Transformer Lecture Notes

Fundamental Definition and Principles of Transformers

• Definition: A transformer is a static device by means of which electrical power at one voltage level is transferred to electrical power at another voltage level while keeping the frequency constant.
• Operating Principle: The transformer works on the principle of electromagnetic induction.
• Physical Basis: The physical basis of a transformer is mutual induction between two circuits linked by a common magnetic flux.
• Coupling and Isolation: The transformer is an electrically isolated but magnetically coupled device.

Constructional Features of Transformers

A transformer mainly consists of two functional parts:

• Laminated Core: In all types of transformers, the core is constructed of silicon sheet steel laminations. These are assembled to provide a continuous magnetic path with a minimum of air gap included.
• Windings:     * The two windings are placed over the core and are insulated from each other as well as from the core.     * Placement: Usually, the low voltage (LV) winding is placed nearer to the core, over which the high voltage (HV) winding is placed.     * Primary Winding: The winding connected to the supply mains.     * Secondary Winding: The winding from which the output voltage is taken.

Classification and Types of Transformers

Transformers can be classified based on various criteria:

Based on Type of Construction

• Core Type: In this construction, the core is surrounded by the windings.
• Shell Type: In this construction, the windings are surrounded by the iron core.

Based on Number of Phases

• Single Phase: For single-phase AC systems.
• Three Phase: For three-phase AC systems.

Based on Type of Application

• Distribution Transformer:     * Ratings: Up to a size of about $200\,kVA$.     * Function: Used to step down distribution voltage to a standard service voltage.     * Duty Cycle: They are kept in operation all 24 hours of the day, whether they are carrying any load or not.
• Power Transformer:     * Ratings: Above $200\,kVA$ (Transcript mentions a rating of above $200\,kVA$ as a threshold).     * Application: Used at generating stations and substations at each end of a power transmission line for stepping up or stepping down the voltage.     * Duty Cycle: They are put in operation during load periods and are disconnected during light load periods.

Characteristics of an Ideal Transformer

An ideal transformer and its core have the following characteristics:

1. Winding resistances are negligible.
2. No magnetic leakage occurs; all flux set by the primary links with the secondary winding.
3. Core losses (hysteresis and eddy current) are negligible.
4. The core has constant permeability, meaning the magnetisation curve for the core is linear.
5. An ideal transformer consists of two purely inductive coils worked on a loss-free core.

EMF Equations and Transformation Ratio

The statically induced emf in a transformer is determined by the following equations:

• RMS Value of Primary Induced EMF ($E_1$):     $E_1 = 4.44 \times f \times \Phi_m \times T_1\,\text{Volts}$
• RMS Value of Secondary Induced EMF ($E_2$):     $E_2 = 4.44 \times f \times \Phi_m \times T_2\,\text{Volts}$
• Flux Dependence: The induced emf depends on the peak value of the magnetic flux ($\Phi_m$).
• Phase Relationship: Inducted emfs in the primary and secondary are in phase with each other, and they lag the corresponding flux by an angle of $90^\circ$.
• Transformation Ratio ($K$):     $K = \frac{E_2}{E_1}$     * The transformation ratio is only valid for phase induced emf.
• Voltage per Turn: This value is identical for both primary and secondary windings:     $\text{Voltage/Turn} = \frac{E_1}{T_1} = \frac{E_2}{T_2} = 4.44 \times f \times \Phi_m$
• No Load Terminal Voltages: Under no-load conditions, terminal voltages ($V$) and induced emfs ($E$) are equal:     $V_1 \approx E_1$$V_2 \approx E_2$
• Step-up vs. Step-down:     * If $K > 1$, the transformer is a step-up transformer.     * If $K < 1$, the transformer is a step-down transformer.

Transformer Operation: No-Load Conditions

• No-Load Current ($I_0$): The current is very small compared to full-load current. As the physical size of the transformer increases, the relative value of the no-load current decreases.
• Lag Angle: The no-load current lags the applied voltage by an angle slightly less than $90^\circ$ (approximately $80^\circ$ to $85^\circ$).
• Input Power: The no-load input power consists of small primary copper losses and core losses. Since copper losses are negligible at no load, the input power effectively gives the core losses.
• Components of No-Load Current:     1. Loss Component ($I_w$): In phase with the applied voltage ($V_1$). $I_w = I_0 \times \cos(\Phi_0)$.     2. Magnetizing Component ($I_\mu$): In phase with the flux or in quadrature (at $90^\circ$) with applied voltage. $I_\mu = I_0 \times \sin(\Phi_0)$.     * Relationship: The magnitude of $I_\mu$ is greater than $I_w$.
• Power Factor: The power factor ($pf$) at no load is very low, approximately $0.1$ to $0.2$ lagging.

Transformer Operation: On-Load Conditions

• Constant Flux Machine: The transformer is a constant flux machine. Load changes do not influence the magnitude of the flux in the core.
• MMF Equality: To maintain constant flux, primary and secondary magnetomotive forces (mmfs) must be equal, neglecting no-load primary mmf:     $I_1 \times T_1 = I_2 \times T_2$
• Current and Load Relationships:     * As secondary load increases, the current drawn from the supply in the primary increases.     * As the load on the transformer increases, its input power factor increases.
• Power Factor Dynamics: The secondary power factor depends on the nature of the load. If no-load current is neglected, primary and secondary have the same power factor; otherwise, primary power factor is slightly less than secondary.
• Leakage Flux: Increases with the load current.
• Primary Voltage Equation:     $V_1 = E_1 + I_1 \times (R_1 + jX_1)$ or $V_1 = E_1 + I_1 \times Z_1$

Referencing Parameters and Equivalent Circuit

Resistance and reactance can be referred from one side to the other. Let $K$ be the transformation ratio ($N_2/N_1$).

Total Parameters Referred to Primary

• $R_{eq1} = R_1 + \frac{R_2}{K^2}$
• $X_{eq1} = X_1 + \frac{X_2}{K^2}$
• $Z_{eq1} = Z_1 + \frac{Z_2}{K^2}$

Total Parameters Referred to Secondary

• $R_{eq2} = R_2 + R_1 \times K^2$
• $X_{eq2} = X_2 + X_1 \times K^2$
• $Z_{eq2} = Z_2 + Z_1 \times K^2$

Approximate Equivalent Circuit Constants

• $R'_2 = \frac{R_2}{K^2}$
• $V'_2 = \frac{V_2}{K^2}$
• $I'_2 = K \times I_2$

Voltage Regulation

• Definition: The ratio of the change in secondary terminal voltage from no load to full load to the rated secondary voltage, expressed as a percentage with primary voltage held constant.
• Formula:     $\%\,\text{Regulation} = \frac{E_2 - V_2}{E_2} \times 100$
• Estimation Formula:     $\%\,\text{Regulation} = D_r \times \cos(\phi) \pm D_x \times \sin(\phi)$     * $D_r$: \% Resistance drop ($\frac{I_2 \times R_{eq2}}{V_2} \times 100$).     * $D_x$: \% Leakage reactance drop ($\frac{I_2 \times X_{eq2}}{V_2} \times 100$).     * Sign Convention: Use $+$ for inductive (lagging) load; use $-$ for capacitive (leading) load.
• Load Nature Implication:     * Regulation is positive for resistive and inductive loads.     * Regulation is negative for capacitive loads (terminal voltage is higher than no-load voltage).
• Zero and Maximum Regulation:     * Load pf for zero regulation is $\frac{X_{02}}{Z_{02}}$ leading.     * Load pf for maximum regulation is $\frac{R_{02}}{Z_{02}}$ lagging.     * The magnitude of maximum regulation equals the per unit value of the equivalent leakage impedance.

Losses in a Transformer

Transformers experience two primary types of losses:

• Copper Losses ($W_{cu}$): These occur due to the resistance of the windings. They are proportional to the square of the load current ($I^2 R$).
• Iron Losses ($W_i$): These occur in the core and are subdivided into:     1. Hysteresis losses.     2. Eddy-current losses.     * Iron losses depend on supply voltage and frequency but are independent of load current and load power factor.

Efficiency of a Transformer

• General Formula:     $\eta = \frac{\text{Output Power}}{\text{Output Power} + \text{Total Losses}}$
• At Fraction $X$ of Full Load:     $\eta = \frac{V_2 \times (X \times I_2) \times \cos(\phi)}{V_2 \times (X \times I_2) \times \cos(\phi) + X^2 \times (\text{Full Load Copper Loss}) + \text{Iron Losses}}$
• Condition for Maximum Efficiency: Variable copper losses = Constant iron losses.
• Loading for Max Efficiency:     $\text{Load at max efficiency} = \text{Full Load} \times \sqrt{\frac{\text{Iron Losses}}{\text{Full Load Copper Losses}}}$
• Load Current for Max Efficiency:     $I_{load} = \sqrt{\frac{\text{Iron Losses}}{\text{Resistance of Transformer}}}$
• Power Factor Effect: For a constant load current, efficiency is highest when the load power factor is unity.
• Typical Peak: Transformers generally reach maximum efficiency at about $3/4$ of full load ($75\%$ load).

All-Day Efficiency

• Scope: This is specifically relevant for distribution transformers.
• Definition:     $\text{All-Day Efficiency} = \frac{\text{KWh output in 24 hours}}{\text{KWh output in 24 hours} + \text{KWh wasted as losses in 24 hours}}$
• Optimization: For high all-day efficiency, transformers should operate near maximum load most of the time. Distribution transformers are designed with low flux density to minimize iron losses and achieve maximum efficiency at roughly half-full load.

Testing Procedures

Open Circuit (O.C.) Test

• Objectives: To estimate iron losses, develop the magnetizing branch of the equivalent circuit, and estimate the no-load power factor.
• Procedure: Conducted on the Low Voltage (LV) side by applying rated voltage at rated frequency while keeping the High Voltage (HV) side open.

Short Circuit (S.C.) Test

• Objectives: To estimate full-load copper losses, determine series elements of the equivalent circuit, and estimate regulation.
• Procedure: Conducted on the HV side by applying a small percentage of rated voltage to circulate the full-load current while shorting the LV side. Power input represents copper losses as iron losses are negligible under these conditions.

Sumpner’s or Back-to-Back Test

• Objectives: To estimate efficiency, regulation, and temperature rise under loaded conditions.
• Requirement: Requires two similar transformers.

Auto Transformers

• Power Transfer Mechanism: Power is transferred both conductively and inductively.
• Transfer Proportions (where $K$ is the turn ratio, $K < 1$):     * Inductive transfer: $(1 - K) \times \text{input}$     * Conductive transfer: $K \times \text{input}$
• Advantages: Higher efficiency and better voltage regulation compared to a 2-winding transformer of the same output.
• Copper Saving:     $\text{Saving of Copper} = K \times (\text{Weight of copper in two-winding transformer})$

Parallel Operation and Load Sharing

• Mandatory Conditions:     1. Transformers must have the same voltage ratio.     2. Proper polarity must be maintained during connection.     3. Per unit impedance should be equal to avoid circulating currents.     4. Equivalent impedance should be inversely proportional to the kVA rating.
• Load Sharing: Load sharing is inversely proportional to the transformers' impedances.

Tap Changers

• Function: Taps are provided for the control of output voltage.
• Placement: Generally provided on the HV side at center turns to maintain constant LV voltage. The voltage per turn remains constant.
• Types:     * Off-load tap changer: Operates only when the transformer is out of circuit.     * On-load tap changer: Designed to operate while the transformer is in circuit.

Three-Phase Transformers

Common Connections and Features

Open Delta (V-V) Characteristics

• Occurs when one transformer of a delta bank is removed.
• Load Capacity: $57.7\%$ of the original 3-phase load, or $86.6\%$ of the rated capacity of the two remaining units.
• Overload: Each unit is overloaded by $73.2\%$ when carrying the full rated 3-phase load.
• Power Factor: Average pf is $86.6\%$ of the balanced load pf. The two transformers operate at different power factors unless the load is balanced unity pf.

Scott Connection Details

• Teaser primary turns are $\frac{\sqrt{3}}{2}$ (approx. $86.6\%$) of the main primary turns.
• The teaser turns ratio is $15\%$ more than that of the main transformer.
• Under balanced conditions, the main transformer rating is $15\%$ greater than the teaser.

Harmonics and Noise

• Cause of Harmonics: High flux densities in the core. Saturation during parts of the sinusoidal wave makes the secondary waveform non-sinusoidal.
• Frequency Rules: Frequency of the $n$-th harmonic is $n \times \text{fundamental frequency}$. RMS value of the $n$-th harmonic is $1/n$ of the fundamental RMS value.
• Impact of Harmonics:     * Current harmonics: High $I^2R$ loss, core loss, magnetic interference, communication interference.     * Voltage harmonics: Increased dielectric loss.
• Transformer Noise Sources:     1. Magnetostriction (change in linear dimensions of core material due to magnetic properties). Reduced by lower flux densities.     2. Mechanical vibrations of laminations due to magnetic forces.

Transformer Fittings

• Bushings: A central conductor surrounded by graded insulation used to take a conductor out through the metallic tank.
• Conservator: A tank connected to the main tank to accommodate expansion and contraction of oil.
• Breather: Contains silica gel to arrest moisture from incoming air, preventing oil contamination.
• Buchholz Relay: A gas-operated device connected between the main tank and the conservator. It operates when an internal fault occurs, evolving gases.

Summary Highlights

• EMF Equation: $E = 4.44 \times f \times \Phi_m \times T$.
• Flux remains constant from no load to full load.
• Efficiency and regulation define performance. Regulation depends on both the nature (pf) and magnitude of the load.
• Maximum efficiency usually occurs near full load ($3/4$ Full Load).
• All-day efficiency is always lower than commercial efficiency.
• Scott connection enables 3-phase to 2-phase conversion.