Electrical Transformers – Formulas and Equations Handbook

Electrical transformers are one of the most widely used electrical machines in different fields of electrical engineering like power systems. So as an electrical engineer, we frequently required to calculate different parameters of a transformer to determine its operating conditions. For this, we require standard formulae, which we have listed in the following sections of this article. This page will serve as an electrical transformer formula handbook for electrical engineering students and professionals.

Definition of Transformer

A transformer is a static ac electrical machine used in electrical power systems for transforming (i.e. increasing or decreasing) the voltage level as per requirement. A transformer changes the level of voltage and current without changing the frequency.

Types of Transformer

Based on operation, a transformer can be of the following three types −

• Step-up Transformer − Increases the voltage level from a lower voltage level.

• Step-down Transformer − Decreases the voltage level from a higher voltage level.

• Isolation Transformer − Does not change the voltage, but separates two electrical circuit electrically. It is also known as 1 to 1 transformer.

EMF Equation of Transformer

The mathematical expression that gives the value of induced EMF in windings of the transformer is known as emf equation of the transformer.

The EMF equation for primary winding is given by,

$$\mathrm{E_{1}=4.44f\phi _{m}N_{1}=4.44fB_{m}AN_{1}}$$

The EMF equation for secondary winding is given by,

$$\mathrm{E_{2}=4.44f\phi _{m}N_{2}=4.44fB_{m}AN_{2}}$$

Where, f is the supply frequency, ϕm is the maximum flux in the core, Bm is the maximum flux density in the core, A is the area of cross-section of the core, 𝑁₁ and 𝑁₂ are the number of turns in the primary and secondary windings.

Turns Ratio of Transformer

The ratio of number of turns in the primary winding to the number of turns in the secondary winding of a transformer is referred to as turns ratio of the transformer. It is usually denoted by the symbol a.

$$\mathrm{Turns\: Ratio, a=\frac{Primary\: winding \: turns (N_{1})}{Secondary\: winding\: turns (N_{2})}}$$

Voltage Transformation Ratio of Transformer

The ratio of the output AC voltage to the input AC voltage of a transformer is known as the voltage transformer ratio of the transformer. It is usually denoted by the symbol K.

$$\mathrm{Voltage\: Transformation\: Ratio, K=\frac{Output\: Voltage (V_{2})}{Input\: Voltage (V_{1})}}$$

Current Transformation Ratio of Transformer

The ratio of the output current (secondary winding current) to the input current (primary winding current) of a transformer is known as current transformation ratio of the transformer.

$$\mathrm{Current\: Transformation\: Ratio, K=\frac{Secondary\: winding\: current (I_{2})}{Primary\: winding\: current (I_{1})}}$$

Relationship among Turns Ratio, Voltage Transformation Ratio, and Current Transformation Ratio

The relationship among turns ratio, voltage transformation ratio, and current transformation ratio is given by the following expression,

$$\mathrm{Turns Ratio, a=\frac{N_{1}}{N_{2}}=\frac{V_{1}}{V_{2}}=\frac{I_{2}}{I_{1}}=\frac{1}{K}}$$

Here, we can see that the current transformation is the reciprocal of the voltage transformation ratio. This is due to the fact that when a transformer increases the voltage, it reduces the current in the same proportion to maintain the constant MMF in the core.

MMF Equation of Transformer

MMF stands for Magnetomotive Force. The mmf is also referred to as ampere-turn rating of a transformer. The mmf is the driving force that establishes a magnetic flux in the core of a transformer. It is given by the product of number of turn in the winding and current through the winding.

For primary winding,

$$\mathrm{MMF=N_{1}I_{1}}$$

For secondary winding,

$$\mathrm{MMF=N_{2}I_{2}}$$

Where, 𝐼₁ and 𝐼₂ are the currents in primary and secondary windings of the transformer respectively.

Equivalent Resistance of Transformer Windings

The primary and secondary windings of a transformer are generally made up of copper wire. Thus, they have a finite resistance, although it is very small. The primary winding resistance is represented by 𝑅₁ and the secondary winding resistance is represented by 𝑅₂.

The equivalent resistance of transformer windings is given by referring the whole circuit of the transformer either on primary side or secondary side.

Thus, the equivalent resistance of transformer windings referred to primary side is given by,

$$\mathrm{R_{01}=R_{1}+R_{2}^{'}=R_{1}+\frac{R_{2}}{K^{2}}}$$

The equivalent resistance of transformer windings referred to secondary side is given by,

$$\mathrm{R_{02}=R_{2}+R_{1}^{'}=R_{2}+R_{1}K^{2}}$$

Where, 𝑅₁ ^′ is the primary winding resistance referred to secondary side, 𝑅₂ ^′ is the resistance of secondary winding referred to primary side, 𝑅₁ is the primary winding resistance, 𝑅₂ is the secondary winding resistance, 𝑅₀₁ is the equivalent resistance of transformer referred to primary side, and 𝑅₀₂ is the equivalent resistance of transformer referred to secondary side.

Leakage Reactance of Transformer Windings

The inductive reactance caused by the leakage magnetic flux in the transformer is referred to as the leakage reactance of the transformer windings.

For primary winding,

$$\mathrm{X_{1}=\frac{E_{1}}{I_{1}}}$$

For secondary winding

$$\mathrm{X_{2}=\frac{E_{2}}{I_{2}}}$$

Where, 𝑋₁ is the primary winding leakage reactance, 𝑋₂ is the secondary winding leakage reactance, 𝐸₁ is the self-induced emf in primary winding, and 𝐸₂ is the self-induced EMF in the secondary winding.

Equivalent Reactance of Transformer Windings

The equivalent reactance is the total reactance offered by both the primary and secondary windings of the transformer.

The equivalent reactance of transformer referred to primary side is,

$$\mathrm{X_{01}=X_{1}+X_{2}^{'}=X_{1}+\frac{X_{2}}{K^{2}}}$$

The equivalent reactance of transformer referred to secondary side is,

$$\mathrm{X_{02}=X_{2}+X_{1}^{'}=X_{2}+K^{2}X_{1}}$$

Where, X₁’ is the leakage reactance of primary winding on secondary side, X₂’ is the leakage reactance of secondary winding on primary side.

Total Impedance of Transformer Windings

The combined opposition offered by the winding resistances and leakage reactances is referred to as the total impedance of the transformer windings.

The impedance of the primary winding of transformer is,

$$\mathrm{Z_{1}=\sqrt{R_{1}^{2}+X_{1}^{2}}}$$

The impedance of the secondary winding of transformer is,

$$\mathrm{Z_{2}=\sqrt{R_{2}^{2}+X_{2}^{2}}}$$

The equivalent impedance of transformer referred to primary side is given by,

$$\mathrm{Z_{01}=\sqrt{R_{01}^{2}+X_{01}^{2}}}$$

The equivalent impedance of transformer referred to secondary side is given by,

$$\mathrm{Z_{02}=\sqrt{R_{02}^{2}+X_{02}^{2}}}$$

Input and Output Voltage Equations of Transformer

The input and output voltage equations of a transformer are found using KVL in the equivalent circuit of the transformer.

The input voltage equation of a transformer is given by,

$$\mathrm{V_{1}=E_{1}+I_{1}R_{1}+jI_{1}X_{1}=E_{1}+I_{1}\left ( R_{1}+jX_{1} \right )=E_{1}+I_{1}Z_{1}}$$

The output voltage equation of a transformer is given by,

$$\mathrm{V_{2}=E_{2}-I_{2}R_{2}-jI_{2}X_{2}=E_{2}-I_{2}\left ( R_{2}+jX_{2} \right )=E_{2}-I_{2}Z_{2}}$$

Transformer Losses

There are two types of losses occur in a transformer− core loss and copper loss.

Core Losses of Transformer

The total core loss of the transformer is the sum of hysteresis loss and eddy current loss, i.e.,

$$\mathrm{Core\: loss=P_{h}+P_{e}}$$

Where, the hysteresis loss is caused due to magnetic reversal in the core.

$$\mathrm{Hysteresis\: loss,P_{h}=\eta B_{max}^{1.6}fV}$$

And, the eddy current occurs due to eddy currents flowing in the core.

$$\mathrm{Eddy \: current\: loss,P_{e}=k_{e} B_{m}^{2}f^{2}t^{2}}$$

Where, η is the Steinmetz coefficient, B_m is the maximum flux density in the core, K_e is the eddy current constant, f is the frequency of magnetic flux reversal, V is the volume of core.

Copper Loss of Transformer

The copper loss occurs due to resistance of the transformer windings.

$$\mathrm{Copper \: loss=I_{1}^{2}R_{1}+I_{2}^{2}R_{2}}$$

Voltage Regulation of Transformer

The voltage regulation of transformer is defined as the change in output voltage from no-load to full load with respect to the no-load voltage.

$$\mathrm{Voltage\: Regualation=\frac{No \: load\: voltage - Full\: load\: voltage}{No \: load\: voltage}}$$

Transformer Efficiency

The ratio of the output power to the input power is called the efficiency of the transformer.

$$\mathrm{Efficiency,\eta =\frac{Output\: power\left ( P_{o} \right )}{Input\: power\left ( P_{i} \right )}}$$

$$\mathrm{Efficiency,\eta =\frac{Output\: power}{Output\: power+Losses}}$$

Transformer Efficiency at Any Load

The efficiency of transformer at an actual load is calculated using the following formula,

$$\mathrm{\eta= \frac{x\times full\: load\: kVA \times power\: factor}{\left (x\times full\: load\: kVA \times power\: factor \right )+Losses}}$$

Where, x is the fraction of loading.

All Day Efficiency of Transformer

The all-day efficiency of a transformer is defined as the ratio of output energy in kWh to the input energy in kWh recorded for 24 hours

$$\mathrm{\eta_{allday}= \frac{Output\: energy\: in\: kWh}{Input\: energy\: in\: kWh}}$$

Condition for Maximum Efficiency of Transformer

When the core losses and copper losses of a transformer are equal, the transformer efficiency is maximum.

Thus, for maximum efficiency of transformer,

$$\mathrm{Copper\: loss = Core\: loss}$$

Load Current Corresponding to Maximum Efficiency of Transformer

The load current or secondary winding current for the maximum efficiency of a transformer is given by,

$$\mathrm{I_{2}=\sqrt{\frac{P_{i}}{R_{02}}}}$$

Conclusion

In this article, we have collected at one place, the most significant formulas of electrical transformer, which are very important for every electrical engineering student and professional.