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 answer parts (b), (c), and (d) of the question, we first need to calculate the impedance and current in the circuit.
(a) Maximum current in the circuit:
Given that the voltage source is V(t) = 310 V cos(150πt), we can find the maximum current by dividing the maximum voltage by the total impedance of the circuit. The impedance, Z, is given by the formula:
Z = √(R² + X_L²)
Where R is the resistance and X_L is the inductive reactance. The inductive reactance can be calculated using the formula:
X_L = 2πfL
Where L is the inductance and f is the frequency. In this case, we are not given the frequency but can assume it to be 150 Hz (based on the cos(150πt) term in the voltage equation.)
Calculating X_L:
X_L = 2πfL
= 2π(150)(20 x 10^-3)
≈ 37.7 Ω
Now, we can calculate the impedance:
Z = √(R² + X_L²)
= √((55)² + (37.7)²)
≈ 66.9 Ω
The maximum current, I_max, is given by:
I_max = V_max / Z
= 310 / 66.9
≈ 4.63 A
So, the maximum current in the circuit is approximately 4.63 A, which differs from the value you previously obtained.
(b) Maximum and RMS voltages across the inductor:
The maximum voltage across an inductor can be calculated by multiplying the maximum current by the inductive reactance, X_L:
V_L(max) = I_max * X_L
= (4.63) * (37.7)
≈ 174.0 V
The RMS voltage across the inductor is given by:
V_L(rms) = V_L(max) / √2
≈ 174.0 / √2
≈ 123.0 V
So, the maximum voltage across the inductor is approximately 174.0 V, and the RMS voltage is approximately 123.0 V.
(c) Average power dissipation:
The average power dissipated in the circuit can be calculated using the formula:
P_avg = (1/2) * V_avg * I_avg * cos(θ)
Where V_avg and I_avg are the average voltage and current, respectively, and θ is the phase angle between the voltage and current waveforms. In this case, the voltage and current waveforms are in phase because there is no mention of any phase shift or reactive component in the resistance.
Since the voltage waveform is symmetrical, the average voltage is 0:
V_avg = 0
The average current, I_avg, is equal to the maximum current divided by √2:
I_avg = I_max / √2
= 4.63 / √2
≈ 3.27 A
Therefore, the average power dissipation, P_avg, is:
P_avg ≈ (1/2) * 0 * (3.27) * cos(0)
= 0
So, the average power dissipation in the circuit is 0.
(d) Maximum and average energy stored in the magnetic field of the inductor:
The energy stored in the magnetic field of an inductor can be calculated using the formula:
W = (1/2) * L * I²
Where L is the inductance and I is the current.
The maximum energy stored in the magnetic field is given when the current is at its maximum (I_max):
W_max = (1/2) * (20 x 10^-3) * (4.63)²
≈ 0.04 J
The average energy stored in the magnetic field can be calculated using the formula:
W_avg = (1/2) * L * I_avg²
Where I_avg is the average current.
W_avg = (1/2) * (20 x 10^-3) * (3.27)²
≈ 0.034 J
So, the maximum energy stored in the magnetic field of the inductor is approximately 0.04 J, and the average energy stored is approximately 0.034 J.