Transformer E.M.F. Equation and Derivation

E.M.F. Equation of a Transformer and Derivation of the Primary and Secondary Winding-Induced Voltage 

The article on Transformer emf equation derivation helps us to calculate the primary and secondary winding-induced voltage and its r.m.s value.  

The derivate relation between primary and secondary windings turns the number and the magnetic flux in the transformer core.

The r.m.s. value of the e.m.f. induced in primary and secondary windings are similar to calculate.

Here, we will show the detailed calculation of the E.M.F. equation for the primary winding, the secondary can get in a similar way.

Let us consider a transformer where

N₁ = No. of turns in the primary;
N₂ = No. of turns in secondary;
Φ_m= Maximum flux in core in Webers
     = B_m×A;
f = Frequency of a.c. input in Hz;

As shown in the Figure below-

The flux increases from its zero value to maximum value Φ_m in one-quarter of the cycle i.e. in 1/4f second.

∴ The average rate of change of flux = Φ_m/(1/4f)     

                                                     = 4fΦ_m Wb/s or volt.

Now, the rate of change of flux per turn means induced e.m.f. in volts.

∴ Average e.m.f./turn = 4fΦ_m volt.

If flux Φ varies sinusoidally, then r.m.s. value of induced e.m.f. is obtained by multiplying the average value with the form factor.

Form factor = (r.m.s. value)/(average value) = 1.11

∴ r.m.s. value of e.m.f./turn = 1.11×4fΦ_m = 4.44fΦ_m volt.

Now, r.m.s. value of the induced e.m.f. in the whole of the primary winding
= (induced e.m.f/turn)×No. of primary turns
E₁ = 4.44fN₁Φ_m = 4.44fN₁B_mA

So, the derivated equation-1 for induced e.m.f in the primary winding is as below:

E₁ = 4.44fN₁Φ_m = 4.44fN₁B_mA

Similarly, r.m.s. value of the e.m.f. induced in secondary is,

E₂ = 4.44fN₂Φ_m = 4.44fN₂B_mA

So, the derivated equation-2 for induced e.m.f in the secondary winding is as below:

E₂ = 4.44fN₂Φ_m = 4.44fN₂B_mA

It is seen from equations 1 and 2 that E₁/N₁ =E₂/N₂ =4.44fΦ_m. 

It means that e.m.f./turn is the same in both the primary and the secondary windings.

In an ideal transformer on no-load, V₁=E₁ and E₂=V₂ where V₂ is the terminal voltage.

You may know the details about the electrical transformer from the following articles:

 

1. Working Principle of Transformer;
2. Transformer Construction;
3. Core-type Transformers;
4. Shell-type Transformers;
5. Elementary Theory of an Ideal Transformer;
6. E.M.F. Equation of Transformer;
7. Voltage Transformation Ratio;
8. Transformer with losses but no Magnetic Leakage;
9. Transformer on No-load;
10. Transformer on Load;
11. Transformer with Winding Resistance but no Magnetic Leakage;
12. Equivalent Resistance;
13. Magnetic Leakage;
14. Transformer with Resistance and Leakage Reactance;
15. Simplified Diagram;
16. Total Approximate Voltage Drop in Transformer;
17. Exact Voltage Drop;
18. Equivalent Circuit Transformer Tests;
19. Open-circuit or No-load Test;
20. Separation of Core Losses;
21. Short-Circuit or Impedance Test;
22. Why Transformer Rating in KVA?;
23. Regulation of a Transformer;
24. Percentage Resistance, Reactance, and Impedance;
25. Kapp Regulation Diagram;
26. Sumpner or Back-to-back-Test;
    
27. The efficiency of a Transformer;
28. Condition for Maximum Efficiency;
29. Variation of Efficiency with Power Factor;
30. All-day Efficiency;
31. Auto-transformer;
32. Conversion of 2-Winding Transformer into Auto-transformer;
33. Parallel Operation of Single-phase Transformers;
34. Questions and Answers on Transformers;
35. Three-phase Transformers;
36. Three-phase Transformer Connections;
37. Star/Star or Y/Y Connection;
38. Delta-Delta or ∆/∆ Connection;
39. Wye/Delta or Y/ Connection;
40. Delta/Wye or ∆/Y Connection;
41. Open-Delta or V-V Connection;
42. Power Supplied by V-V Bank;
43. Scott Connection or T-T Connection;
44. Three-phase to Two-Phase Conversion and vice-versa;
45. Parallel Operation of 3-phase Transformers;
46. Instrument Transformers;
47. Current Transformers;
48. Potential or Voltage Transformers.