Continuous

Derivative Block

Computes the numerical time derivative (rate of change) of the input signal

Description

The Derivative block computes the numerical time derivative (rate of change) of an input signal using backward difference approximation. An optional low-pass filter parameter can be used to reduce noise amplification.

Mathematical Model

Pure derivative (filterCoefficient = 0, default)

dy/dt ≈ (u_n - u_{n-1}) / Δt

Filtered derivative (filterCoefficient = N > 0)

Transfer function: H(s) = N*s / (s + N)
Discrete: D[k] = (N * (u[k] - u[k-1]) + D[k-1]) / (1 + N * Δt)

where:

u_n is the current input
u_{n-1} is the previous input
Δt is the sample time
N is the filter coefficient (higher = less filtering)

Parameters

filterCoefficient

Low-pass filter coefficient for the derivative. Set to 0 (default) for pure differentiation.

N = 0: Pure backward difference (no filtering)
N = 10–100: Moderate to light filtering
Higher N: Less filtering, faster response but more noise
Default: 0

First-Step Behavior

On the very first simulation step, the output is always 0. This prevents a potentially infinite or very large spike that would occur from computing the derivative of an undefined-to-defined transition. On the second step and onward, the normal derivative computation is used.

Noise Sensitivity

Pure numerical differentiation amplifies high-frequency noise. If the input signal is noisy:

Use the filterCoefficient parameter (e.g., N=20) to add low-pass filtering
Alternatively, cascade with a separate low-pass filter block
Or use the PIDController block which has a built-in derivative filter

Examples

Ramp Derivative (1 second simulation)

Linear ramp with slope 2 V/s goes into Derivative block. The output is a constant 2 V, showing that the derivative of a linear signal is its slope.

PID Controller D-Term

Connect the error signal to Derivative, scale with Gain, sum with P and I terms.

Remarks

Pure Differentiation: Backward difference formula, optionally filtered
Numerically Noisy: High-frequency content is amplified without filtering; use filterCoefficient to mitigate
Essential for Control: Forms the D-term in proportional-integral-derivative controllers
Inverse of Integrator: Complementary to the Integrator block for system analysis
First Step: Always outputs 0 on the first step to avoid initialization spikes