--- title: "Two reaction theory; determining Xd and Xq" subject: "Electrical Machines" chapter: 6 key: "semester_4/electrical_machines:two-reaction-theory-xd-xq" source: https://www.bctengineer.com/q/semester_4/electrical_machines/two-reaction-theory-xd-xq site: BCT Engineering --- # Two reaction theory; determining Xd and Xq Electrical Machines · Chapter 6: Three Phase Synchronous Machines - Asked in: 2067 Mangsir (old course), 8 marks - Video: [22=Slip test to find Xd & Xq in salient pole synchronous machine](https://www.youtube.com/watch?v=MDKjzN2NPUA) — CS Boopathi - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/electrical_machines/two-reaction-theory-xd-xq - Whole chapter: https://www.bctengineer.com/md/semester_4/electrical_machines/chapter-6 ## Question Explain two reaction theory of salient pole synchronous machines. Describe a method of determining direct and quadrature axis synchronous reactance of 3 phase synchronous machine. ## Short answer The theory splits armature mmf into d- and q-axis components due to saliency. Direct-axis and quadrature-axis synchronous reactances Xd and Xq are determined by the slip test: drive rotor just below synchronous speed with field open, apply reduced stator voltage; Xd = Vmax/Imin, Xq = Vmin/Imax. ## Answer #### Two-reaction theory In a salient-pole machine the air gap is not uniform: it is small along the pole axis (**direct axis, d-axis**) and large between the poles (**quadrature axis, q-axis**). So the same mmf produces different flux along the two axes, and the single synchronous reactance of a cylindrical rotor cannot be used. Blondel's **two-reaction theory** resolves the armature mmf (and the current) into two components, one along each axis, and treats each separately. - $I_d = I\sin\psi$: **direct-axis component**, in quadrature with the emf $E$, and acting along the pole axis (magnetising or demagnetising). - $I_q = I\cos\psi$: **quadrature-axis component**, in phase with $E$, acting along the inter-polar axis (cross-magnetising). Here $\psi$ is the angle between $I$ and $E$. Each component produces an armature reaction flux, and so a reactance: $$ X_d = X_{ad} + X_l, \qquad X_q = X_{aq} + X_l $$ with $X_d > X_q$ because the d-axis has the lower reluctance (typically $X_q \approx 0.6\text{--}0.7X_d$). #### Phasor equation (generator) $$ \vec E = \vec V + \vec I R_a + j\vec I_d X_d + j\vec I_q X_q $$ ```text E /| jIdXd/ | V at angle delta from E / | / |jIqXq ---+----+-----> V Iq in phase with E Id 90 deg behind/ahead ``` #### Determination of $X_d$ and $X_q$: slip test 1. Drive the rotor by a prime mover at **slightly less than synchronous speed** (slip below 1 %), with the **field winding open**. 2. Apply a **reduced balanced three-phase voltage** (about 25 % of rated, with the same phase sequence) to the stator terminals. 3. The stator rotating field slowly slides past the rotor poles. The effective reluctance varies between the d-axis and q-axis values, so the armature current and terminal voltage oscillate slowly. 4. Record with an oscillograph (or ammeter and voltmeter readings) the maximum and minimum values: - When the field is aligned with the **d-axis**, the reactance is largest, so the current is **minimum** and the voltage is **maximum**. - When aligned with the **q-axis**, the current is **maximum** and the voltage is **minimum**. 5. Then $$ X_d = \frac{V_{max}}{I_{min}}, \qquad X_q = \frac{V_{min}}{I_{max}} $$ (per phase values; $R_a$ is small and neglected). ```text Test connections 3-phase reduced V --> stator field winding: OPEN (voltmeter across it shows induced emf pulsating) prime mover: speed just below Ns ``` $X_d$ is also found from the open-circuit and short-circuit characteristics ($X_d = \frac{V_{oc}}{I_{sc}}$ at the same field current, unsaturated). $X_q$ by this test is typically 60 to 70 % of $X_d$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why can a single synchronous reactance not be used for a salient-pole machine? A. Because the air gap is non-uniform, so the same mmf gives different flux along the d-axis and q-axis B. Because the armature resistance differs along the two axes C. Because the field winding is open-circuited in salient-pole machines D. Because salient-pole machines run at variable frequency Answer: A. The air gap is small on the pole axis and large between poles, so the reluctance and the flux per unit mmf differ on the two axes. (Key point: Why two-reaction theory) 2. In two-reaction theory, which statement about Id and Iq is correct? A. Id is in phase with E and Iq is in quadrature with E B. Id is in quadrature with E and acts along the pole axis; Iq is in phase with E and acts along the inter-polar axis C. Both Id and Iq are in phase with E but act along different axes D. Id = I cos(psi) and Iq = I sin(psi) Answer: B. Id = I sin(psi) is in quadrature with E (magnetising or demagnetising); Iq = I cos(psi) is in phase with E (cross-magnetising). (Key point: Current components) 3. Which relation between Xd and Xq holds for a salient-pole machine, and why? A. Xd < Xq because the d-axis has higher reluctance B. Xd = Xq because leakage reactance is the same C. Xd > Xq because the d-axis has lower reluctance D. Xq > Xd because the q-axis has a smaller air gap Answer: C. The d-axis air gap is small, so reluctance is low and the reactance is larger; typically Xq is about 0.6 to 0.7 of Xd. (Key point: Relation of Xd and Xq) 4. In the slip test, what is the condition of the rotor and the stator supply? A. Rotor at standstill, field shorted, full rated voltage B. Rotor driven just below synchronous speed, field open, reduced balanced three-phase voltage applied C. Rotor driven at twice synchronous speed, field excited with DC, rated voltage D. Rotor driven at synchronous speed, field open, single-phase supply Answer: B. The slip test uses a reduced (about 25 %) balanced 3-phase voltage, slip below 1 % and an open field winding so the stator field slowly slides past the poles. 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