Single-Phase Transformer

Contents

**Define Transformer and explain the working principle of Single-Phase Transformer** 1

**Derive the EMF equation of Single-Phase Transformer** 3

**Describe Equivalent Circuits and Phasor Diagrams of Single-Phase Transformer** 5

**Describe Power Transformer and Distribution Transformer** 7

**Describe the Core Types and Shell Type Transformer** 9

**Recall Cooling and Insulation in Transformer** 10

**Describe Per Unit Calculation in Transformers** 10

**Recall Voltage Regulation of a Single-Phase Transformer** 10

**Calculate the Approximated Voltage Regulation** 10

**Recall the necessary conditions for Maximum, Minimum, and Zero Voltage Regulation** 10

**Recall the Losses that occur in the Transformer** 10

**Describe the Efficiency of a Single-Phase Transformer** 10

**Recall the condition for Maximum Efficiency of Transformer** 10

**Describe All-Day Efficiency of Transformer** 10

**Recall Open-Circuit Test of Single-Phase Transformer** 10

**Recall Short-Circuit Test of Single-Phase Transformer** 10

**Determine Percentage Copper Saving in Autotransformer** 10

**Convert a Two Winding Transformer into an Autotransformer** 10

**Define Transformer and explain the working principle of Single-Phase Transformer**

A transformer is an electrical device that is used to transfer electrical energy between two or more circuits through electromagnetic induction. It consists of two or more coils of insulated wire, known as windings, which are wound around a common magnetic core.

The working principle of a single-phase transformer is based on Faraday’s law of electromagnetic induction. When an alternating current (AC) flows through the primary winding, it creates a changing magnetic field in the core. This changing magnetic field induces a voltage in the secondary winding, which is connected to the load.

The working principle can be summarized in the following steps:

- AC Input: An alternating current is applied to the primary winding of the transformer. The AC input voltage causes an alternating magnetic field to be generated in the core.
- Magnetic Field Induction: The alternating magnetic field induces a voltage in the secondary winding through mutual induction. The magnitude of the induced voltage depends on the turns ratio between the primary and secondary windings.
- Voltage Transformation: The transformer operates based on the principle of voltage transformation. The turns ratio between the primary and secondary windings determines the ratio of the input and output voltages. For example, if the turns ratio is 1:2, a voltage of 100 volts applied to the primary winding will result in an output voltage of 200 volts in the secondary winding.
- Isolation: Transformers provide electrical isolation between the input and output circuits. The primary and secondary windings are physically separated, preventing direct electrical contact between the input and output sides.
- Power Transfer: The transformer efficiently transfers electrical power from the primary winding to the secondary winding. The power transferred is determined by the turns ratio and the load connected to the secondary winding.
- Conservation of Energy: The power transferred from the primary to the secondary winding is nearly equal to the power input, taking into account the efficiency of the transformer.

The efficiency of a transformer is determined by factors such as core material, winding resistance, and design. Transformers are widely used in various applications, including power distribution systems, voltage regulation, electrical equipment, and electronics.

It’s important to note that single-phase transformers are designed to work with single-phase AC power, while three-phase transformers are used for three-phase AC systems.

**Describe Ideal Transformer**

An ideal transformer is a theoretical concept used in electrical engineering to simplify the analysis and calculations of transformer behavior. It is an idealized representation of a transformer that assumes certain ideal conditions.

The characteristics of an ideal transformer include:

- No Core Losses: An ideal transformer assumes that there are no losses associated with the magnetic core material. This means that there is no hysteresis loss or eddy current loss in the core, resulting in a perfect transfer of magnetic flux.
- No Copper Losses: In an ideal transformer, the windings have zero resistance, resulting in no power dissipation due to current flowing through the windings. This means that there are no copper losses in the form of I²R losses.
- No Leakage Flux: An ideal transformer assumes that all the magnetic flux generated by the primary winding is linked completely with the secondary winding, and there is no leakage flux. This ensures a perfect coupling between the windings.
- Perfect Voltage Ratio: The voltage ratio between the primary and secondary windings is assumed to be ideal in an ideal transformer. This means that the ratio of turns in the windings accurately represents the voltage ratio.
- Perfect Efficiency: An ideal transformer has 100% efficiency, meaning that there are no energy losses during the transfer of power from the primary to the secondary winding.

While an ideal transformer is a theoretical concept and does not exist in practice, it serves as a useful tool for understanding the basic principles and behavior of real-world transformers. In practical applications, transformers have losses due to factors such as core losses, winding resistance, and leakage flux, which impact their efficiency and performance. Nonetheless, the ideal transformer provides a useful benchmark for analysis and design calculations.

**Derive the EMF equation of Single-Phase Transformer**

The EMF equation of a transformer relates the voltages and turns ratios of the transformer. The derivation of the EMF equation for a single-phase transformer is as follows:

Consider a transformer with N1 and N2 number of turns in the primary and secondary windings, respectively. Let the applied voltage on the primary side be V1 and the secondary voltage be V2. The current in the primary winding is I1, and the current in the secondary winding is I2.

By Faraday’s law of electromagnetic induction, the emf induced in the primary winding is given by:

e1 = -N1 (dΦ/dt)

where Φ is the magnetic flux in the core.

Now, the magnetic flux linking with both the windings is the same, and it is given by:

Φ = B A

where B is the magnetic flux density, and A is the cross-sectional area of the core.

The voltage induced in the secondary winding can be expressed as:

e2 = -N2 (dΦ/dt)

The mutual flux linking the two windings is given by:

N1 I1 = N2 I2

The transformer is assumed to be ideal, which means that there are no losses, and the core has infinite permeability. Thus, the magnetic flux is entirely confined to the core and does not leak outside. Therefore, the magnetic field generated by the primary winding induces a voltage in the secondary winding that is proportional to the number of turns in the winding.

By substituting the value of Φ in the above equations, we get:

e1 = -N1 (d/dt) (B A) = -N1 A (dB/dt)

e2 = -N2 (d/dt) (B A) = -N2 A (dB/dt)

Dividing both equations by N1 and N2, respectively, we get:

e1/N1 = -A (dB/dt)

e2/N2 = -A (dB/dt)

By dividing the above two equations, we get:

(e1/N1)/(e2/N2) = N2/N1

This is the EMF equation of a single-phase transformer, which relates the turns ratio and the voltages of the transformer.

In summary, the EMF equation of a single-phase transformer is a fundamental equation that describes the relationship between the voltages and turns ratio of a transformer. It is derived from the principles of electromagnetic induction and the assumption of an ideal transformer.

**Describe Equivalent Circuits and Phasor Diagrams of Single-Phase Transformer**

Equivalent Circuit of Single-Phase Transformer:

The equivalent circuit of a single-phase transformer represents the simplified electrical model of the transformer that allows for the analysis of its behavior and performance. It consists of several components that approximate the actual characteristics of the transformer. The primary and secondary windings are represented by their respective resistances (R1 and R2) and leakages reactances (X1 and X2). The magnetic core is represented by the magnetizing inductance (Xm). The ideal transformer is represented by the turns ratio (N1/N2) and the mutual inductance (M). The equivalent circuit of a single-phase transformer is typically shown as follows:

Phasor Diagram of Single-Phase Transformer:

A phasor diagram is a graphical representation that illustrates the phasor relationships between the primary and secondary voltages and currents in a single-phase transformer. It helps visualize the voltage and current relationships and provides insights into the behavior of the transformer.

In a phasor diagram of a single-phase transformer, the primary voltage (V1) is represented as a reference phasor, usually taken as the horizontal axis. The primary current (I1) lags the primary voltage due to the inductive reactance of the primary winding. The secondary voltage (V2) is proportional to the primary voltage based on the turns ratio, and it lags the primary voltage by the same angle. The secondary current (I2) also lags the secondary voltage due to the inductive reactance of the secondary winding.

The phasor diagram shows the magnitudes and relative angles between these phasors, providing a visual representation of the voltage and current relationships in the transformer. It helps in understanding the voltage transformation, phase shifts, and power flow in the transformer.

Overall, the equivalent circuit and phasor diagram of a single-phase transformer are valuable tools for analyzing and understanding the behavior of the transformer under different operating conditions and loads. They assist in calculating parameters such as voltage regulation, efficiency, and power transfer capabilities.

**Explain Approximate Equivalent Circuits referred to Primary and Secondary side of Single-Phase Transformer**

Approximate equivalent circuits of single-phase transformers are used to represent the transformer’s behavior under different load conditions. These equivalent circuits simplify the analysis of the transformer’s performance and allow the calculation of its equivalent parameters, such as resistance, inductance, and leakage reactance.

There are two approximate equivalent circuits referred to the primary and secondary side of a single-phase transformer: the approximate equivalent circuit referred to the primary side and the approximate equivalent circuit referred to the secondary side.

The approximate equivalent circuit referred to the primary side of a single-phase transformer consists of a resistance R1, an inductance X1, and a leakage reactance Xp. R1 and X1 represent the primary winding’s resistance and leakage inductance, respectively, while Xp represents the leakage reactance of the transformer, which is the inductive reactance due to the flux leakage between the primary and secondary windings. The magnetising reactance Xm is also included in this circuit, which represents the inductive reactance required to establish the magnetic flux in the core.

On the other hand, the approximate equivalent circuit referred to the secondary side of a single-phase transformer consists of a resistance R2, an inductance X2, and a leakage reactance Xs. R2 and X2 represent the secondary winding resistance and leakage inductance, respectively, while Xs represents the leakage reactance of the transformer, which is the inductive reactance due to the flux leakage between the secondary and primary windings.

The phasor diagrams of these approximate equivalent circuits show the voltage drops due to the resistive and inductive components of the circuit. These phasor diagrams are used to determine the voltage regulation of the transformer, which is the percentage change in the secondary voltage when the load current changes from no load to full load. The voltage regulation is a measure of the transformer’s ability to maintain a constant secondary voltage under varying load conditions.

In practice, the exact equivalent circuit of a transformer is more complex than the approximate equivalent circuits and includes additional components such as core losses and winding capacitance. However, the approximate equivalent circuits are useful for analyzing the transformer’s behavior under different load conditions and for determining its equivalent parameters.

**Describe Power Transformer and Distribution Transformer**

Power Transformer:

A power transformer is a type of transformer used in high-voltage electrical power transmission and distribution systems. Its main function is to step up or step down the voltage levels to facilitate efficient and safe transmission of electrical power over long distances. Power transformers are typically large and designed to handle high power levels.

Key features of power transformers include:

- High Voltage and Power Ratings: Power transformers are designed to handle high voltage levels, typically ranging from thousands to hundreds of thousands of volts. They are also capable of handling large power ratings, often in the range of several megawatts.
- Step-up and Step-down Operation: Power transformers can step up the voltage from the generator or power source for long-distance transmission and step it down to lower voltage levels suitable for distribution to consumers.
- Core and Windings: Power transformers have a magnetic core made of laminated steel to enhance the magnetic flux and minimize losses. They have two or more sets of windings, including primary windings connected to the high-voltage side and secondary windings connected to the low-voltage side.
- Cooling Systems: Power transformers incorporate cooling systems, such as oil-immersed cooling or forced air cooling, to dissipate heat generated during operation and ensure optimal performance.

Distribution Transformer:

A distribution transformer is a type of transformer used in the electrical distribution network to supply power to residential, commercial, and industrial consumers. It operates at lower voltage levels and is responsible for stepping down the voltage from the transmission level to the distribution level, typically ranging from a few hundred to a few thousand volts.

Key features of distribution transformers include:

- Lower Voltage and Power Ratings: Distribution transformers are designed to operate at lower voltage levels, typically ranging from a few hundred volts to a few thousand volts. They have lower power ratings compared to power transformers, often in the range of a few hundred kilovolt-amperes (kVA) to a few megavolt-amperes (MVA).
- Compact Size: Distribution transformers are relatively smaller in size compared to power transformers. They are often pole-mounted or pad-mounted, making them suitable for urban and residential areas.
- Core and Windings: Distribution transformers also have a magnetic core made of laminated steel to minimize losses. They have two sets of windings, primary windings connected to the higher voltage distribution network and secondary windings connected to the lower voltage consumer side.
- Cooling Systems: Distribution transformers typically use oil-immersed cooling systems to dissipate heat generated during operation.
- Overload and Short Circuit Protection: Distribution transformers incorporate protective devices to prevent damage due to overloads or short circuits, ensuring the safety and reliability of the distribution network.

Overall, power transformers and distribution transformers serve different roles in the electrical power system. Power transformers are responsible for long-distance power transmission, while distribution transformers deliver power to end consumers at lower voltage levels.

**Describe the Core Types and Shell Type Transformer**

Core Types:

- Core Type Transformer: In a core type transformer, the magnetic core is divided into two separate limbs or legs. The primary and secondary windings are wound around these limbs. The magnetic flux generated by the windings passes through the core, resulting in efficient magnetic coupling. Core type transformers are commonly used in low and medium voltage applications.
- Shell Type Transformer: In a shell type transformer, the windings surround the core. The primary and secondary windings are placed on either side of the core, with the core passing through the center. The magnetic flux flows in a circular path, which improves the magnetic coupling between the windings. Shell type transformers are often used in high voltage applications due to their ability to handle higher power levels.

Key features of core type and shell type transformers:

- Core Material: Both core type and shell type transformers use laminated steel as the core material. The laminations help reduce the eddy current losses and enhance the magnetic performance of the transformer.
- Cooling System: Core type transformers usually have a better cooling system due to the exposed surface area of the windings, allowing efficient heat dissipation. Shell type transformers may have a more compact design, resulting in less exposed surface area and potentially requiring additional cooling methods.
- Size and Weight: Shell type transformers are generally larger and heavier compared to core type transformers for the same power rating. The larger size of shell type transformers can make them more suitable for applications where size constraints are not a significant concern.
- Mechanical Strength: Core type transformers offer better mechanical strength due to the separate limbs that provide additional support. Shell type transformers, on the other hand, may be more susceptible to mechanical stress due to the winding arrangement.
- Noise Level: Core type transformers tend to produce less audible noise during operation compared to shell type transformers. The noise generated in shell type transformers is due to the magnetic flux circulating within the winding structure.

The choice between a core type and shell type transformer depends on factors such as the voltage level, power rating, size constraints, and specific application requirements. Both types have their advantages and are suitable for different applications within the power distribution and transmission systems.

**Recall Cooling and Insulation in Transformer**

Cooling and insulation are two essential aspects of transformer design that directly impact its performance and lifespan. The transformer generates heat during operation due to core losses, winding resistance losses, and eddy current losses. Thus, it is essential to remove this heat to ensure reliable operation and prevent damage to the insulation material. Similarly, the transformer’s insulation must be designed to withstand high voltage, mechanical stress, and temperature changes without failing.

Cooling:

The cooling system of a transformer is responsible for removing the heat generated during operation. There are different cooling methods used in transformers, depending on the transformer’s size, type, and application. The following are the most common cooling methods used in transformers:

- Air-cooled (AN): Transformers rated below 5 kVA generally use natural convection for cooling. Larger transformers may use fans or blowers to circulate air across the transformer’s windings and core.
- Oil-Immersed (ONAN): Oil-immersed transformers use the oil as a coolant and insulating medium. The oil’s circulation is either natural (ONAN) or forced (OFAN) using pumps or fans.
- Water-cooled (OW): Water-cooled transformers are used in high-power applications where the transformer’s size is too large for air or oil cooling.

Insulation:

The transformer’s insulation must be designed to withstand high voltage and mechanical stress without breaking down or degrading over time. The insulation system consists of several layers, each designed to provide a specific level of protection. The following are the different types of insulation used in transformers:

- Solid Insulation: Solid insulation materials include cellulose, mica, and ceramic. These materials are used in the transformer’s core, coil, and other structural parts to provide mechanical stability.
- Liquid Insulation: Liquid insulation materials such as transformer oil are used to insulate the transformer’s windings and to provide a medium for heat dissipation.
- Gas Insulation: Gas insulation materials such as nitrogen, sulphur hexafluoride, and air are used in high-voltage transformers to provide insulation.

Transformer insulation systems are classified based on the insulation material used, its temperature rating, and its dielectric strength. The most commonly used insulation systems are class A, B, F, H, and C. The class rating denotes the maximum allowable temperature rise in the transformer’s windings above the ambient temperature. For example, class A insulation is rated for a temperature rise of up to 105°C above the ambient temperature, while class H insulation is rated for a temperature rise of up to 180°C.

**Describe Per Unit Calculation in Transformers**

Per unit (pu) calculation is a widely used method in power system analysis to normalise the parameters of different components such as transformers, generators, and transmission lines. It is a dimensionless quantity that enables easy comparison of equipment with different ratings, operating conditions, and physical properties.

In transformers, per unit values are calculated for various parameters like voltage, current, power, impedance, and admittance. The base values for normalisation are usually the rated values of the transformer, i.e., the nameplate values of voltage and apparent power.

The per-unit value of a parameter is defined as the ratio of its actual value to its base value. For example, the per-unit voltage of a transformer is given by the ratio of its actual voltage to the base voltage. If the rated voltage of the transformer is 11 kV, and the base voltage is taken as 10 kV, then the per-unit voltage is given by:

Per-unit voltage = Actual voltage / Base voltage = 11 kV / 10 kV = 1.1 pu

Similarly, the per-unit impedance of a transformer is given by the ratio of its actual impedance to its base impedance. The base impedance is calculated by dividing the base voltage by the base apparent power. For example, if the base voltage is 10 kV and the base apparent power is 100 MVA, then the base impedance is given by:

Base impedance = (Base voltage)^2 / Base apparent power = (10 kV)^2 / 100 MVA = 1 Ω

If the actual impedance of the transformer is 10 Ω, then the per-unit impedance is given by:

Per-unit impedance = Actual impedance / Base impedance = 10 Ω / 1 Ω = 10 pu

Per unit calculation makes it easier to perform system analysis and to compare different transformers or other equipment. It also enables easy scaling of the system parameters for simulation and analysis purposes.

**Recall Voltage Regulation of a Single-Phase Transformer**

Voltage regulation is an important aspect of transformer performance. It refers to the percentage difference between the no-load voltage and the full-load voltage of a transformer. The voltage regulation of a single-phase transformer can be defined as the ratio of the difference between the no-load voltage and the full-load voltage to the full-load voltage, expressed as a percentage.

Mathematically, the voltage regulation of a transformer can be expressed as:

% Voltage Regulation = [(Vno-load – Full-load) / Full-load] x 100%

where Vno-load is the voltage at the secondary terminals of the transformer when there is no load connected, and Vfull-load is the voltage at the secondary terminals when the transformer is loaded to its full capacity.

The voltage regulation is usually expressed as a percentage of the rated voltage of the transformer. A transformer with good voltage regulation will have a small difference between the no-load and full-load voltages, indicating that it can maintain a constant output voltage under varying load conditions. On the other hand, a transformer with poor voltage regulation will have a large difference between the no-load and full-load voltages, indicating that its output voltage will vary significantly with changes in the load.

Factors that affect the voltage regulation of a transformer include the resistance and reactance of the transformer windings, the power factor of the load, and the transformer’s design characteristics.

For example, a single-phase transformer has a rated voltage of 220V/110V and a rated capacity of 10 kVA. The no-load voltage is measured at 225V, and the full-load voltage is measured at 215V. The voltage regulation of the transformer can be calculated as follows:

% Voltage Regulation = [(225V – 215V) / 215V] x 100% = 4.65%

This means that the output voltage of the transformer will drop by 4.65% when the transformer is loaded to its full capacity.

**Calculate the Approximated Voltage Regulation**

Voltage regulation is an important parameter for measuring the performance of a transformer. It is defined as the difference between the no-load voltage and the load voltage expressed as a percentage of the no-load voltage.

The voltage regulation can be calculated using the following formula:

% Voltage Regulation = [(Vno load – Vload) / Vno load] × 100

where Vno load is the no-load voltage and Vload is the load voltage.

The voltage regulation can also be expressed in terms of per-unit impedance and load current. The per-unit impedance is the ratio of the transformer impedance to its rated voltage. The voltage regulation can be calculated using the following formula:

% Voltage Regulation = (Per-unit impedance × Load Current) × 100

To calculate the approximated voltage regulation, we use the approximate equivalent circuit of the transformer. The approximate equivalent circuit is obtained by neglecting the resistance of the winding and the magnetising reactance of the transformer. The approximate equivalent circuit consists of only the leakage reactance and the core loss resistance.

The voltage regulation can be calculated using the approximate equivalent circuit by the following formula:

% Voltage Regulation = (2 × Per-unit leakage reactance × Load Current) × 100

where Per-unit leakage reactance is the ratio of the leakage reactance of the transformer to its rated voltage.

For example, let’s consider a single-phase transformer with a rated voltage of 10 kV and a rated capacity of 100 kVA. The per-unit impedance of the transformer is 0.05. The per-unit leakage reactance is 0.03. The load current is 50 A.

Using the per-unit impedance formula, the voltage regulation can be calculated as:

% Voltage Regulation = (0.05 × 50) × 100 = 2.5%

Using the approximate equivalent circuit formula, the voltage regulation can be calculated as:

% Voltage Regulation = (2 × 0.03 × 50) × 100 = 3%

Therefore, the approximated voltage regulation for this transformer is 3%.

**Recall the necessary conditions for Maximum, Minimum, and Zero Voltage Regulation**

Voltage regulation is the measurement of the difference between the input voltage and output voltage of a transformer. The voltage regulation of a transformer indicates how well it can maintain a constant voltage output despite varying load conditions. There are certain conditions that must be met in order to achieve maximum, minimum, or zero voltage regulation in a transformer:

- Maximum voltage regulation: This occurs when the load power factor is lagging (inductive). In this case, the secondary voltage drop due to the resistance and leakage reactance of the transformer is maximum, resulting in a higher voltage drop and a decrease in output voltage. The condition for maximum voltage regulation is when the power factor of the load is zero or unity.
- Minimum voltage regulation: This occurs when the load power factor is leading (capacitive). In this case, the secondary voltage drop due to the resistance and leakage reactance of the transformer is minimal, resulting in a lower voltage drop and an increase in output voltage. The condition for minimum voltage regulation is when the power factor of the load is equal to the cosine of the angle of impedance.
- Zero voltage regulation: This occurs when the power factor of the load is such that the voltage drop due to the resistance and leakage reactance of the transformer is equal to the voltage rise due to the magnetising current of the transformer. The condition for zero voltage regulation is when the power factor of the load is equal to the ratio of the transformer’s impedance to its resistance.

It should be noted that achieving zero voltage regulation is ideal but not practical in real-world applications. In practice, the voltage regulation of a transformer is typically kept between the maximum and minimum limits to ensure stable and reliable operation of the system.

**Recall the Losses that occur in the Transformer**

A transformer is a device used to transfer electrical energy from one circuit to another through the principle of electromagnetic induction. During this process, various types of losses occur in the transformer. The following are the losses that occur in the transformer:

- Copper Losses or I^2R Losses: The copper losses or I^2R losses occur in the primary and secondary windings of the transformer due to the flow of current. These losses can be reduced by increasing the size of the conductor.
- Core Losses or Iron Losses: The core losses or iron losses occur in the transformer’s core due to hysteresis and eddy current. Hysteresis loss occurs due to the magnetization and demagnetization of the core material with the variation of the applied magnetic field. Eddy current loss occurs due to the flow of induced current in the core material. These losses can be reduced by using high-grade silicon steel with a low hysteresis loop and by laminating the core.
- Stray Losses: Stray losses occur due to the leakage flux and the capacitance between the windings and between the windings and the core.
- Dielectric Losses: Dielectric losses occur due to the electric field in the insulation between the windings and between the windings and the core. These losses can be reduced by using high-quality insulating materials.
- Load Losses or Ohmic Losses: The load losses or ohmic losses occur in the transformer’s load due to the flow of current through the load resistance. These losses can be reduced by using a high-efficiency transformer.

Therefore, minimising the losses is an essential aspect of transformer design, as the efficiency of the transformer is dependent on the losses.

**Describe the Efficiency of a Single-Phase Transformer**

Transformer efficiency is the ratio of output power to input power, expressed as a percentage. It measures how effectively the transformer can transfer energy from its primary winding to its secondary winding. The efficiency of a single-phase transformer is affected by several factors, including the core and winding losses.

The total losses in a transformer are categorised into two types: core losses and winding losses. Core losses include hysteresis losses and eddy current losses, which occur in the core material of the transformer. Winding losses include copper losses, which occur in the transformer’s windings due to the resistance of the wires.

The efficiency of a transformer is calculated as follows:

Efficiency = Output power / Input power x 100%

Where Output power = Secondary voltage x Secondary current and Input power = Primary voltage x Primary current

The efficiency of a transformer is highest when the copper losses are minimum. The copper losses are minimum when the transformer is loaded to its full rated capacity, and the voltage regulation is low. Therefore, transformers are designed to operate at or near full load, where the efficiency is highest.

Efficiency can be improved by using high-quality core materials, such as laminated silicon steel, to reduce core losses. The winding losses can be minimised by using larger wire sizes, which have lower resistance.

In general, the efficiency of a transformer ranges from 90% to 98%, depending on the size and design of the transformer. High-efficiency transformers are preferred for applications where energy efficiency is critical, such as in power distribution networks or in electric vehicles.

Overall, transformer efficiency is an important factor to consider when designing and selecting transformers for different applications, as it affects energy consumption and operating costs.

**Recall the condition for Maximum Efficiency of Transformer**

Efficiency of a transformer is the ratio of output power to input power. A transformer has maximum efficiency when the copper loss is equal to the iron loss. Copper loss is the power loss due to the resistance of the transformer’s windings, and iron loss is the power loss due to the core’s hysteresis and eddy currents. The efficiency can be calculated using the following formula:

Efficiency = (Output power / Input power) x 100%

To achieve maximum efficiency, the copper loss must be minimised, and the iron loss must be minimised as well. The iron loss can be minimised by using high-quality laminations for the core, reducing the magnetic flux density, and using a core with a low value of hysteresis. The copper loss can be minimised by using larger conductor size for the winding or reducing the current density.

The condition for maximum efficiency can be derived from the efficiency equation as follows:

Efficiency = (Output power / Input power) x 100%

Input power = Output power + Total losses

where Total losses = Copper loss + Iron loss

At maximum efficiency, the derivative of the efficiency with respect to load current is zero, which means that the efficiency is not changing with the load. Mathematically, the condition for maximum efficiency can be expressed as:

dEfficiency / dI = 0

where I is the load current. By differentiating the efficiency equation with respect to load current and equating it to zero, the following expression can be derived:

Output voltage / Input voltage = √(Iron loss / Copper loss)

This expression represents the condition for maximum efficiency of a transformer. The voltage regulation of the transformer is also minimum at this point, which means that the transformer can maintain a constant output voltage even with varying loads.

In summary, to achieve maximum efficiency, the copper loss and iron loss of the transformer must be minimised, and the load current must be adjusted to satisfy the condition given by the expression above.

**Describe All-Day Efficiency of Transformer**

The all-day efficiency of a transformer refers to the average efficiency of a transformer over a 24-hour period. It is an important parameter to consider as the transformer is generally in operation for extended periods, and its efficiency is critical to determine the overall cost of operation. The all-day efficiency of a transformer can be calculated based on the load profile of the transformer and the efficiency at different loads.

To calculate the all-day efficiency of a transformer, the load profile of the transformer needs to be known. The load profile is the variation of the load on the transformer over a 24-hour period. It can be obtained by monitoring the load on the transformer at regular intervals or using load data available from the utility. The load profile is typically represented as a graph of load versus time.

Once the load profile is known, the efficiency of the transformer at each load level can be determined. The efficiency at each load level is generally obtained by conducting efficiency tests at different loads. The efficiency tests involve measuring the input and output power of the transformer at different loads and calculating the efficiency.

The all-day efficiency of the transformer can then be calculated using the following formula:

All-Day Efficiency = Σ (Load Factor × Efficiency) / Σ Load Factor

Where,

Load Factor = Percentage of time the transformer operates at a particular load

Efficiency = Efficiency of the transformer at that load

The load factor is calculated as the percentage of time the transformer operates at a particular load level. The load factor is calculated as follows:

Load Factor = (Number of hours the transformer operates at a particular load / Total number of hours in a day) × 100%

Once the load factor and efficiency at each load level are known, they can be used to calculate the all-day efficiency of the transformer.

In practice, the all-day efficiency is generally lower than the peak efficiency of the transformer, as the transformer operates at different load levels throughout the day. The all-day efficiency is an important parameter to consider while selecting a transformer, as it determines the overall cost of operation. A transformer with a higher all-day efficiency would result in lower energy costs and higher cost savings over its lifetime.

**Recall Open-Circuit Test of Single-Phase Transformer**

The open-circuit test is performed on a transformer to determine the no-load losses and magnetising current. The transformer is disconnected from the supply and one of the windings is kept open while the other winding is connected to a low voltage supply. The rated voltage of the winding to be tested is not applied because it can cause the winding insulation to fail. Instead, a low voltage, typically 5% to 10% of the rated voltage, is applied to the winding.

During the open-circuit test, the voltage, current, and power are measured on the secondary side of the transformer. The primary side is open, and no load is connected. The measured current is the sum of the magnetising current and a small amount of core loss current. The magnetising current is required to establish the magnetic field in the core. The core loss current is the result of hysteresis and eddy current losses.

The open-circuit test can be used to determine the equivalent circuit of the transformer, which is useful for calculating the performance of the transformer under load conditions.

The following formula is used to calculate the power consumed in the open-circuit test:

P0 = V2 x I0

where P0 is the power consumed, V2 is the voltage applied to the secondary winding, and I0 is the current drawn by the secondary winding.

The no-load current and the no-load power factor can also be determined using the open-circuit test. The no-load current is the magnetising current plus a small amount of core loss current. The no-load power factor is the cosine of the angle between the no-load current and the no-load voltage.

The open-circuit test is also called the no-load test. The test is essential to determine the no-load losses and magnetising current, which are essential to determine the efficiency and voltage regulation of the transformer.

**Recall Short-Circuit Test of Single-Phase Transformer**

Short-circuit test, also known as the impedance test, is used to determine the series branch parameters of a transformer. The series branch parameters include the winding resistance (R), leakage inductance (L), and magnetising inductance (M). The short-circuit test is performed by short-circuiting the secondary winding of the transformer and applying a low voltage to the primary winding. The low voltage is used to limit the current flow, as the secondary winding is shorted.

The test setup for the short-circuit test involves the following steps:

- Short-circuit the secondary winding of the transformer
- Apply a low voltage (typically 10-20% of rated voltage) to the primary winding
- Measure the primary current, primary voltage, and the power input to the transformer
- Calculate the resistance, leakage inductance, and magnetising inductance of the transformer using the measured values

The primary current in the short-circuit test is equal to the sum of the magnetising current and the load current, which is negligible as the secondary is shorted. Therefore, the measured primary current is the magnetising current of the transformer. The resistance can be calculated by dividing the power input by the square of the primary current. The leakage inductance can be calculated by subtracting the magnetising inductance from the total inductance measured from the primary side. The magnetising inductance can be calculated by subtracting the leakage inductance from the total inductance measured from the secondary side.

The short-circuit test is a quick and easy method to determine the series branch parameters of a transformer. The data obtained from the test can be used to create an equivalent circuit for the transformer, which is useful in analyzing the transformer’s performance under different operating conditions.

**Recall Autotransformer**

Definition of Autotransformer:

An autotransformer is a transformer that has only one winding which is used as both primary and secondary winding. A portion of the winding is used as the primary winding and another portion of the same winding is used as the secondary winding. The winding is tapped to get the desired output voltage. Autotransformers are used to increase or decrease the voltage level of the input voltage.

Example: Suppose an autotransformer is used to reduce the voltage from 240V to 120V. The winding is tapped at 80% of the total winding length. This means that the primary winding is 80% of the total winding length and the secondary winding is the remaining 20%. The input voltage of 240V is applied across the entire winding. The voltage across the 80% portion of the winding (primary winding) is calculated as 240V x 80% = 192V. The voltage across the 20% portion of the winding (secondary winding) is calculated as 240V – 192V = 48V. Therefore, the output voltage is 48V and the autotransformer has reduced the voltage from 240V to 48V.

Explanation of Autotransformer Operation:

In an autotransformer, the same winding is used as both primary and secondary winding. When an input voltage is applied to the winding, a magnetic flux is created in the core of the transformer. This magnetic flux induces a voltage in the winding. The voltage induced in the winding depends on the number of turns in the winding and the rate of change of magnetic flux. The voltage across a portion of the winding is calculated by multiplying the input voltage with the ratio of the number of turns in that portion of the winding to the total number of turns in the winding.

Advantages of Autotransformer:

- Smaller in size and lighter in weight as compared to the conventional two-winding transformer.
- Lower losses and higher efficiency than conventional transformers.
- Cheaper in cost due to fewer windings.
- Can provide a variable output voltage by tapping the winding at different points.
- Autotransformers are useful in applications where a small voltage adjustment is required.

Disadvantages of Autotransformer:

- Low level of isolation between the primary and secondary winding.
- Cannot be used for applications where galvanic isolation is required.
- Autotransformers have limited voltage ratios compared to conventional transformers.

Applications of Autotransformer:

- Autotransformers are used in the power supply of electronic equipment, such as radios, televisions, and computers.
- Autotransformers are used in the electrical distribution system to regulate the voltage level.
- Autotransformers are used in welding machines to adjust the voltage level.
- Autotransformers are used in voltage regulators for lighting systems.
- Autotransformers are used in variable frequency drives for motor speed control.

**Determine Percentage Copper Saving in Autotransformer**

Definition of Percentage Copper Saving:

Percentage Copper Saving is the amount of copper saved in an autotransformer compared to a conventional two-winding transformer. Since an autotransformer uses only one winding, it requires less copper than a conventional transformer that uses separate windings for primary and secondary.

Example: Suppose a conventional two-winding transformer has a rating of 10 kVA and a voltage ratio of 240V/120V. The primary winding has 100 turns, and the secondary winding has 200 turns. The wire gauge used for the windings is AWG #10. The autotransformer has a voltage ratio of 240V/120V and a rating of 10 kVA. The wire gauge used for the winding is AWG #6. The percentage copper saving in the autotransformer is calculated as follows:

Copper Saving = (Weight of Copper in Conventional Transformer – Weight of Copper in Autotransformer) / Weight of Copper in Conventional Transformer x 100%

Weight of Copper in Conventional Transformer = (100 x 10) + (200 x 10) = 3000 g

Weight of Copper in Autotransformer = (100 x 6) + (100 x 6 x (200/240)) = 2250 g

Copper Saving = (3000 – 2250) / 3000 x 100% = 25%

Therefore, the autotransformer saves 25% of copper compared to the conventional transformer.

Explanation of Percentage Copper Saving:

The percentage copper saving in an autotransformer is due to the fact that it uses a single winding for both primary and secondary, while a conventional transformer uses separate windings for primary and secondary. The amount of copper required for the windings of a transformer is directly proportional to the product of the square of the current and the number of turns in the winding. Since the current flowing through the autotransformer is lower than that of the conventional transformer, due to the fact that it has a higher voltage ratio, the amount of copper required is also lower.

Advantages of Percentage Copper Saving:

- Autotransformers are smaller and lighter in weight than conventional transformers due to the reduced amount of copper required.
- Autotransformers are more efficient than conventional transformers due to lower copper losses.
- Autotransformers are less expensive than conventional transformers due to the reduced amount of copper required.

Disadvantages of Percentage Copper Saving:

- Autotransformers have limited voltage ratios compared to conventional transformers.
- Autotransformers have a low level of isolation between the primary and secondary winding.
- Autotransformers cannot be used for applications where galvanic isolation is required.

Applications of Percentage Copper Saving:

- Autotransformers are used in the electrical distribution system to regulate the voltage level.
- Autotransformers are used in the power supply of electronic equipment, such as radios, televisions, and computers.
- Autotransformers are used in voltage regulators for lighting systems.
- Autotransformers are used in welding machines to adjust the voltage level.
- Autotransformers are used in variable frequency drives for motor speed control.

**Convert a Two Winding Transformer into an Autotransformer**

Definition of Autotransformer:

An autotransformer is a type of transformer that has only one winding, which is used as both the primary and secondary winding. Autotransformers are commonly used in power systems to supply loads at lower voltage levels than the input voltage.

Process to Convert a Two Winding Transformer into an Autotransformer:

A conventional two-winding transformer can be converted into an autotransformer by reconnecting the windings. The process involves connecting the primary and secondary windings of the transformer in series and tapping a point on the winding to obtain the desired output voltage. The voltage ratio of the autotransformer is determined by the tapping point on the winding.

Example:

Consider a two-winding transformer with a primary winding of 100 turns and a secondary winding of 200 turns. The transformer has a voltage ratio of 240V/120V. To convert this transformer into an autotransformer, the following steps can be taken:

- Disconnect the secondary winding from the transformer.
- Connect the primary and secondary windings in series.
- Connect a load between the common connection of the two windings and the tapping point on the winding.
- Select the tapping point to obtain the desired output voltage.

For example, if the desired output voltage is 100V, the tapping point can be selected such that the voltage across the load is 100V. The voltage ratio of the autotransformer is determined by the ratio of the number of turns between the tapping point and the common connection of the two windings.

Advantages of Converting a Two Winding Transformer into an Autotransformer:

- Autotransformers are smaller and lighter in weight than conventional transformers due to the reduced amount of copper required.
- Autotransformers are more efficient than conventional transformers due to lower copper losses.
- Autotransformers are less expensive than conventional transformers due to the reduced amount of copper required.
- Autotransformers are easy to convert from a conventional transformer as the process involves simple rewiring of the windings.

Disadvantages of Converting a Two Winding Transformer into an Autotransformer:

- Autotransformers have limited voltage ratios compared to conventional transformers.
- Autotransformers have a low level of isolation between the primary and secondary winding.
- Autotransformers cannot be used for applications where galvanic isolation is required.

Applications of Autotransformers:

- Autotransformers are used in the electrical distribution system to regulate the voltage level.
- Autotransformers are used in the power supply of electronic equipment, such as radios, televisions, and computers.
- Autotransformers are used in voltage regulators for lighting systems.
- Autotransformers are used in welding machines to adjust the voltage level.
- Autotransformers are used in variable frequency drives for motor speed control.