H(s)

TransferFunction

Continuous

The Transfer Function block implements linear time-invariant (LTI) systems described in the Laplace domain. It takes a transfer function H(s) = num(s)/den(s) and simulates the system's response to input signals using numerical integration methods.

The block accepts numerator and denominator polynomials as text strings (e.g., "1" and "s+1" for a first-order lag) and internally converts them to a state-space representation for efficient computation.

Mathematical Model

H(s) = \frac{\text{num}(s)}{\text{den}(s)}

Where:

num(s) = numerator polynomial coefficients
den(s) = denominator polynomial coefficients
s = Laplace variable

The block uses state-space realization (controllable canonical form) for numerical simulation.

Inputs & Outputs

Direction	ID	Label	Type	Status
→ In	`in`	Input	number	Required
→ In	`reset`	Reset	number	Optional
← Out	`out`	Output	number	Output

Parameters

Parameters	Label	Type	Default
`numerator`	Numerator	polynomial	`1`
`denominator`	Denominator	polynomial	`s+1`
`initialCondition`	Initial Condition	number	`0`

Usage Examples

Low-pass Filter

First-order system with 1 sec time constant:

Step (finalValue: 1) → TransferFunction (num: "1", den: "s+1") → Scope

Damped Oscillator

Second-order system with damping:

Impulse → TransferFunction (num: "100", den: "s^2+10s+100") → Scope

Remarks & Best Practices

State-Space Realization: Internally converted to controllable canonical form for numerically stable integration
Polynomial Syntax: Use s for Laplace variable, s^2 for powers. Supports negative coefficients (e.g., s^2 - 2s + 1)
Order Restriction: The numerator degree must be less than or equal to the denominator degree (proper transfer function)
Time-Step Sensitivity: Higher order systems require smaller simulation time steps for accuracy and stability

Related Components

Integrator: Simple integration (1/s)
StateSpace: Direct state-space system implementation
PIDController: Proportional-Integral-Derivative controller
ZeroPole: Pole-zero form of transfer functions