8.022 Lecture Notes Class 37 - 11/22/2006
Complex impedence instead of diff eq!
Use fact that everything in RLC circuit has same frequency as driving
frequency(?).
V (t) = V̂ eiωt
ˆ iωt
I(t) = Ie
Inductor
V =
dI
=
dt
V
= iωL = χL
I
Capacitor
L dI
dt
iωI
( complex impedence of inductor )
dV
dQ
I
=
=
dt
dt C
C
I
iωV =
C
complex impedence of capacitor
V
1
=
= χc
I
iωC
Resistor
2
V
=R
I
V =I ·z
χL = iωL
χC =
1
iωC
χR = R
RLC Circuit
No derivatives any more! Can sum just like resistors in series.
χtotal = χR + χC + χL = R + iωL +
I=
=
=
V
χtotal
1
R−i(ωL− ωC
)
V
·
1
1
R+i(ωL− ωC ) R−i(ωL− ωC )
1
V (R−i[ωL− ωC
])
1
R2 +(ωL− ωC )2
1
iωC
3
Iˆ =
V
1 2 1/2
[R2 +(ωL− ωC
) ]
tan φ =
1
ωL− ωC
R
Parallel RLC Circuit
Let Y =
1
χ
, I =V ·Y
admittance current
YL =
1
iωL
YC = iωC
YR =
I =V(
1
R
1
1
+ i(ωC −
))
R
ωL
Iˆ = V ( R12 + (ωC +
tan φ =
RωC −
Large ω : L1 , V ωC is important.
V
Small ω : no C , ωL
important.
Can we do equivalent of Thevenin’s?
1 2 1/2
ωL ) )
R
ωL
4
V
= χef f ective
I
zef f = Ref f + iχef f
First term decays, second term oscillates.
Power Dissipation
R does this! (LC circuit just oscillates, even w/o driver no loss of
power).
dV
= RI 2
dt
(= V I)
5
z = R = iχ
z=
iχ
V =
Vˆ eiωt
1
T
< P >avg =
=
iχI
ˆ iωt+ π2
= χIe
�T
0
�T
V · Idt
�T
Iˆ2 R cos2 (ωt)dt − T1 0 χIˆ2 · cos ωt sin ωtdt
0
Iˆ2 R
2
=
Ladder Impedence
z = z1 +
z2 z
z2 + z
Solve:
z1
z=
+
z
�
z12
+ z1 z2
4
6
Let:
z1 = iωL
z2 =
1
iωC
•
iωL
z=
+
2
�
−ω 2 L2 L
+
4
C
•
v<0
for
ω2 >
• for ω 2 <
4
LC
ω 2 L2
L
>
4
C
4
LC
, there’s a real part = resistance! But from only
L = C? It’s because its infinite! Energy keeps traveling out for
certain ω !
Critical Frequency - if you are under, energy will just keep going
oout. Otherwise, will go out and come back.