5.2
无源RC电路
当直流电源（DC）突然与电阻电容（RC）电路断开时，该电路将会变为无源状态。假设在移除电源之前电容器已经变为充满电的状态，则其初始电压（记为V0）可被视为激励电路的初始能量。
在电路的顶部节点处应用基尔霍夫电流定律并代入各元件的电流值，从而得到一阶微分方程。对该方程中的各项进行重新排列、积分，然后在…
当从RC电路中突然移除直流电源时，该电路便成为无源电路。
假设充满电的电容器的初始电压为 V0，则可求得其初始能量，该能量用于激励电路。
在顶部节点应用基尔霍夫电流定律，并代入各元件上的电流值，可得到一个一阶微分方程。
重新整理各项，进行积分并对两边取指数，可得到电路的自然响应，其中积分常数等于初始电压。
电压随时间变化的图像显示，初始电压随时间呈指数衰减。
时间常数 tau 表示电容器放电至其初始电压的 36.8% 所需的时间。
将 tau 的值代入电压响应表达式后，即可确定电阻中消耗的电流和功率。
对时间积分耗散功率可得到电阻吸收的能量。
随着时间趋于无穷大，该能量趋近于电容器中存储的初始能量，表明电容器的初始能量在电阻器中逐渐耗散。
Q1: What happens to an RC circuit when the DC source is suddenly removed?
When a DC source is abruptly disconnected from an RC circuit, the circuit becomes source-free. The fully charged capacitor's initial voltage, V0, acts as the energy source that stimulates the circuit. This triggers a natural response where the capacitor discharges through the resistor, causing the voltage to decay exponentially over time.
Q2: How is the natural response of a source-free RC circuit derived?
Applying Kirchhoff's current law at the circuit node and substituting current values across components yields a first-order differential equation. By rearranging terms, integrating, and taking the exponential on both sides, the natural response is determined. The integration constant equals the initial voltage V0, establishing the exponential decay relationship.
Q3: What does the time constant tau represent in an RC circuit?
The time constant tau (τ) signifies the time required for the capacitor to discharge to 36.8 percent of its initial voltage. This parameter determines the rate at which the capacitor discharges and controls the speed at which the circuit responds to changes. Substituting tau into the voltage response expression allows calculation of current and power dissipated in the resistor.
Q4: How does the voltage decay in a source-free RC circuit over time?
The voltage versus time graph shows that initial voltage decays exponentially with time. As the capacitor discharges, its charge gradually decreases, reducing the voltage across it. This exponential decay continues indefinitely, with the voltage approaching zero as time approaches infinity.
Q5: Where does the capacitor's initial energy go in a source-free RC circuit?
The capacitor's initial energy gradually dissipates in the resistor as heat. Integrating the power dissipated over time provides the total energy absorbed by the resistor. As time approaches infinity, this energy approaches the initial energy stored in the capacitor, completely depleting its charge.
Q6: How can you calculate current and power in a source-free RC circuit?
By substituting the time constant tau into the voltage response expression, the current flowing through the resistor can be determined. Power dissipated in the resistor is calculated as the rate at which energy is lost as heat. These calculations enable analysis of transient response behavior in signal processing and power electronics applications.
Q7: Why is understanding source-free RC circuits important for circuit design?
Understanding the transient response of source-free RC circuits provides valuable insights essential for designing and analyzing circuits in applications such as signal processing, power electronics, and communication systems. The rapid charging and discharging of capacitors is a fundamental process in these fields, making knowledge of exponential decay and time constants critical.