18.03 Class 14
, March 5, 2010
Complex gain
1. Recap
2. Phase lag
3. Driving via the dashpot
4. Complex gain
[1] The story so far:
We're studying solutions of linear constant coefficient
equations
a_n x^(n) + .
.. + a_1 x + a_0 = q(t)
(*)
A key is the characteristic polynomial
p(s) = a_n s^n + .
.. + a_1 s + a_0
For the homogeneous case,
a_n x^(n) + .
.. + a_1 x + a_0 = 0
(*)_h
we found that the roots of
p(s)
give the exponents in exponential solutions,
and that the general solution is a linear combination of these or (these
times a power of
t
in case there are repeated roots).
Euler's formula shows that
e^{(a+bi)t} = e^{at}
so: [Slide]
Transience Theorem:
All homogeneous solutions of (*)_h decay to zero provided that all the
roots of
p(s)
have negative real parts.
In this case the solutions to
(*)_h
are called "transients,"
By superposition, all solutions to (*) converge together
as
t
gets large, and we say that the equation is "stable."
If we have a system modeled by a stable equation, and we are only
interested in what it looks like after the transients have died down,
we can eliminate the initial condition:
____________
input


steady state
>
System
>
signal
____________
output signal
x_p
So we look for a particular solution
x_p .
Sinusoidal input signals are
of particular importance. Experiments indicate that sinusoidal in
gives sinusoidal out. We decide to set our clock so that the input signal is
input = A cos(omega t)
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View Full DocumentExperiments indicate that the steady state output signal is again sinusoidal,
of the same circular frequency:
output = x = B cos (omega t  phi)
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 Spring '09
 vogan
 Exponential Function, Equations, Complex number, Euler's formula, input signal, dashpot Complex

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