--- title: "Ideal transformer and coupled circuit model" subject: "Electrical Machines I" chapter: 2 key: "semester_4/electrical_machines_i:ideal-transformer-coupled-circuit-model" source: https://www.bctengineer.com/q/semester_4/electrical_machines_i/ideal-transformer-coupled-circuit-model site: BCT Engineering --- # Ideal transformer and coupled circuit model Electrical Machines I · Chapter 2: Transformer - Asked in: 2080 Chaitra, 3+5 marks - Video: [EE213 - 06a - Ideal Transformers](https://www.youtube.com/watch?v=lEi4-Sf_lxs) — MAFarooqi - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/electrical_machines_i/ideal-transformer-coupled-circuit-model - Whole chapter: https://www.bctengineer.com/md/semester_4/electrical_machines_i/chapter-2 ## Question What are the characteristics of ideal transformer? Explain the coupled circuit model of transformer with necessary diagrams. ## Short answer An ideal transformer has zero resistances, no leakage flux, infinite permeability, no core loss, 100% efficiency, and V1/V2 = N1/N2 = I2/I1. Coupled circuit model uses self and mutual inductances with equations v1 = R1 i1 + L1 di1/dt + M di2/dt and v2 = R2 i2 + L2 di2/dt + M di1/dt. ## Answer #### Characteristics of an ideal transformer 1. Winding resistances are zero, so there is no copper loss. 2. There is no leakage flux; all the flux links both windings (coupling coefficient k = 1). 3. The core has infinite permeability, so no magnetising current is needed. 4. There is no core loss (no hysteresis or eddy current loss). 5. Hence efficiency is 100%, voltage regulation is zero, and $$ \frac{V_1}{V_2} = \frac{N_1}{N_2} = \frac{I_2}{I_1}, \qquad Z_1 = \left(\frac{N_1}{N_2}\right)^2 Z_2 $$ #### Coupled circuit model of a transformer A real (linear) transformer is treated as two magnetically coupled coils with self inductances $L_1$, $L_2$, mutual inductance $M$, and winding resistances $R_1$, $R_2$. ```text i1 R1 M R2 i2 o-->--/\/\--+ . . +--/\/\--<--o | (L1) (L2) | v1 ) ) ( ( v2 | | o-----------+ +-----------o ``` The flux linking each coil has a self and a mutual part: $$ \lambda_1 = L_1 i_1 + M i_2, \qquad \lambda_2 = M i_1 + L_2 i_2 $$ Voltage equations (with both currents entering the dotted terminals): $$ \begin{aligned} v_1 &= R_1 i_1 + L_1\frac{di_1}{dt} + M\frac{di_2}{dt} \\ v_2 &= R_2 i_2 + L_2\frac{di_2}{dt} + M\frac{di_1}{dt} \end{aligned} $$ In sinusoidal steady state: $V_1 = (R_1 + j\omega L_1)I_1 + j\omega M I_2$ and $V_2 = j\omega M I_1 + (R_2 + j\omega L_2) I_2$. #### Link with leakage and magnetising inductances With $a = N_1/N_2$, the self inductances are split into leakage and magnetising parts: $$ \begin{aligned} L_1 &= L_{l1} + aM, \qquad L_2 = L_{l2} + \frac{M}{a}, \qquad L_m = aM \\ M &= k\sqrt{L_1 L_2}, \quad k \le 1 \end{aligned} $$ Substituting gives the familiar T-equivalent circuit: $R_1$ and $X_{l1} = \omega L_{l1}$ in series, the magnetising reactance $X_m = \omega aM$ in shunt, and $R_2' = a^2R_2$, $X_{l2}' = a^2 X_{l2}$ followed by an ideal transformer of ratio $a$. ```text R1 Xl1 a^2 R2 a^2 Xl2 ideal a:1 o-/\/-mmm--+--------/\/\----mmm-----+--||--o | | || Xm=w aM | || V2 | | || o----------+------------------------+--||--o ``` When $k \to 1$, $R \to 0$ and $L_1, L_2 \to \infty$ with $L_1/L_2 = a^2$, the coupled-circuit model reduces to the ideal transformer. Core loss is added later as a shunt resistance $R_c$ across $X_m$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which describes an ideal transformer? A. Zero resistance, k = 1, infinite permeability, no core loss, 100% efficiency B. Finite resistance and k = 0.5 C. Zero permeability and large loss D. Efficiency of 90% with zero regulation Answer: A. All of these remove loss, leakage and magnetising current. (Key point: Characteristics) 2. What are the flux linkages of coil 1 in the coupled model? A. λ1 = L1 i1 − L2 i2 B. λ1 = M i1 + L2 i2 C. λ1 = L1 i1 + M i2 D. λ1 = L1 i2 + M i1 Answer: C. Self part L1 i1 plus mutual part M i2. (Key point: Flux linkage) 3. Which is the primary voltage equation? A. v1 = R1 i1 + M di1/dt + L1 di2/dt B. v1 = R1 i1 + L1 i1 C. v1 = L1 di1/dt only D. v1 = R1 i1 + L1 di1/dt + M di2/dt Answer: D. Resistive drop plus the self and mutual induced emfs. (Key point: Voltage equation) 4. When does the coupled model reduce to the ideal transformer? A. k → 0 and R large B. L1 = L2 = 0 C. k = 1 with finite L1 and L2 and large R D. k → 1, R → 0, and L1, L2 → ∞ with L1/L2 = a² Answer: D. These are the ideal limits. 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