A certain ac signal at 60 Hz is applied across a 40-mH inductor and a 200-Ù resistor. What is the power factor of this circuit?
To find the power factor of the circuit, we need to calculate the impedance of the inductor and the resistor, and then find the power factor angle.
1. Calculate the inductive reactance (XL) using the formula XL = 2πfL, where f is the frequency (60 Hz) and L is the inductance (40 mH = 0.04 H):
XL = 2π(60)(0.04) = 15.08 Ω
2. The resistance (R) is given as 200 Ω.
3. Calculate the impedance (Z) of the circuit by using the formula Z = √(R^2 + XL^2):
Z = √(200^2 + 15.08^2) = √(40000 + 226.8064) = √40226.8064 = 200.56 Ω
4. The power factor angle (θ) can be found using the tangent of the angle:
θ = arctan(XL/R) = arctan(15.08/200) = 4.28°
5. Finally, calculate the power factor (PF) by taking the cosine of the power factor angle:
PF = cos(θ) = cos(4.28°) ≈ 0.999
Therefore, the power factor of this circuit is approximately 0.999.
To find the power factor of this circuit, we need to understand the relationship between voltage, current, and phase angle in an AC circuit. The power factor is defined as the cosine of the angle between the voltage and current waveforms.
In this case, the 60 Hz AC signal is applied across a 40-mH inductor and a 200-Ù resistor. Since the inductor and resistor are in series, the total impedance of the circuit can be calculated using the formula:
Z = √(R^2 + (XL - XC)^2)
Here, R is the resistance, XL is the inductive reactance, and XC is the capacitive reactance (which is 0 in this case since there is no capacitor in the circuit).
The inductive reactance (XL) can be calculated using the formula:
XL = 2πfL
Where f is the frequency (60 Hz) and L is the inductance (40 mH = 0.04 H).
Now, we can calculate the values:
R = 200-Ù
f = 60 Hz
L = 0.04 H
XL = 2πfL
= 2π(60)(0.04)
≈ 15.08-Ù
Z = √(R^2 + (XL - XC)^2)
= √(200^2 + (15.08)^2)
≈ √(40000 + 226.81)
≈ √40226.81
≈ 200.56-Ù
Next, we need to find the angle between the voltage and current waveforms. This can be done using the formula:
θ = arctan((XL - XC) / R)
θ = arctan((15.08-Ù - 0) / 200-Ù)
= arctan(15.08 / 200)
≈ arctan(0.0754)
≈ 4.31°
Finally, we can find the power factor, which is the cosine of the angle:
power factor (PF) = cos(θ)
= cos(4.31°)
≈ 0.9997
Therefore, the power factor of this circuit is approximately 0.9997.