3.1 Define a phasor diagram with reference to RLC circuits connected across an alternating voltage supply - NSC Electrical Technology Electronics - Question 3 - 2022 - Paper 1

The total supply voltage ( V_t) can be calculated using the formula: $V_t = \sqrt{(V_R - V_C)^2 + (V_L)^2}$ Substituting the values, we have: $V_t = \sqrt{(150 - 90)^2 + (180)^2}$
$= \sqrt{(60)^2 + (180)^2}$
$= \sqrt{3600 + 32400}$

The power factor is considered lagging because the inductive voltage (V_L) is greater than the capacitive voltage (V_C). In RLC circuits, when the inductance dominates, the current lags behind the voltage, thus leading to a lagging power factor.

The total current (I_T) in a parallel circuit can be calculated using the formula: $I_T = \sqrt{I_C^2 + I_L^2 + I_R^2}$ Where:

• I_C = 4 A (capacitive current)
• I_L = 6 A (inductive current)
• I_R = 4 A (resistive current)
  Thus, the calculation is as follows:
  $I_T = \sqrt{(4)^2 + (6)^2 + (4)^2}$
  $= \sqrt{16 + 36 + 16}$

The phase angle (θ) can be calculated using the formula: $\theta = \cos^{-1} \left( \frac{I_R}{I_T} \right)$ Plugging in the values:
$\theta = \cos^{-1} \left( \frac{4}{8.25} \right)$
$\approx \cos^{-1}(0.4858) \approx 61.65°$.

The phasor diagram for the parallel RLC circuit will show the capacitive current (I_C), inductive current (I_L), and resistive current (I_R) with their respective phase angles relative to the total current (I_T). The vectors should be arranged in such a way that the angle θ between I_T and I_R indicates the phase difference, with I_C leading and I_L lagging. (A sketched diagram is recommended if possible.)

The circuit is predominantly inductive because the inductive current (I_L = 6 A) is greater than the capacitive current (I_C = 4 A). This indicates that the circuit behavior is influenced more by the inductance.

At resonance, the quality factor (Q) can be calculated using the formula: $Q = \frac{R}{X_L}$ Where:

• R = 2200 Ω
• X_L = 150 Ω
  Substituting the values:
  $Q = \frac{2200}{150} \approx 14.67$.

The bandwidth (BW) can be calculated using the formula:
$BW = \frac{f_r}{Q}$ Where:

• f_r = 2,387 kHz
• Q ≈ 14.67
  Thus:
  $BW = \frac{2,387 \times 10^3}{14.67} \approx 162.82 Hz$.

The reactance of the capacitor (X_C) can be determined using:
$X_C = \frac{1}{2 \pi f_r C}$ Rearranging gives:
$C = \frac{1}{2 \pi f_r X_C}$ Where:

• f_r = 2,387 kHz
• X_C = 150 Ω
  Calculating gives:
  $C = \frac{1}{2 \pi \times 2,387 \times 10^3 \times 150} \approx 4.45 \times 10^{-7} F => 444.51 nF$.

Selectivity is a measure of how well a resonant circuit responds to a range of frequencies while excluding others. A higher selectivity indicates that the circuit can better discriminate between its resonant frequency and other frequencies.

The average power in an AC circuit can be calculated using:
$P = V_{rms} I_{rms} \cos(\phi)$ Where:

• V_{rms} is the root mean square voltage,
• I_{rms} is the root mean square current,
• \phi is the phase angle between the voltage and current. The average power represents the effective power consumed by the circuit over a period.