Summarizing Rl Circuits


  • When a series connection of a resistor and an inductor—an RL circuit—is connected to a voltage source, the time variation of the current is

    \(I={I}_{0}(1-{e}^{-t/\tau })\text{(turning on).}\)

    where \({I}_{0}=V/R\) is the final current.

  • The characteristic time constant \(\tau \) is \(\tau =\cfrac{L}{R}\) , where \(L\) is the inductance and \(R\) is the resistance.
  • In the first time constant \(\tau \), the current rises from zero to \(0\text{.}\text{632}{I}_{0}\), and 0.632 of the remainder in every subsequent time interval \(\tau \).
  • When the inductor is shorted through a resistor, current decreases as

    \(I={I}_{0}{e}^{-t/\tau }\text{(turning off).}\)

    Here \({I}_{0}\) is the initial current.

  • Current falls to \(0\text{.}\text{368}{I}_{0}\) in the first time interval \(\tau \), and 0.368 of the remainder toward zero in each subsequent time \(\tau \).


characteristic time constant

denoted by \(\tau \), of a particular series RL circuit is calculated by \(\tau =\cfrac{L}{R}\), where \(L\) is the inductance and \(R\) is the resistance

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