--- title: "Voltage regulation expression for inductive load" subject: "Electrical Machines" chapter: 2 key: "semester_4/electrical_machines:voltage-regulation-inductive-load" source: https://www.bctengineer.com/q/semester_4/electrical_machines/voltage-regulation-inductive-load site: BCT Engineering --- # Voltage regulation expression for inductive load Electrical Machines · Chapter 2: Transformer - Asked in: 2076 Baisakh, 8 marks - Video: [PHASOR DIAGRAM ( INDUCTIVE LOAD) FOR A SINGLE PHASE TRANSFORMER](https://www.youtube.com/watch?v=IKx5yHPn3V8) — Xplanator - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/electrical_machines/voltage-regulation-inductive-load - Whole chapter: https://www.bctengineer.com/md/semester_4/electrical_machines/chapter-2 ## Question Define voltage regulation of a transformer. Derive an expression for voltage regulation if a transformer is loaded with an inductive load. ## Short answer Voltage regulation is (E2 – V2)/E2 × 100%, the secondary voltage change from no-load to full-load. For inductive load with lagging power factor cosφ, referred impedance R02, X02, the approximate regulation is VR = (I2 R02 cosφ + I2 X02 sinφ)/E2 × 100%. ## Answer **Voltage regulation** is the change in secondary terminal voltage from no load to full load (at the same primary voltage and a given power factor), expressed as a fraction of the no-load voltage: $$ \text{VR} = \frac{E_2 - V_2}{E_2}\times100\% $$ where $E_2$ is the no-load secondary voltage and $V_2$ the full-load terminal voltage. #### Derivation (lagging power factor) Refer all quantities to the secondary side: $R_{02} = R_2 + R_1/a^2$ and $X_{02} = X_2 + X_1/a^2$. With the exciting branch ignored, the circuit is a source $E_2$ in series with $R_{02} + jX_{02}$, feeding the load current $I_2$ at angle $\phi$ lagging $V_2$. ```text Phasor diagram (inductive load, V2 as reference) E2 /| I2X02 / | / | V2 /___| O---------> I2R02 \ phi \ I2 (lags V2) ``` Using $V_2$ as reference: $V_2 = V_2\angle0°$, $I_2 = I_2\angle-\phi$. $$ \begin{aligned} \bar E_2 &= \bar V_2 + \bar I_2(R_{02} + jX_{02}) \\ &= (V_2 + I_2R_{02}\cos\phi + I_2X_{02}\sin\phi) + j(I_2X_{02}\cos\phi - I_2R_{02}\sin\phi) \end{aligned} $$ The quadrature term is small, so $$ E_2 \approx V_2 + I_2R_{02}\cos\phi + I_2X_{02}\sin\phi $$ $$ E_2 - V_2 = I_2(R_{02}\cos\phi + X_{02}\sin\phi) $$ Hence $$ \boxed{\text{VR} = \frac{I_2R_{02}\cos\phi + I_2X_{02}\sin\phi}{E_2}\times100\%} $$ For the exact value: $E_2 = \sqrt{(V_2 + I_2R_{02}\cos\phi + I_2X_{02}\sin\phi)^2 + (I_2X_{02}\cos\phi - I_2R_{02}\sin\phi)^2}$. For a leading load, the sign of the $X_{02}\sin\phi$ term changes (VR $= I_2(R_{02}\cos\phi - X_{02}\sin\phi)/E_2$), and the regulation can become negative. The maximum regulation occurs when $\cos\phi = R_{02}/Z_{02}$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Voltage regulation is: A. (V2 - E2)/V2 x 100% B. (E2 + V2)/E2 x 100% C. (E2 - V2)/E2 x 100%, the fall in secondary voltage from no load to load as a fraction of no-load voltage D. Output divided by input Answer: C. E2 is the no-load voltage and V2 the loaded terminal voltage. (Key point: Definition) 2. Referred to the secondary, R02 equals: A. R2 + a^2 R1 B. R2 + R1/a C. R1 + R2 D. R2 + R1/a^2 Answer: D. Primary impedance is divided by a^2 when moved to the secondary. (Key point: Referred values) 3. For a lagging load, E2 - V2 is approximately: A. I2(R02 cosphi + X02 sinphi) B. I2(R02 cosphi - X02 sinphi) C. I2(R02 sinphi + X02 cosphi) D. I2 Z02 Answer: A. Taking the in-phase component of the drop; the quadrature term is small. (Key point: Approximate expression) 4. For a leading load the X02 sinphi term: A. Doubles B. Changes sign, so regulation can be negative C. Is unchanged D. Disappears Answer: B. VR = I2(R02 cosphi - X02 sinphi)/E2. (Key point: Leading load) 5. Maximum regulation occurs when: A. cosphi = 1 B. cosphi = X02/R02 C. cosphi = R02/Z02 D. cosphi = 0 Answer: C. This is the stated condition for maximum VR. 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