31. electromagnetic oscillations & alternating...
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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/−−
-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