RL Circuits

Core Sentence / 核心句

An RL circuit changes exponentially because the inductor opposes changes in current.

中文理解：电感产生的 self-induced EMF 随 $d i / d t$ 改变，使 current 不能跳变，只能以 exponential transient 接近新的 steady state。

τ_{L} = \frac{L}{R _{seen}}

$R_{seen}$ 是 inductor terminals 看到的 equivalent resistance。

The time constant sets the pace of the transition, not the final value.

simple series RL with zero initial current：

i (t) = \frac{E}{R} (1 - e^{- t / τ_{L}})

at $t = τ_{L}$ ，current reaches about $63.2%$ of its final value。

i (t) = I_{0} e^{- t / τ_{L}}

at $t = τ_{L}$ ，current retains about $36.8%$ 。

energy：

U_{L} (t) = U_{0} e^{- 2 t / τ_{L}}

solve $0^{-}$ steady state；
enforce $i_{L} (0^{+}) = i_{L} (0^{-})$ ；
redraw the $0^{+}$ circuit；
solve $t \to \infty$ with ideal inductor replaced by a wire；
use $V_{L} = L d i / d t$ for initial slope；
use $τ_{L} = L / R_{seen}$ only after finding the resistance seen by the inductor。

capacitor voltage cannot jump；inductor current cannot jump；
initially uncharged capacitor behaves as a short；initially unenergized inductor behaves as an open；
DC steady capacitor behaves as an open；DC steady ideal inductor behaves as a short；
$τ_{C} = R C$ ； $τ_{L} = L / R$ 。