Transformer concept | Transformer formula | Transformer principle.

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

By Faraday’s law of induction:

${displaystyle V_{text{P}}=-N_{text{P}}{frac {mathrm {d} Phi }{mathrm {d} t}}}$ . . . (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 energy, apparent, real and reactive power are each conserved in the input and output:

${displaystyle S=I_{text{P}}V_{text{P}}=I_{text{S}}V_{text{S}}}$ . . . . (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.