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 this problem, we need to use the concepts of AC circuit analysis, including impedance calculations, phasor diagrams, and power calculations.
(a) Determining the maximum current in the circuit:
The maximum current in an AC circuit can be calculated using Ohm's law in the phasor domain. In this case, the impedance (Z) of the circuit is given by:
Z = √(R² + (XL - XC)²)
Where:
R is the resistance,
XL is the inductive reactance,
XC is the capacitive reactance.
In this problem, the inductive reactance XL is given by the equation XL = ωL, where L is the inductance of the inductor (20 mH) and ω is the angular frequency (π rad/s).
Using this information, let's calculate the inductive reactance XL and the impedance Z:
XL = ωL = π * 20 mH = 0.02 π ohms
R = 55 ohms
Z = √(55² + (0.02 π - 0)²)
Now, calculate the maximum current (Imax) using Ohm's law in the phasor domain:
Imax = Vmax / Z
Since the maximum voltage (Vmax) is given as 310 V, substitute this value and the calculated impedance Z to find the maximum current.
(b) Determining the maximum and rms voltages across the inductor:
In an AC circuit, the voltage across an inductor (VL) can be calculated using the formula:
VL = I * XL
Substituting the values calculated earlier for the maximum current (Imax) and the inductive reactance (XL), we can find the maximum voltage (Vmax) across the inductor.
The root mean square (rms) voltage across the inductor can be calculated as:
VRMS = Vmax / √2
(c) Determining the average power dissipation:
In an AC circuit, the average power dissipated (Pavg) can be calculated using the formula:
Pavg = VRMS * IRMS * cos(θ)
Where:
VRMS is the rms voltage,
IRMS is the rms current, and
cos(θ) is the power factor angle.
To calculate Pavg, we need to find the rms current (IRMS) and the power factor angle (θ).
(d) Determining 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 calculated using the formula:
Emax = (1/2) * L * Imax²
Substituting the values of L and Imax, we can calculate Emax.
The average energy stored in the magnetic field of the inductor can be calculated as:
Eavg = (1/2) * L * IRMS²
Substituting the value of L and the calculated IRMS, we can calculate Eavg.