# Transformer concept | Transformer formula | Transformer principle.

Transformer

An apparatus for reducing or increasing the voltage of an alternating current

A transformer is a passive electrical device that transfers electrical energy from one electrical circuit to another, or multiple circuits. A varying current in any one coil of the transformer produces a varying magnetic flux in the transformer's core, which induces a varying electromotive force across any other coils wound around the same core. Electrical energy can be transferred between separate coils without a metallic (conductive) connection between the two circuits.

Ideal transformer equations
${\displaystyle V_{\text{P}}=-N_{\text{P}}{\frac {\mathrm {d} \Phi }{\mathrm {d} t}}}$ . . . (eq. 1)
${\displaystyle V_{\text{S}}=-N_{\text{S}}{\frac {\mathrm {d} \Phi }{\mathrm {d} t}}}$ . . . (eq. 2)
Where ${\displaystyle V}$ is the instantaneous voltage${\displaystyle N}$ is the number of turns in a winding, dΦ/dt is the derivative of the magnetic flux Φ through one turn of the winding over time (t), and subscripts P and S denotes primary and secondary.
Combining the ratio of eq. 1 & eq. 2:
Turns ratio ${\displaystyle ={\frac {V_{\text{P}}}{V_{\text{S}}}}={\frac {N_{\text{P}}}{N_{\text{S}}}}=a}$ . . . (eq. 3)

Where for a step-down transformer a > 1, for a step-up transformer a < 1, and for an isolation transformer a = 1.
By law of conservation of energyapparentreal and reactive power are each conserved in the input and output:
${\displaystyle S=I_{\text{P}}V_{\text{P}}=I_{\text{S}}V_{\text{S}}}$ . . . . (eq. 4)
Where ${\displaystyle S}$ is conserved power and ${\displaystyle I}$ is current.
Combining eq. 3 & eq. 4 with this end note gives the ideal transformer identity:
${\displaystyle {\frac {V_{\text{P}}}{V_{\text{S}}}}={\frac {I_{\text{S}}}{I_{\text{P}}}}={\frac {N_{\text{P}}}{N_{\text{S}}}}={\sqrt {\frac {L_{\text{P}}}{L_{\text{S}}}}}=a}$ . (eq. 5)
Where ${\displaystyle L}$ is winding self-inductance.
By Ohm's law and ideal transformer identity:
${\displaystyle Z_{\text{L}}={\frac {V_{\text{S}}}{I_{\text{S}}}}}$ . . . (eq. 6)
${\displaystyle Z'_{\text{L}}={\frac {V_{\text{P}}}{I_{\text{P}}}}={\frac {aV_{\text{S}}}{I_{\text{S}}/a}}=a^{2}{\frac {V_{\text{S}}}{I_{\text{S}}}}=a^{2}{Z_{\text{L}}}}$ . (eq. 7)
Where ${\displaystyle Z_{\text{L}}}$ is the load impedance of the secondary circuit & ${\displaystyle Z'_{\text{L}}}$ is the apparent load or driving point impedance of the primary circuit, the superscript ${\displaystyle '}$ denoting referred to the primary.