A 20 mH inductor in series with a resistance of 55 ohms is connected to a source whose voltage is = 310 V cos 150πt, where t is in seconds.
(a) Determine the maximum current in the circuit. (I solved this already, I=5.63A)
(b) Determine the maximum and rms voltages across the inductor.
(c) Determine the average power dissipation.
(d) Determine the maximum and average energy stored in the magnetic field of the inductor.
To solve parts (b), (c), and (d) of the problem, we need to find the current through the inductor in terms of time.
We know that the voltage across an inductor is given by the formula:
V(t) = L * di/dt
where V(t) is the voltage across the inductor, L is the inductance, and di/dt is the rate of change of current with respect to time.
To find the current through the inductor, we can integrate the voltage equation with respect to time:
∫V(t) dt = ∫L * di/dt dt
By integrating both sides, we get:
∫V(t) dt = L * ∫di
Integrating the left side of the equation gives us the total change in voltage over a period of time, which we can calculate. On the right side, the integral of di represents the change in current.
Now, let's proceed to solve the problem:
(b) Determine the maximum and rms voltages across the inductor.
To find the maximum voltage across the inductor, we can use the formula:
Vmax = L * (dImax/dt)
We have already solved part (a) and obtained the maximum current in the circuit, which is 5.63 A. To determine the rate of change of current, we need to differentiate the equation for current with respect to time (t):
i(t) = (1/L) * ∫V(t) dt
Differentiating both sides with respect to t gives us:
di(t)/dt = (1/L) * dV(t)/dt
Since we are given the equation for V(t) as 310 V cos(150πt), we can differentiate it to find dV(t)/dt. After differentiating, we substitute the values into the equation above to find dImax/dt.
To find the rms voltage across the inductor, we use the formula:
Vrms = Vmax / √2
(c) Determine the average power dissipation.
The average power dissipation in the circuit can be found using the formula:
Pavg = IVrms * cosθ
Where I is the rms current flowing through the circuit, Vrms is the rms voltage across the inductor, and θ represents the phase angle between the current and voltage waveforms.
(d) Determine the maximum and average energy stored in the magnetic field of the inductor.
The maximum energy stored in the magnetic field of the inductor can be found using the equation:
Emax = (1/2) * LI^2
Where L is the inductance and I is the maximum current.
The average energy stored in the magnetic field of the inductor can be found using the formula:
Eavg = (1/2) * LIrms^2
Where L is the inductance and Irms is the rms current.
Please note that for the calculations, you need to substitute the appropriate values into the formulas to determine the values for parts (b), (c), and (d) of the problem.