5.4
ソースなしの RL 回路
DC 電源が RL (抵抗インダクター) 回路から突然切り離されると、回路の電源がなくなります。 インダクタの初期電流が I0 であると仮定する事で、インダクタに蓄えられる初期エネルギーを求めることができます。
キルヒホッフの電圧則を回路のループに適用し、インダクタと抵抗の両端の電圧を代入すると、一…
DC電源がRL回路から突然切断されると、ソースフリーになります。
インダクタの初期電流がi0であると仮定すると、インダクタに蓄積された初期エネルギーを決定できます。
ループの周囲にキルヒホッフの電圧則を適用し、インダクタと抵抗の両端の電圧を代入すると、1次微分方程式が得られます。
項を並べ替え、積分し、制限を適用すると、対数方程式が得られます。
両側で指数を取ることにより、回路の自然な応答の最終的な表現が決定されます。
電流対時間グラフは、初期電流の指数関数的な減少を示しています。
電流応答は、インダクタンスと抵抗の比である時定数で表すことができます。
電流式は、抵抗器の両端で消費される電圧と電力を決定するために使用されます。
時間の経過に伴う消費電力の統合により、抵抗によって吸収されるエネルギーの式が得られます。
時間が無限大に近づくと、抵抗によって吸収されたエネルギーはインダクタに蓄積された初期エネルギーに近づき、初期エネルギーが抵抗内で徐々に消費されることを意味します。
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.