Calculator · Dynamics & Control
Transfer Function Calculator
Enter the numerator and denominator of G(s) = N(s)/D(s) in descending powers of s. The calculator reports order, properness, DC gain, poles, zeros and the stability classification.
Input
G(s) = (1) / (s^2 + 2s + 1)
Results
| Numerator order deg(N) | 0 |
|---|---|
| System order deg(D) | 2 |
| Relative degree | 2 |
| Classification | Strictly proper — deg(N) < deg(D) |
| DC gain G(0) | 1 |
| Poles | -1, -1 |
| Zeros | None |
| Stability | Asymptotically stable — every pole is in the open left half-plane |
| Time simulation | Supported |
Technical reference
G(s) = N(s) / D(s)
deg(N) < deg(D) ⇒ strictly proper · deg(N) = deg(D) ⇒ biproper · deg(N) > deg(D) ⇒ improper
- N(s)
- numerator polynomial, descending powers of s
- D(s)
- denominator polynomial, descending powers of s
- deg(D)
- system order
- G(0)
- DC gain, N(0)/D(0)
Strictly proper and biproper systems are both proper; only improper (deg(N) > deg(D)) is flagged as not realizable as written — the engine reports it rather than truncating it.
G(0) is undefined, not zero, when D(0) = 0 (e.g. a free integrator in the loop); the result then reads “Undefined” instead of a number.
G(0) = N(0)/D(0) is evaluated directly on the coefficients as entered; common numerator and denominator factors are not symbolically cancelled first. An unreduced representation such as s/s therefore keeps its D(0) = 0 singularity, even though the cancelled form would be 1.
Assumptions
- · Continuous-time, single-input single-output, real coefficients.
- · Coefficients are read in descending powers of s, exactly as displayed.
- · Poles and zeros come from the shared numerical root solver; conditioning warnings appear in the diagnostics rather than being hidden.
- · An improper transfer function is reported as improper, never silently truncated.
Continue
The whole transfer function travels with you: time, frequency and stability workspaces.