Circuits with Resistors and Inductors (LR Circuits) · 含电阻和电感的电路（LR电路）
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| inductor/ɪnˈdʌktə/ | 电感器 | diàn gǎn qì |
Flip the switch and the current takes its time to rise
- Close a switch on a plain resistor and the current jumps up instantly.
- Add an inductor 电感器 and the current climbs gradually instead.
- The inductor's back-EMF fights the sudden change, slowing the rise.
- A resistor and inductor together make an LR circuit — the magnetic cousin of RC.
一按开关,电流慢慢地上升
- 对一个普通电阻合上开关,电流立刻跳上去。
- 加一个电感器,电流反而渐渐爬升。
- 电感器的反电动势对抗突然的变化,减缓上升。
- 电阻和电感器合在一起构成一个 LR 电路——RC 的磁学表亲。
The current rises exponentially
- Close the switch and the current climbs: $I(t) = \dfrac{\varepsilon}{R}\left(1 - e^{-t/\tau}\right)$.
- It starts at zero, rises fast, then eases toward its final value $\varepsilon/R$.
- The inductor's opposition is strongest at the start and fades as $I$ steadies.
- The shape is the same exponential curve as an RC charging capacitor.
电流指数地上升
- 合上开关,电流爬升:$I(t) = \dfrac{\varepsilon}{R}\left(1 - e^{-t/\tau}\right)$。
- 它从零开始,快速上升,然后向最终值 $\varepsilon/R$ 缓和。
- 电感器的对抗在开始时最强,随着 $I$ 稳定而消退。
- 这个形状和 RC 电容器充电的指数曲线一样。
Long after the switch closes, the LR current settles at $\varepsilon/$ ____. · 开关闭合很久之后，LR电流稳定在$\varepsilon/$ ____。
The final steady current is $\varepsilon/R$ (inductor acts like a wire). · 最终稳态电流为$\varepsilon/R$（电感表现得像导线）。
The time constant τ = L/R
- The natural timescale here is $\tau = \dfrac{L}{R}$.
- In one $\tau$, the current reaches about $63\%$ of its final value.
- After about $5\tau$ it is essentially at $\varepsilon/R$.
- A bigger $L$ (more inertia) or smaller $R$ makes the rise slower.
时间常数 τ = L/R
- 这里的自然时间尺度是 $\tau = \dfrac{L}{R}$。
- 在一个 $\tau$ 内,电流达到最终值的约 $63\%$。
- 大约 $5\tau$ 之后,它基本上到达 $\varepsilon/R$。
- 更大的 $L$(更多惯性)或更小的 $R$ 使上升更慢。
The time constant of an LR circuit is: · LR电路的时间常数为：
For an LR circuit $\tau = L/R$. · 对于LR电路$\tau = L/R$。
An LR circuit has $L = 10\ \text{H}$ and $R = 5\ \Omega$. Find $\tau$ (in s). · LR电路具有$L = 10\ \text{H}$和$R = 5\ \Omega$。求$\tau$（单位s）。
$\tau = L/R = 10/5 = 2\ \text{s}$.
The inductor's two extremes
- At the first instant ($t = 0$): the inductor blocks the change, acting like an open gap (zero current).
- After a long time ($t \to \infty$): the current is steady, so the inductor acts like a plain wire.
- Notice this is the opposite of a capacitor's behaviour.
- Those two limits let you check any LR answer quickly.
电感器的两个极端
- 在最初一刻($t = 0$):电感器阻挡变化,像一个断开的缺口(零电流)。
- 经过很长时间后($t \to \infty$):电流稳定,所以电感器像一根普通导线。
- 注意这与电容器的行为相反。
- 那两个极限让你能快速检验任何 LR 答案。
An LR circuit switching on · LR电路的开关接通
An inductor resists sudden change. Sort each fact by the moment it describes. · 电感抵抗突变。按描述的时刻对每个事实进行分类。
At the first instant a switch closes, an inductor acts like: · 开关闭合的第一瞬间，电感表现得像：
It blocks the sudden change, so at $t=0$ the current is zero (open gap). · 它阻挡突变，因此在$t=0$时电流为零（断路间隙）。
An inductor's early and late behaviour is the reverse of a capacitor's. · 电感的早期和晚期行为与电容器相反。
Inductor: open then wire. Capacitor: wire then open. They are reversed. · 电感：先断路后导线。电容器：先导线后断路。它们正好相反。
Switching off can spark
- Open the switch and the current tries to drop to zero at once.
- The inductor fights that huge $dI/dt$ with a large back-EMF.
- That surge can jump the switch gap as a spark.
- It is why circuits with big coils need protection when switched off.
断电时可能打火
- 断开开关,电流试图立刻降到零。
- 电感器用一个大的反电动势对抗那个巨大的 $dI/dt$。
- 那个电涌能以火花跳过开关的缝隙。
- 这就是为什么有大线圈的电路在断电时需要保护。
Select all · 所有 true statements about LR circuits. · 选择关于LR电路的所有正确陈述。
Exponential rise, τ = L/R, blocked at t = 0. The rise is gradual, not instant. · 指数上升，τ=L/R，在t=0时被阻挡。上升是渐进的，不是瞬时的。
An LR circuit has $L = 6\ \text{H}$ and $R = 2\ \Omega$. Find its time constant.
- $\tau = \dfrac{L}{R} = \dfrac{6}{2} = 3\ \text{s}$.
- After $3\ \text{s}$ the current is about $63\%$ of $\varepsilon/R$.
一个 LR 电路 $L = 6\ \text{H}$,$R = 2\ \Omega$。求它的时间常数。
- $\tau = \dfrac{L}{R} = \dfrac{6}{2} = 3\ \text{s}$。
- $3\ \text{s}$ 后电流约为 $\varepsilon/R$ 的 $63\%$。
An inductor's limits are the reverse of a capacitor's. At $t = 0$ an inductor acts like an open gap (blocks current), and at $t \to \infty$ like a plain wire. Swapping these — or confusing them with the capacitor's — is the classic LR mistake.
电感器的极限与电容器的相反。在 $t = 0$ 时电感器像一个断开的缺口(阻挡电流),在 $t \to \infty$ 时像一根普通导线。交换这两者——或与电容器的混淆——是经典的 LR 错误。
In an LR circuit, the current rises as $I = \tfrac{\varepsilon}{R}(1 - e^{-t/\tau})$ with time constant $\tau = L/R$ (about $63\%$ per $\tau$). At $t=0$ the inductor acts like an open gap, and at $t\to\infty$ like a wire — the reverse of a capacitor.
在 LR 电路中,电流以 $I = \tfrac{\varepsilon}{R}(1 - e^{-t/\tau})$ 上升,时间常数 $\tau = L/R$(每个 $\tau$ 约 $63\%$)。在 $t=0$ 时电感器像一个断开的缺口,在 $t\to\infty$ 时像一根导线——与电容器相反。