1

11/2/12 Lecture XIX 1

Physics 122

AC circuits

11/2/12 Lecture XIX 2

LC circuit •  Two forms of energy:

–  Electric – in a capacitor –  Magnetic - in solenoid

•  Oscillator! •  When energy in the

capacitance is at its maximum, energy in the inductance is at its minimum

•  No energy is lost, it is just changing its form

constLICVUtotal =+=22

22

11/2/12 Lecture XIX 3

LC circuit •  Total drop of voltage over a

closed circuit must be 0:

•  Take derivative over time:

•  Second derivative of function is proportional to the negative function

0 =VL +VC =QC+ L dI

dt=1C

I dt∫ + L dIdt

0 = 1CI + L d

2Idt2

d 2Idt2

= −1LC

I

I = I0 sin(ωt)

ω =1/ LC

Q0 -Q0

2

11/2/12 Lecture XVIII 4

LCR circuit •  Resistors dissipate energy

–  Convert electrical energy into thermal energy

•  Resistor acts like friction for a weight on a spring

•  Damped oscillator!

11/2/12 Lecture XIX 5

•  Magnetic field in a solenoid

•  It creates a magnetic flux through itself

•  Changing magnetic flux generates emf (voltage gained) = -drop of Voltage (voltage lost)

Inductance

IB ∝

dtdI

dtdB

dtdBA ∝∝Φ

=Φ

dtdIL

dtdNVemf L −=Φ

−=−=

11/2/12 Lecture XIX 6

AC circuits •  External source of alternating

voltage V(t)=V0sinωt – Forced oscillator

•  Frequency f, cyclic frequency ω=2πf •  Active elements

–  Inductors –  Capacitors

•  Passive elements –  Resistors

3

11/2/12 Lecture XIX 7

Phasor diagram •  Graphical way to present current

and voltage in AC circuit •  Present current and voltage in a

certain element (resistor, inductance, capacitance) as vectors rotating in (x,y) plane

•  Vector length is equal to maximum possible value of I or V: I0 and V0

•  Vector projection on y-axis is equal to the value of current or voltage at the moment t

tωx

y I0 )sin()( 0 tItI ω⋅=

11/2/12 Lecture XIX 8

Current and voltage in AC circuit

00 RIV =

Drop of voltage over resistor (V) follows I

Current and voltage in phase

tωx

y I0 )sin()( 0 tItI ω⋅=

Vo )sin()( 0 tVtVR ω=

11/2/12 Lecture XIX 9

tωx

y I0 )sin()( 0 tItI ω⋅=

Current and voltage in AC circuit

ω

ωω

ωωω

LXIXV

tILV

tILdt

tdLIV

dtdILV

L

L

L

L

L

=

=

+=

==

=

00

00

00

)90sin(

)cos()sin(

Vo )cos()( 0 tVtVR ω=

Inductor: current lags voltage

4

11/2/12 Lecture XIX 10

Current and voltage in AC circuit

VC =QC=1C

I dt∫

VC =1C

I0 sin(ω t)dt =∫ −1ωC

I0 cos(ω t)

V0 =1ωC

I0 = XCI0

XC =1ωC

Capacitor: voltage lags current

tωx

y I0 )sin()( 0 tItI ω⋅=

Vo

)cos()( 0 tVtVR ω−=

11/2/12 Lecture XIX 11

LCR circuit

V0R = I0R, V0L = I0XL, V0C = I0XC

R = R, XL =ω L, XC =1ωC

"V "= "

VR "+"

VL "+"

VC "= ZI

Z − impedance

tωx

y

I0 )sin()( 0 tItI ω⋅=

VR VL

VC

Ohm’s law!!

11/2/12 Lecture XIX 12

Impedance

V0 =| Z | I0| Z |= R2 + (XL − XC )

2

tanφ = (XL − XC ) / R

y

x

I0

VR VL

VC

V0

This equation relates maximum voltage to maximum current. Mind, that they do not happen at the same time. There is a phase shift between Imax and Vmax -φ

5

11/2/12 Lecture XIX 13

Phase angle and power factor

ZRRXX CL

=

−=

ϕ

ϕ

cos

tan

φcos"" rmsrms

rmsrms

IVIVP

ZIV

=⋅=

=

Cosφ – power factor NB. V and I are not real vectors, only in phasor diagram representation.

x

I0

VR VL

VC

V0

φ

11/2/12 Lecture XIX 14

Energy in AC circuit Total average power in AC Power dissipated only in resistors:

RIP rms2=

φcosrmsrmsIVP =

11/2/12 Lecture XIX 15

Resonance in LCR circuit

LCf

π21

=

22 )( CL

rmsrms

XXRZZVI

−+=

=

• Largest current when impedance is the smallest • XL, XC are functions of frequency • At a certain frequency

LC

CL

XX CL

1

1

=

=

=

ω

ωω