Transfer Function Calculator
Solve H = Y / X for the gain, output, or input amplitude — then build H(jω) for first- and second-order systems and see the Bode magnitude and phase plots, block diagram, converters and references.
Gain depends on frequency
For a real system |H(jω)| changes with frequency, so the same input amplitude gives different output amplitudes at different frequencies.
Phase is the other half
H(jω) is complex: its magnitude is the gain and its angle is the phase shift between output and input.
Poles set the dynamics
The roots of the denominator (poles) determine speed and stability — a pole at s = −1/τ gives a first-order response with time constant τ.
Cascaded blocks multiply
Systems in series have H = H₁·H₂, which is why gains in decibels simply add.
System Builder & Bode Plots
Pick a standard system, set its parameters and a frequency, and get |H(jω)|, its gain in dB, and its phase — plotted on a Bode diagram.
Frequency response H(jω)
Bode magnitude (dB) and phase (°)
Unit Conversions
Gain ↔ decibels
Frequency
Common Gain Values
| Linear gain | Decibels | Meaning |
|---|---|---|
| 0.01 | −40 dB | Heavy attenuation |
| 0.1 | −20 dB | One-tenth amplitude |
| 0.5 | −6.02 dB | Half amplitude |
| 0.707 | −3.01 dB | Corner / cut-off point |
| 1 | 0 dB | Unity gain |
| 2 | +6.02 dB | Doubled amplitude |
| 10 | +20 dB | Ten times amplitude |
| 100 | +40 dB | One hundred times |
Standard Forms
| System | H(s) | |H(jω)| |
|---|---|---|
| First-order low-pass | K / (1 + s/ω₀) | K / √(1 + (ω/ω₀)²) |
| First-order high-pass | K(s/ω₀) / (1 + s/ω₀) | K(ω/ω₀) / √(1 + (ω/ω₀)²) |
| Second-order low-pass | Kωₙ² / (s² + 2ζωₙs + ωₙ²) | K / √((1−r²)² + (2ζr)²), r = ω/ωₙ |
How to Use This Calculator
The core relation links the input amplitude, the gain, and the output amplitude at a single frequency; the builder supplies the gain for a real system.
Choose what to solve for
Pick Y, X, or |H| — the formula rearranges automatically.
Enter amplitudes or a gain preset
Presets cover common gains such as −3 dB, unity and +20 dB.
Build a real system
Choose a first- or second-order system, set K, ω₀ (or ωₙ, ζ) and ω to get |H(jω)| and phase.
Read the Bode plots
Magnitude in dB and phase in degrees are plotted against frequency on a log axis, with your ω marked.
Send the gain back
Use the button to load the computed |H| into the calculator and find the resulting output for a given input.
Go Advanced
Adds sensitivity, dB values, and time-domain metrics such as overshoot and settling time.
What This Tool Offers
Solve any variable
Compute output, input, or gain.
Block diagram
X(s) → H(s) → Y(s) at a glance.
Bode plots
Magnitude (dB) and phase (°) vs. frequency, with your ω marked.
System builder
First-order low/high-pass and second-order low-pass responses.
Converters & tables
Gain ↔ dB, rad/s ↔ Hz, and a gain reference table.
Sensitivity check
See how a 1% change in an input moves the result.
What Does H(s) = Y(s)/X(s) Measure?
A transfer function describes how a linear, time-invariant system turns an input into an output. In the Laplace domain it is simply the ratio of the output transform to the input transform, assuming zero initial conditions.
Y(s) = H(s) · X(s)
X(s) = Y(s) / H(s)
Sinusoidal steady state (s = jω):
|H(jω)| = |Y| / |X|, gain in dB = 20·log₁₀|H|
Frequently Asked Questions
H(s) is the general transfer function in the complex frequency variable s. Setting s = jω gives the frequency response H(jω), which describes how the system treats steady sinusoids of angular frequency ω.
Because gains of cascaded systems multiply, and the logarithm turns multiplication into addition. A 20 dB stage followed by a −6 dB stage simply gives +14 dB overall, and very large or small ratios become manageable numbers.
The frequency at which the gain falls to about 0.707 of its passband value (half the power). For a first-order system it is exactly the corner frequency ω₀ = 1/τ.
It controls how oscillatory a second-order system is. Low ζ gives a pronounced resonance peak and overshoot, ζ = 1 is critically damped, and larger values are overdamped and sluggish.
No. Transfer functions apply to linear, time-invariant systems. Nonlinear systems may be linearised around an operating point, but then the transfer function is only valid for small deviations.
References
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| H(s) | Transfer function | dimensionless |
| Y(s) | Laplace transform of output | variable·s |
| X(s) | Laplace transform of input | variable·s |
| s | Laplace variable | 1/s |
What it means
The transfer function H(s) is the ratio of the Laplace transform of the output signal Y(s) to the input signal X(s), assuming zero initial conditions: H(s) = Y(s) / X(s). It completely characterises the linear time‑invariant (LTI) system in the frequency domain. The transfer function contains all the information about the system’s gain and phase response. It is used in control theory, filter design, and circuit analysis. The poles and zeros of H(s) determine stability and frequency response. Example: For a simple RC low‑pass filter, the transfer function is H(s) = 1/(1 + sRC). The frequency response is obtained by evaluating H(jω). This allows the design of filters with desired cut‑off frequencies.
Worked example
Transfer Function – Practical Example
Real‑World| Parameter | Value |
|---|---|
| R | 1 kΩ |
| C | 1 µF |
| Formula | H(s) = 1/(sRC+1) |
Common mistakes
Watch unit consistency and the assumptions behind the formula; misapplying it outside its valid conditions is the most frequent error.Applications
Transfer function H(s) = Y(s)/X(s) characterises the input‑output relationship of a linear system in the s‑domain. It is used to analyse system behaviour, design filters, and evaluate stability. Engineers use it to model circuits, control systems, and signal processing chains. The transfer function is a central concept in systems engineering.
- System modelling and analysis
- Filter design and frequency response evaluation
- Control system design and stability analysis
- Signal processing and system identification
- Educational understanding of system representation
Related calculators · Control Systems
Recently updated formulas
Suggest an improvement
Found a mistake? Have a better example? Let us know.
You may send up to 3 suggestions per 24 hours.