--- title: "Power invariance of symmetrical transformation" subject: "Power System Analysis II" chapter: 4 key: "semester_5/power_system_analysis_ii:power-invariance-zero-sequence" source: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/power-invariance-zero-sequence site: BCT Engineering --- # Power invariance of symmetrical transformation Power System Analysis II · Chapter 4: Unbalanced System Analysis - Asked in: 2075 Chaitra, 6+2 marks - Video: [POWER INVARIANCE IN SYMMETRICAL COMPONENTS /KTU / POWER SYSTEM ANALYSIS](https://www.youtube.com/watch?v=GJHJYKfeaEQ) — praveen raj - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/power-invariance-zero-sequence - Whole chapter: https://www.bctengineer.com/md/semester_5/power_system_analysis_ii/chapter-4 ## Question Prove that the symmetrical component transformation is power invariant. Discuss also the significance of zero sequence circuit. ## Short answer The transformation gives S = 3(Va0Ia0*+Va1Ia1*+Va2Ia2*); with the unitary form it is exactly invariant. Zero sequence currents are in phase, require ground/neutral return, so the zero sequence network governs earth faults, grounding impedance, and protection coordination. ## Answer 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. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. With the usual A, S = 3(Va0 Ia0* + ...) so the transformation is: A. Not related to phase power B. Exactly invariant C. Power invariant apart from the factor 3 D. Invariant only for balanced Answer: C. A is not unitary. (Key point: Matrix A) 2. Exact power invariance is obtained with: A. T = 3A B. T = A^-1 C. T = A/3 D. T = A/sqrt(3) Answer: D. T^T T* = U. (Key point: Exact invariance) 3. The zero sequence network is essential because: A. It decides ground fault current and neutral grounding effects B. It is the only network with sources C. It carries load power D. It affects only line-to-line faults Answer: A. Most faults are L-G. (Key point: Zero sequence importance) 4. A delta winding in the zero sequence network: A. Passes it to lines B. Blocks zero sequence current from lines but lets it circulate C. Amplifies it D. Shorts it to ground outside Answer: B. Used to trap third harmonics. 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