5.2
Circuito RC sem Fonte
Quando uma fonte DC é desconectada abruptamente de um circuito RC (Resistor-Capacitor), o circuito fica sem fonte. Assumindo que o capacitor estava to…
Quando uma fonte CC é removida abruptamente de um circuito RC, o circuito torna-se livre de fonte.
Supondo que a tensão inicial do capacitor totalmente carregado seja V0, sua energia inicial, que estimula o circuito, pode ser obtida.
Aplicando a lei atual de Kirchhoff no nó superior e substituindo os valores atuais entre os componentes, obtém-se uma equação diferencial de primeira ordem.
Reorganizar os termos, integrar e obter o exponencial em ambos os lados produz a resposta natural do circuito, onde a constante de integração é igual à tensão inicial.
O gráfico de tensão versus tempo mostra que a tensão inicial decai exponencialmente com o tempo.
A constante de tempo, tau, significa o tempo necessário para o capacitor descarregar até 36,8% de sua tensão inicial.
Ao substituir o valor de tau na expressão de resposta de tensão, a corrente e a potência dissipadas no resistor podem ser determinadas.
A integração da potência dissipada ao longo do tempo fornece a energia absorvida pelo resistor.
À medida que o tempo se aproxima do infinito, essa energia se aproxima da energia inicial armazenada no capacitor, o que implica que a energia inicial do capacitor se dissipa gradualmente no resistor.
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.