EMF Equation of a Transformer: Theory, Practical Experiments, and Solved Examples

In Electrical Engineering, a transformer is a fundamental device employed in power transmission and distribution systems to transfer electrical energy between circuits through electromagnetic induction. This article presents a rigorous mathematical formulation of the electromotive force (EMF) induced in a transformer, which is essential knowledge for any practicing electrical engineer.

Prior to engaging in transformer design or analysis, it is imperative to understand its operating principles. A transformer facilitates power transfer from the primary to the secondary circuit via mutual inductance. The core generates a magnetic flux that changes with time. This changing flux induces an EMF in both the primary and secondary windings, as described by Faraday’s law..

This article explores the derivation of the EMF equation for a transformer, analyzes its physical significance, and demonstrates its practical relevance through illustrative solved examples.

As our primary objective is to study the EMF equation of a transformer, Here is a brief overview of the transformer itself. Fundamentally, a transformer is an electromagnetic induction device used to transfer electrical energy between two or more circuits.

It consists of a magnetic core on which both the primary and secondary windings are wound. In core-type transformers, both windings are placed on the same limb of the core, whereas in shell-type transformers, the windings are distributed on separate limbs. The core is typically constructed from high-permeability silicon steel to reduce core losses and improve magnetic efficiency.

Two-winding transformers, consisting of electrically isolated primary and secondary windings, are widely used in power transmission and distribution networks.

In certain applications, such as voltage boosting or voltage regulation, a single-winding transformer known as an autotransformer is used. Unlike conventional two-winding transformers, an autotransformer has a common winding for both the primary and secondary, resulting in improved efficiency and reduced size and cost for specific voltage conversion tasks

The working principle of a transformer is based on Faraday’s Law of Electromagnetic Induction,

Faraday’s First Law states that when a conductor is placed in a varying magnetic field, an electromotive force (EMF) is induced in it

The magnitude of the induced EMF is proportional to the rate of change of magnetic flux and the number of turns of the winding exposed to the magnetic field

1. Applying Alternating Supply Voltage to Transformer Primary Winding

When an external alternating supply voltage is applied to the transformer’s primary winding, current starts to flow through the winding. This current induces a time-varying (i.e., alternating) magnetic field. The behavior of this magnetic field corresponds to that of the applied supply voltage.

2. Producing Magnetic Flux in the Core

The alternating magnetic field produces alternating magnetic flux lines in the core, which act as a common medium for both the primary and secondary windings. Since the core is designed using a high-permeability (low-reluctivity) material, the magnetic flux lines travel through the core and couple with the secondary winding. The core and the magnetic flux lines are common to both the primary and secondary windings.

3. EMF Induced in Secondary Winding

According to Faraday’s Law of Electromagnetic Induction, the alternating magnetic flux lines cut across the conductors in the secondary winding, thereby inducing an EMF in the secondary winding. The magnitude of the induced EMF depends on the number of turns in the secondary winding. Similarly, the amount of magnetic flux produced depends on the ampere-turns, i.e., the product of the number of turns and the current flowing through them.

4. Voltage Transformation Between Primary and Secondary Windings

The voltage transformation between the primary and secondary windings depends on the number of turns in each winding.

If the number of turns in the primary winding is greater than that in the secondary winding, the secondary voltage will be lower than the primary voltage. This is because of the higher ampere-turns in the primary winding. Such a transformer is known as a step-down transformer.

Conversely, if the number of turns in the secondary winding is greater than that in the primary winding, the secondary voltage will be higher. Such a transformer is called a step-up transformer.

\[ Transformation Ratio = K = \frac{N_s}{N_p} = \frac{V_s}{V_p}\]

Where:

N_s​: Number of turns in the secondary winding

N_p​: Number of turns in the primary winding

V_s: Voltage in the secondary winding

V_p​: Voltage in the primary winding

In this case:

1. For a step-up transformer, K>1
2. For a step-down transformer, K<1

5.No Electrical Connection

Generally, both primary and secondary windings are wound on separate limbs or on a common limb but are electrically insulated from each other. Therefore, there is no direct electrical connection between them. They are electrically isolated. The core acts as a common magnetic medium between the two windings.

Let us consider that a sinusoidal alternating input voltage V_m​ is applied to the primary winding of a transformer. As a result, a primary current I_m flows through the winding.

\[ Voltage applied to winding (V_p) = V_m sin \omega t\]

\[ Primary current I_p = I_m sin \omega t\]

This primary current I_p​ produces a magnetic flux in the primary winding. The magnetic flux is also sinusoidal in nature, just like the primary current.

\[ Magnetic flux produced \phi = \phi_m sin \omega t\]

These magnetic flux lines are linked to the secondary winding through the transformer core, which acts as a common medium for both windings. As the magnetic flux lines are alternating in nature, an EMF is induced in the secondary winding of the transformer.

According to Faraday’s Law of Electromagnetic Induction, the EMF induced in the secondary coil is proportional to the rate of change of magnetic flux

\[ EMF induced -e = \frac{d \phi}{dt}\]

Substituting the expression for magnetic flux into this equation

\[ -e = \frac{d \phi_m sin \omega t}{dt}\]

Differentiating with respect to time (t)

\[ -e = \phi_m \omega cos \omega t \]

Since cos⁡(ωt)varies continuously between ±1, the induced EMF also reaches both positive and negative peak values.

For the positive peak value, substitute cos⁡(ωt)=1

\[e_{max} = \omega \phi_m\]

For the negative peak value, substitute cos⁡(ωt) = −1

\[-e_{max} = \omega \phi_m\]

However, for practical purposes, we consider the positive peak value. So the maximum EMF induced in the secondary winding is:

\[e_{max} = \omega \phi_m\]

Since angular velocity ω=2πf, where f is the frequency of the supply

\[ e_{max} = 2 \pi f \phi_m\]

This equation shows the EMF induced per turn in the secondary winding. If the transformer has N turns in the secondary winding, then multiplying by N gives:

\[ e_{max} = 2 \pi f N \phi_m\]

This is the maximum EMF induced in the secondary winding.

However, in practical applications, we usually deal with the RMS (Root Mean Square) value of the EMF. The RMS value is:

\[ V_{rms} = \frac{V_{max}}{\sqrt{2}}\]

\[ V_{rms} = \frac{e_{max}}{\sqrt{2}}\]

\[ V_{rms} = \frac{2 \pi f N \phi_m}{\sqrt{2}}\]

This is the final expression for the RMS value of EMF induced in the secondary winding of a transformer.

\[E_{rms} = 4.44 \phi f N volts\]

To observe and verify the induced EMF in the secondary winding of a transformer when an alternating voltage is applied to the primary winding, and to confirm the transformation ratio of the transformer.

1. ingle-phase transformer with known primary and secondary turns — 1 No.
2. AC power supply — 1 No.
3. Variac (variable autotransformer) to vary AC input voltage — 1 No.
4. Voltmeter (for measuring primary input voltage and secondary output voltage) — 2 Nos.
5. Ammeter (for measuring AC input current and secondary output current) — 2 Nos.
6. Wattmeter (optional, to measure input and output power) — 2 Nos.
7. Connecting wires (as required, suitable sizes)
8. Switch — 1 No.
9. Electrical load (known rated) — 1 No.

1. Connect the primary winding of the transformer to the AC power supply through a Variac and a switch.
2. Connect a voltmeter across the primary winding to measure the input voltage.
3. Connect an ammeter in series with the primary winding to measure the primary input current.
4. Connect a voltmeter across the secondary winding to measure the induced EMF.
5. Connect an ammeter in series with the secondary winding and the external electrical load.
6. Connect a wattmeter on the primary side to measure input power (Optional) .
7. Connect a wattmeter on the secondary side to measure output power (Optional).
8. Close the switch and gradually apply the rated alternating voltage to the primary winding using the Variac.
9. Observe the voltmeter reading on the secondary side; this voltage represents the induced EMF.
10. Vary the supply voltage (or frequency, if possible) and observe the changes in the induced secondary voltage.
11. Measure the current and power on the primary side for further analysis (Optional).

1. Set the Variac to its maximum (zero output voltage) position so that zero voltage is applied to the primary winding at the start of the experiment.
2. Switch ON the power supply.
3. Gradually increase the voltage at the primary side by adjusting the Variac.
4. Set the voltage at a level where the current just starts to flow at a minimum measurable value.
5. Record the input voltage, and the primary and secondary parameters such as voltage, current, and power using the connected measuring instruments.
6. Increase or decrease the voltage to the next desired level and record all corresponding parameters again.
7. Repeat this process for several voltage levels, recording all readings at each step in the observation table.

• Ensure all connections are insulated and secure.
• Do not exceed the rated voltage of the transformer.
• Handle equipment carefully to avoid electric shocks.

The obtained parameters are need to be reordered as follows,

S.No	Primary Turns (Np)	Primary Voltage (VP)	Primary Current(Vs)	Primary Power (PP)	Secondary Turns (NS)	Secondary Voltage (Vs)	Secondary Current(Is)	Secondary Power(Ps)	voltage Ratio (Vs/Vp)	Turns Ratio (Ns/NP)	Current Ratio (IP/IS)
1	1000	0	0	0	250	0	0	0	0	0.25	0
2	1000	25	0.2	5	250	6.25	0.8	5	0.25	0.25	4
3	1000	50	0.4	20	250	12.5	1.6	20	0.25	0.25	4
4	1000	75	0.6	45	250	18.75	2.4	45	0.25	0.25	4
5	1000	100	0.8	80	250	25	3.2	80	0.25	0.25	4
6	1000	125	1.0	125	250	31.5	4	125	0.25	0.25	4
7	1000	150	1.2	180	250	37.5	4.8	180	0.25	0.25	4
8	1000	175	1.4	245	250	43.75	5.6	245	0.25	0.25	4
9	1000	200	1.6	320	250	50	6.4	320	0.25	0.25	4
10	1000	250	2	500	250	62.5	8	500	0.25	0.25	4

Finding secondary Induced EMF based on Input Parameters

S.No	Primary Voltage (Vp)	Supply frequency (f Hz)	Primary Turns (NP)	Calculated Flux (Φm​ (Wb))	Secondary Turns (NS)	Calculated Secondary Voltage (Vs)
1	0	50	1000	0	250	0
2	25	50	1000	0.000112613	250	6.25
3	50	50	1000	0.000225225	250	12.5
4	75	50	1000	0.000337838	250	18.75
5	100	50	1000	0.00045045	250	25
6	125	50	1000	0.000563063	250	31.25
7	150	50	1000	0.000675676	250	37.5
8	175	50	1000	0.000788288	250	43.75
9	200	50	1000	0.000900901	250	50
10	150	50	1000	0.000675676	250	62.5

The following parameters are observed during experiment taken for calculation

Supply Voltage = 250 Voltage maximum and 25 voltage in steps

Number of turns in primary winding =1000 turns

Number of turns in secondary winding = 500 turns

Supply frequency = 50 Hz

1. For 100 voltage input
   \[ = \phi_m = \frac{100}{4.44 x 50 x 1000}\]
   \[=0.00045 wb\]
   
2. For 200 voltage input
   \[ = \phi_m = \frac{200}{4.44 x 50 x 1000}\]
   \[=0.000900901 wb\]

1. For 100 Voltage Input
   \[ V_s = 4.44 x 0.00045 x 50 x 250 \]
   \[25 Volts\]
2. For 200 Voltage Input
   \[ V_s = 4.44 x 0.000900901 x 50 x 250 \]
   \[50 Volts\]

1. For 100 Voltage Input
   \[ V_s = 4.44 x 0.00045 x 50 x 250 \]
   \[12.5 Volts\]
2. For 200 Voltage Input
   \[ V_s = 4.44 x 0.000900901 x 50 x 250 \]
   \[25 Volts\]

• The secondary voltage increases in proportion to the primary voltage according to the transformation ratio
  \[ \frac{V_s}{V_p} = \frac{N_s}{N_p}\]
  where Vs​ and Vp are the secondary and primary voltages, and Ns​ and Np are the number of turns in the secondary and primary windings, respectively.
• The induced magnetic flux is common to both the primary and secondary windings.
• The secondary induced EMF also depends on the supply frequency; if the supply frequency changes, the induced EMF in the secondary winding changes accordingly

1. Predict the Induced Voltage:
   The EMF equation helps us calculate the voltage induced in the transformer windings (primary or secondary) due to the changing magnetic flux in the core. This voltage is what allows the transformer to transfer electrical energy from the primary winding to the secondary winding.
2. Relate Voltage to Design Parameters:
   The equation links electrical parameters such as:
   • Number of turns in the winding (N)Maximum magnetic flux (Φ_m​)Frequency of the applied voltage (f)
   This helps engineers design transformers to achieve the desired voltage transformation.
3. Determine Transformer Ratings:
   Using the EMF equation, designers can specify the size, core material, and number of turns needed to handle a particular voltage and frequency.
4. Explain Voltage Transformation Ratio:
   The EMF equation shows how voltages on primary and secondary windings relate proportionally to the ratio of their turns, which is the core operating principle of transformers (step-up or step-down voltage).

A Distribution Transformer supplied with 220 Voltage AC input supply with 50 HZ frequency . The number of primary and secondary turns are 1200 and 350 respectively. Assume core flux as 0.3088 mWb. Find The EMF induced in secondary winding

Given:

Input Supply voltage Voltage V1 = 220V AC

Supply Frequency = 50 Hz

Number of Turns in Primary winding : 1200

Number of Turns in Secondary winding : 350

Core Flux = 0.3088 mWb

To Find:

Find EMF Induced In Secondary winding

Solution:

Formula to find EMF induced in secondary winding,

\[EMF Induced e = 4.44 \phi f N Volts\]

Substitite given values in this equation, we get

\[ 4.44 . 0.3088 X 10^{-3} . 50 . 350\]

\[ 24 Volts\]

Result:

The EMF induced in secondary winding is 24 Volts

In this article, we have explored the fundamental operating principles of transformers, focusing on the rigorous mathematical derivation of the electromotive force (EMF) equation. Understanding how alternating magnetic flux induces voltage in the transformer windings is essential for designing and analyzing transformer performance in power systems.

The EMF equation establishes a clear relationship between the induced voltage, number of turns, magnetic flux, and supply frequency, providing a critical tool for engineers to predict voltage transformation and optimize transformer design. Additionally, the experimental setup discussed confirms the theoretical findings and highlights the practical behavior of transformers under varying operating conditions.

By mastering these concepts and equations, electrical engineers can effectively design transformers for a wide range of applications, ensuring efficient power transmission and reliable electrical energy conversion.