1. AC power. Resonance. Transformers.
Lecture 30
AC power. Resonance. Transformers.
Transformers

2. Recap: Phasors The entire thing rotates CCW.
Magnitude of the phasors: VL I VR
The entire thing rotates CCW.
Real voltages = horizontal projections of the phasors. VC

3. Recap: Impedance and Phase angleVR VL VC
VL−VC E ϕ R XL XC
XL−XC Z ϕ
Impedance
But also:
Etc.
ALWAYS DRAW THE DIAGRAM!!!

4. ACT: eRL 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 I VR VL E
And this is the current through all elements.

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

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

7. More filters e e High ω ⇒ large XL ⇒ VL ~ ε ⇒ VR ~ 0 and I ~ 0
Vout
High ω ⇒ large XL ⇒ VL ~ ε ⇒ VR ~ 0 and I ~ 0
High-pass filter is Vout = VL
Low-pass filter is Vout = VR L e R C
Band pass filter (resonance)
Low ω ⇒ I ~ 0 due to capacitor
High ω ⇒ I ~ 0 due to inductor

8. ACT: Bring in phase (I) ε Increase ω i Decrease ω
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? t i ε
Increase ω
Decrease ω
Current and driving voltage cannot be in phase. IR e
IXC
IXL
From the figure, current leads driving voltage ⇒ ϕ < 0
⇒ XC > XL ⇒ to make them equal, frequency needs to increase.

9. ACT: Bring in phase (II)
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? t i ε I A. B. I
i (t) is the horizontal projection of the phasor.
From the figure:
At t = 0, i ~ 2/3I (>0)
And it should be increasing. C. I

10. Resonance L e R C
Current amplitude in a series RLC circuit driven by a source of amplitude E :
Maximum current when impedance is minimum
Resonance:
Driving frequency = natural frequency

11. Band pass filter Maximum current maximum cosϕ (cosϕ = 1)
2ωo Z E I
Maximum current
maximum cosϕ
(cosϕ = 1)
ϕ = 0 (circuit in phase)
Resonance
Low ω ⇒ I ~ 0 due to capacitor
High ω ⇒ I ~ 0 due to inductor

12. 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 ⇒ decrease frequency ω

13. Power in AC circuits Instantaneous power supplied to the circuit:
Often more useful: Average power
Define:

14. Maximum power ⇔ ϕ = 0 ⇔ Resonant circuit
Power factor
Power factor (PF)
Maximum power ⇔ ϕ = 0 ⇔ Resonant circuit
All energy dissipation happens at the resistor(s).

15. The Q factor How “sharp” is the resonance? (ie, the resonance peak)
Umax is max energy stored in the system
ΔU is the energy dissipated in one cycle
For RLC circuit,
Losses only come from R :
period

16. 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

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

18. 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? 10
278
1667
10,000
278,000
DEMO: Jacob’s ladder

19. Tesla Coil HV capacitor is charged
at high enough VC a spark across the space gap allows a current through the primary coil and HV capacitor (LC circuit)
current in primary coil induces emf in secondary coil
with each cycle energy gets transferred to secondary coil
torus acts like a capacitor with earth and forms LC circuit with secondary coil
when enough energy builds up in secondary circuit it discharges to ground through a big spark