RLC Energization and Effects of L-stable Solver

This model shows the increased precision and numerical stability of ARTEMiS art5 solver compared to trapezoidal or Tustin solvers.

This simple demo involves an RLC circuit with a breaker.

Explanation

The next figure compares the precision of the art5 solver of ARTEMiS and the Tustin solver. The test consists on a simple energization from null initial conditions. L = 20 mH, C = 1 uF: LC resonance frequency is 1.12 kHz and the simulation sample time of 100us correspond to a sampling frequency of 10 kHz. We can observe on the curves that the Tustin solver exhibits a phase shift while the art5 solver matches the result obtained with a variable step solver with error control.

The reason for this improved precision over Tustin solver (or trapezoidal) is that the art5 solver is an order 5 approximation to the matrix exponential while Tustin is an order 2.

The next figure compares the numerical stability of the art5 solver of ARTEMiS to the Tustin solver. In the test, the RLC circuit is opened from steady-state. We can observe that the Tustin solver now exhibits a very large oscillation compared to the art5 solver, even with a time step 4 times larger.

The reason for this good numerical behavior is that the art5 solver is L-stable [1].

Demonstration

When the model is first opened, run the model with art5 solver and a time step of 100 µs. Plot the capacitor voltage by double-clicking the block named Double-click to plot results.

Change the ARTEMiS GUIde solver to 'trapezoidal'. This will switch the simulation solver to the native SPS 'Tustin' solver. Again, plot the result by double-clicking on the block named 'Double-click to plot results'. The trace should appear in a different color and with a certain phase shift.

Now, keep the trapezoidal solver and reduce the time step to 10 us. The plot again the result. You observe that the Tustin-10us and art-100us match well while the Tustin-100µs is phase-shifted.

This phenomenon is caused by the fact that the trapezoidal or Tustin method approximates the exact matrix exponential with an order 2 approximation while the ARTEMiS art5 solver is ordered 5 (meaning that the art5 discretization formula approximates the Taylor series expansion up to the 5th order terms).

References

[1] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer-Verlag, 1993