Math 2280-001
Mon Mar 2
, Use last Friday's notes to understand damped forced oscillations,
m x##C c x#C k x = F0 cos w t
i.e. with c O 0. As we discussed briefly on Friday, in this case the undetermined coefficients particular
solution is called the "steady periodic" solution, because this sinusoidal solution persists as t/N,
whereas the general homogeneous solution always decays exponentially to zero.
And then do something completely different, and entirely the same:
, 3.7 Electrical circuits.
It turns out that the mathematics of RLC circuits is identical to that for damped mass spring configurations:
http://cnx.org/content/m21475/latest/pic012.png
Charge Q t coulombs accumulates on the capacitor, at a rate (current) I t
amperes (coulombs/sec), i.e Q # t = I t .
i t in the diagram above)
Kirchoff's Law: The sum of the voltage drops around any closed circuit loop equals the applied voltage
V t (volts). The units of voltage are energy units - Kirchoff's Law says that a test particle traversing any
closed loop returns with the same potential energy level it started with:
1
For Q t :
L Q ## t C R Q # t C
Q t =V t
C
1
For I t :
L I## t C R I# t C
I t = V# t
C
If we specify that the voltage is given by a periodic sine function then V# t will be a cosine function:
1
For Q t :
L Q ## t C R Q # t C
Q t = V t = E0 sin w t
C
1
For I t :
L I## t C R I# t C
I t = V# t = E0 w cos w t .
C
1
We can just transcribe all of our work from mass-spring systems. Just substitute L for m, R for c,
for
C
k, and I t for x t . (There could be other situations where you want to study Q t instead, depending on
the context.)
1
I t = V# t = E0 w cos w t
C
For example, for RLC circuits with R O 0 the general solution for I t is
I t = Isp t C Itr t .
L I## t C R I# t C
p
.
2
(One difference from the forced mass-spring configuration it makes sense to express Isp t as
Isp t = I0 cos w t K a = I0 sin w t K g , g = a K
I0 sin w t K g , since the applied voltage is E0 sin w t . Since 0 % a % p, it follows that
p
p
%g%
. Transcribing ...
2
2
K
F0
C w =
kKm w
2 2
C c2w
0 I0 w =
The denominator
1
KL w
Cw
2
2
E0 w
0 I0 w =
1
2
KL w
C
E0
1
KL w
Cw
2
C R2 w
2
.
2
C R2
C R2 of I0 w is called the impedance Z w of the circuit (because
the larger the impedance, the smaller the amplitude of the steady-periodic current that flows through the
circuit). Notice that for fixed resistance, the impedance is minimized and the steady periodic current
1
amplitude is maximized when
= L w , i.e.
Cw
1
C=
if L is fixed and C is adjustable (old radios).
2
Lw
1
L=
if C is fixed and L is adjustable
2
Cw
Both L and C are adjusted in this M.I.T. lab demonstration:
http://www.youtube.com/watch?v=ZYgFuUl9_Vs.