Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current...

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Chapter 28 – AC Circuits

Transcript of Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current...

Page 1: Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current and emf, induced current would be in same direction as I(t) half the time, and

Chapter 28 – AC Circuits

Page 2: Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current and emf, induced current would be in same direction as I(t) half the time, and

LC Circuit •  Suppose switch closed at t = 0 with capacitor initially

charged to Q. What is q(t)?

VC =q

C

Ld2q

dt2+

q

C= 0

q(t) = Q cos(ωt)

ω =�

1LC

q(t = 0) = Q

I(t = 0) = 0

Initial conditions:

+q -q

I

��E · �dl = −L

dI

dt= −

i

∆Vi = −Vc

I = −dq

dt

Page 3: Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current and emf, induced current would be in same direction as I(t) half the time, and

•  Initially, energy is stored by capacitor in the form of an electric field.

•  When capacitor is drained (q=0), what form is the energy? •  Notice that

•  What form is energy in inductor? •  Magnetic! for solenoid:

•  Magnetic energy density:

•  Always true (general result is based on a much more formal derivation)

when q=0, |I| = Imax = Qω = Q

�1

LC

U =12CV 2 =

12

Q2

C

All energy in inductor!

12LI2 = uB × volume

uB =B2

2µ0

U =1

2

Q2

C=

1

2LI2max

Page 4: Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current and emf, induced current would be in same direction as I(t) half the time, and

•  Energy in an LC circuit oscillates between electric and magnetic •  similar to mechanical harmonic oscillator (kinetic and potential)

Page 5: Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current and emf, induced current would be in same direction as I(t) half the time, and

Damped LC oscillations •  With resistance in the circuit, LC oscillations damp out.

•  underdamped case:

•  Similar to a damped mass on spring

I

��E · d� = VC + IR = −L

dI

dt

Ld2q

dt2+ R

dq

dt+

1C

q = 0

q(t) = q0 exp(−R/(2L)t) cos

��1

LC− R2

4L2t

+q

-q I

I =dq

dt

Page 6: Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current and emf, induced current would be in same direction as I(t) half the time, and

Driven AC Circuits •  Driven resistor:

•  Current in phase with driving voltage source.

Vemf (t)− IR = 0 I =Vemf

R

Page 7: Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current and emf, induced current would be in same direction as I(t) half the time, and

I

+ -

I =dq

dt= C

dVemf

dt= VpCω cos(ωt) = VpCω sin(ωt + π/2)

Imax =Vp

XC

=⇒ XC =1

ωCCapacitive reactance

Vemf − VC = 0

VC =q

C= Vemf

Driven Capacitor

Page 8: Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current and emf, induced current would be in same direction as I(t) half the time, and

•  A light bulb is attached in series to a capacitor and an AC voltage source with an adjustable frequency. Keeping the peak voltage constant, the light bulb will be brightest for

1.  low frequencies 2.  high frequencies 3.  same for all frequencies

Page 9: Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current and emf, induced current would be in same direction as I(t) half the time, and

C

R

(t)

3Driven RC Circuit

ξ(t)− IR− q

C= 0

ξ = Vp sin(ωt)

dq

dtR+

q

C= Vp sin(ωt)

I =dq

dt=

Vp�( 1ωC )2 +R2

cos(ωt+ δ)δ = tan−1(−ωRC)

Page 10: Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current and emf, induced current would be in same direction as I(t) half the time, and

•  Faraday’s Law of induction

Driven Inductor

��E · d� = −L

dI

dt= −Vemf

I(t) = − Vp

ωLcos(ωt)

Imax =Vp

XL

XL = ωL Inductive reactance

Page 11: Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current and emf, induced current would be in same direction as I(t) half the time, and

Driven LR Circuit

−V (t) + IR = −LdI

dt

I(t) =V0�

R2 + (Lω)2cos(ωt− φ)

φ = tan−1 wL

R

if V (t) = V0 cos(ωt)

Page 12: Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current and emf, induced current would be in same direction as I(t) half the time, and

Jumping ring demo •  why does ring always repel from solenoid (with AC

circuit)?

http://www.youtube.com/watch?v=Pl7KyVIJ1iE

Page 13: Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current and emf, induced current would be in same direction as I(t) half the time, and

Jumping Ring Analysis

•  Suppose current in coils goes as I(t) = I0 sin(ωt) •  The emf in loop will go as

•  The induced current will therefore go as

•  Repulsive force when Iring and I are in opposite directions

I(t)

φ = tan−1 wL

R

Vemf ∝ −dI/dt ∝ − cos(ωt)

Iring ∝ − cos(ωt− φ)

Page 14: Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current and emf, induced current would be in same direction as I(t) half the time, and

Jumping Ring Cont’d •  If there were no phase difference between induced

current and emf, induced current would be in same direction as I(t) half the time, and the ring would not jump!

I(t)

Vemf

Page 15: Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current and emf, induced current would be in same direction as I(t) half the time, and

Jumping Ring Cont’d •  Now add phase shift between induced emf and induced

curent, given by

•  For large inductance/and or high frequencies, phase shift is near 90°. Here is a phase shift of 70°:

φ = tan−1 wL

R

I(t) Vemf

Iind

Now currents are in opposite directions more often than in same direction!

Page 16: Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current and emf, induced current would be in same direction as I(t) half the time, and

speaker crossover

Page 17: Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current and emf, induced current would be in same direction as I(t) half the time, and
Page 18: Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current and emf, induced current would be in same direction as I(t) half the time, and

Driven LRC Circuit

•  An RLC circuit driven by an AC voltage source exhibits frequency-dependent behavior.

−V (t) + IR +q

C= −L

dI

dt

Ip =Vp�

R2 + (XL −XC)2=

Vp

Z

I =dq

dt

+q -q

Ld2q

dt2+R

dq

dt+

q

C= Vp sin(ωdt)

Ip =Vp�

R2 + (ωL− 1ωC )2

I(t) = Ip sin(ωdt− φ)

tanφ =XL −XC

R

V (t) = Vp sin(ωdt)

Page 19: Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current and emf, induced current would be in same direction as I(t) half the time, and

Amplitude and phase in the driven RLC circuit

•  Peak current has a maximum at resonance: •  resonant frequency

•  Ip = Vp/R •  At resonance, voltage and

current are in phase

(XL = XC =�

L

C)

Page 20: Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current and emf, induced current would be in same direction as I(t) half the time, and

Transformers and power supplies •  A transformer is a pair of

coils linked by mutual inductance.

•  An AC current in the primary induces a current in the secondary.

•  The secondary voltage differs from the primary voltage by the ratio of the number of turns.

•  Both step-up and step-down transformers are possible.

ξ1 = −N1dΦB

dt(ΦB is flux per winding)

ξ2 = −N2dΦB

dt

ξ2

ξ1=

N2

N1=

V2

V1

|V1|max = |ξ1|max

|V2|max = |V1|maxN2

N1

Page 21: Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current and emf, induced current would be in same direction as I(t) half the time, and

Question 34.18 Transformers I

Page 22: Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current and emf, induced current would be in same direction as I(t) half the time, and

A 6 V battery is connected to one

side of a transformer. The output

voltage across coil B is:

Question 34.20 Transformers III

A B 6 V

Page 23: Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current and emf, induced current would be in same direction as I(t) half the time, and

•  Transformers are used to produce low voltages for electronic equipment.

•  Then they’re combined with diodes that convert AC to DC and capacitors to smooth the DC voltage.

Page 24: Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current and emf, induced current would be in same direction as I(t) half the time, and

Input Power Vs Output Power •  How does the input power compare to the output power?

•  Ignoring self inductance, P1 = V1I1

P1 = P2

I2 = I1V1

V2= I1

N1

N2

Page 25: Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current and emf, induced current would be in same direction as I(t) half the time, and

Clicker Question If the resistance R is decreased what happens to I1?

a) Increases b) Decreases c) Remains the same

Page 26: Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current and emf, induced current would be in same direction as I(t) half the time, and

Transmission of Electrical Power

Slide 26-11

•  Electric power is most efficiently transmitted at high voltages. •  This reduces I2R energy losses in the power lines. •  But most end uses require lower voltages. •  Transformers accomplish voltage changes throughout the power grid.

Page 27: Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current and emf, induced current would be in same direction as I(t) half the time, and

Two-Phase Power to Your Home

Slide 26-12

center tap is connected to ground

rms = root mean squared:

Page 28: Chapter 28 – AC Circuits - University of Colorado … AC Circuits • Driven resistor: ... current and emf, induced current would be in same direction as I(t) half the time, and