Digital Signal Processing Reference
In-Depth Information
i
(
t
)
x
(
t
)
f
(
t
)
v
(
t
)
M
L
(a)
(b)
Figure 1.15
Analogous systems.
i
(
t
)
C
L
Figure 1.16
LC
circuit
.
We see that the model for the mass in (1.16) and for the circuit in (1.18) are of the
same mathematical form; these two systems are called
analogous systems
. We de-
fine analogous systems as systems that are modeled by equations of the same math-
ematical form.
As a second example, consider the
LC
circuit in Figure 1.16, which is excited
by initial conditions. The loop equation for this circuit is given by
t
L
di(t)
dt
1
C
L
+
i(t)dt = 0.
(1.19)
-
q
Expressing this equation as a function of charge
q
(
t
) yields
d
2
q(t)
dt
2
1
LC
q(t) = 0.
+
(1.20)
Recall the linearized equation for a simple pendulum:
d
2
u(t)
dt
2
g
L
u(t) = 0.
[eq(1.15)]
+
Comparing the last two equations, we see that the pendulum and the
LC
circuit are
analogous systems.
In the two preceding examples, analogous electrical circuits are found for
two mechanical systems. We can also find analogous thermal systems, analogous
fluidic systems, and so on. Suppose that we know the characteristics of the
LC
cir-
cuit; we then know the characteristics of the simple pendulum. We can transfer
our knowledge of the characteristics of circuits to the understanding of other types
