PHY 114 A General Physics II 11 AM-12:15 PM TR Olin 101 Plan …users.wfu.edu/natalie/s12phy114/lecturenote/Lecture15.pdf · 2012-03-22 · 3/22/2012 PHY 114 A Spring 2012 -- Lecture

3/22/2012 PHY 114 A Spring 2012 -- Lecture 15 1

PHY 114 A General Physics II 11 AM-12:15 PM TR Olin 101

Plan for Lecture 15 (Chapter 33):

Alternating current circuits

1.AC EMF

2.R, C, L behaviors in AC circuit

3.RLC circuits


Remember to send in your chapter reading questions…

3rd exam will be scheduled for evenings during the week of 4/2/2012


Homework hint:

“The switch S is closed for a long time, and no voltage is measured across the capacitor.” Which of the following statements is false?

A. The voltage across the inductor and the capacitor must always be the same.

B. After a long time the current through the inductor is E/R.

C. After a long time the current through the resistor is E/R.

D. After a long time, the current through the inductor is 0.


Homework hint:

After the switch S is opened (t=0) which of the following statements are true?

A. The circuit is now an LC circuit with the initial condition that Q(t=0)=CE.

B. The circuit is now an LC circuit with the initial condition that I(t=0)=E/R.

C. The current will oscillate forever with an angular frequency of

./1 LC=ω


DC - LRC circuit:

01

0

2

2

=−++

=−−−

E

E

QCdt

dQRdt

QdL

RIdtdIL

CQ

t

Q/CE

C

( )2

2/

21'

)'cos(1)(:0)0( assumingSolution

−=

−=

==−

LR

LC

teCtQtQ

LRt

ω

ωE


AC EMF source

)2sin()sin()()( maxmax ftVtVtvt πω ∆=∆=∆≡E

t

∆Vmax

∆v

1/2πf


AC EMF source

)2sin()sin()()( maxmax ftVtVtvt πω ∆=∆=∆≡E

t

∆v

[ ] 22max

/1

0

2max

2

/1

0max

21)2sin()(

0)2sin()(

rms

f

f

VVdtftVtv

dtftVtv

∆≡∆=∆=∆

=∆=∆

∫

∫

π

π


In US the standard wall voltage is ∆Vrms =120 V with f =60 Hz In Europe the standard wall voltage is ∆Vrms =220 V with f =50 Hz

http://users.telenet.be/worldstandards/electricity.htm#voltage

Voltage standards throughout the world:

http://users.telenet.be/worldstandards/electricity.htm


DC - LRC circuit: AC - LRC circuit:


Which of the following statements most accurately reflect the relationships between the behaviors of LRC circuits with a DC versus an AC EMF source.

A. The differential equations are different and the solutions are different.

B. The differential equations are the same – but the solutions are different.

C. The differential equations are the same – so the solutions have to be the same.

In fact: The DC solutions to the differential equations are generally valid representations of transient behaviors. Often we are interested in the “steady state” solutions in the presence of AC voltage sources.


)sin()(

0)()sin(0

max

max

tR

VtI

tRItVRIv

ω

ω∆

=⇒

=−∆=−∆

AC EMF with resistor:


AC EMF with capacitor:

)cos()()(

)sin()(

0)()sin(

0

max

max

max

tVCdt

tdQtI

tVCtQC

tQtV

CQv

ωω

ω

ω

∆==⇒

∆=⇒

=−∆

=−∆


AC EMF with inductor:

)cos()(

)sin()(

0)()sin(

0

max

max

max

tL

VtI

tL

Vdt

tdIdt

tdILtV

dtdILv

ωω

ω

ω

∆−=⇒

∆=⇒

=−∆

=−∆


Summary of results for each circuit element:

)sin()( :onlyresistor With max φω −∆

= tR

VtI

)sin()(:form theof EMF AC General

max φω −∆=∆ tVtv

)cos()( :onlycapacitor With max φωω −∆= tVCtI

)cos()( :onlyinductor With max φωω

−∆

−= tL

VtI


Comment about phase factor φ: Plots of sin(2πt+φ)

φ=0

φ=0.5

φ=1.57


Series AC LRC circuit

C

)sin()(:form thehave to

)(solution Assume)sin(

0

max

max

φω

ω

−=

∆=

=−−−

tItI

tItV

RIdtdIL

CQ

E

E

valueconsistent a has valueconsistent a has

equations aldifferenti satisfies )( :if success Declare

max

φ•••

ItI



C

)sin()( :Assuming

0)sin(

max

max

φω

ω

−=

=−−−∆

tItI

RIdtdIL

CQtV

0)sin(

)cos()cos()sin(

max

maxmax

max

=−−

−−−

+∆

φω

φωωω

φωω

tRI

tLIC

tItV

Must be true at all t; special values of Imax and φ



C

( ){ } 0)sin()cos()sin(

0)sin()cos(1)sin(

maxmax

maxmax

=−+−+−−∆

=

−+−

+−−∆

φωφωω

φωφωωω

ω

tRtXXItV

tRtLC

ItV

LC

( )( )

2222

22

os and sin if

sinsincos

sincoscossinsinyour text) of 12-A pg. (see identities trigSome

BABc

BAA

BABA

+≡

+≡

++=+

+=+

αα

βαββ

βαβαβα


( ){ }( )

( )( ) ( )

0)sin()cos(

)sin(

0)sin()cos()sin(

2222

22maxmax

maxmax

=

−++−

+−++−

+−×

++−−∆

=−+−+−−∆

φωφω

ω

φωφωω

tRXX

RtRXX

XX

RXXItV

tRtXXItV

LCLC

LC

LC

LC

( )( )

( )

( ){ }{ }

ZVI

RXXtZItV

ttZItVRXX

RRXX

XXRXXZ

CL

LC

LC

LC

LC

maxmax

maxmax

maxmax

22

22

22

tan

0)sin()sin(0)sin(cos)cos(sin)sin(

cos

sin

Let

∆=⇒

−=⇒=⇒

=−+−∆=−+−−∆

++−≡

++−

+−≡

++−≡

φαφ

φωαωφωαφωαω

α

α



C

)sin()( :Assuming

0)sin(

max

max

φω

ω

−=

=−−−∆

tItI

RIdtdIL

CQtV

( )

ZVI

RXX

RXXZ

CXLX

CL

CL

CL

maxmax

22

tan

1

∆=

−=

+−≡

≡≡

φ

ωω


Example AC LRC circuit: Consider the series resistor R and inductor L connected to an AC EMF. The voltage amplitude Vmax is held fixed while the frequency ω is varied. The bulb is brightest when

Α. ω is highest Β. ω is lowest C. Bulb brightness is

independent of ω


Example AC LRC circuit: Consider the series resistor R and capacitor C connected to an AC EMF. The voltage amplitude Vmax is held fixed while the frequency ω is varied. The bulb is brightest when


independent of ω


Example AC LRC circuit: Consider the series resistor R and capacitor C connected to an AC EMF. The voltage amplitude Vmax is held fixed while the frequency ω is varied. The bulb is brightest when


independent of ω


Power in an AC circuit

( )

( ) ( )φωωφω

ωφω

ωφω

sin)sin()cos(cos)(sin

)sin()sin(

)sin( )sin(

22

max

2max

maxmax

tttZ

V

ttZ

V

tVtIvIP

−∆

=

−∆

=

∆−=∆=

( ) φcos21 2

max

ZVP

avg

∆=

∫∫ ==ωπωπ

ωωω/2

0

/2

0

2 0)cos()sin( 21)(sin

:Note

dtttdtt


Power in an AC circuit -- continued

( ) φcos21 2

max

ZVP

avg

∆=

( )

( )

( ) RIRZVP

ZR

RXX

RRXXZ

avg

CL

LL

2max

2max

22

22

21

21

cos

:Recall

=

∆

=⇒

=++

=

+−≡

φ


Frequency dependence of impedance in AC circuit

( )

22

2

2022

22

22

0

1

1

1

frequency resonance a found wecircuit, pureFor

RLZ

RC

LRXXZ

LC

LC

LL

+

−=

+

−=+−=

=

ωωω

ωω

ω

( )2

2

2

2022

2max

2max

121

21

RL

RVRZVP

avg

+

−

∆=

∆

=

ωωω



DC - LRC circuit: AC - LRC circuit:

( )2

2/

21'

)'cos(1)(:0)0( assumingSolution

−=

−=

==−

LR

LC

teCtQtQ

LRt

ω

ωE )sin()(

)sin(for solution state"-Steady"

max

max

φω

ω

−∆

=

∆=

tZVtI

tVE

( ) 22

tan ;1 ;

RXXZ

RXX

CXLX

CL

CLCL

+−≡

−=≡≡ φ

ωω


AC transformer devices:

dtdNv

vdt

dNv

v

B

B

Φ−=∆

∆

Φ−=∆

∆

22

2

11

1

:for law sFaraday'

:for law sFaraday'

11

22

:2 and 1for same theis

ion,approximat good a To

vNNv

dtd B

∆=∆⇒

Φ


AC transformer example:

101

:12012for need / determinecan we

,amplitudes RMS theoequality t theapplyingin time; goscillatin are and Both

1

2

1

2

1

2 12

21

11

22

==

=

∆∆

∆=∆

rms

rms

rms

rms

VV

NN

VVNN

vv

vNNv

If ∆v1 were a DC voltage source: A. The transformer would work in the same way. B. The transformer would not work at all.

Documents

PHY 114 A General Physics II 11 AM-12:15 PM TR Olin 101 Plan …users.wfu.edu/natalie/s12phy114/lecturenote/Lecture15.pdf · 2012-03-22 · 3/22/2012 PHY 114 A Spring 2012 -- Lecture