This document covers several topics related to AC circuits: 1) It discusses AC circuits with capacitors and inductors, including how they act as filters that pass either high or low frequencies. It also introduces the concept of resonance in RLC circuits. 2) It examines transformers and how they can be used to change voltages by exploiting Faraday's law of induction and the ratio of coils between the primary and secondary windings. 3) It covers concepts like impedance, phase, power factor, and the quality factor Q as they relate to AC circuits, particularly resonant circuits near the natural frequency.

1. Lecture 30
AC power. Resonance.
Transformers.

2. ACT: εC circuit
A capacitor is connected to an AC generator as
shown. If the frequency of the generator is
C
doubled, the amplitude of the current in the
circuit will
ε
A. increase by a factor of 2
B. not change
C. decrease by a factor of 2
I =
E
= EC ω
Z
Z =
(X
L
− XC
)
2
+ R 2 = XC =
1
Cω

3. Filters
Vout
Vout depends on frequency:
High ω ⇒ smaller reactance ⇒ VC = Vout → 0
C
Low ω ⇒ larger reactance ⇒ no current flows
through R ⇒ smaller VR ⇒ VC = Vout → ε
VL
ε
This is a circuit that only passes
low frequencies: low-pass filter
 Bass knob on radio
ω0 = 1
RC
R
ε
ω
If instead we look at the voltage through the resistor: high-pass filter
 Treble control

4. More filters
L
High ω ⇒ large XL ⇒ VL ~ ε ⇒
R
High-pass filter is Vout = VL
Low-pass filter is Vout = VR
ε
L
R
C
ε
VR ~ 0 and I ~ 0
VL
ε
Low ω → I ~ 0 due to capacitor
High ω → I ~ 0 due to inductor
Band pass filter (back to this
later: resonance)
ω0 = 1
RC
Vout
ε
ω0
ω

5. ACT: εRL circuit
An RL circuit is driven by an AC generator
as shown in the figure. The current
through the resistor and the generator
voltage are:
A. always out of phase
B. always in phase
C. sometimes in phase and sometimes out
of phase
And this is the current through all
elements, by the way…
E
VL
I
VR

6. ACT: εRC circuit
A series RC circuit is driven by emf ε =E sinωt. Which of the
following could be an appropriate phasor diagram?
VL
VC
E
E
VC
VR
VR
VR
A
B
~
VC
C
For this circuit, which of the following is true?
(a) The drive voltage is in phase with the current.
(b) The drive voltage lags the current.
(c) The drive voltage leads the current.
E

7. ACT: Bring in phase
The current and driving voltage in an RLC circuit are shown in the graph.
How should the frequency of the power source be changed to bring these
two quantities in phase?
A. Increase ω
B. Decrease ω
C. Current and driving voltage
cannot be in phase.
0
i
ε
t
From the figure, current leads driving voltage ⇒ ϕ < 0
⇒ XC > XL ⇒ to make them equal, frequency needs to
increase.
IXL
IR
IXC
ε

8. ACT: Bring in phase
The current and driving voltage in an RLC circuit are shown in the graph.
Which of the following phasor diagrams represents the current at t = 0?
A.
B.
I
I
t
i (t) is the horizontal projection of the phasor.
I
C.
0
i
ε
From the figure:
At t = 0, i ~ 2/3I (>0)
And it should be increasing.

9. Resonance
E
Current amplitude in a series RLC circuit driven I =
Z
by a source of amplitude E :
Maximum current when impendance Z =
i.e., when X L = XC
1
Lω =
Cω
ω0 =
E
R
I =
I
0
0
1
LC
(X
L
− XC
)
2
+ R 2 is minimum
Resonance:
Driving frequency =
natural frequency
E
E
= cos ϕ
Z
R
Maximum current ⇔ maximum cosϕ ⇔
cosϕ ~ 1 ⇔ ϕ ~ 0 (circuit in phase)
ω
2ωo
It’s the bandpass filter!

10. ACT: Resonance
This circuit is being driven __________
its resonance frequency.
A. above
B. below
C. exactly at
To achieve resonance, we need to decrease XL and to
increase XC ⇒ to decrease frequency ω

11. Power in AC circuits
Instantaneous power supplied to the circuit:
(
P (t ) ≡ ε (t ) i (t ) = ( E cos ω ) I cos ( ω − ϕ )
t
t
Often more useful: Average power
P (t ) = EI cos ω cos ( ω −ϕ)
t
t
1
P = EI cos ϕ
2
cos ω cos ( ω −ϕ) = cos ω ( cos ω cos ϕ − sin ω sin ϕ)
t
t
t
t
t
1 2π
1
2
∫0 cos xdx = 2
2π
1 2π
cos x sin x =
∫0 cos x sin xdx = 0
2π
cos2 x
Define:
E rms =
E
2
=
I rms =
I
2
P = E rms Irms cos ϕ
)

12. Power factor
P = E rms Irms cos ϕ
Power factor (PF)
Maximum power ⇔ ϕ = 0 ⇔
I =
Resonant circuit
E
E
=
cos ϕ
Z
R
2
P = I rms R
All energy dissipation happens
at the resistor(s).

13. The Q factor
How “sharp” is the resonance? (ie, the resonance peak)
Umax
Q ≡ 2π
∆U
• For RLC circuit, Umax =
Umax is max energy stored in the system
ΔU is the energy dissipated in one cycle
1
LI 2
2
period
1 2
1 2  2π
• Losses only come from R : ∆ U = I RT = I R 
ω
2
2
 0
ωL
0
Q =
ω0 =
1
LC
Q =
R
1
X L ,0
=
R
ωCR
0
=
XC ,0
R

÷
÷

X L ,0
XC ,0
Q =
=
R
R

14. X L ,0
XC ,0
Q =
=
R
R
Large Q ⇒ sharp peak ⇒ better “quality”
L and C control how much energy is stored.
R controls how much energy is lost.
Small resistance ⇒ Large Q

15. Transformers
Application of Faraday’s Law
• Changing EMF in primary coil creates changing flux
• Changing flux creates EMF in secondary coil.
V =N
Magnetic flux remains
mostly in the core.
Core “directs” B lines
d ΦB
for both coils
dt
d ΦB V1
V
=
= 2
dt
N1 N2
V1 V2
=
N1 N2
V1
Efficient method to change
voltage for AC.
N1
N2
d ΦB
dt
If no energy is lost in the coils, power on both sides must be the same
V1I1 = V2I2
V2

16. In-class example: Jacob’s ladder
A transformer outputs Vrms = 20,000 V when it is plugged into a wall
source (Vrms = 120 V). If the primary coil (coil hooked to the wall)
has 1667 loops, how many loops does the secondary coil have?
A. 10
B. 278
C. 1667
D. 10,000
V1 V2
=
N1 N2
N2 = N1
V2
20, 000 V
= ( 1667 )
= 278, 000
V1
120 V
E. 278,000
DEMO:
Jacob’s ladder