Physics 202, Lecture 18 Today’s Topics

!  Reminder !  RL Circuits !  RC Circuits !  LC (RLC) Circuits and Electromagnetic Oscillations

!  AC Circuits with AC Source !  Phasors

1

Turn on RL Circuit (reminder)

Note: the time constant is !=L/R

!

I =V0R(1" e

"t

L /R )

2

Turn off RL Circuit (reminder)

Note: the time constant is !=L/R

!

I = I0e"

tL /R

3

Charging a Capacitor in RC Circuit

Charging Note: ! RC is called the time constant

!

4

Discharging a Capacitor in RC Circuit

discharging !

I(t) = "QRC

e"t /RC

Note the time constant !=RC

!

q(t) /C + R dq(t)dt

= 0

5

LC Circuit and Oscillation " Exercise: Find the oscillation frequency of a LC circuit

!

!

"

!q(t) /C ! L dI(t)dt

= 0

q(t) + 1!02

d 2q(t)dt 2

= 0

!0 =1LC

q = Qmax cos(!0t +" )I = #!0Qmax sin(!0t +" )

eq. of Harmonic Oscillation

Total Energy is conserved

I

6

Show that energy is conserved in LR circuit:

E = 12Q2

C+12LI 2

=12C

Q02 cos2 (!0t +" ) +

12L #!0Q0 sin(!0t +" )[ ]2

=Q02

2Ccos2 (!0t +" ) +

12!02LQ0

2 sin2 (!0t +" )

Now use !0 = 1 / LC

E = Q02

2Ccos2 (!0t +" ) +

12LC

LQ02 sin2 (!0t +" )

=Q02

2Ccos2 (!0t +" ) + sin

2 (!0t +" )#$ %&

=Q02

2C independent of time 7

AC Power Source "  "V = "Vmax sin(#t+$0) = "Vmax sin(#t)

Initial phase, usually set $0=0

T=2%/#&

#: angular frequency #=2%f

t=0

8

AC Circuit " Find out current i and voltage difference "VR, "VL, "VC.

Notes: •  Kirchhoff’s rules still apply ! •  A technique called phasor analysis is convenient.

i

"Vmax Sin(#t)

9

Phasor "  A sinusoidal function y= Asin$ can be represented

graphically as a phasor vector with length A and angle $ (w.r.t. to horizontal)

$&

Asi

n$&

10

Resistors in an AC Circuit " Ohm’s Law: "V=IR at any time

i

iR="V/R=Imax sin#t, Imax="Vmax/R

# The current through an resistor is in phase with the voltage across it Phasor view

Function view

11

Inductors in an AC Circuit "  "V - Ldi/dt =0

" iL=Imax sin(#t-%/2) Imax="Vmax/XL, XL= #L ! inductive reactance # The current through an inductor is 90o behind the voltage across it.

Phasor view

Function view i

12

Capacitors in an AC Circuit

"  "V - q/C=0, dq/dt =i

i

" iL=Imax sin(#t+%/2) Imax="Vmax/XC, XC= 1/(#C) ! capacitive reactance # The current through a capacitor is 90o ahead of the voltage across it. Phasor view

Function view

13

Summary: Inductor and capacitor, and lagging versus leading

" For an inductor: Current lags the voltage

" For a capacitor: Current leads the voltage

14

As the frequency in this simple ac circuit increases, while the rms voltage is held

constant, the rms current through the resistor

A. increases. B. does not change. C. may increase or decrease depending on the magnitude

of the original frequency. D. may increase or decrease depending on the magnitude

of the resistance. E. decreases.

15

As the frequency in this simple ac circuit increases, while the rms voltage is held

constant, the rms current through the resistor

A. increases. B. does not change. C. may increase or decrease depending on the magnitude

of the original frequency. D. may increase or decrease depending on the magnitude

of the resistance. E. decreases.

16

The figure shows the voltage and current for a(n) A.  inductor.

B.  capacitor.

C.  resistor.

The figure shows the voltage and current for a(n) A.  inductor.

B.  capacitor.

C.  resistor.

Which phasor diagram correctly represents the phase relationship between the voltage across the resistor and the voltage across the inductor in a circuit containing an inductor L, a capacitor C, and a resistor R all connected in series?

Which phasor diagram correctly represents the phase relationship between the voltage across the resistor and the voltage across the inductor in a circuit containing an inductor L, a capacitor C, and a resistor R all connected in series?

The switch S was at position a for a long time when it is switched to b. Describe the current flow in the right loop. Assume that the inductor and the wires have zero resistance.

The switch S was at position a for a long time when it is switched to b. Describe the current flow in the right loop. Assume that the inductor and the wires have zero resistance.

A 5-!F capacitor is charged to 30 V and is then connected in series with a 10-!H inductor and a 50-W resistor. The current in this circuit after a long time has passed will be

A.  0 B.  8.83 A C.  15.4 A D.  21.2 A E.  some value that cannot be determined

from the given information.

A 5-!F capacitor is charged to 30 V and is then connected in series with a 10-!H inductor and a 50-W resistor. The current in this circuit after a long time has passed will be

A.  0 B.  8.83 A C.  15.4 A D.  21.2 A E.  some value that cannot be determined

from the given information.