Operating Point for Van der Pol Oscillator

Open Model

This example shows advantage of initialization with a Simscape operating point (OP) for a Van der Pol oscillator. OP initialization enables pushing the Van der Pol oscillator directly into its periodic steady state.

Model

A Van der Pol oscillator is a self-maintained electrical circuit made up of an inductor, a capacitor, and a nonlinear resistor. Van der Pol oscillator evolves in time according to the second-order differential equation:

$\ddot{x}-\mu(1-x^2)\dot{x} + x = 0$

where $x$ is the normalized current, and $\mu$ is a scalar parameter indicating the strength of nonlinear damping.

When $\mu$ >0, the system dynamic result in a periodic motion called limit cycle. Near the origin $x=0$ , the system is unstable, and far from the origin, the system is damped.

Limit Cycle Behavior Without An Operating Point

The simulation runs long enough for the oscillator to reach a periodic steady state. The left figure illustrates the dynamic behavior of current which converges to the steady state at some point. Another way to visualize the state dynamics is to plot a phase portrait of $\dot{I}$ vs $I$ (see plot on the right). Observe that Van der Pol oscillator converges to a limit cycle.

Create Simscape Operating Point

Extract a steady-state Simscape operating point for the model by using simscape.op.create and the Simscape log result from the previous simulation. Use '30' as the time because the simulation had visually reached the steady state by that time.

Limit Cycle Behavior With An Operating Point

Validate the operating point by initializing the Van der Pol oscillator model with the operating point. The resulting behavior is the steady-state limit cycle without the transient spiral.