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Mathematics for Physics

88 Transformers

An electrical device of magnetically coupled coils that is used to transfer energy between circuits while stepping alternating voltage up or down.

The transformer voltage ratio. An ideal transformer shares the same changing flux in the primary and secondary windings. This principle is used to relate the two voltages to the turn counts.

The transformer voltage ratio is

\[ \dfrac{V_{s}}{V_{p}} = \dfrac{N_{s}}{N_{p}} \]

where

  • \(V_{s}\) is the secondary voltage.
  • \(V_{p}\) is the primary voltage.
  • \(N_{s}\) is the number of secondary turns.
  • \(N_{p}\) is the number of primary turns.

The current ratio of an ideal transformer. Power is conserved in an ideal transformer, so the currents stand in the inverse ratio of the turns. This principle is used to step current down when voltage is stepped up.

The transformer current ratio is

\[ \dfrac{I_{s}}{I_{p}} = \dfrac{N_{p}}{N_{s}} \]

where

  • \(I_{s}\) is the secondary current.
  • \(I_{p}\) is the primary current.
  • \(N_{s}\) is the number of secondary turns.
  • \(N_{p}\) is the number of primary turns.

Coupling by mutual inductance. The coupling is mutual inductance. This principle is used to write the induced secondary emf from the primary current.

The secondary emf from mutual inductance is

\[ \mathcal{E}_{s} = -M\dfrac{dI_{p}}{dt} \]

where

  • \(\mathcal{E}_{s}\) is the secondary emf.
  • \(M\) is the mutual inductance.
  • \(I_{p}\) is the primary current.
  • \(t\) is time.