5.4
Circuito RL sem Fonte
Quando uma fonte DC é repentinamente desconectada de um circuito RL (Resistor-Indutor), o circuito fica sem fonte. Supondo que o indutor tenha uma cor…
Quando uma fonte CC é repentinamente desconectada de um circuito RL, ela se torna livre de fonte.
Supondo que o indutor tenha uma corrente inicial i0, a energia inicial armazenada no indutor pode ser determinada.
Aplicando a lei de tensão de Kirchhoff ao redor do loop e substituindo as tensões no indutor e no resistor produz uma equação diferencial de primeira ordem.
Reorganizar termos, integrar e aplicar os limites fornece uma equação logarítmica.
Ao tomar exponencial em ambos os lados, a expressão final da resposta natural do circuito é determinada.
O gráfico de corrente versus tempo mostra uma diminuição exponencial na corrente inicial.
A resposta atual pode ser expressa em termos da constante de tempo, que é a razão entre a indutância e a resistência.
A expressão de corrente é usada para determinar a tensão e a potência dissipadas pelo resistor.
A integração da potência dissipada ao longo do tempo fornece a expressão da energia absorvida pelo resistor.
À medida que o tempo se aproxima do infinito, a energia absorvida pelo resistor se aproxima da energia inicial armazenada no indutor, o que implica que a energia inicial é gradualmente dissipada no resistor.
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.