91Ó°ÊÓ

In an RL circuit, the time constant, denoted by \( \tau \), is a crucial factor that influences how quickly the current reaches its maximum value. It is defined as the ratio between the inductance \( L \) and the resistance \( R \) in the circuit, given by the formula:\[ \tau = \frac{L}{R} \]The time constant represents the time it takes for the current to reach approximately 63.2% of its steady-state value. A smaller time constant means the system responds more quickly, while a larger time constant indicates a slower response. In the problem, we needed to halve \( \tau \), meaning the circuit should respond twice as fast. This change is accomplished by adjusting the resistance, since the inductance \( L \) remains constant. By doubling the resistance from \( 2200 \Omega \) to \( 4400 \Omega \), the new time constant \( \tau' \) becomes:\[ \tau' = \frac{L}{2R} \]

Steady-state current is the constant flow of electric charge that occurs after a circuit has reached equilibrium. Once the initial response has settled, and for long time after the switch is closed, the current remains steady, maintained solely by the voltage source overcoming the circuit's resistance. The steady-state current \( I \) in an RL circuit is calculated using Ohm's Law:\[ I = \frac{V_0}{R} \]For the given RL circuit, with resistance \( R = 2200 \Omega \) and voltage \( V_0 = 240 \text{ V} \), the steady-state current is approximately \( 0.109 \text{ A} \). Even with a change in resistance to \( 4400 \Omega \) to modify the time constant, it is important to keep the current constant since that matches the desired outcome, so the voltage must be increased to \( 480 \text{ V} \) to keep \( I \) unchanged according to \( I = \frac{480}{4400} \approx 0.109 \text{ A} \).

Resistance and voltage play fundamental roles in the behavior of an RL circuit. Resistance \( R \), measured in ohms \( \Omega \), is the opposition that the circuit offers to the flow of current. A higher resistance results in less current flow with the same voltage, and it directly affects the time constant \( \tau \) of the circuit.

• Resistance: Doubled in this scenario to adjust the time constant. Increasing the resistance makes the circuit reach its steady-state faster.
• Voltage: The driving force that pushes the current through the circuit. Must be adjusted to keep the steady-state current constant when the resistance changes.

In modifying the original circuit set-up, increasing \( R \) to \( 4400 \Omega \) required us to also adjust the voltage to \( 480 \text{ V} \), ensuring the steady-state current remains consistent with the original configuration. This interplay between resistance and voltage is a classic application of Ohm’s Law, showing the delicate balance needed to achieve specific electrical characteristics in a circuit.