Differential Equation Applications
Kirchoff’s Law
The current, i in a series circuit with constant inductance, L , constant resistance, R and a constant voltage, V applied is described by:
L di/dt + Ri = V
If we have such a circuit with L = 3 henrys, R = 8 ohms, V = 20 volts, find the current after 60 seconds.
What is known?
This is a nice problem, because the equation is given, we need only to plug in the given values:
3 di/dt + 8i = 20
We are looking for i = ? when t = 60.
Separate variables and integrate:
3 di/(20-8i) = dt
Integrating: -3ln(20-8i)/8 = t + C
Now, solve for the constants:
ln (20-8i) = -8t/3 + C
20 – 8i = C e-8t/3
But we need some initial conditions to solve for C. Since you all are Electrical Technology majors, you know that when t = 0, i = 0, right?
Using this:
C = 20
Then i = 2.5(1 – e-8t/3)
So when t = 60, i = 2.5A
