# Documentation

## How Simscape Models Represent Physical Systems

### Representations of Physical Systems

This section describes important characteristics of the mathematical representations of physical systems, and how Simscape™ software implements such representations. You might find this overview helpful if you:

• Require details of such representations to improve your model fidelity or simulation performance.

• Are constructing your Simscape model or its components with the Simscape language.

• Need to troubleshoot Simscape modeling or simulation failures.

Mathematical representations are the foundation for physical simulation. For more information about simulation, see How Simscape Simulation Works.

### Differential, Differential-Algebraic, and Algebraic Systems

The mathematical representation of a physical system contains ordinary differential equations (ODEs), algebraic equations, or both.

• ODEs govern the rates of change of system variables and contain some or all of the time derivatives of the system variables.

• Algebraic equations specify functional constraints among system variables, but contain no time derivatives of system variables.

• Without algebraic constraints, the system is differential (ODEs).

• Without ODEs, the system is algebraic.

• With ODEs and algebraic constraints, the system is mixed differential-algebraic (DAEs).

A system variable is differential or algebraic, depending on whether or not its time derivative appears in the system equations.

### Stiffness

A mathematical problem is stiff if the solution you are seeking varies slowly, but there are other solutions within the error tolerances that vary rapidly. A stiff system has several intrinsic time scales of very different magnitude [1].

A stiff physical system has one or more components that behave "stiffly" in the ordinary sense, such as a spring with a large spring constant. Mathematical equivalents include quasi-incompressible fluids and low electrical inductance. Such systems often exhibit high frequency oscillations in some of their components or modes.

### Events and Zero Crossings

Events are discontinuous changes in system state or dynamics as the system evolves in time; for example, a valve opening, or a hard stop.

A zero crossing is a specific event type, represented by the value of a mathematical function changing sign.

### Working with Simscape Representation

A Simscape model is equivalent to a set of equations representing one or more physical systems as physical networks.

#### Creating and Detecting Zero Crossings in Simscape Models

Simulink® and Simscape software have specific methods for detecting and locating zero-crossing events. For general information, see Zero-Crossing Detection in the Simulink documentation.

Your model can contain zero-crossing conditions arising from several sources:

• Simscape and normal Simulink blocks copied from their respective block libraries

• Expressions programmed in the Simscape language

You can disable zero-crossing detection on individual blocks, or globally across the entire model. Zero-crossing detection often improves simulation accuracy, but can slow simulation speed.

 Tip   If the exact times of zero crossings are important in your model, then keep zero-crossing detection enabled. Disabling it can lead to major simulation inaccuracies.

Enabling and Disabling Zero-Crossing Conditions in Simscape Language.  In the Simscape language, you can create or avoid Simulink zero-crossing conditions in your model by switching between different implementations of discontinuous conditional expressions. You can:

• Use relational operators, which create zero-crossing conditions. For example, programming the operator relation: ```a < b``` creates a zero-crossing condition.

• Use relational functions, which do not create zero-crossing conditions. For example, programming the functional relation: `lt(a,b)` does not create a zero-crossing condition.