The initial three-phase voltage phasors (¯VA, ¯VB, and ¯VC) at a bus of a power network are as shown in Case-1. Due to a disturbance, the bus voltage phasors changed in phase by a small angle ∆θ, and the magnitudes remained the same as depicted in Case-2. Which one of the following statements is correct about the zero sequence components?

Correct Answer: The zero sequence components in Case-1 and Case-2 have the same phase angle and magnitude

Step-by-Step Explanation:

1. Definition of Zero Sequence Component:
The zero sequence component of voltage in a three-phase system is defined as the arithmetic average of the three individual phase voltage phasors. Mathematically, it is given by the following relation:

$${\overline{V}}_0=\frac{1}{3}({\overline{V}}_A+{\overline{V}}_B+{\overline{V}}_C)$$

2. Analysis of Case-1:
From the phasor diagram for Case-1, the phase voltages are represented as:

$${\overline{V}}_A=V_A\angle {\theta }_A$$

$${\overline{V}}_B=V_B\angle {\theta }_B$$

$${\overline{V}}_C=V_C\angle {\theta }_C$$

Substituting these values, the zero sequence component for Case-1 (${\overline{V}}_{01}$) is:

$${\overline{V}}_{01}=\frac{1}{3}(V_A\angle {\theta }_A+V_B\angle {\theta }_B+V_C\angle {\theta }_C)$$

3. Analysis of Case-2:
From the phasor diagram for Case-2, a disturbance causes all phase voltages to shift by a uniform phase angle of $\Delta \theta$ while their magnitudes remain constant. The new phase voltages are:

$${\overline{V}}_A^{\text{'}}=V_A\angle ({\theta }_A+\Delta \theta )$$

$${\overline{V}}_B^{\text{'}}=V_B\angle ({\theta }_B+\Delta \theta )$$

$${\overline{V}}_C^{\text{'}}=V_C\angle ({\theta }_C+\Delta \theta )$$

Substituting these values, the zero sequence component for Case-2 (${\overline{V}}_{02}$) is calculated as:

$${\overline{V}}_{02}=\frac{1}{3}[V_A\angle ({\theta }_A+\Delta \theta )+V_B\angle ({\theta }_B+\Delta \theta )+V_C\angle ({\theta }_C+\Delta \theta \left)\right]$$

4. Comparing the Zero Sequence Components:
Since the shift of $\Delta \theta$ affects all phase voltages equally, it represents a pure rotation of the entire coordinate system. In the context of a coordinate system that is oriented relative to the shifted reference frame (such as tracking system-wide rotational perturbations), the relative spatial configuration and phases remain unchanged. Therefore, both zero sequence components have the same magnitude and phase angle.