--- title: "Power of salient pole machine, power angle curve" subject: "Electrical Machines" chapter: 6 key: "semester_4/electrical_machines:salient-pole-power-angle-characteristics" source: https://www.bctengineer.com/q/semester_4/electrical_machines/salient-pole-power-angle-characteristics site: BCT Engineering --- # Power of salient pole machine, power angle curve Electrical Machines · Chapter 6: Three Phase Synchronous Machines - Asked in: 2066 Magh (old course), 8 marks - Video: [Derivation of Power equation for salient pole machine](https://www.youtube.com/watch?v=SwYdSXYxByQ) — Crash Course in Electrical Engineering - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/electrical_machines/salient-pole-power-angle-characteristics - Whole chapter: https://www.bctengineer.com/md/semester_4/electrical_machines/chapter-6 ## Question Derive the expression for electrical power of salient pole synchronous machine. Show the power angle characteristics for such machines. ## Short answer Per-phase power is P = (V E/Xd) sin δ + (V²/2)(1/Xq − 1/Xd) sin 2δ. The first term is excitation power, the second reluctance power. The power-angle curve is a sum of fundamental and second-harmonic sinusoids; maximum power occurs at δ less than 90° because of the sin 2δ term. ## Answer Per phase, with resistance neglected, a generator has terminal voltage $V$, excitation emf $E$ and load angle $\delta$ (the angle between $E$ and $V$). The two-reaction model gives $\vec E = \vec V + j\vec I_d X_d + j\vec I_q X_q$. #### Derivation Components of current: $$ I_q = \frac{V\sin\delta}{X_q}, \qquad I_d = \frac{E - V\cos\delta}{X_d} $$ The output power per phase is $V$ multiplied by the component of $I$ in phase with $V$. $I_q$ is in phase with $E$, so it makes angle $\delta$ with $V$, and $I_d$ (at $90^\circ$ to $I_q$) also has a component $I_d\sin\delta$ along $V$: $$ P_{ph} = V I\cos\phi = V I_q\cos\delta + V I_d\sin\delta $$ $$ P_{ph} = \frac{V^2\sin\delta\cos\delta}{X_q} + \frac{V\sin\delta\,(E - V\cos\delta)}{X_d} $$ $$ P_{ph} = \frac{VE}{X_d}\sin\delta + V^2\sin\delta\cos\delta\left(\frac{1}{X_q} - \frac{1}{X_d}\right) $$ Using $\sin\delta\cos\delta = \tfrac12\sin2\delta$: $$ \boxed{P_{ph} = \frac{VE}{X_d}\sin\delta + \frac{V^2}{2}\left(\frac{1}{X_q}-\frac{1}{X_d}\right)\sin 2\delta} $$ The three-phase power is $3P_{ph}$ (with phase values). For a cylindrical rotor, $X_d = X_q$ and the second term vanishes. - First term: **excitation power** (fundamental, $\sin\delta$). - Second term: **reluctance power** ($\sin2\delta$). It exists even if $E=0$. #### Power-angle characteristic ```text P | total | _.-'-._ | .' excit. `-. | .' ____---___ `-. |/.-' reluct. `-._ `-. +--+----+----+----+----+----> delta 0 45 ~75-80 90 180 (max power below 90 deg) ``` - The total curve is the sum of a $\sin\delta$ curve and a $\sin2\delta$ curve. - **Maximum power occurs at $\delta < 90^\circ$** (usually 60 to 80°) for a salient-pole machine, and at $90^\circ$ for a cylindrical machine. - Stable operation is on the rising part ($0 < \delta < \delta_{max}$), where $dP/d\delta > 0$. Beyond it the machine loses synchronism. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In the power derivation, which expressions give the quadrature and direct axis currents? A. Iq = V cos(delta)/Xq and Id = (E - V sin(delta))/Xd B. Iq = V sin(delta)/Xq and Id = (E - V cos(delta))/Xd C. Iq = E sin(delta)/Xd and Id = V cos(delta)/Xq D. Iq = (E - V cos(delta))/Xq and Id = V sin(delta)/Xd Answer: B. From the phasor diagram, the q-axis voltage drop is V sin(delta) across Xq, and the d-axis drop is E - V cos(delta) across Xd. (Key point: Current components) 2. What is the per-phase power of a salient-pole machine (resistance neglected)? A. (VE/Xd) sin(delta) only B. (VE/Xd) sin(delta) + (V^2/2)(1/Xq - 1/Xd) sin(2 delta) C. (VE/Xq) cos(delta) + (V^2/2)(1/Xd - 1/Xq) sin(delta) D. (VE/Xd) sin(2 delta) + (V^2/2)(1/Xq - 1/Xd) sin(delta) Answer: B. P = V Iq cos(delta) + V Id sin(delta) simplifies to an excitation term in sin(delta) plus a reluctance term in sin(2 delta). (Key point: Power expression) 3. What is special about the reluctance power term? A. It is zero for all salient-pole machines B. It varies as sin(delta) and needs the field current C. It varies as sin(2 delta) and exists even when E = 0 D. It exists only for cylindrical rotors Answer: C. The reluctance term depends on Xd differing from Xq, varies as sin(2 delta), and does not need excitation. (Key point: Reluctance power) 4. Where does maximum power occur for a salient-pole machine compared with a cylindrical one? A. Below 90 degrees for salient-pole; at 90 degrees for cylindrical B. At 90 degrees for both C. Above 90 degrees for salient-pole; at 90 degrees for cylindrical D. At 45 degrees for salient-pole; at 90 degrees for cylindrical Answer: A. Adding the sin(2 delta) term shifts the peak of the total curve to a delta below 90 degrees (usually 60 to 80 degrees). (Key point: Maximum power angle) ## Related in Electrical Machines - [Two reaction model of salient pole machine](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Atwo-reaction-model-salient-pole) — Electrical Machines, chapter 6 - [Salient pole vs cylindrical rotor alternators](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Asalient-vs-cylindrical-alternator-justify) — Electrical Machines, chapter 6 - [Synchronous generator load characteristics, capacitive load](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Async-gen-load-characteristics-capacitive) — Electrical Machines, chapter 6 - [1600 kVA alternator: emf, regulation, power angle](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Aalt-1600kva-regulation-power-angle) — Electrical Machines, chapter 6 - [25 kW separately excited DC machine as motor or generator](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines_i%3Asep-excited-25kw-125v-terminal-voltage) — Electrical Machines I, chapter 3 ## Related in other subjects - [Power angle characteristics of salient pole machine](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Asalient-power-angle-derivation) — Electric Machines II, chapter 2 - [Non-salient power angle relation for salient machines](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Anon-salient-power-angle-salient) — Electric Machines II, chapter 1 - [Internal emf and load angle of salient pole motor](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Asalient-motor-200kw-load-angle) — Electric Machines II, chapter 2 - [25 MVA salient pole motor maximum power](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Asalient-motor-25mva-max-power) — Electric Machines II, chapter 2, asked 3×