Power invariance of symmetrical transformation

A transformation is power invariant if the power calculated from the transformed (sequence) quantities equals the power calculated from the original (phase) quantities. The symmetrical component transformation gives $S = 3(V_{a0}I_{a0}^* + V_{a1}I_{a1}^* + V_{a2}I_{a2}^*)$, which is power invariant apart from a constant factor 3, and exactly invariant in its unitary form.

Proof

Complex power in phase quantities:

$$S = V_aI_a^* + V_bI_b^* + V_cI_c^* = V_p^TI_p^*$$

With $V_p = AV_s$, $I_p = AI_s$, where $A = \begin{bmatrix} 1&1&1\\1&a^2&a\\1&a&a^2 \end{bmatrix}$:

$$S = (AV_s)^T(AI_s)^* = V_s^T\,A^TA^*\,I_s^*$$

$A$ is symmetric, so $A^T = A$; and $a^* = a^2$, so $A^* = \begin{bmatrix} 1&1&1\\1&a&a^2\\1&a^2&a \end{bmatrix}$. Using $1+a+a^2 = 0$, $a^3 = 1$:

$$A^TA^* = \begin{bmatrix} 3 & 1+a+a^2 & 1+a^2+a \\ 1+a^2+a & 1+a^3+a^3 & 1+a^4+a^2 \\ 1+a+a^2 & 1+a^2+a^4 & 1+a^3+a^3 \end{bmatrix} = \begin{bmatrix} 3&0&0\\0&3&0\\0&0&3 \end{bmatrix}$$

Hence

$$S = 3V_s^TI_s^* = 3V_{a0}I_{a0}^* + 3V_{a1}I_{a1}^* + 3V_{a2}I_{a2}^*$$

Exact power invariance

If the transformation matrix is taken as $T = \frac{1}{\sqrt3}A$, then $T^TT^* = \frac13\cdot3U = U$ (unit matrix) and

$$S = V_p^TI_p^* = V_s^TI_s^* = V_{0}I_{0}^* + V_{1}I_{1}^* + V_{2}I_{2}^*$$

So the power is the same in both reference frames: the transformation is power invariant. With the usual matrix $A$ the sequence power of phase a is multiplied by 3 (three phases), and there are still no cross terms.

Significance of the zero sequence circuit

• Zero sequence currents are equal and in phase in all three phases, so they need a return path through neutral or ground: $I_n = 3I_{a0}$. The zero sequence network therefore depends on transformer winding connections (star grounded, star ungrounded, delta) and on neutral grounding.
• It decides the current in ground faults (L-G, L-L-G). Since about 70–80% of faults are L-G, the zero sequence network is essential for finding earth fault current, choosing neutral grounding impedance and setting earth fault relays.
• A neutral impedance $Z_n$ appears as $3Z_n$ only in the zero sequence circuit, so grounding methods change only this network.
• Delta windings block zero sequence current from the lines but let it circulate inside; this is used to trap third harmonics and to isolate ground faults between voltage levels.
• Zero sequence currents in lines induce voltages in nearby telephone lines (interference) and are used for ground-fault detection.