--- title: "Referred quantities in a transformer" subject: "Electrical Machines I" chapter: 2 key: "semester_4/electrical_machines_i:transformer-referred-quantities" source: https://www.bctengineer.com/q/semester_4/electrical_machines_i/transformer-referred-quantities site: BCT Engineering --- # Referred quantities in a transformer Electrical Machines I · Chapter 2: Transformer - Asked in: 2081 Chaitra (new course), 1+5 marks - Video: [Equivalent Circuit of Transformer Referred to Primary and Secondary - Single Phase Transformer](https://www.youtube.com/watch?v=3Jc8MN8HiQI) — Ekeeda - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/electrical_machines_i/transformer-referred-quantities - Whole chapter: https://www.bctengineer.com/md/semester_4/electrical_machines_i/chapter-2 ## Question Define referred quantities in transformer. Derive the expression for the referred quantities (voltage, current, impedance) in a transformer. ## Short answer Referred quantities represent one winding’s values on the other side keeping power and losses unchanged. With K = N₂/N₁: voltage V₂' = V₂/K, current I₂' = K I₂, impedance Z₂' = Z₂/K². Impedance from LV to HV multiplies by turns ratio squared. ## Answer **Referred quantities** are values of voltage, current and impedance of one winding expressed as equivalent values on the other side, chosen so that power, losses and per-unit drops are unchanged. Referring lets the two windings be joined into a single equivalent circuit without the ideal transformer. Let $K = N_2/N_1$ (transformation ratio). #### Referred voltage Emf per turn is the same in both windings: $E_1/N_1 = E_2/N_2$. So a secondary voltage seen on the primary side is $$ V_2' = \frac{N_1}{N_2}V_2 = \frac{V_2}{K} $$ and a primary voltage referred to the secondary is $V_1' = K V_1$. #### Referred current The secondary load current is balanced by the primary load component: $N_1 I_2' = N_2 I_2$ (mmf balance). Hence $$ I_2' = \frac{N_2}{N_1}I_2 = K I_2 $$ and $I_1' = I_1 / K$ when the primary current is referred to the secondary. Volt-amperes are unchanged: $V_2' I_2' = (V_2/K)(K I_2) = V_2 I_2$. #### Referred resistance The copper loss must be the same before and after referring: $$ \begin{aligned} I_2'^2 R_2' &= I_2^2 R_2 \\ R_2' &= R_2\left(\frac{I_2}{I_2'}\right)^2 = \frac{R_2}{K^2} \end{aligned} $$ #### Referred reactance and impedance The reactive power (or the voltage drop as a fraction of voltage) must be the same: $$ \begin{aligned} I_2'^2 X_2' &= I_2^2 X_2 \Rightarrow X_2' = \frac{X_2}{K^2} \\ Z_2' &= \frac{V_2'}{I_2'} = \frac{V_2/K}{K I_2} = \frac{Z_2}{K^2} \end{aligned} $$ Likewise, primary impedance referred to the secondary: $Z_1' = K^2 Z_1$. #### Summary | Quantity | Secondary referred to primary | Primary referred to secondary | |---|---|---| | Voltage | $V_2/K$ | $K V_1$ | | Current | $K I_2$ | $I_1/K$ | | Impedance (R, X, Z) | $Z_2/K^2$ | $K^2 Z_1$ | Rule of thumb: to move an impedance from LV to HV multiply by (turns ratio)², i.e. HV side always has the larger impedance. **Example:** for 2000/200 V ($K = 0.1$), $R_2 = 0.02\ \Omega$ referred to HV is $0.02/0.01 = 2\ \Omega$, so $R_{01} = R_1 + 2\ \Omega$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What are referred quantities? A. Equivalent values on the other side chosen so power, losses and per-unit drops are unchanged B. Values multiplied by turns ratio, with power changed C. Values measured at no load D. Values referred only for resistances Answer: A. Referring lets the two windings merge into a single equivalent circuit. (Key point: Definition) 2. How is secondary current referred to the primary? A. I2' = I2/K B. I2' = K I2 where K = N2/N1, from mmf balance C. I2' = K² I2 D. I2' = I2 Answer: B. N1 I2' = N2 I2 gives I2' = (N2/N1) I2. (Key point: Current) 3. How is secondary impedance referred to primary? A. Z2' = K Z2 B. Z2' = Z2/K C. Z2' = Z2/K², found by keeping I²R loss unchanged D. Z2' = K² Z2 Answer: C. I2'² R2' = I2² R2 gives R2' = R2/K²; likewise X and Z. (Key point: Impedance) 4. For 2000/200 V (K = 0.1), what is R2 = 0.02 Ω referred to the HV side? A. 0.2 Ω B. 0.0002 Ω C. 20 Ω D. 2 Ω Answer: D. 0.02/0.01 = 2 Ω; impedance referred to the HV side is larger. 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