Calculator Inputs
Formula Used
The calculator models a continuous-time transfer function in Laplace form.
General form: H(s) = K × N(s) / D(s)
Numerator polynomial: N(s) = bmsm + bm-1sm-1 + ... + b0
Denominator polynomial: D(s) = ansn + an-1sn-1 + ... + a0
Zeros: Solve N(s) = 0.
Poles: Solve D(s) = 0.
DC gain: H(0) = K × b0 / a0, when a0 ≠ 0.
Frequency response: Substitute s = jω, then compute magnitude 20 log10|H(jω)| and phase ∠H(jω).
How to Use This Calculator
- Enter numerator coefficients in descending powers of s.
- Enter denominator coefficients in descending powers of s.
- Set the overall gain K for the system.
- Choose minimum and maximum angular frequency values.
- Set plot points for the response graph resolution.
- Click the calculate button to generate the transfer function.
- Review poles, zeros, stability, DC gain, and bandwidth estimates.
- Download CSV or PDF output for reports or documentation.
Example Data Table
| Field | Example Value | Meaning | Expected Output |
|---|---|---|---|
| Numerator coefficients | 2, 5 | Represents 2s + 5 | Zero near -2.5 |
| Denominator coefficients | 1, 4, 5 | Represents s² + 4s + 5 | Poles near -2 ± 1i |
| System gain K | 1 | Scales the numerator response | DC gain equals 1 |
| Frequency range | 0.1 to 100 rad/s | Logarithmic analysis interval | Bode-style plot updates |
| Plot points | 200 | Smooth response sampling | Dense graph trace |
Frequently Asked Questions
1. What does this calculator return?
It returns the transfer function, poles, zeros, DC gain, stability class, approximate bandwidth, and a frequency response plot using your coefficient lists and gain.
2. How should I enter coefficients?
Enter coefficients from the highest power of s to the constant term. Separate values with commas, spaces, or semicolons.
3. What is the meaning of poles and zeros?
Zeros make the numerator equal zero. Poles make the denominator equal zero. Their locations shape stability, transient behavior, and frequency response.
4. Why is the DC gain sometimes undefined?
If the denominator constant term is zero, the system has a pole at the origin. Then H(0) is not finite, so the calculator marks DC gain as undefined.
5. How is stability determined?
The calculator checks pole real parts. All negative real parts indicate stability. Any positive real part indicates instability. Zero real parts give marginal stability.
6. What does the bandwidth estimate mean?
It is a simple approximate cutoff where magnitude first falls about 3 dB below the peak value within the scanned frequency range.
7. Are complex poles supported?
Yes. The solver returns complex roots numerically, then reports real part, imaginary part, magnitude, damping ratio, and time constant when applicable.
8. When should I use more plot points?
Use more points when poles or zeros are closely spaced, when the frequency range is very wide, or when you want a smoother graph.