2.1 Define the term impedance with reference to RLC circuits - NSC Electrical Technology Power Systems - Question 2 - 2018 - Paper 1

In the case of a pure capacitive circuit, the voltage lags the current by 90 degrees, while in a pure inductive circuit, the current lags the voltage by 90 degrees. These can be represented in graphical waveforms where the sine wave of voltage is shifted.

• For a capacitive circuit:
  • Voltage Waveform: Starts at 0 and peaks after current.
• For an inductive circuit:
  • Current Waveform: Starts after the voltage peaks.

To find the total impedance (Z) in the RLC circuit, we use the formula:

$Z = \sqrt{R^2 + (X_L - X_C)^2}$

Substituting the given values:

$Z = \sqrt{12^2 + (22 - 36)^2} = \sqrt{12^2 + (-14)^2}$ $= \sqrt{144 + 196}$ $= \sqrt{340}$ $= 18.44 \Omega$

The current (I) can be calculated using Ohm's law:

$I = \frac{V_s}{Z}$

Substituting the values:

$I = \frac{60}{18.44} \approx 3.25 A$

The Q-factor is calculated using the formula:

$Q = \frac{V_s}{I \times \sin(\theta)}$

Here, we know:

• $\theta$ is the angle corresponding to the phase difference (50° in this case), hence:

$Q = \frac{60 \times 3.25}{\sin(50^\circ)} \approx 149.83$

The value of the inductive reactance (!X_L) is directly proportional to the frequency (f) and can be calculated using:

$X_L = 2\pi f L$

This means that as frequency increases, the inductive reactance also increases, leading to greater opposition to current.

The resonant frequency ($f_0$) is the frequency at which the inductive reactance ($X_L$) equals the capacitive reactance ($X_C$). At this frequency, the circuit can achieve maximum current with minimum resistance. This can be expressed mathematically, and it is a critical point for analyzing RLC circuits.