Explain the term inductance with reference to RLC circuits connected to an AC supply - NSC Electrical Technology Power Systems - Question 3 - 2022 - Paper 1

To calculate the impedance ($Z$) of the circuit, we use the formula:

$Z = \sqrt{R^2 + (X_L - X_C)^2}$ Substituting the given values:

• $R = 25 \Omega$
• $X_C = 13 \Omega$
• $X_L = 94 \Omega$

$Z = \sqrt{25^2 + (94 - 13)^2} = \sqrt{25^2 + 81^2} \approx 84.77 \Omega$

The phase angle ($\theta$) can be calculated using the formula:

$\theta = \tan^{-1}\left(\frac{X_L - X_C}{R}\right)$ Substituting the given values:

$\theta = \tan^{-1}\left(\frac{94 - 13}{25}\right) = \tan^{-1}\left(\frac{81}{25}\right) \approx 72.85^\circ$

The value of the inductor ($L$) can be calculated with the formula:

$L = \frac{X_L}{2\pi f}$ Substituting the values:

• $X_L = 94 \Omega$
• $f = 60 Hz$

$L = \frac{94}{2 \times \pi \times 60} \approx 0.25 H = 250 mH$

A lagging power factor occurs when the current lags behind the voltage in an RLC circuit. This is typically due to the inductive nature of the circuit, where the inductive load causes the current to delay relative to the applied voltage.

At resonance, the inductive reactance ($X_L$) and capacitive reactance ($X_C$) cancel each other out. This results in the circuit becoming purely resistive, causing the current and voltage waveforms to be in phase with each other.

To calculate the total current ($I_T$) in the circuit, use the formula:

$I_T = \sqrt{I_R^2 + (I_L - I_C)^2}$ Substituting the values:

• $I_R = 11 A$
• $I_C = 7 A$
• $I_L = 9 A$

$I_T = \sqrt{11^2 + (9 - 7)^2} = \sqrt{11^2 + 2^2} \approx 11.18 A$

The power factor ($\cos \phi$) can be calculated using:

$\cos \phi = \frac{I_R}{I_T}$ Substituting:

$\cos \phi = \frac{11}{11.18} \approx 0.98$

The total power ($P$) can be calculated with:

$P = V_T \times I_T \times \cos \phi$ Substituting:

$P = 110 \times 11.18 \times 0.98 \approx 1205.20 W$

The circuit has a lagging power factor because the inductive current is greater than the capacitive current, indicating that the current lags the voltage.

A parallel RLC circuit produces the response at points A and B in Figure 3.5.

The impedance ($Z$) represents the total opposition to the flow of alternating current in a circuit, while the current ($I$) is the actual flow of electric charge. Impedance, being a complex quantity, considers both resistance and reactance.

As the frequency increases, the impedance decreases in Figure 3.5 A. This is because the inductive reactance increases with frequency, while the capacitive reactance decreases, thus affecting the total impedance of the circuit.