31. Electromagnetic Oscillations & Alternating Current31. Electromagnetic Oscillations & Alternating Current31-2. LC Oscillations, Qualitatively

-RC circuit : Energy stored Capacitor

- RL circuit : Energy stored Inductor

CqU E 2

2

= RCc =τ

2

2LiU B = LRL /=τ

-LC circuit : Energy transfer: Capacitor ↔ Inductor

dtdqi /=

dtiq ∫ ⋅=

31-4. LC Oscillations, Quantitatively • The Block-Spring Oscillator

( )φω += tXx cos

2

2

,dt

xddtdv

dtdxv ==

• The LC Oscillator

22

2212

21

21

21 kx

dtdxm

kxmvUUU sb

+⎟⎠⎞

⎜⎝⎛=

+=+=Total Energy

22

2212

21

121

21

/

qCdt

dqL

CqLiUUU EB

+⎟⎠⎞

⎜⎝⎛=

+=+=Total Energy

0=+=dtdxkx

dtdvmv

dtdU

02

2

=+ kxdt

xdm

mk /=ω Angular velocity

0=+=dtdq

Cq

dtdiLi

dtdU

2

2

,dt

qddtdi

dtdqi ==

012

2

=+ qCdt

qdL

( )φω += tQq cos

LC/1=ω

( )φωω +−== tQdtdqi sin

• Electrical and Magnetic Energy Oscillation

Electrical Energy : ( )φω +== tC

QC

qUE2

22

cos22

Magnetic Energy :

)(sin2

)(sin

22

222212

21

φω

φωω

+=

+==

tC

Q

tCQLLiUB

31-5. Damped Oscillations in an RLC Circuit

Total energy is not conserved.

CqLiUUU EB /2212

21 +=+=

dtdq

Cq

dtdiLiRi

dtdU

+=−= 2

2

2

,dt

qddtdi

dtdqi ==

012

2

=++ qCdt

dqRdt

qdL

( )φω +′= − tQeq LRt cos2/

( )22 2/ LR−=′ ωω

Solution:

LC/1=ω

( )φω +′== − teC

QC

qU LRtE

2/22

cos22

LRte 2/−

LRte 2/−−

qQ

-Q

εmax

-εmax

t

ε

tdωε sinmaxε=

31-6. Alternating Current

( )φω −= tIi dsin

ωd: driving angular velocity

31-7. Forced Oscillations

LC/1=ω : Natural angular velocity

When ωd = ω, Resonance: current i becomes maximum.

31-8. Three Simple Circuits• A Resistive Load

tdωεε sinmax=

tR

Vi dR

R ωsin=

tR

RiP dωε 2

2max2 sin==Power

0=− RvεtVRiv dRRR ωsin==

( )φω −= tIi dRR sinConsider

RVI R

R =

tVv dRR ωsin=

maxε=RV

0=and φ

1. Angular speed: i and vR have the same speed ωd.

2. Length: Amplitude IR and VR.

3. Projection: iR and vR.

4. Rotation angle: the same angle ωdt

• A Capacitive Load tdωεε sinmax=

tCVdt

dqi ddCC

C ωω cos==

0=− CvεtV

Cqv dC

CC ωsin==

tVv dCC ωsin=

Capacitive reactance :C

Xd

C ω1

=

( )o90sincos +== tXVt

XVi d

C

Cd

C

CC ωω

( )φω −= tIi dCC sin

CCC XIVConsider

= and o90−=φ

1. Angular speed: iC and vC have the same speed ωd.

2. Length: Amplitude IC and VC.

3. Projection: iC and vC.

4. Rotation angle of iC is 90º (π/2) advance of that of vC.

• A Inductive Load tdωεε sinmax=

tL

Vdtdi

dLL ωsin=

0=− LvεtV

dtdiLv dL

LL ωsin==

tVv dLL ωsin=

Inductive reactance : LX dC ω=

( )o90sincos −== tXVt

XVi d

L

Ld

L

LL ωω

( )φω −= tIi dLL sin

LLL XIVConsider

= and o90+=φ

1. Angular speed: iC and vC have the same speed ωd.

2. Length: Amplitude IL and VL.

3. Projection: iL and vL.

4. Rotation angle of iL is 90º (π/2) behind of that of vL.

tL

VdttL

Vi dd

Ld

LL ω

ωω cossin ⎟⎟

⎠

⎞⎜⎜⎝

⎛−=⋅= ∫

31-9. The Series RLC Circuittdωεε sinmax=

( )φω −= tIi dsin• The Current Amplitude

Resistor: iR and vR have are in phase; the same phase.

Capacitor: iC advance vC in phase; φ = -90º.

Inductor: iL behind vL in phase; φ = +90º.

LCR vvv ++=ε ( ) ( ) ( )22222max CLCLR IXIXIRVVV −+=−+=ε

( ) ZXXRI

CL

max22

max εε=

−+= ( )22

CL XXRZ −+= Impedance

( )22

max

/1 CLRI

dd ωω

ε−+

= Current amplitude

• The Phase Constant

IRIXIX

VVV CL

R

CL −=

−=φtan

RXX CL −=φtan Phase constant

XL > XC : more inductive than capacitive.

XL < XC : more capacitive than inductive.

XL = XC : in resonance.

• Resonance

CL XX =C

Ld

d ωω 1

=

LCd1

=ω Maximum I( ) RXXR

ICL

max22

max εε=

−+=

LCd1

==ωω Natural angular velocity

31-10. Power in Alternating-Current Circuit

( )[ ] ( )φωφω −=−== tRIRtIRiP dd2222 sinsinPower

Average Power RIRiPavg

22

22⎟⎠

⎞⎜⎝

⎛== Root-mean-square (rms)

2IIrms =

RIP rmsavg2=

22maxεε == rmsrms andVV

( )22CL

rmsrmsrms

XXRZI

−+==

εε

Average Power

ZRIRI

ZRIP rmsrmsrms

rmsrmsavg εε

=== 2

ZR

IZIRVR ===

max

cos εφ

φε cosrmsrmsavg IP =

Power factor : cosφ