Which parameter describes the rate of exponential approach to the final value in a first-order system?

Which parameter describes the rate of exponential approach to the final value in a first-order system?

• A. Dead time
• B. Gain
• C. Time constant
• D. Damping ratio

Answer: (C)

The time constant sets how fast a first-order system approaches its final value. For a step input, the response follows y(t) = y_final + (y_initial − y_final) e^(−t/τ). The term e^(−t/τ) is the exponential that decays at a rate governed by τ: smaller τ means a faster approach, larger τ means a slower one. A handy rule is that after a time equal to τ, the response is about 63% of the way from its initial value to the final value. The other options describe different aspects: dead time is simply a delay before the response starts, gain determines the final magnitude, and damping ratio belongs to second-order systems and describes oscillatory behavior rather than the rate of exponential approach in a first-order system.

The time constant sets how fast a first-order system approaches its final value. For a step input, the response follows y(t) = y_final + (y_initial − y_final) e^(−t/τ). The term e^(−t/τ) is the exponential that decays at a rate governed by τ: smaller τ means a faster approach, larger τ means a slower one. A handy rule is that after a time equal to τ, the response is about 63% of the way from its initial value to the final value. The other options describe different aspects: dead time is simply a delay before the response starts, gain determines the final magnitude, and damping ratio belongs to second-order systems and describes oscillatory behavior rather than the rate of exponential approach in a first-order system.