Chapter 5: First and Second-Order Circuits

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5.2:

RC Circuit without Source

JoVE 핵심
Electrical Engineering

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JoVE 핵심 Electrical Engineering

RC Circuit without Source

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5.1: First-Order Circuits

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5.3: RC Circuit with Source

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When a DC source is removed abruptly from an RC circuit, the circuit becomes source-free. Assuming the fully charged capacitor's initial voltage is V0, its initial energy, which stimulates the circuit, can be obtained. Applying Kirchhoff's current law at the top node and substituting current values across the components gives a first-order differential equation. Rearranging terms, integrating, and taking the exponential on both sides, yields the natural response of the circuit, where the integration constant equals the initial voltage. The voltage versus time graph shows that the initial voltage decays exponentially with time. The time constant, tau, signifies the time required for the capacitor to discharge to 36.8 percent of its initial voltage. By substituting tau's value into the voltage response expression, the current and power dissipated in the resistor can be determined. Integrating the dissipated power over time provides the energy absorbed by the resistor. As the time approaches infinity, this energy approaches the initial energy stored in the capacitor, implying that the capacitor's initial energy gradually dissipates in the resistor.

5.2:

RC Circuit without Source

When a DC source is abruptly disconnected from an RC (Resistor-Capacitor) circuit, the circuit becomes source-free. Assuming that the capacitor was fully charged before the source was removed, its initial voltage, denoted as V₀, can be considered as the initial energy that stimulates the circuit.

Applying Kirchhoff's current law at the top node of the circuit and substituting the current values across the components, a first-order differential equation is obtained. By rearranging the terms in this equation, integrating, and then taking the exponential on both sides, the natural response of the circuit is determined. The integration constant in this equation equals the initial voltage.

The voltage versus time graph shows that the initial voltage decays exponentially with time. This means that the charge on the capacitor gradually decreases, which in turn reduces the voltage across it.

The time constant of the circuit, represented by the Greek letter tau (τ), signifies the time required for the capacitor to discharge to 36.8% of its initial voltage. This time constant plays a critical role in determining the rate at which the capacitor discharges and, as a result, the speed at which the circuit responds to changes.

By substituting the value of tau into the voltage response expression, the current flowing through the resistor, as well as the power dissipated in the resistor, can be calculated. The power dissipated in the resistor is the rate at which energy is lost in the form of 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. This implies that the initial energy of the capacitor gradually dissipates in the resistor, eventually depleting the capacitor's charge.

In conclusion, understanding the behavior of RC circuits when the DC source is removed provides valuable insights into the transient response of these circuits. This knowledge is essential for designing and analyzing circuits in applications such as signal processing, power electronics, and communication systems, where the rapid charging and discharging of capacitors is a fundamental process.