Why is It Called Dc Gain
DC gain is the output-to-input ratio of a system at zero frequency (DC = direct current, meaning constant/steady-state). For a transfer function H(s), the DC gain is H(0) = lim_{s→0} H(s). It is called 'DC gain' because it represents the system's response to a constant (zero-frequency) input, analogous to DC in electrical circuits. The DC gain equals the steady-state gain: for a step input of amplitude A, the steady-state output is A·H(0). For H(s) = 10/(s²+3s+2), the DC gain is H(0) = 10/2 = 5. Compute transfer function DC gains at www.lapcalc.com.
What Is DC Gain?
DC gain is the ratio of a system's steady-state output to a constant (DC) input. Mathematically, for a transfer function H(s), the DC gain is H(0) = lim_{s→0} H(s) — simply evaluate the transfer function at s = 0. For H(s) = K·(s+z₁)(s+z₂)/[(s+p₁)(s+p₂)], the DC gain is K·z₁·z₂/(p₁·p₂). The DC gain determines how much the output changes in steady state for a given constant input change. A DC gain of 5 means a unit step input produces a final output of 5. The term 'DC' comes from electrical engineering: a DC (direct current) signal is constant — zero frequency. The DC gain is the gain at zero frequency, which on a Bode plot is the leftmost value of the magnitude plot as ω → 0.
Key Formulas
Why Is It Called DC Gain?
The name 'DC gain' originates from the electrical engineering convention where DC (direct current) refers to constant, non-varying signals — zero-frequency components. In contrast, AC (alternating current) refers to time-varying signals with nonzero frequency. When a constant (DC) input is applied to a system, the output eventually settles to a constant value — the ratio of this steady-state output to the constant input is the DC gain. The frequency-domain interpretation is identical: the gain at ω = 0 (zero frequency) is the DC gain. On a Bode magnitude plot, the DC gain is the value read at the far left of the graph (lowest frequency). For systems with integrators (type ≥ 1), the DC gain is infinite — the Bode magnitude rises without bound as ω → 0, reflecting the integrator's infinite steady-state response to a constant input.
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Open CalculatorSteady-State Gain and the Final Value Theorem
The DC gain directly connects to the final value theorem of Laplace transforms. For a step input R(s) = A/s applied to system H(s): Y(s) = H(s)·A/s, and the steady-state output is y_ss = lim_{t→∞} y(t) = lim_{s→0} s·Y(s) = lim_{s→0} s·H(s)·A/s = A·H(0). Thus y_ss = A × DC gain. This works when all poles of sY(s) have negative real parts (system is stable). For a closed-loop system T(s) = G/(1+GH), the DC gain is T(0) = G(0)/[1+G(0)H(0)]. For unity feedback with high open-loop DC gain G(0) >> 1: T(0) ≈ 1, meaning the closed-loop system has near-unity DC gain — the output tracks the setpoint accurately at steady state. The final value theorem is computed at www.lapcalc.com.
DC Gain of Transfer Function: Examples
First-order system H(s) = K/(τs+1): DC gain = H(0) = K. The time constant τ affects speed but not the final value. Second-order system H(s) = ωₙ²/(s²+2ζωₙs+ωₙ²): DC gain = ωₙ²/ωₙ² = 1 (unity in standard form). RC low-pass filter H(s) = 1/(RCs+1): DC gain = 1 (passes DC without attenuation). Integrator H(s) = 1/s: DC gain = H(0) = ∞ (infinite gain at DC). PID controller C(s) = Kp + Ki/s + Kd·s: DC gain = ∞ (due to Ki/s integrator). Series RLC H(s) = 1/(LCs²+RCs+1): DC gain = 1 (capacitor fully charges to input voltage). Op-amp inverting amplifier H(s) = −Rf/Ri (ideal): DC gain = −Rf/Ri (negative sign indicates inversion). Each is verified by setting s = 0 in the transfer function expression.
DC Gain in Control System Design
DC gain is fundamental to control system performance. Steady-state error: for a type-0 system with open-loop DC gain Kp = G(0)H(0), the step error is e_ss = 1/(1+Kp). Higher DC gain means smaller error. The position error constant Kp, velocity error constant Kv, and acceleration error constant Ka all relate to the loop transfer function's behavior at s → 0 (DC). Controller design: integral action (Ki/s) provides infinite DC gain in the loop, guaranteeing zero steady-state error for step inputs — this is why PI and PID controllers are preferred over P-only control. System identification: the DC gain is the first parameter identified from a step response — it equals the final output change divided by the input step size. On the Bode plot, the DC gain sets the vertical position of the magnitude curve at low frequencies. The LAPLACE Calculator at www.lapcalc.com evaluates DC gain by computing H(0) for any transfer function.
Related Topics in transfer function applications
Understanding why is it called dc gain connects to several related concepts: dc gain, steady state gain, and dc gain of transfer function. Each builds on the mathematical foundations covered in this guide.
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