DCGain

DC gain, short for direct-current gain, is the steady-state gain of a system when the input is held constant. In linear time-invariant systems, the DC gain is defined as the transfer function evaluated at zero frequency: G_dc = G(0) = lim s→0 G(s). It describes how a constant input is amplified or attenuated once all transient dynamics have died out, and it is independent of time-varying behavior such as settling time or overshoot.

In continuous-time systems, if G(s) = K/(τs + 1), the DC gain is K. If the transfer function contains an integrator, for example G(s) = K/s, the limit as s→0 diverges, indicating an unbounded response to a constant input in the absence of saturation or other nonlinear effects. If a system includes a zero at the origin, the DC gain may be zero.

In discrete-time systems, the equivalent concept is the transfer function evaluated at z = 1, provided the system is causal and stable. The DC gain equals the sum of the impulse response sequence h[n] over all n ≥ 0, and it determines the steady-state response to a constant input (a unit step) in the discrete domain.

DC gain is used to characterize amplifiers, filters, sensors, and control loops, guiding design choices to ensure proper steady-state behavior. It is computed by substituting the appropriate variable (s for continuous time, z for discrete time) with its DC value and, in rational transfer functions, by evaluating the limit or substituting directly when permissible. Cautions include interpreting DC gain correctly in the presence of poles at or near the origin or in nonlinear or time-varying systems.