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Inductance models

Inductance relates current change to induced voltage and stores energy in a magnetic field. It turns current into a dynamic state variable: an ideal inductor does not permit an arbitrary instantaneous current jump under a finite voltage model.

Think like a programmer

Represent inductance as a positive parameter and branch current as state. Derive its rate from the signed voltage convention, then integrate with an explicit step and solver choice. Calculate magnetic energy from the same current state; log voltage, current derivative, step size, and energy so a transient plot can be checked rather than merely watched.

Model checklist

Inputs
Positive inductance L, signed current, signed inductor voltage, resistance/source context, time step, integrator, and initial current.
State
Inductor current, current derivative, magnetic energy, elapsed time, and numerical diagnostics.
Rule
Use dI/dt = V/L under the declared sign convention, integrate current, then derive energy from the updated or documented time-level state.
Output
Current transient, induced voltage, magnetic energy, and energy/power consistency data.
Check
Constant current gives zero ideal induced voltage; larger L gives a smaller derivative at fixed V; energy ½LI² is nonnegative; reversing current reverses voltage derivative sign but not stored energy; step refinement changes a stated numerical tolerance.
\[V=L\frac{dI}{dt}\]

Under a constant ideal voltage, dI/dt = V/L: double L and the current slope halves. This is a local update rule, not a license to take an arbitrarily large numerical time step. For a lossless ideal inductor, power P = VI is the rate of change of magnetic energy,

\[U_L= rac12LI^2,qquad P=VI= rac{dU_L}{dt}\]

Current sign matters in the voltage law, while the squared energy is unchanged by reversing current. A useful regression fixture starts with constant current samples: finite differences should return zero derivative and ideal induced voltage should be zero. Another holds voltage and L fixed while comparing two time steps against a known linear current ramp. Record which numerical current time level is used for energy, because inconsistent timing can manufacture an apparent energy error.

Try this experiment

Prediction: A larger inductance slows current change under the same voltage, while energy remains nonnegative for either current direction.

Set V = 6 V, compare L = 2 H and 4 H, and predict each current slope. Write a constant-current/zero-voltage test, then calculate energy for L = 2 H, I = 3 A, and I = −3 A. State which integration metadata must accompany a transient result.

Where this model breaks

Real inductors have winding resistance, core loss, saturation, hysteresis, parasitic capacitance, temperature response, limited current ratings, and radiated fields. Mutual inductance couples circuits, so one scalar L may be insufficient. Fast transients can violate a lumped-element assumption and need distributed electromagnetic fields rather than a single current state.

Summary

Treat inductor current as integrated state, derive its rate from an explicit sign convention, and test the voltage–derivative and energy relationships. Record integration settings and material limits before interpreting a transient as physical.

Glossary

Sources

Model contract

Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.

\[\text{observable output} = f(\text{inputs},\,\text{state})\]
Inputs
Quantities you set or measure, with units and useful bounds.
State
Values the program must retain to reproduce the next result.
Rule
The relationship or update that turns inputs and state into a result.
Check
A known limit, unit check, invariant, or measured result that can expose a bad model.

Implement the idea as a model

For Inductance Models, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Magnetism and Induction, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.

Guided experiment

Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.

Where this model breaks

This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.

Summary

Treat Inductance Models as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.

Glossary

  • Input: a measured value or chosen parameter supplied to a model.
  • State: the smallest set of values needed to continue or reproduce a model.
  • Validation: comparing an output with a known result, limit, invariant, or measurement.

Self-check

  1. Which values are inputs, and which values must remain state?
  2. What observable result would tell you the model is behaving as expected?
  3. Which assumption would you test first before applying the model to a real system?

Model review: turn Inductance Models into a test

Integrate inductor current as state and verify its voltage–current-derivative relationship.

  1. Name the inputs and units that the magnetism and induction model needs.
  2. Separate the state you must keep from values you can calculate when needed.
  3. Write one rule that maps the current state and inputs to an observable result.
  4. Choose a limiting case, unit check, invariant, or known result before trusting an output.
  5. State one assumption you would change before using this simplified model for a real decision.
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