At time t=0, a potential difference of 475 V is applied to a coil with an inductance L=0.642 H and a resistance R=36 ohms. How long does it take to reach 20 percent of maximum energy in the inductor?
V = L di/dt + i R
i is of form i = (V/R)(1 - e^-kt)
di/dt = (V/R)k e^-kt
so
V = (L V/R )k e^-kt +R(V/R)(1-e^-kt)
1 = (L/R)k e^-kt + 1 - e^-kt
0 = (L/R)k -1
so
k = R/L
so
i = (V/R) (1 - e^-(R/L) t )
U = energy stored in magnetic field of inductor = (1/2) L i^2
U = (1/2) L V^2/R^2 (1-e^-kt)^2
this is maximum when i is maximum which is as t--> oo
U max = (1/2)L(V/R)^2
.2 Umax = 0.1 L(V/R)^2
so when is that true?
0.1 L(V/R)^2 = .5 L(V/R)^2 (1-e^-kt)^2
.2 = (1-e^-kt)^2
1 - e^-kt = sqrt .2 = .447
e^-kt = 1-.447 = .552786
1/e^kt = .552786
e^kt = 1.81
kt = ln 1.81 = .593
t = .593/k
but k = R/L = 36/.642
so
t = .593 (.642/36) = .0106 s
Thanks!
You are welcome but ALWAYS check my arithmetic :)
To determine the time it takes for the inductor to reach 20 percent of its maximum energy, we need to calculate the energy stored in the inductor at maximum and then use the exponential nature of energy buildup in an inductor to find the time.
The energy stored in an inductor can be calculated using the formula:
E = (1/2) * L * I^2
where E is the energy stored in the inductor, L is the inductance, and I is the current flowing through the inductor.
In this case, we need to find the current flowing through the inductor when it reaches 20 percent of its maximum energy. Let's say this current is I20. We can calculate it using the formula:
I20 = sqrt(2E / L)
where E is 20 percent of the maximum energy and L is the inductance.
Now, let's calculate the maximum energy stored in the inductor:
Emax = (1/2) * L * Imax^2
Since we know the applied potential difference and the resistance of the coil, we can calculate the maximum current, Imax, using Ohm's law:
Imax = V / R
where V is the potential difference and R is the resistance.
Now that we have the maximum energy, Emax, and the current at 20 percent energy, I20, we can use the exponential decay of energy in an inductor to find the time it takes to reach 20 percent energy.
The energy buildup in an inductor follows the equation:
E(t) = Emax * (1 - e^(-t / Tau))
where E(t) is the energy at time t, Emax is the maximum energy, t is the time, and Tau is the time constant given by L / R.
We want to find the time, t, when the energy equals 20 percent of the maximum energy, 0.2 * Emax.
0.2 * Emax = Emax * (1 - e^(-t / Tau))
Now we can solve this equation for t.
Let's plug in the given values and calculate the time it takes to reach 20 percent of maximum energy in the inductor.