--- title: "Complex power, impedance and power triangles" subject: "Power System Analysis I" chapter: 3 key: "semester_4/power_system_analysis_i:complex-power-impedance-power-triangle" source: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/complex-power-impedance-power-triangle site: BCT Engineering --- # Complex power, impedance and power triangles Power System Analysis I · Chapter 3: Computational Technique - Asked in: 2069 Poush, 4 marks - Video: [Power System Analysis - Complex Power and the Power Triangle](https://www.youtube.com/watch?v=axS7MncCQ84) — Romero Engineering Company - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/complex-power-impedance-power-triangle - Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-3 ## Question What is complex power? Show and explain the symmetry between the impedance triangle and power triangle. ## Short answer Complex power is S=VI*=P+jQ, where P is active power, Q reactive power, and |S| apparent power. For a load Z=R+jX carrying current I, S=|I|^2Z=|I|^2R+j|I|^2X, so the power triangle is similar to the impedance triangle, scaled by |I|^2 with the same angle and power factor. ## Answer **Complex power** is $S = VI^* = P + jQ$. Its real part P (W) is the active power, its imaginary part Q (var) is the reactive power, and its magnitude $|S|$ (VA) is the apparent power. The angle of S is the power-factor angle $\phi = \theta_v - \theta_i$. #### Symmetry with the impedance triangle For a load $Z = R + jX = |Z|\angle\phi$ carrying current I, $V = IZ$. So $$ S = VI^* = (IZ)I^* = |I|^2Z = |I|^2R + j|I|^2X $$ So $P = |I|^2R$, $Q = |I|^2X$ and $|S| = |I|^2|Z|$. Every side of the impedance triangle is multiplied by the same real number $|I|^2$. The power triangle is therefore **similar** to the impedance triangle: ```text Impedance triangle Power triangle /| /| |Z| / | X |S| / | Q = |I|^2 X / | / | /phi| /phi| +----+ +----+ R P = |I|^2 R ``` | Impedance triangle | Power triangle | Link | |---|---|---| | R | P | $P = \lvert I\rvert{}^2R$ | | X | Q | $Q = \lvert I\rvert{}^2X$ | | $\lvert Z\rvert{}$ | $\lvert S\rvert{}$ | $\lvert S\rvert{} = \lvert I\rvert{}^2\lvert Z\rvert{}$ | | angle $\phi = \tan^{-1}(X/R)$ | angle $\phi = \tan^{-1}(Q/P)$ | same angle | | $\cos\phi = R/\lvert Z\rvert{}$ | $\cos\phi = P/\lvert S\rvert{}$ | same power factor | Significance: - The two triangles have the same angle. The power factor of a load can be found either from its impedance or from its powers. - Inductive X (+) gives +Q (vars absorbed). Capacitive X (−) gives −Q (vars supplied). Both triangles flip below the axis together. - This symmetry holds only because S is defined as $VI^*$. With $V^*I$, Q would be $-|I|^2X$ and the power triangle would be the mirror image of the impedance triangle. Example: $Z = 6 + j8\ \Omega$ with $|I| = 10$ A gives $S = 600 + j800$ VA. Both triangles have angle 53.13° and pf 0.6 lagging. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is complex power? A. S = V*I = P + jQ B. S = VI* = P + jQ with |S| the apparent power C. S = V + I D. S = P x Q Answer: B. Defined as V times conjugate of I. (Key point: Definition) 2. Why are the impedance and power triangles similar? A. S = |I|^2 Z, so every side of the impedance triangle is multiplied by the same real number |I|^2 B. Because V is constant C. Because current is zero D. Because R equals X Answer: A. Scaling by a real factor keeps the angle. (Key point: Why triangles are similar) 3. Which pairing is correct? A. R <-> Q, X <-> P B. R <-> |S|, X <-> P C. |Z| <-> P D. R <-> P, X <-> Q, |Z| <-> |S| Answer: D. P = I^2R, Q = I^2X, |S| = I^2|Z|. (Key point: Correspondence) 4. What would happen with V*I? A. P would change sign B. S would be unchanged C. Q = -I^2X, so the power triangle would be the mirror image of the impedance triangle D. Impedance triangle would disappear Answer: C. Using V*I flips the sign of Q. 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