Bioinstrumentation - Introduction to Biomedical Engineering - page 549

EXAMPLE PROBLEM 9.13
Find

i for the following circuit.

i

v (V)

1 +

−

v

2 F

0 1 2 t (s)

Solution
For

t <

0 and

t >

2,

v ¼

0 V, and therefore

i ¼

0 in this interval : For nonzero values, the voltage

waveform is described with two different functions:

v ¼ t

V for 0 t 1, and

v ¼ t

ð 2 Þ V

for 1 < t 2 : Equation (9.24) is used to determine the current for each interval as follows.

For 0 < t < 1

i ¼ C dv

d
dt

dt ¼

2

ðÞ¼

2A

For 1 t 2

i ¼ C dv d

dt ¼

2

dt t

ð ð 2 Þ Þ ¼ 2A

Voltage cannot change instantaneously across a capacitor. To have a step change in volt-
age across a capacitor, an infinite current must flow through the capacitor, and that is not
physically possible. Of course, this is mathematically possible using a Dirac delta function.
Equation (9.24) defines the current through a capacitor for a given voltage. Suppose one
is given a current through a capacitor and asked to find the voltage. To find the voltage, we
start from Eq. (9.24) by multiplying both sides by dt, giving

i ðÞ dt ¼ Cdv

Integrating both sides yields

Z t Z v ð t Þ

i ðÞ d l ¼ C dv

t o

v ð t o Þ