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

To find the total supply voltage (V_T), use the formula:

$V_T = \sqrt{V_R^2 + (V_L - V_C)^2}$

Substituting the given values: $V_T = \sqrt{150^2 + (180 - 90)^2} = \sqrt{150^2 + 90^2}$

Calculating: $V_T = \sqrt{22500 + 8100} = \sqrt{30600} \approx 174.93 \text{ V}$

The power factor in this circuit is lagging because the inductive voltage (V_L) is greater than the capacitive voltage (V_C). Hence, the overall effect leads to a situation where the voltage lags behind the current.

Using the formula for total current in a parallel RLC circuit:

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

Substituting the values: $I_T = \sqrt{4^2 + (6 - 4)^2} = \sqrt{16 + 4} = \sqrt{20} \approx 4.47 A$

The phase angle (θ) can be calculated using:

$\theta = \cos^{-1}\left(\frac{I_R}{I_T}\right)$

Substituting the values: $\theta = \cos^{-1}\left(\frac{4}{4.47}\right) \approx 26.49^{\circ}$

In a phasor diagram for FIGURE 3.3, represent the currents I_C, I_L, and I_R as vectors, and the total current I_T will be the resultant of these vectors. I_T is typically represented at an angle θ from the horizontal reference line.

The circuit is inductive because the inductive current (I_L) is greater than the capacitive current (I_C), indicating that the overall effect is determined more by the inductor.

At resonance, X_L = X_C. Using the formula for quality factor (Q):

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

Substituting known values: $Q = \frac{2200}{150} \approx 14.67$

The bandwidth (BW) can be calculated using:

$BW = \frac{f_r}{Q}$

Substituting its value: $BW = \frac{2,387 \times 10^3}{14.67} \approx 162.82 \text{ Hz}$

Using the resonant frequency equation:

$X_C = \frac{1}{2 \pi f_r C}$

Rearranging gives: $C = \frac{1}{2 \pi f_r X_C}$

Substituting: $C = \frac{1}{2 \pi \times 2,387 \times 10^3 \times 150} \approx 4.445 \times 10^{-7} F \text{ or } 444.51 nF$

Selectivity is a measure of how well a resonant circuit responds to a range of frequencies and excludes others. A high selectivity indicates a narrow bandwidth, meaning the circuit is more sensitive to resonant frequencies.

The waveform in FIGURE 3.5 (A) is produced by an inductor, as indicated by the phase relationship between the voltage and the current, where voltage leads.

The average power in FIGURE 3.5 (B) can be identified by calculating the product of the RMS voltage and current with the power factor. This indicates the real power being dissipated, while the peak values represent the instantaneous power.