For the first-order transfer function G(s) = K/(τs+1), what is the steady-state gain and what does τ represent?

For the first-order transfer function G(s) = K/(τs+1), what is the steady-state gain and what does τ represent?

• A. Steady-state gain is K; τ is the time constant indicating the speed of response.
• B. Steady-state gain is 1; τ is the damping ratio.
• C. Steady-state gain is 0; τ is the rising time.
• D. Steady-state gain is Kτ; τ is the frequency.

Answer: (A)

In a first-order system, the long-term (steady-state) response to a constant input is determined by the DC gain, which is G(0). For G(s) = K/(τs+1), G(0) = K, so the steady-state gain is K—the output scales with the input by that factor regardless of τ. The parameter τ is the time constant. It sets how fast the system responds: the step response is y(t) = K[1 − e^(−t/τ)], so after a time equal to τ, the output has reached about 63% of its final value. The pole sits at s = −1/τ, so larger τ means a slower response. It doesn’t represent damping or frequency, which belong to different concepts.

In a first-order system, the long-term (steady-state) response to a constant input is determined by the DC gain, which is G(0). For G(s) = K/(τs+1), G(0) = K, so the steady-state gain is K—the output scales with the input by that factor regardless of τ.

The parameter τ is the time constant. It sets how fast the system responds: the step response is y(t) = K[1 − e^(−t/τ)], so after a time equal to τ, the output has reached about 63% of its final value. The pole sits at s = −1/τ, so larger τ means a slower response. It doesn’t represent damping or frequency, which belong to different concepts.