6.1 Transformers Fundamentals

6.1 Transformers Fundamentals

Introduction to Transformers

A transformer is a static electromagnetic device that transfers electrical energy between two or more circuits through the principle of mutual induction. Its primary function is to change the voltage and current levels while maintaining the same frequency and power (neglecting losses). Transformers are the cornerstone of modern AC power systems, enabling efficient long-distance transmission by stepping up voltages to reduce current and associated $I^2R$ losses, and stepping down voltages to safe, usable levels for industrial, commercial, and residential consumption. This section explores their construction, fundamental theory, performance characteristics, and various configurations essential for power system engineering.

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1. Importance, Applications, and Types

1.1 Importance of Transformers

1. Efficient Power Transmission: By increasing voltage (stepping up) at the generating station, the current for a given power level is reduced. Since transmission line losses are proportional to the square of the current ($P_{loss} = I^2R$), this dramatically increases transmission efficiency.

2. Voltage Transformation for Utilization: Allows safe and practical voltage levels for end-use equipment (e.g., 11kV/433V for industry, 230V/110V for homes).

3. Electrical Isolation: Provides galvanic isolation between primary and secondary circuits, enhancing safety and protecting equipment.

4. Impedance Matching: Used in electronics to match source and load impedance for maximum power transfer.

1.2 Key Applications

1. Power System Core: Step-up transformers at power plants, step-down transformers at substations, and distribution transformers.

2. Industrial: Furnace transformers, rectifier transformers for variable speed drives.

3. Commercial/Residential: Distribution transformers on poles or in pad-mounted enclosures.

4. Instrumentation: Potential Transformers (PT) and Current Transformers (CT) for measurement and protection.

5. Electronics: Low-power transformers in power supplies, audio transformers, and isolation transformers.

1.3 Classification of Transformers

1. Based on Construction:

   • Core Type: Windings surround a substantial part of the core (wrapped around limbs). Commonly used for high-voltage applications.

   • Shell Type: Core surrounds a substantial part of the windings (encases them). Often used for low-voltage, high-current applications.

   • Berry Type (Distributed Shell): Special type with magnetic circuit distributed like spokes.

2. Based on Phases:

   • Single-Phase Transformers

   • Three-Phase Transformers (or bank of three single-phase units).

3. Based on Function:

   • Step-Up Transformer: Secondary voltage > Primary voltage ($N_2 > N_1$).

   • Step-Down Transformer: Secondary voltage < Primary voltage ($N_2 < N_1$).

   • Isolation Transformer: $V_1 = V_2$, used solely for isolation.

4. Based on Cooling Medium:

   • Oil-Immersed (ONAN, ONAF): Oil for insulation and cooling.

   • Dry Type (Air-Cooled): Use air as cooling and insulating medium.

   • SF6 Gas-Insulated: For fire-hazard environments.

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2. Construction (Core and Windings)

The transformer consists of two main parts: the magnetic core and the electrical windings.

2.1 Core

1. Function: Provides a low-reluctance path for the magnetic flux, linking the primary and secondary windings.

2. Material: Made of high-permeability, low-hysteresis loss silicon steel laminations (typically 0.3-0.5 mm thick).

3. Lamination: The core is laminated (thin sheets insulated from each other) to reduce eddy current losses.

4. Types: As per classification above (Core Type, Shell Type). The core cross-section is usually stepped or cruciform to utilize the circular coil space efficiently.

2.2 Windings

1. Function: Conductors where the input (primary) and output (secondary) electrical energy is applied and extracted.

2. Material: Copper or aluminum conductors, insulated with paper, enamel, or varnish.

3. Types:

   • Cylindrical Windings: Concentric layers of circular coils. Used for core-type transformers. The LV winding is placed nearer to the core to minimize insulation requirements.

   • Sandwich (Pancake) Windings: Disc-type coils stacked alternately (HV-LV-HV). Used for shell-type transformers and better short-circuit strength.

4. Insulation: Critical for reliable operation. Includes conductor insulation, inter-layer insulation, and major insulation between windings and between winding and core.

2.3 Additional Components

1. Tank: Contains the core-winding assembly and insulating/cooling oil.

2. Bushings: Insulated terminals that bring out the winding connections through the tank.

3. Conservator: An expansion tank that allows for oil expansion/contraction.

4. Breather: Contains silica gel to absorb moisture from air entering the conservator.

5. Radiators/Fans: Increase cooling surface area for oil.

6. Buchholz Relay: A gas-actuated relay mounted on the pipe connecting the tank to the conservator, providing protection against internal faults.

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3. Single-Phase Transformer Theory

3.1 EMF Equation and Vector Diagram

1. Principle: A sinusoidal alternating flux ($\phi = \phi_m \sin \omega t$) linking a coil of $N$ turns induces an EMF according to Faraday's law: $e = -N \frac{d\phi}{dt}$.

2. EMF Equation:

   • Let $\phi_m$ be the maximum value of the sinusoidally varying core flux.

   • The RMS value of the induced EMF in each winding is given by: $E = 4.44 \ f \ \phi_m \ N$ Where:

     • $E$ is the RMS voltage (V)

     • $f$ is the supply frequency (Hz)

     • $\phi_m$ is the maximum flux in the core (Wb)

     • $N$ is the number of turns in the winding.

   • This applies to both primary ($E_1 = 4.44 f \phi_m N_1$) and secondary ($E_2 = 4.44 f \phi_m N_2$) windings.

3. Turns Ratio (a): $a = \frac{N_1}{N_2}$ For an ideal transformer: $\frac{V_1}{V_2} = \frac{E_1}{E_2} = \frac{N_1}{N_2} = a$ and $\frac{I_1}{I_2} = \frac{N_2}{N_1} = \frac{1}{a}$ (assuming 100% efficiency).

4. Vector Diagram at No-Load:

   • The mutual flux ($\phi_m$) is taken as the reference.

   • The induced EMFs ($E_1$ and $E_2$) lag the flux $\phi_m$ by $90^\circ$.

   • The no-load current ($I_0$) leads the flux slightly due to core loss, or is in phase with it for an ideal lossless core. $I_0$ has two components:

     • Magnetizing Component ($I_m$): Creates the flux, lags $V_1$ by $90^\circ$.

     • Core Loss Component ($I_c$): Supplies hysteresis and eddy current losses, in phase with $V_1$.

   • $I_0 = \sqrt{I_m^2 + I_c^2}$

3.2 No-Load and On-Load Operation

1. No-Load (Open Circuit) Operation:

   • Condition: Secondary winding open-circuited, primary connected to rated voltage.

   • Primary Current: A small current called No-Load Current ($I_0$) flows (typically 2-6% of full-load current).

   • Function: $I_0$ supplies the core losses (hysteresis & eddy current) and establishes the mutual flux $\phi_m$ in the core.

   • Power Drawn: Equal to the core losses (Iron Losses).

2. On-Load Operation:

   • Condition: Secondary winding connected to a load impedance $Z_L$.

   • Effect: Secondary current $I_2$ flows. According to Lenz's law, this current creates an MMF ($N_2I_2$) that opposes the main flux. To maintain the flux (and thus $E_1$) nearly constant, the primary draws an additional load component of current ($I_1'$).

   • MMF Balance: $N_1I_1 = N_1I_0 + N_2I_2$. For large transformers, $I_0$ is negligible, so $N_1I_1 \approx N_2I_2$.

3.3 Voltage Regulation, Losses, and Efficiency

1. Voltage Regulation:

   • Definition: The change in secondary terminal voltage from no-load to full-load, expressed as a percentage of the full-load voltage.

   • Formula: $\% \text{Voltage Regulation} = \frac{V_{2(NL)} - V_{2(FL)}}{V_{2(FL)}} \times 100$ Where $V_{2(NL)} = E_2$ (for an ideal transformer).

   • Significance: Measures the ability of a transformer to maintain constant secondary voltage under varying load. Good regulation is desired (low %).

   • Approximate Formula: For lagging power factor loads, regulation is positive. For leading power factor loads, it can be zero or even negative.

2. Losses in a Transformer:

   • Core Losses (Iron Losses):

     • Hysteresis Loss: Due to reversal of magnetization in the core. $P_h \propto f \ B_m^{1.6 \text{ to } 2}$.

     • Eddy Current Loss: Due to induced currents in the core laminations. $P_e \propto f^2 \ B_m^2 \ t^2$, minimized by lamination.

     • Iron losses are constant for a given voltage and frequency (occur in the magnetic core).

   • Copper Losses ($I^2R$ Losses):

     • Heat loss due to current flow in the resistance of primary and secondary windings.

     • $P_{cu} = I_1^2 R_1 + I_2^2 R_2 = I_1^2 R_{01} = I_2^2 R_{02}$

     • Copper losses vary with the square of the load current.

3. Efficiency:

   • Definition: Ratio of output power to input power.

   • Formula: $\eta = \frac{\text{Output Power}}{\text{Input Power}} = \frac{\text{Output}}{\text{Output + Losses}}$

   • Full-load Efficiency: $\eta_{FL} = \frac{V_2 I_{2FL} \cos \phi_2}{V_2 I_{2FL} \cos \phi_2 + P_{i} + P_{cu(FL)}}$

   • Condition for Maximum Efficiency: Occurs when Copper Loss = Iron Loss. $I_2^2 R_{02} = P_i \quad \text{or} \quad \text{Load for max } \eta = \sqrt{\frac{P_i}{P_{cu(FL)}}} \times \text{Full Load}$

3.4 Equivalent Circuit and Tests

To analyze performance, the transformer is represented by an equivalent circuit referred to either primary or secondary side.

1. Exact Equivalent Circuit:

   • Referred to Primary: Combines parameters $R_1, X_1$ (primary winding), $R_c, X_m$ (core loss & magnetizing branches), and $R_2', X_2'$ (secondary winding referred to primary).

   • Referred to Secondary: Similar arrangement with primary parameters referred to secondary.

   • Simplified Equivalent Circuit: For practical calculations, the shunt branch ($R_c || X_m$) is moved to the input terminals, and the winding impedances are combined. $R_{01} = R_1 + R_2' = R_1 + a^2 R_2$ $X_{01} = X_1 + X_2' = X_1 + a^2 X_2$ $Z_{01} = \sqrt{R_{01}^2 + X_{01}^2}$

2. Open-Circuit (OC) Test:

   • Purpose: To determine core losses ($P_i$) and shunt circuit parameters ($R_c$ & $X_m$).

   • Procedure: LV side is energized at rated voltage and frequency, HV side left open. Measure $V_{1(OC)}$, $I_{0(OC)}$, and $P_{0(OC)}$ on the LV side.

   • Calculations: $\cos \phi_0 = \frac{P_0}{V_1 I_0}$ $I_c = I_0 \cos \phi_0, \quad I_m = I_0 \sin \phi_0$ $R_c = \frac{V_1}{I_c}, \quad X_m = \frac{V_1}{I_m}$ $P_i = P_0$ (measured power is the iron loss).

3. Short-Circuit (SC) Test:

   • Purpose: To determine full-load copper losses and equivalent resistance and reactance ($R_{01}, X_{01}$).

   • Procedure: LV side is short-circuited, and a reduced voltage (5-10% of rated) is applied to the HV side until rated current flows. Measure $V_{SC}$, $I_{SC}$, and $P_{SC}$ on the HV side.

   • Calculations: $Z_{01} = \frac{V_{SC}}{I_{SC}}$ $R_{01} = \frac{P_{SC}}{I_{SC}^2}$ $X_{01} = \sqrt{Z_{01}^2 - R_{01}^2}$ $P_{cu(FL)} \approx P_{SC}$ (copper loss at the current at which test was performed, typically full-load current).

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4. Three-Phase Transformer Connections

Three-phase transformation can be achieved using a three-phase transformer unit (one core with three limbs) or a bank of three single-phase transformers.

4.1 Common Connection Groups

The connection is denoted by the HV winding connection followed by the LV winding connection (e.g., Delta-Star).

1. Delta-Delta (Δ-Δ) Connection:

   • Advantages: Suitable for unbalanced loads, no problem with third harmonics, can operate with one transformer removed (open-delta or V-V connection at 57.7% capacity).

   • Disadvantages: No neutral point available on either side.

   • Application: Industrial installations.

2. Star-Star (Y-Y) Connection:

   • Advantages: Economical for high-voltage windings, neutral available for grounding on both sides.

   • Disadvantages: Prone to third harmonic voltages, requires a tertiary delta winding or neutral grounding for stability. Sensitive to unbalanced loads.

   • Application: Rarely used alone; often with a tertiary winding.

3. Delta-Star (Δ-Y) Connection:

   • Most Common Configuration.

   • Advantages: Neutral available on the LV (star) side for 3-phase 4-wire distribution. The delta on HV side provides a path for third harmonic currents, preventing distortion.

   • Applications: Step-down distribution transformers (HV Δ, LV Y with neutral). Standard for supplying 3-phase 4-wire systems (415V/240V).

4. Star-Delta (Y-Δ) Connection:

   • Advantages: Neutral can be grounded on HV side. Delta on LV side suppresses third harmonics.

   • Applications: Step-up transformers at generating stations (HV Y, LV Δ).

4.2 Phase Shift and Vector Groups

1. Phase Displacement: The connection creates a phase angle difference between the HV and LV line voltages. This is standardized by vector group notation (e.g., Dyn11, YNd1).

2. Clock Notation: The HV line voltage phasor is taken as the reference (minute hand at 12 o'clock). The corresponding LV line voltage phasor indicates the hour (e.g., 11 means -30° lag, 1 means +30° lead).

3. Importance: Essential for parallel operation of transformers. Transformers to be paralleled must belong to the same vector group.

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5. Special Transformers

5.1 Pulse Transformers

1. Function: Designed to transmit electrical pulses with minimal distortion. The output pulse should be a faithful replica of the input pulse.

2. Key Requirements:

   • Fast Rise Time: Core material must have high saturation flux density and low losses at high frequencies (e.g., ferrite, permalloy).

   • Low Leakage Inductance: Achieved by interleaving or close coupling of windings.

   • Low Winding Capacitance: To prevent pulse rounding and oscillations.

3. Applications:

   • Triggering of thyristors and SCRs in power electronics.

   • Coupling stages in digital logic circuits and radar systems.

   • Gate drive circuits for MOSFETs and IGBTs.

5.2 Autotransformers

1. Construction: A single winding acts as both primary and secondary. Part of the winding is common to both circuits.

2. Principle and Turns Ratio:

   • If $N_1$ is total turns and $N_2$ are the common turns, then: $\frac{V_1}{V_2} = \frac{N_1}{N_2} = a$

   • The transformation is partly conductive and partly inductive. The induced voltage handles only a fraction of the total power.

3. Power Transfer Relationship:

   • Output Power ($S_{out}$) = Transformed Power ($S_{ind}$) + Conducted Power ($S_{cond}$).

   • $S_{ind} = \left(1 - \frac{1}{a}\right) S_{out}$

   • For a small voltage ratio (e.g., 2:1), the transformed power is only half the output power.

4. Advantages:

   • Smaller Size & Cost: For a given rating, uses less copper and core material than a two-winding transformer (for voltage ratios close to 1).

   • Higher Efficiency: Lower $I^2R$ losses due to reduced conductor material.

   • Better Voltage Regulation: Lower leakage impedance.

5. Disadvantage:

   • No Electrical Isolation: Primary and secondary are connected, posing a safety hazard if the common connection fails. Not suitable for step-down applications where isolation is critical (e.g., household supply).

6. Applications:

   • Variable AC Supplies (Variac): Continuously variable autotransformers.

   • Starting of Induction Motors: To reduce starting current.

   • Interconnecting Power Systems with slightly different voltage levels.

   • Voltage Boosting in distribution lines.

Conclusion: The transformer is a quintessential component in AC power systems. From its fundamental EMF equation and equivalent circuit to its various three-phase connections and specialized forms like the autotransformer, a deep understanding of its principles, performance characteristics, and construction is vital for any power engineer. This knowledge enables the proper selection, operation, and maintenance of transformers, ensuring efficient, reliable, and safe power delivery.