--- title: "Transformer equivalent circuit, referring parameters" subject: "Electrical Machines" chapter: 2 key: "semester_4/electrical_machines:equivalent-circuit-primary-secondary-derivation" source: https://www.bctengineer.com/q/semester_4/electrical_machines/equivalent-circuit-primary-secondary-derivation site: BCT Engineering --- # Transformer equivalent circuit, referring parameters Electrical Machines · Chapter 2: Transformer - Asked in: 2074 Bhadra (old course) - 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/equivalent-circuit-primary-secondary-derivation - Whole chapter: https://www.bctengineer.com/md/semester_4/electrical_machines/chapter-2 ## Question Draw the equivalent circuit of a transformer with their parameters as it is in primary side and secondary side. How all parameters can be transferred to primary side - explain with mathematical derivation. ## Short answer The equivalent circuit includes primary R1, X1, secondary R2, X2, and shunt Rc, Xm. To refer secondary to primary, multiply impedances by a^2 (a=N1/N2): R2'=a^2 R2, X2'=a^2 X2. Total referred series impedance is R01=R1+a^2 R2, X01=X1+a^2 X2. ## Answer An equivalent circuit replaces the magnetically coupled windings with an electric network that has the same terminal behaviour. Winding resistance and leakage reactance are put in series, and the core is represented by a shunt branch. #### Equivalent circuit (ideal transformer $N_1:N_2$ in the centre) ```text R1 X1 R2 X2 o-/\/\--UUU--+---+----+--/\/\--UUU--o | | | ideal V1 Rc Xm ) N1:N2 ( V2 Load | | | o------------+---+----+-------------o ``` - $R_1, X_1$: primary resistance and leakage reactance. $R_2, X_2$: the secondary values. - $R_c$: core-loss resistance. $X_m$: magnetising reactance, with $I_c = E_1/R_c$ and $I_\mu = E_1/X_m$ (where $E_1 \approx V_1$). Together they form $I_0$. - $a = N_1/N_2$. #### Referring all parameters to the primary side The power in each element must remain the same after referring. Secondary quantities are referred to the primary as follows: $$ V_2' = aV_2,\qquad I_2' = \frac{I_2}{a} $$ For the secondary impedance, equate the power (and reactive power) before and after referring: $$ I_2'^{\,2}R_2' = I_2^2R_2 \Rightarrow R_2' = R_2\left(\frac{I_2}{I_2'}\right)^2 = a^2R_2 $$ $$ X_2' = a^2X_2,\qquad Z_L' = a^2Z_L $$ The referred circuit has all the elements on the primary side: ```text R1 X1 R2' X2' o-/\/\--UUU--+--/\/\--UUU--+---o | | V1 Rc Xm ZL' (= a^2 ZL) | | o------------+------------+---o ``` The total referred values are then $$ R_{01} = R_1 + a^2R_2,\qquad X_{01} = X_1 + a^2X_2,\qquad Z_{01} = \sqrt{R_{01}^2 + X_{01}^2} $$ The shunt branch ($R_c, X_m$) is often moved to the input terminals (approximate circuit) because $I_0$ is small. With this approximation the circuit is $V_1$ across $R_c \parallel X_m$, followed by $R_{01} + jX_{01}$ in series with $Z_L'$. #### Referring to the secondary side Primary quantities are referred to the secondary by dividing by $a$ (voltage) and $a^2$ (impedance): $$ R_1' = \frac{R_1}{a^2},\quad X_1' = \frac{X_1}{a^2},\quad R_{02} = R_2 + \frac{R_1}{a^2},\quad X_{02} = X_2 + \frac{X_1}{a^2} $$ Similarly $R_c' = R_c/a^2$, $X_m' = X_m/a^2$, and the primary current becomes $aI_1$. **Rule:** an impedance is multiplied by $a^2$ when moved from the secondary to the primary, and divided by $a^2$ when moved from the primary to the secondary, where $a = N_1/N_2$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The core is represented by: A. A shunt branch with Rc (core loss) and Xm (magnetising) B. A series resistance C. A series capacitor D. Nothing Answer: A. Ic = V1/Rc and I_mu = V1/Xm. (Key point: Shunt branch) 2. A secondary resistance referred to the primary is: A. R2/a^2 B. a^2 R2 C. a R2 D. R2 a^3 Answer: B. Equating I^2R: I2'^2 R2' = I2^2 R2 with I2' = I2/a gives R2' = a^2 R2. (Key point: Referring to primary) 3. Total resistance referred to the primary is: A. R1 + R2/a^2 B. R1 + a R2 C. R01 = R1 + a^2 R2 D. R1 + R2 Answer: C. Add the referred secondary resistance to R1. (Key point: Total referred) 4. Moving an impedance from primary to secondary: A. Multiply by a^2 B. Divide by a C. Multiply by a D. Divide by a^2 Answer: D. a = N1/N2; to the primary multiply by a^2, to the secondary divide. 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