EMF Equation of a transformer
The emf equation of a transformer is derived by considering a single-phase transformer. When a single-phase AC supply is given to a transformer, the emf gets induced in the secondary winding by means of electromagnetic induction.
Let,
As shown in fig, flux ϕm increases from zero value to maximum value in one-quarter of the cycle, i.e., in 1/4f second.
![\[\text{Average Rate of change of flux} = \frac{\phi_m}{1/4f} = 4f\phi_m\]](https://elpedia.com/wp-content/ql-cache/quicklatex.com-43202540b5f131c45ffff70bf025e54b_l3.png "Rendered by QuickLaTeX.com")
By faraday’s law of electromagnetic induction, rate of change of flux is nothing but induced emf,
![\[\text{Average emf per turn} = 4f\phi_m \, \, \text{volt}\]](https://elpedia.com/wp-content/ql-cache/quicklatex.com-3e0c24cfcacfbf313a0557bf5ea03d12_l3.png "Rendered by QuickLaTeX.com")
If the flux varies sinusoidally, then the rms value of induced emf is obtained by multiplying the average value with form factor.
![\[\text{Form Factor} = \frac{\text{RMS Value}}{\text{Average Value}} = 1.11 \]](https://elpedia.com/wp-content/ql-cache/quicklatex.com-30d42bc90531c4ce468e0fcc9dc2505c_l3.png "Rendered by QuickLaTeX.com")
![\[\text{RMS Value of emf per turn} = 1.11 \times 4f\phi_m = 4.44f\phi_m \, \, \text{volt} \]](https://elpedia.com/wp-content/ql-cache/quicklatex.com-52bfcc767b71e09030ded07392394fb9_l3.png "Rendered by QuickLaTeX.com")
The rms value of induced emf in the primary winding is given by,
![\[ = \text{Induced emf per turn} \times \text{No. of Primary Winding} \]](https://elpedia.com/wp-content/ql-cache/quicklatex.com-214da6970cd376b9fd2f136c8075f4d1_l3.png "Rendered by QuickLaTeX.com")
(1) 
Similarly, rms value of induced emf in the secondary winding is given by,
(2) 
The above two equations are the emf equation of a transformer in the primary and secondary winding respectively.
From equations (1) and (2), we get the emf induced per turn in the primary and secondary winding of a transformer.
![\[\frac{E_1}{N_1} = \frac{E_2}{N_2} = 4.44f\phi_m \]](https://elpedia.com/wp-content/ql-cache/quicklatex.com-5f5ed1ee9fecffe39935c24db8c5826a_l3.png "Rendered by QuickLaTeX.com")
Dividing the equations (1) by (2), we get,
![\[\frac{E_1}{E_2} = \frac{N_1}{N_2} = K \]](https://elpedia.com/wp-content/ql-cache/quicklatex.com-8899212a8dda30716d6f2ad7fe624e96_l3.png "Rendered by QuickLaTeX.com")
where K is a constant, called transformation ratio.
For an ideal transformer, E1 = V1 & E2 = V2
where
V1 – AC Supply voltage of transformer in the primary side.
V2 – Terminal voltage of transformer in the secondary side.
Thus, the transformation ratio K of a transformer is
![\[\frac{E_1}{E_2} = \frac{N_1}{N_2} = \frac{V_1}{V_2}= K \]](https://elpedia.com/wp-content/ql-cache/quicklatex.com-aa23d1f94a64de083dba6d6b635de571_l3.png "Rendered by QuickLaTeX.com")
In the above equation,
If N2 > N1 & E2 > E1 then the transformer is a step up transformer.
If N1 > N2 & E1 > E2 then the transformer is a step down transformer.
