Transfer Function Calculator

Rational Transfer Function, Poles and Zeros

G(s) = N(s)/D(s) = (b₀·sᵐ + b₁·sᵐ⁻¹ + … + bₘ) / (a₀·sⁿ + a₁·sⁿ⁻¹ + … + aₙ) | D(pᵢ) = 0 → poles pᵢ | N(zⱼ) = 0 → zeros zⱼ | system order = n

Variables: G(s) = transfer function (output units per input unit), s = complex frequency σ + jω (rad/s), b₀…bₘ = numerator coefficients entered highest power first (units of the numerator quantity multiplied by sᵐ⁻ᵏ, i.e. by (rad/s)^(m−k) for the k-th entry), a₀…aₙ = denominator coefficients entered highest power first (units of the denominator quantity multiplied by (rad/s)^(n−k)), m = numerator degree (dimensionless count), n = denominator degree after any exact leading zeros are removed (dimensionless count), pᵢ = i-th pole (rad/s), zⱼ = j-th zero (rad/s). Roots come from an Aberth–Ehrlich iteration (closed form for n ≤ 2) refined against the original polynomial; repeated roots are recognised and listed once with their multiplicity, e.g. −1 (×3); poles, zeros and the DC gain are shown to 6 significant figures. Exact leading zeros are ignored, so '0 1 3 2' is analysed as s² + 3s + 2. Source: Ogata, Modern Control Engineering, 5th ed., Ch. 2 (Mathematical Modeling of Control Systems — transfer function definition)

DC Gain (Zero-Frequency Gain)

G(0) = bₘ / aₙ | if the system is stable, a unit step gives y(∞) = lim(s→0) s·G(s)·(1/s) = G(0)

Variables: G(0) = DC (zero-frequency) gain (output units per input unit), bₘ = constant term (last entry) of the numerator, units = numerator quantity units, aₙ = constant term (last entry) of the denominator, units = denominator quantity units, y(∞) = steady-state step response (output units). The ratio is formed from the two constant terms directly, with no pole-zero cancellation: G = s/(s² + 3s) is reported as ∞ rather than 1/3, and a genuine integrator (aₙ = 0) is likewise reported as ∞. The final-value step result holds only when every pole satisfies Re(pᵢ) < 0; for an unstable system G(0) is still displayed but y(∞) does not exist. Source: Ogata, Modern Control Engineering, 5th ed. (Laplace transform — final value theorem)

Stability Classification from Pole Location

Asymptotically stable ⇔ Re(pᵢ) < 0 for every pole | unstable ⇔ Re(pᵢ) > 0 for at least one pole, or repeated poles on the jω axis | marginally stable ⇔ no pole with Re(pᵢ) > 0 and one or more simple poles with Re(pᵢ) = 0. | This tool treats a pole as on the jω axis when |Re(pᵢ)| ≤ 1×10⁻⁹·|pᵢ|

Variables: pᵢ = i-th system pole (rad/s), Re(pᵢ) = real part of the pole (rad/s; for a complex pair Re(pᵢ) = −ζ·ωn, where ζ = damping ratio (dimensionless) and ωn = undamped natural frequency (rad/s)), Im(pᵢ) = imaginary part (rad/s), count of poles with Re(pᵢ) > 0 = number of right-half-plane poles (dimensionless). Simple poles on the jω axis (s² + 1, a single free integrator s) are reported 'Marginally stable'; repeated ones (s², (s² + 1)²) are reported 'Unstable', because their response grows without bound. Such poles are marked '(Im axis)' and drawn in yellow on the s-plane plot. Source: Nise, Control Systems Engineering, 7th ed., Ch. 6 (Stability)

System Type (Number of Free Integrators)

Type N = number of poles at s = 0 | G(s) = K · Π(s − zⱼ) / [ sᴺ · Π(s − pᵢ) ]

Variables: N = system type = number of poles exactly at s = 0, i.e. trailing zero coefficients of the denominator (dimensionless count), s = complex frequency (rad/s), K = leading gain constant of the factored form, equal to b₀/a₀ (output units per input unit, times (rad/s)^(n−m)), zⱼ = j-th finite zero (rad/s), pᵢ = i-th nonzero pole (rad/s). Factors are written (s − zⱼ) and (s − pᵢ) so that zⱼ and pᵢ are the root locations themselves, matching the poles and zeros listed above. Type governs steady-state error only for the open-loop function of a unity-feedback loop (Type 0 → finite position error, Type 1 → zero position error and finite velocity error, and so on); this tool counts origin poles of whatever G(s) is entered. Source: Nise, Control Systems Engineering, 7th ed., Ch. 7 (Steady-State Errors — system type and static error constants)