Chapter 28: Alternating-Current Circuits Thursday November 3rd

• Review of energy and oscillations in LC circuits • Alternating current theory

• Defn of terms, e.g., rms values • Resistance • Capacitive reactance • Inductive reactance

• Putting it all together – LRC circuits • Voltage/phase relations • Impedance • Resonance

Reading: up to page 501 in the text book (Ch. 28)

U =UB+U

E=12LI 2 +

12Q2

C! =

1LC

Ch. 28: Electromagnetic oscillations

Q(t) =Qm

cos!t

I(t) =!!Qm

sin!t

Ch. 28: Alternating Current V(t)=V

Psin !t +!

V( ); I(t)= IP sin !t +!I( )

Sine curve starts at ωt = -π/6 or -30o

In this example: ϕV = +π/6 or 30o

Root-Mean-Square:

Vrms

=V

P

2

Irms

=I

P

2

! = 2!fAngular frequency:

V(t)=VPsin!t

I(t)=VR=VP

Rsin!t

IP VP

I(t) V(t)

AC circuits: the resistive term

For a resistor ONLY: Current and voltage in phase

! IP

=VP

/R

Irms

=Vrms

/R

Voltage (driving term):

Current response:

AC circuits: the capacitive term Current peaks ¼ cycle before voltage

V(t)

I(t)

VP IP

!t

Q(t) =CV(t) =CVPsin!t

Charge (driving term):

I(t) =dQdt

=CVP

ddt

(sin!t)

= !CVPcos!t = !CV

Psin(!t + "

2)

Current response:

AC circuits: the capacitive term Current peaks ¼ cycle before voltage

V(t)

I(t)

VP IP

!t

Q(t) =CV(t) =CVPsin!t

Charge (driving term):

I(t)= !CVPsin(!t + !

2)

Current response: Current leads voltage by 90o

IP=VP

1/ !C=VP

XC

Capacitive reactance: XC

= 1/ !C (units - !)

AC circuits: the inductive term Voltage peaks ¼ cycle before current

!tI(t)

V(t)

VP IP

VL=!LdI /dt =!V

Psin!t

Current (driving term):

I(t)=VP

Lsin!t! dt

=!VP

!Lcos!t =

VP

!Lsin(!t ! !

2)

Current response:

AC circuits: the inductive term Voltage peaks ¼ cycle before current

!tI(t)

V(t) Current (driving term):

I(t) =

VP

!Lsin(!t! "

2)

Current response:

Inductive reactance: XL= !L (units - !)

Voltage leads current by 90o

I

P=

VP

!L=

VP

XL

VL=!LdI /dt =!V

Psin!t

VP IP

Summary

v

Phasor Diagrams Resistor Capacitor Inductor

VRp= I

pR

V

Cp= I

pX

CVLp= I

pXL

V, I in phase V lags I by 90o V leads I by 90o

http://en.wikipedia.org/wiki/Phasor_(sine_waves)

V(t)= sin!t; I(t)= sin(!t !!)

Phasor Diagrams: Adding the Voltages

tan! =VLp!V

Cp

VRp

Phasor Diagrams: Adding the Voltages

tan! =

XL!X

C

R="L!1/ "C

R

V

p= I

pR2 + I

pX

L! I

pX

C( )2

! Ip=

Vp

R2 + XL!X

C( )2=Vp

Z

Z = R2 + X

L!X

C( )2

Modified Ohm’s law:

Impedance:

Phase:

Units:ohms!

"##

$

%&&

Resonance

Ip

=V

p

R2 + XL!X

C( )2

XL= X

C

XL> X

C

XL<X

C

V leads V lags

!

o=

1LC

At resonance, Z = R, and ϕ = 0 (just like a DC circuit)

Resonance XL= X

C

XL> X

C

XL<X

C

V leads V lags

P = 1

2I

pV

pcos! = I

rmsV

rmscos!

Power delivered to the circuit: