Page 85 -

ric function of x? What modification to the curve is observed if the degree of
the polynomial is increased?

Application to a Simple RC Circuit
The solution giving the voltage across the capacitor in Figure 3.2 following
the closing of the switch can be written in the following form:



Vt( ) = V ( )exp −0  t  + V 1 − exp − t  (3.15)

s
c c   RC     RC
V (t) is called the time response of the RC circuit, or the circuit output result-
c
ing from the constant input V . The time constant RC of the circuit has the
s
units of seconds and, as you will observe in the present analysis and other
problems in subsequent chapters, its ratio to the characteristic time of a given
input potential determines qualitatively the output of the system.

FIGURE 3.2
The circuit used in charging a capacitor.

In-Class Exercise

Pb. 3.26 A circuit designer can produce outputs of various shapes by select-
ing specific values for the circuit time constant RC. In the following simula-
tions, you can examine the influence of this time constant on the response of
the circuit of Figure 3.2.
Using V (0) = 3 volts, V = 10 volts (capacitor charging process), and RC = 1 s:
s
c
a. Sketch a graph of V (t). What is the asymptotic value of the solu-
c
tion? How long does it take the capacitor voltage to reach the value
of 9 volts?
b. Produce an M-file that will plot several curves of V (t) correspond-
c
ing to:

© 2001 by CRC Press LLC