5.4
无源电阻-电感器电路
当直流电源突然与电阻-电感器（RL）电路断开时，那么该电路将会变为无源状态。假设电感器中具有表示为I0的初始电流，则能够以此来确定电感器中所存储的初始能量。
在电路环路中应用基尔霍夫电压定律并代入到电感器和电阻器两端的电压中，便会得到一阶微分方程。对数方程是通过对方程中的各项进行重新排列、积分并应用…
当直流电源突然从RL电路断开时，该电路便成为无源电路。
假设电感器具有初始电流 i0，则可确定电感器中存储的初始能量。
对回路应用基尔霍夫电压定律，并代入电感和电阻两端的电压，可得到一个一阶微分方程。
重新整理项、积分并应用极限，可得到一个对数方程。
通过对等式两边取指数，可确定该电路自然响应的最终表达式。
电流随时间变化的曲线显示初始电流呈指数下降。
该电流响应可以用时间常数来表示，时间常数为电感与电阻之比。
当前表达式用于确定电阻两端的电压和消耗的功率。
对时间积分耗散功率可得到电阻吸收能量的表达式。
随着时间趋近于无穷大，电阻吸收的能量趋近于电感器中存储的初始能量，这意味着初始能量逐渐在电阻中耗散。
Q1: What happens to an RL circuit when the DC source is suddenly disconnected?
When a DC source is disconnected from an RL circuit, the circuit becomes source-free. The inductor, which initially stores energy based on its current, begins to release this energy. Applying Kirchhoff's voltage law around the loop yields a first-order differential equation that describes how the circuit responds to the sudden removal of the source.
Q2: How is the natural response of a source-free RL circuit expressed mathematically?
The natural response is derived by applying Kirchhoff's voltage law and solving the resulting first-order differential equation. After rearranging, integrating, and applying limits, a logarithmic equation is obtained. Taking the exponential of both sides yields the final expression showing current as an exponential function of time.
Q3: What is the time constant in an RL circuit and why does it matter?
The time constant is the ratio of inductance to resistance (L/R) and represents the speed at which the circuit responds to changes. A larger time constant means the current decays more slowly, while a smaller time constant indicates faster decay. This parameter is fundamental for predicting how quickly the circuit reaches steady state.
Q4: Why does current decrease exponentially in a source-free RL circuit?
When the source is removed, the inductor's magnetic field collapses, driving current through the resistor. The resistor dissipates energy as heat, reducing the current exponentially over time. The rate of decay depends on the time constant; larger resistance or smaller inductance causes faster exponential decay of the initial current.
Q5: How can you calculate the power dissipated in the resistor of a source-free RL circuit?
Power dissipated in the resistor is calculated using the current expression derived from the natural response. Since power equals I²R, substituting the exponential current function gives the instantaneous power. This power represents the rate at which energy stored in the inductor is converted to heat in the resistor.
Q6: What is the relationship between initial inductor energy and energy absorbed by the resistor?
The initial energy stored in the inductor equals one-half LI₀². As time approaches infinity, the total energy absorbed by the resistor approaches this initial value. This demonstrates energy conservation: all magnetic energy initially stored in the inductor is gradually dissipated as heat in the resistor until the inductor's energy is depleted.
Q7: How does a source-free RL circuit differ from other first-order circuits?
Like other first-order circuits, the source-free RL circuit exhibits exponential response governed by a single time constant. However, RL circuits store energy in magnetic fields, while RC circuits store energy in electric fields. Understanding source-free RL behavior provides insights applicable to analyzing first-order circuits across various applications.