This article describes the Transformer Equations and Transformer Formulas, including their applications in step-up and step-down transformers. It also provides solved examples, all types of transformer formulas, and FAQs.
What is a Transformer?
A transformer is an electrical device that changes the voltage level of an alternating current (AC). It operates on the principle of electromagnetic induction, where a varying magnetic field induces an electromotive force (EMF) in a nearby coil.
Key Components:
- Primary Coil: Connected to the input voltage source.
- Secondary Coil: Delivers the transformed voltage.
- Core: A magnetic material that channels the magnetic flux.
Uses of Transformers:
- Voltage step-up in power transmission.
- Voltage step-down for household appliances.
- Isolation transformers for safety.
Transformer Equations and Formulas
Transformer EMF Equation
The EMF equation of a transformer relates the electromotive force (EMF) induced in the windings to the number of turns, flux, and frequency of the supply.
EMF Equation of Transformer :
Where:
- E: RMS value of induced EMF in the winding (volts).
- f: Frequency of the AC supply (Hz).
- N: Number of turns in the winding.
- ϕm: Maximum flux in the core (Weber).
If the number of turns in the primary winding is N1, the emf equation of the primary winding is,
Where E1 is Induced EMF in the primary winding.
If the number of turns in the secondary winding is N2, the emf equation of the secondary winding is,
Where E2 is Induced EMF in the secondary winding.
Transformer Voltage Ratio Formula
The primary principle of transformers is based on Faraday’s Law of Electromagnetic Induction. The relationship between the voltage and the number of turns in the coils is given by:
The Transformer Voltage Ratio Formula is,
Where:
- V1: Voltage across the primary coil.
- V2: Voltage across the secondary coil.
- N1: Number of turns in the primary coil.
- N2: Number of turns in the secondary coil.
Voltage Transformation Ratio of Transformer Formula
The voltage Transformation Ratio (VTR) of a transformer is the ratio of the secondary voltage to the primary voltage.
The formula for the voltage ratio of the transformer(K) is.
Where K: Voltage transformation ratio (unitless).
The formula for the voltage ratio of the transformer(K) in terms of transformer current is,
Cases:
- Step-Up Transformer: When K>1 (N2>N1), the secondary voltage V2 is greater than the primary voltage V1. The formula for a step-up transformer for secondary voltage is V2= V1(N2/N1). Here, the ratio of N2 and N1 is more than 1, therefore, the secondary voltage is more than the primary voltage.
- Step-Down Transformer: When K<1 (N2<N1), the secondary voltage V2 is less than the primary voltage V1. The formula for a step-down transformer for secondary voltage is V2= V1(N2/N1). Here, the ratio of N2 and N1 is less than 1, therefore, the secondary voltage is less than the primary voltage.
- Isolation Transformer: When K= (N2=N1), the secondary and primary voltages are equal.
Transformer Turn Ratio Formula
The Transformer Turns Ratio (TTR) is the ratio of the number of turns in the primary winding to the number of turns in the secondary winding of a transformer. It determines the relationship between the input (primary) and output (secondary) voltages and currents.
The Transformer Turns Ratio (TTR) is defined as:
Turn ratio can also be defined in terms of voltage and current and the formula for turn ratio is,
Cases:
- Step-Up Transformer: N2>N1, so V2>V1.
- Step-Down Transformer: N2<N1, so V2<V1.
Input & Output Voltage Equations
The input voltage equation of the transformer is,
Where,
- V1: Voltage across the primary winding
- E1: Induced EMF across the primary winding
- R1: Resistance of the primary winding
- X1: Reactance of the primary winding
- Z1: Impedance of the primary winding
- I1: Current in the primary winding
The output voltage equation of the transformer is,
Where,
- V2: Voltage across the secondary winding
- E2: Induced EMF across the secondary winding
- R2: Resistance of the secondary winding
- X2: Reactance of the secondary winding
- Z2: Impedance of the secondary winding
- I2: Current in the secondary winding
Equivalent Resistance of Transformer Windings:
The equivalent resistance of transformer windings is the combined resistance of the primary and secondary windings as reflected to either side (primary or secondary).
Primary Side Equivalent Resistance
The formula for calculating the primary side equivalent resistance when the secondary winding resistance is reflected to the primary side:
Where:
- R1: Resistance of the primary winding.
- R2: Resistance of the secondary winding.
- R01: Equivalent resistance of transformer at the primary side
- K: Turns ratio (K=N1/N2).
Secondary Side Equivalent Resistance
The formula for calculating the secondary side equivalent resistance When the primary winding resistance is reflected to the secondary side:
Where,
- R2: Resistance of the secondary winding.
- R1: Resistance of the primary winding.
- R02: Equivalent resistance of transformer at the secondary side
- K: Turns ratio (K=N1/N2).
Equivalent Reactance of Transformer Windings:
The equivalent reactance of transformer windings accounts for the combined leakage reactance of the primary and secondary windings, reflected to one side of the transformer (either primary or secondary). It represents the opposition to current flow due to the leakage flux not linking both windings completely.
Primary Side Equivalent Reactance
The formula for calculating the primary side equivalent reactance when the reactance of the secondary winding is reflected to the primary side:
Where:
- X1: Leakage reactance of the primary winding.
- X2: Leakage reactance of the secondary winding.
- X01: Equivalent reactance of transformer at the primary side
- K: Turns ratio (K=N1/N2).
The formula for calculating the secondary side equivalent reactance when the reactance of the primary winding is reflected to the secondary side:
Where,
- X2: Leakage reactance of the secondary winding.
- X1: Leakage reactance of the primary winding.
- X02: Equivalent reactance of transformer at the secondary side
- K: Turns ratio (K=N1/N2).
Impedance of Transformer Windings:
The impedance of transformer windings is the total opposition offered by the transformer windings to the flow of alternating current.
Impedance of Primary Winding:
The impedance of the primary winding of a transformer is the combined resistance and reactance of the primary winding, representing its total opposition to the flow of alternating current.
The impedance of the primary winding can be expressed as:
Impedance of Secondary Winding:
The impedance of the secondary winding of a transformer is the combined resistance and reactance of the secondary winding that represents the total opposition to the flow of alternating current.
The impedance of the secondary winding can be expressed as:
Equivalent Impedance of Primary Winding:
The equivalent impedance of a transformer is the combined impedance of the primary and secondary windings, reflected to either the primary or the secondary side.
Primary Side Equivalent Impedance
The formula for calculating the primary side equivalent impedance when the impedance of the secondary winding is reflected to the primary side:
Secondary Side Equivalent Impedance
The formula for calculating the secondary side equivalent impedance when the impedance of the primary winding is reflected to the secondary side:
Losses In Transformer
Losses in transformers can be broadly categorized into two main types: core losses and copper losses.
Core Losses:
Core losses occur in the transformer’s magnetic core due to alternating magnetic flux. These are further divided into Hysteresis Losses and Eddy Current Losses.
Hysteresis Loss:
The hysteresis loss occurs because of the lagging of magnetic flux density behind the magnetizing force in the core material.
The formula for calculating hysteresis loss in a transformer is,
Where:
- Ph: Hysteresis Loss (in watts)
- η: Hysteresis constant (depends on the core material)
- Bm: Maximum flux density in the core (in teslas)
- x: Steinmetz constant (ranges between 1.5 and 2.5 for most materials)
- f: Frequency of magnetization (in hertz)
- V: Volume of the core (in cubic meters)
Eddy Current Loss:
Eddy current loss in a transformer occurs due to circulating currents induced in the core material.
The formula for Eddy Current Loss in a transformer is given by:
Where:
- Pe: Eddy Current Loss (in watts)
- Ke: Eddy current constant (depends on the material’s resistivity and design)
- Bm: Maximum flux density in the core (in teslas)
- f: Frequency of the magnetic field (in hertz)
- t: Thickness of the core laminations (in meters)
- V: Volume of the core (in cubic meters)
Copper Loss
Copper losses occur in a transformer when the current flows in its winding. This loss happens due to conductor resistance. These losses vary with the square of the load current (I2R) and are directly proportional to the loading on the transformer.
The formula for Copper Loss in a transformer is :
Where:
- Pc: Copper Loss (in watts)
- I1: Current in the primary winding (in amperes)
- R1: Resistance of the primary winding (in ohms)
- I2: Current in the secondary winding (in amperes)
- R2: Resistance of the secondary winding (in ohms)
Dielectric Loss
Dielectric loss in a transformer occurs due to the energy loss in the insulating material when an alternating electric field is applied.
The formula for Dielectric Loss in a transformer is :
Where:
- ω=2πf: Angular frequency (in radians/second)
- C: Capacitance of the insulation (in farads)
- V: Voltage across the dielectric (in volts)
- tanδ: Dielectric loss factor (related to cosδ)
- Pd: Dielectric Loss (in watts)
- cosδ: Power factor of the dielectric, where δ is the dielectric loss angle
Transformer Efficiency
The efficiency of a transformer is the ratio of its output power to the input power. The formula for transformer efficiency is:
Here, Total losses = Cu loss + Iron Loss + Dielectric Loss
Efficiency at Load
The copper loss in the transformer depends on the loading. So, efficiency changes with loading. You can calculate the efficiency of a transformer at a particular load using the following formula.
All Day Efficiency
All-day efficiency is the efficiency of a transformer over a 24-hour period. The transformer supplies varying loads throughout the day. All-day efficiency is important for distribution transformers that operate under varying load patterns.
The formula for all-day efficiency is:
Condition for Maximum Efficiency
The condition for maximum efficiency of a transformer is when the core (iron) losses are equal to the copper losses.
Load Current at Maximum Efficiency
The formula for the load current corresponding to the maximum efficiency of a transformer is:
Percentage Resistance, Reactance & Impedance
The percentage resistance, reactance, and impedance of a transformer are calculated using the rated parameters of the transformer. These values are expressed as a percentage of the rated voltage or full-load current.
1. Percentage Resistance (%R)
The formula for the Percentage Resistance (%R) of the transformer is:
2. Percentage Reactance (%X)
The formula for the Percentage Reactance (%X) of the transformer is:
3. Percentage Impedance (%Z)
The formula for the Percentage Reactance (%Z) of the transformer is:
kVA Rating of the Transformer
The kVA rating of a transformer is a measure of its apparent power capacity, It is defined as the product of voltage and current at full load.
The formula of kVA rating for single-phase transformers is:
Where:
- V1: Rated Primary voltage (in volts).
- V2: Rated Secondary voltage (in volts).
- I1: Rated Primary current (in amperes).
- I2: Rated Secondary current (in amperes).
kVA rating formula for three-phase transformers is:
Voltage Regulation Of Transformer
Voltage regulation of a transformer is defined as the change in secondary terminal voltage when the load varies from no-load to full-load while keeping the primary voltage constant. It is expressed as a percentage of the full-load secondary voltage.
The voltage Regulation Formula of the transformer is:
Where:
- Vno-load: Secondary voltage at no load
- Vfull-load: Secondary voltage at full load
The voltage regulation formula for lagging power factor load is:
Where:
- I2: Load current
- R02: Equivalent resistance referred to the secondary side
- X02: Equivalent reactance referred to the secondary side
- cosϕ2: Power factor of the load
- sinϕ2: Sine of the power factor angle
- V2: Rated secondary voltage
The voltage regulation formula for leading power factor load is:
Solved Examples
Example 1: Step-up Transformer
Problem: A step-up transformer has 200 turns in the primary coil and 800 turns in the secondary coil. The primary voltage is 230V. Find the secondary voltage.
Solution:
The secondary voltage can be calculated using the step-up transformer formula:
Answer: The output voltage is 920V.
Example 2: Step-down Transformer
Problem: A step-down transformer has 500 turns in the primary coil and 100 turns in the secondary coil. The primary voltage is 440V, find the secondary voltage.
The secondary voltage can be calculated using the step-down transformer formula:
Answer: The output voltage is 88V.
Example 3: Efficiency Calculation
Problem: A transformer supplies 12 kW of output power while drawing 14 kW of input power. Determine the efficiency of the transformer.
Solution: Using the transformer efficiency formula:
Answer: The transformer efficiency is 85.71%.
FAQs on Transformer Formula
1. What is the Transformer Equation?
The transformer equation relates the voltage and turns ratio:
2. How do you calculate transformer efficiency?
Efficiency is calculated using the following formula:
3. What is the use of the turns ratio formula?
The turns ratio formula is used to identify the type of transformer (step-up or step-down). The formula for the turn ratio of the transformer is:
4. What is the formula for a step-up transformer?
5. What is the formula for a step-down transformer?
Conclusion
Understanding Transformer Equations and Formulas is essential for working with transformers. This article explains simple formulas like step-up and step-down transformer equations, efficiency formulas, and current ratios.