Chapter 7 • 6 Hours • Expected Marks: 6M Always Tested in Q7 (5–6 Marks)

Single-Phase Transformers

Static magnetic energy conversion, EMF derivation, core flux constancy, exact equivalent circuits, open-circuit/short-circuit test analysis, efficiency optimization, and autotransformers.

7.1 & 7.2 Operating Principle, EMF Equation & Core Flux Constancy

Operating Principle & Construction

A transformer is a static electromagnetic device that transfers alternating electrical energy between circuits through mutual electromagnetic induction, without changing frequency.
• Core-Type: Windings surround a considerable part of the laminated iron core. Two limbs with cylindrical coils. Preferred for high-voltage transmission.
• Shell-Type: The iron core surrounds a major portion of the windings. Three limbs with sandwich coils on the central limb. Better mechanical protection.

EMF Equation Derivation

Let mutual flux in the core be sinusoidal: \(\Phi(t) = \Phi_m \sin(\omega t)\). According to Faraday's Law, instantaneous induced EMF in primary of \(N_1\) turns is: \[e_1 = -N_1 \frac{d\Phi}{dt} = -N_1 \frac{d}{dt}[\Phi_m \sin(\omega t)] = -N_1 \omega \Phi_m \cos(\omega t) = N_1 \omega \Phi_m \sin\left(\omega t - 90^\circ\right)\] Maximum value: \(E_{1m} = N_1 \omega \Phi_m = N_1 (2\pi f) \Phi_m = 2\pi f N_1 \Phi_m\). The RMS value is: \[E_1 = \frac{E_{1m}}{\sqrt{2}} = \frac{2\pi}{\sqrt{2}} f N_1 \Phi_m = \sqrt{2}\pi f N_1 \Phi_m \approx 4.44 f N_1 \Phi_m\] Similarly, for the secondary winding: \[E_2 = 4.44 f N_2 \Phi_m\]

Constancy of Main Core Flux Under Varying Load (Exam Favorite)

Because primary induced EMF \(E_1 \approx V_1\) (neglecting small winding drop), we have: \[\Phi_m = \frac{V_1}{4.44 f N_1}\] When supply voltage \(V_1\) and frequency \(f\) are constant, \(\Phi_m\) is strictly constant. When a load draws secondary current \(I_2\), its demagnetizing MMF \(N_2 I_2\) is dynamically counteracted by the primary drawing a compensating load component current \(I_1' = (N_2/N_1) I_2\), maintaining constant core flux!
(See full derivation in Derivation Sheet Section 2).

7.3 & 7.4 Equivalent Circuit & Transformer Testing (OC and SC Tests)

Approximate Equivalent Circuit Referred to Primary

Transformation ratio: \(K = \frac{N_2}{N_1} = \frac{V_2}{V_1} = \frac{I_1}{I_2}\).
• Secondary resistance referred to primary: \(R_2' = \frac{R_2}{K^2}\).
• Secondary reactance referred to primary: \(X_2' = \frac{X_2}{K^2}\).
• Total equivalent resistance: \(R_{eq1} = R_1 + R_2'\).
• Total equivalent reactance: \(X_{eq1} = X_1 + X_2'\).

Open-Circuit (OC) Test

• Conducted on: Low Voltage (LV) side with High Voltage (HV) side open-circuited.
• Applied Voltage: Rated LV voltage.
• Purpose: Measure core/iron loss (\(P_i = W_0\)) and determine shunt magnetizing parameters (\(R_0, X_0\)).

\[\cos\phi_0 = \frac{W_0}{V_0 I_0}, \quad I_w = I_0 \cos\phi_0, \quad I_\mu = I_0 \sin\phi_0 = \sqrt{I_0^2 - I_w^2}\] \[R_0 = \frac{V_0}{I_w}, \quad X_0 = \frac{V_0}{I_\mu}\]

Short-Circuit (SC) Test

• Conducted on: High Voltage (HV) side with Low Voltage (LV) side safely short-circuited by a heavy conductor.
• Applied Voltage: Reduced voltage (typically 5–10% of rated) adjusted until rated full-load current \(I_{sc}\) flows.
• Purpose: Measure full-load copper loss (\(P_{cu,FL} = W_{sc}\)) and determine series equivalent parameters (\(R_{eq}, X_{eq}\)).

\[R_{eq1} = \frac{W_{sc}}{I_{sc}^2}, \quad Z_{eq1} = \frac{V_{sc}}{I_{sc}}, \quad X_{eq1} = \sqrt{Z_{eq1}^2 - R_{eq1}^2}\]

7.5 & 7.6 Voltage Regulation, Efficiency & Autotransformers

Voltage Regulation & Efficiency Formulations

\[\text{Approximate Voltage Regulation: } \%VR = \frac{I_2 R_{eq2} \cos\phi_2 \pm I_2 X_{eq2} \sin\phi_2}{V_{20}} \times 100\%\] \[\text{Efficiency: } \eta = \frac{x S \cos\phi}{x S \cos\phi + P_i + x^2 P_{cu,FL}} \times 100\%\]
In \(\%VR\), use \(+\) for lagging power factor, and \(-\) for leading power factor.
Condition for Maximum Efficiency

Differentiating \(\eta\) with respect to load current \(I_2\) and setting to zero yields: \[x^2 P_{cu,FL} = P_i \implies \text{Variable Copper Loss} = \text{Constant Iron Loss}\] Load fraction for maximum efficiency: \[x = \sqrt{\frac{P_i}{P_{cu,FL}}}\]

Auto-Transformer Copper Saving

An autotransformer utilizes a single continuous tapped winding. Power is transferred between primary and secondary through both induction and direct electrical conduction: \[\text{Conducted Power} = K \times \text{Input Power}, \quad \text{Inducted Power} = (1 - K) \times \text{Input Power}\] \[\frac{\text{Weight of Cu in Auto}}{\text{Weight of Cu in 2-Winding}} = 1 - K\] (See full derivation in Derivation Sheet Section 9).

Master Worked Example: Transformer OC/SC Test (PYQ 2082 Baishakh Q7)

Parameter Evaluation Referred to Secondary Side PYQ 2082 Baishakh Q7 (5 Marks)
Given Test Data:
Rating: 20 kVA, 2200/220 V, 50 Hz. \(K = \frac{220}{2200} = 0.1\).
OC test on LV (secondary): \(V_0 = 220\text{ V}, I_0 = 1.1\text{ A}, W_0 = 125\text{ W}\).
SC test on HV (primary): \(V_{sc} = 52.7\text{ V}, I_{sc} = 8.4\text{ A}, W_{sc} = 287\text{ W}\).
Calculations:
1. Shunt Parameters (Already on Secondary LV side): \[\cos\phi_0 = \frac{125}{220 \times 1.1} = \frac{125}{242} \approx 0.5165 \implies \sin\phi_0 = 0.8563\] \[I_w = 1.1 \times 0.5165 = 0.568\text{ A}, \quad I_\mu = 1.1 \times 0.8563 = 0.942\text{ A}\] \[R_0' = \frac{220}{0.568} \approx 387.3\,\Omega, \quad X_0' = \frac{220}{0.942} \approx 233.5\,\Omega\] 2. Series Parameters on Primary HV side: \[R_{eq1} = \frac{287}{8.4^2} \approx 4.067\,\Omega, \quad Z_{eq1} = \frac{52.7}{8.4} \approx 6.274\,\Omega\] \[X_{eq1} = \sqrt{6.274^2 - 4.067^2} = \sqrt{22.82} \approx 4.777\,\Omega\] 3. Referring Series Parameters to Secondary (LV): \[R_{eq2} = K^2 R_{eq1} = (0.1)^2 \times 4.067 = 0.0407\,\Omega\] \[X_{eq2} = K^2 X_{eq1} = (0.1)^2 \times 4.777 = 0.0478\,\Omega\]
Final Answer: All parameters referred to secondary: \[R_{eq2} = 0.0407\,\Omega, \quad X_{eq2} = 0.0478\,\Omega, \quad R_0' = 387.3\,\Omega, \quad X_0' = 233.5\,\Omega\]
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