Transformer

 

A transformer is an electrical device that transfers electrical energy between two or more circuits through electromagnetic induction.

 

    Ideal transformer equations (eq.)

By Faraday's law of induction:
${\displaystyle V_{\text{S}}=-N_{\text{S}}{\frac {\mathrm {d} \Phi }{\mathrm {d} t}}}$ . . . (1)^[a]
${\displaystyle V_{\text{P}}=-N_{\text{P}}{\frac {\mathrm {d} \Phi }{\mathrm {d} t}}}$ . . . (2)
Combining ratio of (1) & (2)
Turns ratio ${\displaystyle ={\frac {V_{\text{P}}}{V_{\text{S}}}}={\frac {-N_{\text{P}}}{-N_{\text{S}}}}=a}$ . . . (3) where

for step-down transformers, a > 1

for step-up transformers, a < 1

By law of conservation of energy, apparent, real and reactive power are each conserved in the input and output
${\displaystyle S=I_{\text{P}}V_{\text{P}}=I_{\text{S}}V_{\text{S}}}$ . . . (4)
Combining (3) & (4) with this endnote^[b][4] yields the ideal transformer identity
${\displaystyle {\frac {V_{\text{P}}}{V_{\text{S}}}}={\frac {I_{\text{S}}}{I_{\text{P}}}}={\frac {N_{\text{P}}}{N_{\text{S}}}}={\sqrt {\frac {L_{\text{P}}}{L_{\text{S}}}}}=a}$ . (5)
By Ohm's law and ideal transformer identity
${\displaystyle Z_{\text{L}}={\frac {V_{\text{S}}}{I_{\text{S}}}}}$ . . . (6)
Apparent load impedance Z'_L (Z_L referred to the primary)
${\displaystyle Z'_{\text{L}}={\frac {V_{\text{P}}}{I_{\text{P}}}}={\frac {aV_{\text{S}}}{I_{\text{S}}/a}}=a^{2}{\frac {V_{\text{S}}}{I_{\text{S}}}}=a^{2}{Z_{\text{L}}}}$ . (7)