2.1 Define the following with reference to RLC circuits: 2.1.1 Power factor 2.1.2 Q-factor of an inductor in a resonant circuit - NSC Electrical Technology Electronics - Question 2 - 2019 - Paper 1

The Q-factor (Quality Factor) of an inductor in a resonant circuit refers to the ratio of the inductor's inductive reactance ( $X_L$) to its internal resistance (R). It can be expressed as:

$Q = \frac{X_L}{R}$

This factor indicates the efficiency of the inductor; a higher Q-factor represents lower energy losses.

1. Wattless voltage dividers: These circuits are used to distribute voltage without any power loss.

2. Oscillating circuits: RLC circuits are commonly used to create oscillators for signal generation in electronics.

To find the total voltage ( $V_T$) in the series circuit, we can apply the Pythagorean theorem:

$V_T = \sqrt{V_R^2 + (V_C - V_L)^2}$ Substituting the given values:

$= \sqrt{12^2 + (24 - 16)^2}$ $= \sqrt{12^2 + 8^2}$ $= \sqrt{144 + 64}$ $= \sqrt{208}$ $= 14.42 V$

Using the formula for inductive reactance:

$X_L = \frac{V_L}{I_T}$ Substituting the values:

$X_L = \frac{16 V}{3 A} = 5.33 Ω$

At resonance, $X_L = X_C$. Therefore, we can express the capacitance as follows:

$C = \frac{1}{2\pi f X_C}$ Substituting the known values:

$X_C = 50 Ω$ $f = 2 kHz = 2000 Hz$

So,

$C = \frac{1}{2\pi \cdot 2000 \cdot 50} = 1.59 \mu F$

Using Ohm's law and the resonance condition where $Z = R$:

$I = \frac{V_T}{Z} = \frac{120 V}{12 Ω} = 10 A$

If the resistance is doubled, the current in the circuit will be halved, as current is inversely proportional to resistance when voltage is constant.

1. The impedance (Z) is at minimum, equal to the resistance (R).

2. The voltage across the inductor and capacitor are equal in magnitude but opposite in phase, leading to resonant conditions.

3. The current is at its maximum value for a given applied voltage.