Transitoires RC et RL, impédance et résonance
GRE Subject Test · GRE Physics Sujet 30 6:18 Narration en anglais · Sous-titres anglais + 中文 incrustés
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A capacitor's voltage and an inductor's current cannot jump under an ordinary finite drive, yet the two settle to opposite steady states.
在普通的有限激励下，电容器的电压和电感器的电流都不能突变，但两者最终达到的稳态却相反。
This topic solves RC and RL switching from stated initial conditions, tracks the stored energy, and uses complex impedance for phase and resonance.
本主题从给定的初始条件求解 RC 和 RL 电路的开关过程，追踪储存的能量，并用复阻抗处理相位和谐振。
For a resistor in series with a capacitor and a constant source, Kirchhoff's law gives R dq by dt plus q over C equals V s, with time constant tau equal to R C.
对于与电容器和恒定电源串联的电阻，基尔霍夫定律给出 R dq/dt 加 q/C 等于 Vs，时间常数 τ 等于 RC。
The capacitor voltage moves from its initial value toward V s, closing the gap by a factor e every time constant.
电容器电压从初始值趋向 Vs，每经过一个时间常数，差距缩小为原来的 1/e。
An uncharged capacitor follows V s times one minus e to the minus t over tau, while its current starts at V s over R and decays.
未充电的电容器电压为 Vs 乘以一减 e 的负 t/τ 次方，而电流从 Vs/R 开始衰减。
Capacitor voltage is continuous across the switch; the resistor current may jump.
电容器电压在开关前后连续；电阻电流可以突变。
In a series RL circuit, Kirchhoff's law gives L dI by d t plus R I equals V sub s, and tau equals L divided by R.
在RL串联电路中，基尔霍夫定律给出L乘以dI/dt加上R乘以I等于V_s，且tau等于L除以R。
The current after a constant drive is I of t equals V sub s divided by R plus the quantity I of zero minus V sub s divided by R, times e to the minus t over tau.
恒定驱动后的电流为I(t)等于V_s除以R加上(I(0)减去V_s/R)乘以e的负t/tau次方。
Inductor current is continuous for a finite voltage; its voltage may change abruptly as switching changes dI by d t.
对于有限电压，电感电流是连续的；其电压可能在开关改变dI/dt时发生突变。
At late times, an ideal inductor in a DC circuit acts as a zero-voltage connection, while an ideal capacitor has zero DC current.
在较晚时刻，直流电路中的理想电感表现为零电压连接，而理想电容则无直流电流。
Stored energies are L I squared over 2 and C V sub C squared over 2.
储存能量分别为LI²/2和CV_C²/2。
During decay into a resistor, the initially stored energy becomes resistor heat rather than disappearing when the source is disconnected.
在向电阻放电过程中，初始储存的能量转化为电阻热量，而非在电源断开时消失。
Use the declared phasor convention exp of i omega t.
使用声明的相量约定 e^(iωt)。
A resistor has impedance R, an inductor i omega L, and a capacitor 1 over i omega C, which equals negative i over omega C.
电阻的阻抗为 R，电感为 iωL，电容为 1/(iωC)，即 −i/(ωC)。
A series RLC circuit therefore has Z equals R plus i X, with X equals omega L minus 1 over omega C.
因此，串联 RLC 电路的阻抗 Z = R + iX，其中 X = ωL − 1/(ωC)。
Divide the source voltage phasor by Z to obtain current.
将源电压相量除以 Z 即可求得电流。
Its magnitude is V sub rms divided by the square root of R squared plus X squared when rms quantities are used.
当使用均方根量时，其幅值为 V_rms 除以 √(R²+X²)。
The impedance phase phi satisfies tan phi equals X over R. Current lags voltage for X greater than 0 and leads it for X less than 0.
阻抗相位 φ 满足 tanφ = X/R；当 X > 0 时电流滞后于电压，当 X < 0 时电流超前于电压。
Adding the scalar magnitudes R, omega L, and 1 over omega C loses the vector phase information.
将标量幅值 R、ωL 和 1/(ωC) 直接相加会丢失矢量相位信息。
Series resonance occurs at omega zero equals one over the square root of L C, when X equals zero.
串联谐振发生在 ω₀ = 1/√(LC) 且 X = 0 时。
Current is maximal for a fixed voltage in this ideal series model, and the voltage current phase difference is zero.
在此理想串联模型中，固定电压下电流达到最大值，且电压与电流的相位差为零。
Mean real power is V sub R M S times I sub R M S times cosine phi, which equals I sub R M S squared times R.
平均实功率为 V_rms × I_rms × cosφ = I_rms² × R。
Ideal L and C exchange energy but dissipate no average power.
理想的电感和电容交换能量但不耗散平均功率。
This differs from transient natural frequency: a damped series circuit can oscillate at the square root of one over L C minus R over 2 L squared, when underdamped.
这与瞬态自然频率不同：欠阻尼的阻尼串联电路可在 √(1/(LC) − (R/(2L))²) 处振荡。
Large resistance removes such free oscillation but does not change the condition X equals zero of the ideal driven series impedance.
大电阻会消除这种自由振荡，但不会改变理想受迫串联阻抗中 X = 0 的条件。
State whether a question asks for a step response, free decay or sinusoidal steady state.
请明确题目要求的是阶跃响应、自由衰减还是正弦稳态。
An RC circuit switches to a fixed source V s.
RC 电路切换到固定电源 Vs。
Kirchhoff's law gives R C times the rate of change of V C, plus V C, equals V s, so V C equals V s plus the initial difference times e to the minus t over R C.
基尔霍夫定律给出 RC 乘 VC 的变化率加 VC 等于 Vs，所以 VC 等于 Vs 加上初始差值乘以 e 的负 t/RC 次方。
Take 2 kilohms, 1 millifarad and a 12 volt source, starting uncharged.
取 2 千欧、1 毫法和 12 伏电源，电容器初始不带电。
The time constant is 2000 times 0.001: 2 seconds.
时间常数等于 2000 乘 0.001：2 秒。
After one time constant the voltage is 12 times one minus e to the minus 1: 7.59 volts.
经过一个时间常数后，电压等于 12 乘一减 e 的负一次方：7.59 伏。
The time constant sets how fast the remaining gap to 12 volts shrinks.
时间常数决定了与 12 伏之间剩余差距缩小的快慢。
An RC capacitor starts at eight volts and switches to a negative four volt ideal source through R.
一个RC电容初始电压为8伏，通过电阻R切换至负4伏理想电源。
Find its voltage at arbitrary t, the first zero crossing and the initial current defined into its positive voltage plate.
求其在任意时刻t的电压、第一次过零点以及流入其正极板的初始电流。
Explain voltage continuity.
解释电压连续性。
With tau equals R C, V sub C equals negative 4 plus 12 e to the minus t over tau volts.
当 τ = RC 时，V_C = -4 + 12e^(-t/τ) 伏。
Zero crossing satisfies e to the minus t over tau equals one third, so t equals tau times natural log of 3.
过零点满足 e^(-t/τ) = 1/3，因此 t = τ ln 3。
I of 0 equals V sub s minus V sub C of 0 all over R equals negative 12 volts over R.
I(0) = (V_s - V_C(0)) / R = -12 V / R。
Finite current cannot change capacitor charge instantaneously, so voltage begins at 8 volts.
有限电流无法瞬间改变电容电荷，因此电压从 8 V 开始。
Award solution, endpoint check, crossing, signed current and continuity.
给出解答、端点检查、过零点、带符号电流及连续性。
An RC circuit has time constant tau equal to R times C, while an RL circuit has tau equal to L over R.
RC 电路的时间常数 tau 等于 R 乘以 C，而 RL 电路的时间常数 tau 等于 L 除以 R。
Do not confuse current amplitude with rms current, or resonance of a driven impedance with damped free-oscillation frequency.
不要混淆电流幅值与有效值电流，也不要将受迫阻抗的谐振与阻尼自由振荡频率混为一谈。
time constant: The exponential response scale, RC for an RC circuit and L/R for an RL circuit.
时间常数：指数响应的时间尺度，RC电路为RC，RL电路为L除以R。
impedance: The complex voltage-to-current phasor ratio in sinusoidal steady state.
阻抗：正弦稳态下的复电压与电流相量之比。
Retrieve the meanings without reading the definitions, then change a parameter in the original transfer and test the limiting case.
在不阅读定义的情况下回忆其含义，然后更改原始传递函数中的一个参数，并检验极限情况。