Calculator Inputs
Formula Used
The calculator uses a two state nonlinear model:
dx/dt = f(x, y, t) and dy/dt = g(x, y, t)
Trajectory integration uses the fourth order Runge Kutta method:
s(n+1) = s(n) + h(k1 + 2k2 + 2k3 + k4) / 6
Equilibrium points satisfy f(x, y) = 0 and g(x, y) = 0. Local stability is estimated from the Jacobian matrix trace and determinant.
How to Use This Calculator
- Select the nonlinear electrical model that best matches your circuit.
- Enter the initial states for voltage, current, phase, or normalized variables.
- Set the simulation time and time step. Smaller steps improve accuracy.
- Adjust the plot limits to focus on the expected operating region.
- Enter model parameters such as damping, forcing, resistance, inductance, and nonlinearity.
- Press the calculate button to view the phase portrait and stability results.
- Use the CSV and PDF buttons to export values for reports.
Example Data Table
| Example | Model | x0 | y0 | Main parameters | Expected behavior |
|---|---|---|---|---|---|
| Nonlinear tank | Nonlinear RLC | 1.2 | 0.15 | R=0.35, L=1, C=1, k=0.25 | Damped motion toward an operating point |
| Self oscillator | Van der Pol | 0.5 | 0.1 | μ=1.2 | Limit cycle style motion |
| Driven resonator | Duffing | 1 | 0 | α=1, β=0.25, δ=0.2 | Nonlinear resonance path |
| Negative resistance | Tunnel diode | -1 | 1 | a=1, b=0.6, c=0.2, d=0.7 | Multiple stability regions |
Nonlinear System Phase Portrait Guide
Why phase portraits matter
Nonlinear electrical systems rarely move in straight paths. A diode, magnetic core, oscillator, or switching element can change behavior with state. A phase portrait turns those state equations into a visual map. The horizontal axis often represents voltage. The vertical axis can represent current, speed, or a normalized derivative. Arrows show the local direction of motion. A traced path shows how one starting condition moves through the field.
Electrical use cases
This calculator helps when a circuit cannot be judged by one transfer curve. It is useful for nonlinear RLC tanks, negative resistance oscillators, Duffing style resonators, and control loops. It can show whether a starting point settles, grows, or circles around a limit region. The plot can also reveal separatrix behavior. That boundary separates safe and unstable regions. Engineers use this view before simulation refinement.
How the calculator works
The form accepts initial states, time range, integration step, plot bounds, and model parameters. The solver applies a fourth order Runge Kutta method. This gives stable numerical paths for many smooth systems. A grid is also sampled. Each grid point becomes a small direction segment. Equilibrium points are found with repeated Newton searches. The local Jacobian is then used to estimate trace, determinant, eigenvalues, and stability class.
Reading the results
A sink suggests damping or convergence. A source suggests growth away from the operating point. A saddle means the operating point is direction dependent. A center or spiral points toward oscillation. The numerical class is local, so it should be combined with the full portrait. Large step sizes can hide fast motion. Very wide plot limits can make important details look small. Start with modest bounds, then expand after the first run.
Practical workflow
Begin with a known circuit model. Enter realistic component values. Run one portrait around the expected operating point. Then change one parameter at a time. Compare the exported data in a report. Use the graph as a guide, not as the final sign off. For safety critical designs, confirm results with a dedicated circuit simulator and laboratory testing. Document assumptions clearly, especially scaling choices, state units, simplified nonlinear terms, and reviewer notes.
FAQs
1. What is a nonlinear phase portrait?
It is a state space plot that shows how a nonlinear system moves over time. Each arrow shows local direction. A trajectory shows the path from one starting point.
2. Why is this useful in electrical analysis?
Many electrical circuits include nonlinear parts. Diodes, magnetic cores, oscillators, and switching circuits can show behavior that simple linear plots miss.
3. What do x and y represent?
They are two state variables. In circuit work, x may represent voltage. y may represent current, phase speed, or a normalized derivative.
4. What is an equilibrium point?
It is a point where both derivatives are zero. The system can stay there unless disturbed. Its local type helps describe stability.
5. What does a saddle mean?
A saddle is stable in one direction and unstable in another. Small changes in starting point can produce very different paths.
6. Why use a smaller time step?
A smaller time step usually improves numerical accuracy. It is important when the circuit changes quickly or has strong nonlinear terms.
7. Can this replace a circuit simulator?
No. It is a design and learning aid. Use it to explore behavior, then confirm important designs with detailed simulation and testing.
8. What exports are included?
The CSV file includes trajectory values. The PDF includes the chart, summary data, model name, and stability information for documentation.