Inductance models

An electric current produces a magnetic field. When that current changes, its magnetic flux changes and induction produces a voltage that opposes the change. Inductance measures the flux linkage produced per unit current in a linear system. In a circuit model, it makes current a stored state that cannot jump under a finite voltage.

Inductance depends on geometry, number of turns, magnetic material, and coupling to other circuits. A single constant is an excellent approximation for many coils, but real inductors also have winding resistance, parasitic capacitance, core loss, and saturation.

Flux linkage and self-inductance

If a coil has \(N\) and each turn links the same magnetic flux \(\Phi_B\), the flux linkage is

\[\lambda=N\Phi_B\]

For a linear inductor, flux linkage is proportional to current:

\[\lambda=LI\]

Therefore

\[L=\frac{N\Phi_B}{I}\]

The SI unit is the henry:

\[1\ \mathrm H=1\ \mathrm{Wb/A}=1\ \mathrm{V\,s/A}\]

The proportional relation assumes geometry and magnetic permeability do not change with current. When a core saturates or shows hysteresis, flux linkage is not simply a constant times current.

Inductor voltage law and sign

Faraday's law says the induced emf associated with a coil is

\[\mathcal E_{\mathrm{ind}}=-\frac{d\lambda}{dt}\]

For constant inductance,

\[\mathcal E_{\mathrm{ind}}=-L\frac{dI}{dt}\]

This negative sign expresses Lenz's law: the induced emf opposes the current change. In circuit analysis, voltage is often defined with the passive sign convention, where current enters the positive-voltage terminal. The terminal voltage across the ideal inductor is then

\[V_L=L\frac{dI}{dt}\]

The two equations do not conflict. One describes the induced emf around the coil; the other describes the external terminal voltage needed to drive the chosen positive current change. Writing the polarity next to the equation prevents an apparent sign disagreement.

Current continuity

Integrate the terminal law over a short interval:

\[\Delta I=\frac{1}{L}\int_{t_1}^{t_2}V_L(t)\,dt\]

A finite voltage applied for a vanishing time produces a vanishing current change. Therefore ideal inductor current is continuous unless the model includes a voltage impulse:

\[I_L(0^+)=I_L(0^-)\]

An ideal switch may change circuit topology instantly, but it cannot erase the magnetic energy implied by the current just before switching. Real circuits respond with high voltage, arcing, parasitic capacitance, or protective clamp conduction.

Inductance of a long solenoid

For an ideal long solenoid of length \(\ell\), cross-sectional area \(A\), and \(N\), the internal field is approximately

\[B=\mu\frac{N}{\ell}I\]

The flux through each turn is approximately \(\Phi_B=BA\). Substitution into the definition of inductance gives

\[L=\mu\frac{N^2A}{\ell}\]

Inductance grows with the square of turn count because more turns both create more field for the same current and link more flux. Increasing core permeability or area raises inductance; increasing length lowers it in this ideal geometry.

The expression neglects fringing at the ends, leakage flux, gaps, winding thickness, and nonlinear magnetic response. It is a geometry model, not a universal formula for every coil.

Series RL response

Connect a constant voltage source, resistor, and inductor in series. Kirchhoff's voltage law gives

\[L\frac{dI}{dt}+RI=V_s\]

For arbitrary initial current, the solution is

\[I(t)=\frac{V_s}{R} +\left[I(0)-\frac{V_s}{R}\right]e^{-tR/L}\]

The time constant is

\[\tau=\frac{L}{R}\]

At the switching instant, current retains its initial value. At long times under a constant DC source, an ideal inductor has zero voltage and acts as a short circuit, so current approaches \(V_s/R\). After one time constant, the difference from the final current is reduced to \(e^{-1}\) of its initial value.

RL decay

If the source is removed while the inductor current has a continuous path through a resistor,

\[I(t)=I(0)e^{-tR/L}\]

The resistor converts the inductor's stored magnetic energy into internal energy. If the path is broken without a protective route, the inductor produces whatever voltage the real system permits in its attempt to keep current continuous. This is why flyback diodes, snubbers, and other clamps are used with coils.

Worked slope and time constant

Apply a constant \(6\ \mathrm V\) to an ideal \(2\ \mathrm H\). The current slope is

\[\frac{dI}{dt}=\frac{V_L}{L} =\frac{6\ \mathrm V}{2\ \mathrm H} =3\ \mathrm{A/s}\]

Doubling the inductance to four henries halves the slope to \(1.5\ \mathrm{A/s}\). If the two-henry inductor is instead in series with \(4\ \Omega\), the time constant is

\[\tau=\frac{2\ \mathrm H}{4\ \Omega}=0.50\ \mathrm s\]

The constant-voltage linear ramp applies only when the inductor terminal voltage really stays constant. In the series RL circuit, the growing resistor voltage reduces inductor voltage, producing the exponential rather than an unlimited ramp.

Mutual inductance

A changing current in one coil can link magnetic flux through another. In a linear two-coil model,

\[\lambda_2=MI_1,\qquad |\mathcal E_2|=M\left|\frac{dI_1}{dt}\right|\]

The sign depends on winding orientation and terminal conventions, often recorded with dot notation. Reciprocity gives the same mutual inductance in either direction for ordinary passive reciprocal media:

\[M_{12}=M_{21}=M\]

The coupling coefficient is defined by

\[M=k\sqrt{L_1L_2},\qquad 0\le k\le1\]

Perfect coupling is an idealization. Leakage flux makes \(k<1\) in real transformers.

Nonlinear inductance

For a nonlinear magnetic core, it is safer to store flux linkage as a function of current. The terminal voltage is

\[V=\frac{d\lambda(I)}{dt} =\frac{d\lambda}{dI}\frac{dI}{dt}\]

The incremental inductance \(d\lambda/dI\) can vary with current and history. Hysteresis means the state may require more than instantaneous current. Treating a saturating core as one constant inductance can greatly underpredict current peaks.

Time update in Python

For a constant ideal inductance under the passive sign convention,

inductance, voltage, current, dt = 0.1, 2.0, 0.0, 0.001
assert inductance > 0
next_current = current + voltage / inductance * dt
stored_energy = 0.5 * inductance * next_current**2
print(f"Current: {next_current:.6f} A; stored energy: {stored_energy:.6g} J")

The Python makes the time level explicit. If power uses the old current while stored energy uses the new current, the comparison needs an integration rule consistent with that staggering. Halve the step and compare current at identical physical times.

Verification and limits

Constant current must produce zero ideal inductor voltage. At fixed voltage, doubling inductance must halve current slope. At long DC time, recover the series resistance limit. Reversing current leaves stored energy unchanged but changes signs in mutually coupled voltage relations according to the declared orientation.

Real inductors include winding resistance, core hysteresis and eddy-current loss, saturation, parasitic capacitance, leakage flux, temperature dependence, acoustic noise, and finite voltage and current ratings. At high frequencies, a coil can cease to behave inductively near self-resonance. Fast or physically large systems may need distributed electromagnetic modeling instead of one lumped current state.

A worked Python notebook

Apply two volts across an ideal 0.1 henry inductor initially carrying zero current. The question for this calculation is precise: how does inductor current change when we change time under constant voltage, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

Constant voltage drives a ramp only while the ideal source can maintain that voltage. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 0 to 0.01 s. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for time under constant voltage, expressed in s. The output is inductor current, expressed in A. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = 20 q\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 0.005 s, the expression gives 0.1 A. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("20*q", locals={"q": q})
reference_input = 0.005
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "s")
print("Predicted inductor current:", reference_output, "A")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 0 A. The midpoint, at 0.005 s, gives 0.1 A. The upper endpoint gives 0.2 A. Read these together with the assumptions: they describe different choices of time under constant voltage within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([0.0, 0.005, 0.01])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} s -> {output_value:.6g} A")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows time under constant voltage; the vertical axis shows inductor current. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(0.0, 0.01, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Time under constant voltage (s)")
ax.set_ylabel("Inductor current (A)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does inductor current change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 0.005 s is 20 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = 20\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 0.00505
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "A")
print("Full expression:", full_prediction, "A")
print("Approximation difference:", full_prediction - linear_prediction, "A")

Here the changed input is 0.00505 s. The tangent estimate is 0.101 A, while the full expression gives 0.101 A. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 0.01
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of time under constant voltage inside the stated interval, predict whether inductor current will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: Constant voltage drives a ramp only while the ideal source can maintain that voltage. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of A: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(0.01) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 0.01, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

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