An AC voltage of the form Δv = 100 sin 1 000t, where Δv is in volts and t is in seconds, is applied to a series RLC circuit. Assume the resistance is 475 Ω, the capacitance is 5.50 µF, and the inductance is 0.500 H. Find the average power delivered to the circuit.
Question
An AC voltage of the form Δv = 100 sin 1 000t, where Δv is in volts and t is in seconds, is applied to a series RLC circuit. Assume the resistance is 475 Ω, the capacitance is 5.50 µF, and the inductance is 0.500 H. Find the average power delivered to the circuit.
Solution
The average power delivered to a circuit is given by the formula:
P = Vrms * Irms * cos(Φ)
where:
- Vrms is the root mean square voltage,
- Irms is the root mean square current, and
- cos(Φ) is the power factor, which is the cosine of the phase angle between the current and the voltage.
First, we need to find the root mean square voltage (Vrms). For a sinusoidal AC voltage, Vrms is given by Vpeak/√2. The peak voltage (Vpeak) is the amplitude of the sinusoidal voltage, which is 100V in this case. So,
Vrms = Vpeak/√2 = 100V/√2 = 70.7V
Next, we need to find the root mean square current (Irms). The current in a series RLC circuit under sinusoidal steady state is given by the voltage divided by the impedance (Z) of the circuit. The impedance of a series RLC circuit is given by:
Z = √(R² + (XL - XC)²)
where:
- R is the resistance,
- XL is the inductive reactance, given by 2πfL, and
- XC is the capacitive reactance, given by 1/(2πfC).
The frequency (f) of the AC voltage is given by 1000 rad/s. Converting this to Hz, we get f = 1000/(2π) = 159.15 Hz.
So, XL = 2πfL = 2π * 159.15 Hz * 0.500 H = 500 Ω, and XC = 1/(2πfC) = 1/(2π * 159.15 Hz * 5.50 µF) = 183.26 Ω.
Substituting these values into the formula for Z, we get:
Z = √((475 Ω)² + (500 Ω - 183.26 Ω)²) = 570.09 Ω.
Now we can find Irms = Vrms/Z = 70.7V/570.09 Ω = 0.124 A.
Finally, we need to find the power factor cos(Φ). The phase angle (Φ) in a series RLC circuit is given by:
Φ = arctan((XL - XC)/R)
Substituting the values we found earlier, we get:
Φ = arctan((500 Ω - 183.26 Ω)/475 Ω) = 34.19°.
So, cos(Φ) = cos(34.19°) = 0.829.
Substituting all these values into the formula for P, we get:
P = Vrms * Irms * cos(Φ) = 70.7V * 0.124 A * 0.829 = 7.27 W.
So, the average power delivered to the circuit is approximately 7.27 Watts.
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