Lecture 30 AC power. Resonance. Transformers. Transformers.

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Lecture 30 AC power. Resonance. Transformers. Transformers

Transcript of Lecture 30 AC power. Resonance. Transformers. Transformers.

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

Lecture 30AC power.

Resonance. Transformers.

Transformers

Page 2: Lecture 30 AC power. Resonance. Transformers. Transformers.

Recap: Phasors

ε =E cos ωt +ϕ( ) i =I cos ωt( )

vR=V

Rcos ωt( ) VR

=RI

vC=V

Ccos ωt −

π2

⎝⎜

⎠⎟ VR

=XCI X

C=

1

vL=V

Lcos ωt +

π2

⎝⎜

⎠⎟ VL

=XLI X

L=Lω

I

tωVR

VL

VC

Magnitude of the phasors:

R

C C

L L

V RI

V X I

V X I

The entire thing rotates CCW.

Real voltages = horizontal projections of the phasors.

Page 3: Lecture 30 AC power. Resonance. Transformers. Transformers.

Recap: Impedance and Phase angle

I

tωVR

VL

VC

VL−VC

E

ϕ

R

XL

XC

XL−XC

Z

ϕ

tan L CX X

R

tan L C

R

V V

V

sin L CX X

Z

cos

RZ

But also: Etc.

ALWAYS DRAW THE DIAGRAM!!!

2 2L C RE V V V

R

C C

L L

V RI

V X I

V X I

2 2L CE I X X R

E =I Z 2 2L CZ X X R

Impedance

Page 4: Lecture 30 AC power. Resonance. Transformers. Transformers.

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: εRL circuit

I

VR

VL

E

And this is the current through all elements.

Page 5: Lecture 30 AC power. Resonance. Transformers. Transformers.

A series RC circuit is driven by an emf ε =E sinωt. 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

VL

VC

E

VR

VC

E

VR

VC

ACT: εRC circuit

VR

Page 6: Lecture 30 AC power. Resonance. Transformers. Transformers.

Low- and high-pass filters

C

ε

R

Vout depends on frequency:

High ω ⇒ smaller reactance ⇒ VC = Vout → 0

Vout

Low ω ⇒ larger reactance ⇒ no current flows through R ⇒ smaller VR ⇒ VC = Vout → ε

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

outV

ε

0

1

RCω ω

Page 7: Lecture 30 AC power. Resonance. Transformers. Transformers.

More filters

L

ε

RC

L

ε

R

Vout

Band pass filter (resonance)

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

εoutV

0

1

RCω

outV

ε

ω

High ω ⇒ large XL ⇒ VL ~ ε ⇒

VR ~ 0 and I ~ 0

High-pass filter is Vout = VL

Low-pass filter is Vout = VR0

RL

ω

Page 8: Lecture 30 AC power. Resonance. Transformers. Transformers.

ACT: Bring in phase (I)

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

t

From the figure, current leads driving voltage ⇒ ϕ < 0

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

IR

εIXC

IXL

Page 9: Lecture 30 AC power. Resonance. Transformers. Transformers.

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?

0

t

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.

Page 10: Lecture 30 AC power. Resonance. Transformers. Transformers.

Resonance

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

EI

Z

Maximum current when impedance is minimum

2 2L CZ X X R

i.e., when L CX X

1L

ω

ω ω 0

1

LCResonance:Driving frequency = natural frequency

L

ε

RC

Page 11: Lecture 30 AC power. Resonance. Transformers. Transformers.

Band pass filter

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

EI

Z cos

ER

00

2ωo

Z

E

I Maximum current

ω 0

1

LC

maximum cosϕ

(cosϕ = 1)

ϕ = 0 (circuit in phase)

Resonance

ω 1

LC

Page 12: Lecture 30 AC power. Resonance. Transformers. Transformers.

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

Page 13: Lecture 30 AC power. Resonance. Transformers. Transformers.

Power in AC circuits

P t( ) ≡ε t( )i t( ) = E cos ωt +ϕ( )( ) I cosωt( )

Instantaneous power supplied to the circuit:

Often more useful: Average power

ω ω cos cosP t EI t t

ω ω ω ω ω cos cos cos cos sin sin cost t t t t

22 2

0

1 1cos cos

2 2x xdx

π

π

2

0

1cos sin cos sin 0

2x x x xdx

π

π

1cos

2P EI

Define:

2rms

EE

2rms

II cosrms rmsP E I

Page 14: Lecture 30 AC power. Resonance. Transformers. Transformers.

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 ⇔ ϕ = 0 ⇔ Resonant circuit

Page 15: Lecture 30 AC power. Resonance. Transformers. Transformers.

The Q factor

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

Q =2π 1

2LI 2

⎝⎜

⎠⎟

2

I 2R

ω0

⎝⎜⎜

⎠⎟⎟=ω

0L

R=

XL,0

R

Umax is max energy stored in the

system

ΔU is the energy dissipated in one

cycle

max2U

QU

π

• For RLC circuit, 2max

12

U LI

• Losses only come from R : 2 2

0

1 1 22 2

U I RT I Rπ

ω

period

,0 ,0L CX XQ

R R ,0

0

1 CXQ

CR Rω

0

1

LCω

Page 16: Lecture 30 AC power. Resonance. Transformers. Transformers.

,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

Page 17: Lecture 30 AC power. Resonance. Transformers. Transformers.

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

Page 18: Lecture 30 AC power. Resonance. Transformers. Transformers.

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

Page 19: Lecture 30 AC power. Resonance. Transformers. Transformers.

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