
I
the current in the circuit i
I
We find:
I
The voltage lags by 90◦
Alternating Current
(AC)
Power
What are EM Oscillations?
An analogy: block on a spring
I
Potential energy is like the energy in a capacitor
U → UE
I
Kinetic energy is like the energy in the inductor
K → UB
I
ω=
p
p
k/m → 1/LC
Chapter 31
EM Oscillations
What are EM
Oscillations?
Passive Circuits
Alternating Current
(AC)
Power
Chapter 31
EM Oscillations
Passive Circuits
Let’s look more closely at this LC circuit:
I
The total energy: U =
I
Therefore,
Li2
2
=
=
=
=
d2 q
dt2
I
2C
= const
What are EM
Oscillations?
Passive Circuits
Alternating Current
(AC)
dU
dt
I
+
q2
+
q
LC
Li2
q2
+
2
2C
di
q dq
Li +
dt C dt
dq d2 q
q dq
L
+
2
dt dt
C dt
0
d
dt
= 0
Solution: q(t) = Q cos(ωt + φ), i(t) = −I sin(ωt + φ)
p
with I = ωQ and ω = 1/LC
Power
Chapter 31
EM Oscillations
Passive Circuits
What are EM
Oscillations?
For an LC circuit:
Passive Circuits
q(t) = Q cos(ωt + φ)
(1)
i(t) = −I sin(ωt + φ)
(2)
Therefore, the energy stored is:
I
UE =
q2
2C
=
I
UB =
Li2
2
= 12 LI 2 sin2 (ωt + φ) =
Q2
2C
cos2 (ωt + φ)
[note: I 2 =
I
Q2
2
2C sin (ωt
(ωQ)2 = Q2 /LC]
+ φ)
Adding them up, we get
U = UE + UB =
Q2
2
2
2C [sin (ωt + φ) + cos (ωt + φ)]
=
Q2
2C
Alternating Current
(AC)
Power
Passive Circuits
For an LC circuit:
q(t) = Q cos(ωt + φ)
i(t) = −I sin(ωt + φ)
Q2
UE =
cos2 (ωt + φ)
2C
Q2
UB =
sin2 (ωt + φ)
2C
Q2
Utotal =
2C
Chapter 31
EM Oscillations
What are EM
Oscillations?
Passive Circuits
Alternating Current
(AC)
Power
Passive Circuits
Chapter 31
EM Oscillations
What are EM
Oscillations?
Question 1
Passive Circuits
Alternating Current
(AC)
The current in an oscillating LC circuit is zero. Which one
of the following statements is true?
(a) The charge on the capacitor is equal to zero coulombs.
(b) Charge is moving through the inductor.
(c) The energy is equally shared between the electric and
magnetic fields.
(d) The energy in the electric field is maximized.
(e) The energy in the magnetic field is maximized.
Power
Passive Circuits
Chapter 31
EM Oscillations
What are EM
Oscillations?
Example 1
Passive Circuits
Alternating Current
(AC)
Power
A 1.5 µF capacitor is charged to 57 V by a battery, which is
then removed. At time t = 0 s, a 12 mH coil is connected in
series with the capacitor.
(a) What is the potential difference across the inductor as a
function of time?
(b) What is the maximum rate at which the current
changes?
Chapter 31
EM Oscillations
Passive Circuits
What if we throw a resistor in the mix?
What are EM
Oscillations?
Passive Circuits
Alternating Current
(AC)
Power
How does the previous analysis change?
I
Energy is no longer constant, but decreases over time
I
The power dissipated by a resistor is just P = iV = i2 R
I
Therefore,
dU
di
q dq
= Li +
= −i2 R
dt
dt C dt
(3)
Passive Circuits
The equation for an RLC circuit:
di
q dq
Li +
= −i2 R
dt C dt
q
di
Li + i = −i2 R
dt C
d2 q
q
L 2 + + iR = 0
dt
C
dq
q
d2 q
L 2 +R + =0
dt
dt
C
The solution:
q(t) = Qe−Rt/2L cos(ω 0 t + φ)
Frequency:
p
I ω0 =
ω 2 − (R/2L)2
p
I ω =
1/LC
Chapter 31
EM Oscillations
What are EM
Oscillations?
Passive Circuits
Alternating Current
(AC)
Power
Chapter 31
EM Oscillations
Passive Circuits
For an RLC circuit:
−Rt/2L
0
q(t) = Qe
cos(ω t + φ)
2
2
q
Q −Rt/L
UE (t) =
=
e
cos2 (ω 0 t + φ)
2C
2C
The energy in the circuit dissipates over a time with a time
constant τL = L/R
What are EM
Oscillations?
Passive Circuits
Alternating Current
(AC)
Power
Passive Circuits
Chapter 31
EM Oscillations
What are EM
Oscillations?
Passive Circuits
Example 2
Alternating Current
(AC)
Power
An RLC circuit has inductance L = 12 mH, capacitance
C = 1.6 µF, resistance R = 1.5 Ω and begins to oscillate at
a time t = 0 s.
(a) How long will it take for the charge oscillations to
reach 50% of their initial value?
(b) How many oscillations are completed within this time?
Alternating Current (AC)
Generators create an oscillating emf:
Chapter 31
EM Oscillations
What are EM
Oscillations?
Passive Circuits
Alternating Current
(AC)
Power
I
As the flux varies, so does the enduced emf E
I
E(t) = E sin(ωd t)
I
Therefore, i(t) = I sin(ωd t − φ)
Alternating Current (AC)
A general RLC circuit with an oscillating emf:
Chapter 31
EM Oscillations
What are EM
Oscillations?
Passive Circuits
Alternating Current
(AC)
Power
I
The oscillations in the circuit are at frequency ωd
I
This is true even if ωd 6= ω 0
I
The current moves back and forth through each
element
Alternating Current (AC)
Resistive load:
Chapter 31
EM Oscillations
What are EM
Oscillations?
Passive Circuits
Alternating Current
(AC)
Power
For this case,
I
E − iR = 0
I
i = E/R = Em sin(ωd t)/R
I
In general, we expect i(t) = I sin(ωd t − φ)
I
For a resistive load, there is no phase delay (φ = 0◦ )
Alternating Current (AC)
We can graph the applied voltage and the resulting current:
Chapter 31
EM Oscillations
What are EM
Oscillations?
Passive Circuits
Alternating Current
(AC)
Power
On the right, we have a phasor diagram
I
The angle of the phasor gives the argument of the sine
I
The length of the phasor is the max value of v or i
I
The projection onto the y-axis gives the instantaneous v
or i
Alternating Current (AC)
Chapter 31
EM Oscillations
What are EM
Oscillations?
Example 3
Passive Circuits
Alternating Current
(AC)
Power
For a resistive load, if R = 200 Ω and Em = 36.0 V with
fd = 60 Hz,
(a) what is the potential difference across the resistor
vR (t)?
(b) what is the current in the resistor iR (t)?
Alternating Current (AC)
Capacitive load:
Chapter 31
EM Oscillations
What are EM
Oscillations?
Passive Circuits
Alternating Current
(AC)
Power
For this case,
I
q = Cv = CV sin(ωd t)
I
i = dq/dt = ωd CV cos(ωd t) = (V/XC ) cos(ωd t)
I
The Capacitive Reactance: XC = 1/ωd C
I
XC has units of ohms and V = IXC .
Alternating Current (AC)
We can graph the applied voltage and the resulting current:
Chapter 31
EM Oscillations
What are EM
Oscillations?
Passive Circuits
Alternating Current
(AC)
Power
On the right, we have a phasor diagram
I
Notice how the current leads the voltage by 90◦
I
q = CV sin(ωd t)
I
i = (V/XC ) sin(ωd t + 90◦ )
Alternating Current (AC)
Chapter 31
EM Oscillations
What are EM
Oscillations?
Example 4
Passive Circuits
Alternating Current
(AC)
Power
For a capacitive load, if C = 15.0 µF and Em = 36.0 V with
fd = 60 Hz,
(a) what is the potential difference across the capacitor
vC (t)?
(b) what is the current in the circuit iC (t)?
Alternating Current (AC)
Chapter 31
EM Oscillations
What are EM
Oscillations?
Inductive load:
Passive Circuits
Alternating Current
(AC)
Power
For this case,
I
v = Ldi/dt = V sin(ωd t)
I di
dt
= (V/L) sin(ωd t)
Alternating Current (AC)
Chapter 31
EM Oscillations
What are EM
Oscillations?
I
Solving for i(t),
Passive Circuits
Alternating Current
(AC)
di = (V/L) sin(ωd t)dt
Z
V
sin(ωd t)dt
i(t) =
L
V
= −
cos(ωd t)
ωd L
V
=
sin(ωd t − 90◦ )
XL
I
The Inductive Reactance: XL = ωd L
I
XL has units of ohms and V = IXL .
Power
Alternating Current (AC)
We can graph the applied voltage and the resulting current:
Chapter 31
EM Oscillations
What are EM
Oscillations?
Passive Circuits
Alternating Current
(AC)
Power
On the right, we have a phasor diagram
I
Notice how the current lags the voltage by 90◦
I
v = V sin(ωd t)
I
i = (V/XL ) sin(ωd t − 90◦ )
Alternating Current (AC)
Question 2
Chapter 31
EM Oscillations
What are EM
Oscillations?
Passive Circuits
An inductor circuit operates at a frequency f = 120 Hz. The
Alternating Current
(AC)
Power
peak voltage is 120 V and the peak current through the
inductor is 2.0 A. What is the inductance of the inductor in
the circuit?
(a) 0.040 H
(b) 0.080 H
(c) 0.160 H
(d) 0.320 H
(e) 0.640 H
Alternating Current (AC)
Chapter 31
EM Oscillations
What are EM
Oscillations?
Example 5
Passive Circuits
Alternating Current
(AC)
Power
For a capacitive load, if L = 230 mH and Em = 36.0 V with
fd = 60 Hz,
(a) what is the potential difference across the inductor
vL (t)?
(b) what is the current in the circuit iL (t)?
Alternating Current (AC)
Chapter 31
EM Oscillations
What are EM
Oscillations?
Summary of Simple Circuits
Passive Circuits
Alternating Current
(AC)
Power
Alternating Current (AC)
Putting them together:
Chapter 31
EM Oscillations
What are EM
Oscillations?
Passive Circuits
Alternating Current
(AC)
Power
Each component operates as before:
I
resistor: i(t) and vR (t) are in phase
I
capacitor: i(t) leads vC (t) by 90◦
I
inductor: i(t) lags vL (t) by 90◦
Alternating Current (AC)
The result:
Chapter 31
EM Oscillations
What are EM
Oscillations?
Passive Circuits
Alternating Current
(AC)
Power
I
E is equal to the vector sum of the three potential
differences
Alternating Current (AC)
The vectors:
Chapter 31
EM Oscillations
What are EM
Oscillations?
Passive Circuits
Alternating Current
(AC)
Power
Em2 = VR2 + (VL − VC )2
= I 2 [R2 + (XL − XC )2
Em
I = p
2
R + (XL − XC )2
I = Em /Z
p
Impedence: Z = R2 + (XL − XC )2
Alternating Current (AC)
We can now find the max current. But what is the phase
between the applied voltage and the current?
Chapter 31
EM Oscillations
What are EM
Oscillations?
Passive Circuits
Alternating Current
(AC)
Power
tan(φ) =
=
=
VL − VC
VR
IXL − IXC
IR
XL − XC
R
Alternating Current (AC)
tan(φ) =
XL −XC
R
Chapter 31
EM Oscillations
What are EM
Oscillations?
Passive Circuits
Alternating Current
(AC)
Power
Alternating Current (AC)
Notice that the max current depends on the impedence:
Em
p
2
R + (XL − XC )2
= Em /Z
I =
For a fixed R, I is max if XL = XC , so
ωd L = 1/ωd C
√
ωd = 1/ LC
But if XL = XC , then φ = 0.
This is called resonance!
Chapter 31
EM Oscillations
What are EM
Oscillations?
Passive Circuits
Alternating Current
(AC)
Power
Alternating Current (AC)
Chapter 31
EM Oscillations
What are EM
Oscillations?
Passive Circuits
Alternating Current
(AC)
Power
Chapter 31
EM Oscillations
Power
The power dissipated in the RLC circuit leaves through the
resistor only:
What are EM
Oscillations?
Passive Circuits
P = i2 R
2
= [I sin(ωd t − φ)] R
= I 2 R sin2 (ωd t − φ)
I
The average power is just Pavg = I 2 R/2
√
Define: Irms = I/ 2 (root-mean-squared)
I
2 R
This gives Pavg = Irms
I
Alternating Current
(AC)
Power
Power
Example 6
Chapter 31
EM Oscillations
What are EM
Oscillations?
Passive Circuits
Alternating Current
(AC)
Power
An RLC circuit is driven by Erms = 120 V at fd = 60 Hz.
The circuit has R = 200 Ω, XL = 80.0 Ω and XC = 150 Ω.
(a) What is the phase constant of the circuit?
(b) What is the average rate Pavg at which energy is
dissipated?
(c) What new capacitance is needed to maximize Pavg if
the other parameters are held constant?
Power
A transformer changes one AC voltage to another AC
voltage:
Chapter 31
EM Oscillations
What are EM
Oscillations?
Passive Circuits
Alternating Current
(AC)
Power
I
A primary coil is wrapped around an iron core
I
An emf is induced in a secondary coil also on the core
I
The relationship between the two voltages is
Ns
Vs = Vp
Np
Chapter 31
EM Oscillations
Power
Ns
Vs = Vp
Np
What are EM
Oscillations?
Passive Circuits
Alternating Current
(AC)
Power
I
Step-up transformer: Ns > Np
I
Step-down transformer: Np > Ns
I
Energy is conserved: Ip Vp = Is Vs → Is = Ip Np /Ns
Power
Example 7
Chapter 31
EM Oscillations
What are EM
Oscillations?
Passive Circuits
Alternating Current
(AC)
A utility pole transformer operates at Vp = 8.5 kV on the
primary side and supplies electricity to houses at Vs = 120
V.
(a) What is the number of turns ratio Np /Ns ?
(b) If the houses use a total of 78 kW of power, what are
the rms currents in the primary and secondary of the
transformer?
(c) What is the resistive load in the primary and secondary
circuit?
Power