Lecture 30AC power. Resonance.

Transformers.

ACT: C circuit

C

A capacitor is connected to an AC generator as shown. If the frequency of the generator is doubled, the amplitude of the current in the circuit will

A. increase by a factor of 2

B. not change

C. decrease by a factor of 2

EI

Z

2 2 1L C CZ X X R X

C

EC

Filters

C

R

Vout depends on frequency:

High smaller reactance VC = Vout 0

Vout

Low larger reactance no current flows through R smaller VR VC = Vout

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

LV

0 1 RC

More filters

LV

0 1 RC

High large XL VL ~

VR ~ 0 and I ~ 0

High-pass filter is Vout = VL

Low-pass filter is Vout = VR

L

R

L

RC

Low I ~ 0 due to capacitorHigh I ~ 0 due to inductor

Band pass filter (back to this later: resonance)

outV

0

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

ACT: RL circuit

I

VR

VL

E

And this is the current through all elements, by the way…

A series RC circuit is driven by emf =E sint. Which of the following could be an appropriate phasor diagram?

~

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

A CB

VR

VL

VC

E

VR

VC

E

VR

VC

ACT: RC circuit

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

B. Decrease

C. Current and driving voltage cannot be in phase.

0

t

i

From the figure, current leads driving voltage < 0

XC > XL to make them equal, frequency needs to increase.

IR

IXC

IXL

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?

0

t

i

C.

IA.

B.I

IFrom the figure:

At t = 0, i ~ 2/3I (>0)

And it should be increasing.

i (t) is the horizontal projection of the phasor.

Resonance

Current amplitude in a series RLC circuit driven by a source of amplitude E :

EI

Z

Maximum current when impendance is minimum

2 2L CZ X X R

i.e., when L CX X

1L

C

0

1

LC

Resonance:Driving frequency = natural frequency

EI

Z cos

ER

00

2o

R

E

IMaximum current maximum cos cos ~ 1 ~ 0 (circuit in phase)

It’s the bandpass filter!

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

Power in AC circuits

cos cosP t t i t E t I t

Instantaneous power supplied to the circuit:

Often more useful: Average power

cos cosP t EI t t

cos cos cos cos cos sin sint t t t t

22 2

0

1 1cos cos

2 2x xdx

2

0

1cos sin cos sin 0

2x x x xdx

1cos

2P EI

Define:

2rms

EE

2rms

II cosrms rmsP E I

Power factor

cosrms rmsP E I Power factor (PF)

cosE E

IZ R

2rmsP I R All energy dissipation

happens at the resistor(s).

Maximum power = 0 Resonant circuit

The Q factor

How “sharp” is the resonance? (ie, the resonance peak)

,00 LXLQ

R R

Umax is max energy stored in the

system

ΔU is the energy dissipated in one

cycle

max2U

QU

• For RLC circuit, 2max

12

U LI

• Losses only come from R : 2 2

0

1 1 22 2

U I RT I R

period

,0 ,0L CX XQ

R R

,0

0

1 CXQ

CR R

0

1

LC

,0 ,0L CX XQ

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

Transformers

1 2

1 2

V V

N N

Application of Faraday’s Law• Changing EMF in primary coil creates changing flux• Changing flux creates EMF in secondary coil.

Efficient method to change voltage for AC.

f or both coilsBdV N

dt

1 2

1 2

Bd V V

dt N N

If no energy is lost in the coils, power on both sides must be the same

1 1 2 2VI V I

N1V1 N2 V2

Bd

dt

Magnetic flux remains mostly in the core. Core “directs” B lines

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

E. 278,000

1 2

1 2

V V

N N

22 1

1

20,000 V1667 278,000

120 V

VN N

V

DEMO: Jacob’s ladder