ECE 209:
Circuits and Electronics Laboratory
Review of Circuits as LTI Systems
*
Math Background: ODE’s, LTI Systems, and Laplace Transforms
Engineers must have analytical machinery to understand how systems change over time.
For example,
springs and dampers in car suspension systems absorb kinetic energy after road disturbances and dissipate
the energy
gently
over time. Electronic filters protect circuits from large input transients in a similar way.
Hence, engineers need to know how to choose components in
dynamical
systems.
LTI Systems and ODEs:
The motion of many physical systems can be modeled knowing only a few
of its derivatives.
For example, a car’s position over time can be described well knowing its velocity or
acceleration. So we focus on the
linear timeinvariant
(LTI) system
x
(
t
)
LTI System
y
(
t
)
which has input
x
(
t
) (e.g., height of terrain under the car) and output
y
(
t
) (e.g., height of passenger’s seat).
By our assumption that all we need are a few derivatives of the input and the output, then a typical
ordinary
differential equation
(ODE) that might model such a system is
y
′′
+ 3
y
′
+ 2
y
=
x
′′
−
x
(1)
where
y
′
and
y
′′
are the first and second derivatives of signal
y
(
t
). Given a known input
x
(
t
), we would like
to
integrate
this differential equation to find an expression of
y
(
t
) that is only in terms of
t
. Then we can see
how to adjust our design parameters so that
y
(
t
) has a desirable shape (e.g., even with a bumpy
x
(
t
) road
surface, the car seat
y
(
t
) stays in one place). So we need a convenient way to
solve
this differential equation.
Linear Decomposition:
Assume that
x
can be broken into two parts so that
x
(
t
) =
x
1
(
t
) +
x
2
(
t
). By
inspection, if I can solve
Equation (1)
for the response
y
1
to
x
1
alone and the response
y
2
to
x
2
alone, then
the response
y
to
x
would be
y
1
+
y
2
. Further, imagine that I could find a set of functions such that:
•
When a function in the set is an input to
Equation (1)
, the output is simple to find.
•
Every useful input
x
(
t
) can be expressed as a sum (or integral) of functions from this set.
In this case, solving ODE’s like
Equation (1)
would be trivial. Before even knowing the input
x
, I could find
the solution to the ODE for every function in this special set. Then when I finally do know my input, I just
sum up the relevant prototypical solutions.
Consider functions of the form
t
mapsto→
e
st
where
s
=
σ
+
jω
is a
complex number
with real part
σ
and
imaginary part
ω
. Notice that for a function
f
(
t
) =
Ae
st
(where
A
is any constant complex number),
f
′
(
t
) =
sAe
st
=
sf
(
t
)
and
integraldisplay
f
(
t
) d
t
=
1
s
Ae
st
=
1
s
f
(
t
)
.
(2)
That is, for a
complex exponential
, differentiation is identical to multiplication by
s
and integration is identical
to division by
s
. So complex exponentials turn ODE calculus into simple algebra.
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 Fall '08
 Staff
 Complex number, Theodore P. Pavlic

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