Significance of components; phasor voltages

Significance of symmetrical components

Symmetrical components replace an unbalanced set of three phasors by three balanced sets (zero, positive and negative sequence).

• For symmetrical equipment the three sequence networks are independent, so unbalanced faults and loads are solved with three single-phase circuits instead of coupled three-phase equations.
• Unsymmetrical faults (L-G, L-L, L-L-G) are solved by simple interconnection of sequence networks.
• Each sequence has a physical meaning: negative sequence causes rotor heating of machines; zero sequence shows ground currents. Many protective relays work on these quantities.

Phase voltages in terms of symmetrical components

With $a = 1\angle120^\circ$ ($a^2 = 1\angle240^\circ$, $1+a+a^2 = 0$), the components of phases b and c are expressed through phase a:

• Zero sequence: $V_{b0} = V_{c0} = V_{a0}$
• Positive sequence: $V_{b1} = a^2V_{a1}$, $V_{c1} = aV_{a1}$
• Negative sequence: $V_{b2} = aV_{a2}$, $V_{c2} = a^2V_{a2}$

Each phase voltage is the sum of its components:

$$\begin{aligned} V_a &= V_{a0} + V_{a1} + V_{a2} \\ V_b &= V_{a0} + a^2V_{a1} + aV_{a2} \\ V_c &= V_{a0} + aV_{a1} + a^2V_{a2} \end{aligned}$$ $$\begin{bmatrix} V_a\\V_b\\V_c \end{bmatrix} = \begin{bmatrix} 1&1&1\\1&a^2&a\\1&a&a^2 \end{bmatrix}\begin{bmatrix} V_{a0}\\V_{a1}\\V_{a2} \end{bmatrix}$$