5.4
소스가 없는 RL 회로
RL(저항-유도자) 회로에서 DC 소스의 연결이 갑자기 끊어지면 회로에 소스가 없게 됩니다. 유도자에 I0 로 표시된 초기 전류가 있다고 가정하면 유도자에 저장된 초기 에너지가 결정될 수 있습니다.
회로 루프 주위에 키르히호프의 전압 법칙을 적용하고 유도자와 저항기의…
DC 소스가 RL 회로에서 갑자기 분리되면 소스가 없어집니다.
인덕터가 초기 전류 i0를 갖는다고 가정하면 인덕터에 저장된 초기 에너지를 결정할 수 있습니다.
루프 주위에 Kirchhoff의 전압 법칙을 적용하고 인덕터와 저항기의 전압을 대체하면 1차 차동 방정식이 생성됩니다.
항을 재정렬하고, 적분하고, 한계를 적용하면 로그 방정식이 생성됩니다.
양쪽에서 지수를 취함으로써 회로의 자연 응답의 최종 표현이 결정됩니다.
전류 대 시간 그래프는 초기 전류의 기하급수적인 감소를 보여줍니다.
전류 응답은 인덕턴스와 저항의 비율인 시간 상수로 표현할 수 있습니다.
전류 표현은 저항기에서 발산되는 전압과 전력을 결정하는 데 사용됩니다.
시간이 지남에 따라 소산되는 전력의 통합은 저항기가 흡수하는 에너지에 대한 표현을 제공합니다.
시간이 무한대에 가까워짐에 따라 저항기에 의해 흡수된 에너지는 인덕터에 저장된 초기 에너지에 접근하며, 이는 초기 에너지가 저항기에서 점차적으로 소산됨을 의미합니다.
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.