Bioinstrumentation - Introduction to Biomedical Engineering - page 546

EXAMPLE PROBLEM 9.11
Find

v in the following circuit.

i (A)

+
v
-

1

i 2 H

0 1 2 3 t (s)

Solution
The solution to this problem is best approached by breaking it up into time intervals consistent
with the changes in input current. Clearly, for t < 0 and t > 2, the current is zero and therefore

v ¼ 0. We use Eq. (9.19) to determine the voltage in the other two intervals as follows.

For 0 < t < 1
In this interval, the input is

i ¼ t

, and

v ¼ L di 2 d ð t Þ

dt ¼ dt ¼

2V

For 1 t 2
In this interval, the input is

i ¼ t

ð 2 Þ , and

v ¼ L di 2 d t

ð ð 2 Þ Þ

dt ¼

¼ 2V

dt

Equation (9.19) defines the voltage across an inductor for a given current. Suppose one is
given a voltage across an inductor and asked to find the current. We start from Eq. (9.19) by
multiplying both sides by giving

dt,

v ðÞ dt ¼ Ldi

Integrating both sides yields

Z t Z i ð t Þ

v ðÞ d l ¼ L

a ð 9 : 20 Þ

d

t 0

i ð t 0 Þ

or

Z t

1
L

i ð t Þ¼ v ðÞ d l þ i ð t Þ

0

t 0