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Circuit simulation

Circuit simulation is the numerical solution of a circuit model. It does not replace physical reasoning or measurement. It converts a description of nodes, branches, components, sources, and initial conditions into equations, solves those equations, and reports voltages, currents, powers, and diagnostic residuals.

This lesson explains the model pipeline without embedding an interactive simulation. The important ideas are the same ones used in a trustworthy solver: topology is stored as data, every sign convention is explicit, each component contributes equations, and every result is checked against conservation laws and known limiting cases.

A diagram becomes a graph

A lumped circuit contains nodes and branches. A node is a connected conducting region treated as having one potential. A branch is a component between two nodes. The visual position, curve of a wire, or direction in which a symbol is drawn does not by itself determine voltage or current.

Each branch record needs at least:

One node is chosen as the zero-voltage reference. That removes the arbitrary common offset in node potentials. Choosing another reference changes absolute node labels but not voltage differences, branch currents, or powers.

import numpy as np

circuit = {"supply_v": 5.0, "R1_ohm": 1000.0, "R2_ohm": 2000.0}
assert circuit["R1_ohm"] > 0 and circuit["R2_ohm"] > 0
print("Voltage-divider parameters:", circuit)

Validation should reject duplicate identifiers, missing endpoints, nonphysical passive parameters, and mixed units before matrix assembly. A disconnected network may be valid only if each floating part has enough constraints to define its voltages.

Equations come from conservation and components

At each ideal node, Kirchhoff's current law requires

\[\sum_k I_k=0\]

For a closed loop in the lumped quasistatic model, Kirchhoff's voltage law requires

\[\sum_k \Delta V_k=0\]

Component models connect branch voltage and current. An ideal resistor obeys

\[I=\frac{V_a-V_b}{R}\]

An ideal capacitor and inductor introduce state:

\[I_C=C\frac{dV_C}{dt},\qquad V_L=L\frac{dI_L}{dt}\]

An ideal independent voltage source constrains its terminal voltage, while an ideal independent current source constrains its branch current. The solver must combine these different relations without confusing a known source value with an unknown response.

Linear DC analysis

For a resistor network with independent sources, the steady operating point is often a linear system:

\[\mathbf A\mathbf x=\mathbf b\]

The unknown vector may contain non-reference node voltages and selected source currents. The matrix contains conductances and topology. The right-hand side contains independent source values. Modified nodal analysis is a common formulation because it handles ideal voltage sources by adding their currents as unknowns and their voltage relations as equations.

For a resistor of conductance \(G=1/R\) between two non-reference nodes, its nodal matrix contribution has the pattern

\[G \begin{bmatrix} 1 & -1\\ -1 & 1 \end{bmatrix}\]

The positive diagonal entries describe current leaving each node through the resistor; the negative off-diagonal entries couple the two endpoint voltages. If one endpoint is the reference, the reduced system keeps only the other diagonal contribution.

R1, R2, supply = circuit["R1_ohm"], circuit["R2_ohm"], circuit["supply_v"]
matrix = np.array([[1 / R1 + 1 / R2]])
rhs = np.array([supply / R1])
node_voltage = np.linalg.solve(matrix, rhs)[0]
print(f"Divider output: {node_voltage:.6f} V")

Avoid explicitly computing a matrix inverse. A factorization-based linear solve is more accurate and efficient. The solver should report singular or poorly conditioned systems instead of returning arbitrary large numbers.

A small stamped example

Suppose a node with unknown voltage \(V\) is connected to a \(10\ \mathrm V\) through \(2\ \mathrm{k\Omega}\), and to the zero-volt reference through \(3\ \mathrm{k\Omega}\). KCL gives

\[\frac{V-10\ \mathrm V}{2\ \mathrm{k\Omega}} +\frac{V}{3\ \mathrm{k\Omega}}=0\]

In conductance form,

\[\left(\frac{1}{2000\ \Omega} +\frac{1}{3000\ \Omega}\right)V =\frac{10\ \mathrm V}{2000\ \Omega}\]

The solution is \(V=6\ \mathrm V\). The two-kiloohm branch carries \(2\ \mathrm{mA}\) into the node and the three-kiloohm branch carries the same current out. The example checks both the matrix solution and node conservation.

Reconstructing signed branch results

Matrix unknowns are not the complete report. After solving node potentials, calculate every branch voltage from its stored polarity and every branch current from its component law. If a result is negative, the physical value points opposite the chosen reference. Do not flip the branch definition just to display a positive number.

Power under the passive sign convention is

\[P_k=V_kI_k\]

Positive power means absorption, and negative power means delivery. For a closed ideal circuit,

\[\sum_k P_k=0\]

This ledger catches many orientation and stamping errors. It complements, rather than replaces, KCL and component-law checks.

Residuals expose numerical and modeling mistakes

For the solved linear system, compute

\[\mathbf r=\mathbf A\mathbf x-\mathbf b\]

A small residual means the numerical solution satisfies the assembled equations. It does not show that the intended circuit was assembled. Therefore calculate several layers of diagnostics:

Use tolerances relative to the scale of the circuit. A fixed threshold can label a tiny low-power circuit as failed or hide a significant error in a high-power circuit.

current1 = (supply - node_voltage) / R1
current2 = node_voltage / R2
assert np.isclose(current1, current2)
power_residual = current1**2 * R1 + current2**2 * R2 - supply * current1
assert np.isclose(power_residual, 0.0)
print("KCL and power residuals:", current1 - current2, power_residual)

Transient analysis

Capacitor voltages and inductor currents carry state from one time to the next. A transient solver repeatedly combines the current state, source values, component equations, and a time-integration rule.

For a simple RC circuit,

\[\frac{dV_C}{dt}=\frac{V_s(t)-V_C}{RC}\]

An explicit Euler update is

\[V_C^{n+1}=V_C^n +\Delta t\frac{V_s^n-V_C^n}{RC}\] 
C, dt, voltage = 1e-6, 1e-5, 0.0
for _ in range(1000):
    rate = ((supply - voltage) / R1 - voltage / R2) / C
    voltage += dt * rate
print("Capacitor voltage after 0.01 s:", voltage)

Step size is part of the model record. Too large a step can distort a transient or make an otherwise stable physical circuit numerically unstable. Repeat a reference problem with smaller steps and compare at the same physical times. Networks with very different time constants may be stiff and benefit from implicit integration.

Nonlinear devices require iteration

A diode or transistor does not generally produce one constant linear matrix. A nonlinear solver begins with a guess, forms a local linear approximation, solves for a correction, and iterates. Convergence requires more than a small change in the unknown vector; the physical equation residual must also be small.

# A diode and series resistor, solved by bracketed bisection.
saturation, thermal_voltage, resistance, supply = 1e-12, 0.02585, 1000.0, 1.0
low, high = 0.0, supply
for _ in range(80):
    voltage = (low + high) / 2
    residual = (
        saturation * np.expm1(voltage / thermal_voltage)
        - (supply - voltage) / resistance
    )
    if residual > 0:
        high = voltage
    else:
        low = voltage
print("Diode voltage (V):", voltage)
assert abs(residual) < 1e-10

The solver must distinguish convergence failure from a valid physical result. Damping, continuation, or small parasitic conductances can help numerically, but any regularization changes the mathematical model and should be disclosed.

Failure modes should be explicit

A trustworthy circuit calculation can fail. Common causes include a floating network, conflicting ideal voltage sources, an ideal voltage-source loop, an ideal current-source cut set, zero resistance where the formulation cannot support it, missing initial conditions, nonfinite parameters, and nonlinear nonconvergence.

Report the implicated nodes and elements where possible. “Singular matrix” is a numerical symptom; “nodes A and B form an unconstrained floating subnetwork” is an actionable model diagnosis. Never replace failure with a plausible trace or silently chosen ground node.

Verification ladder

Begin with circuits whose answers are known independently: one source and one resistor, a two-resistor divider, series and parallel equivalents, and a first-order RC response. Then check invariances. Reversing a branch reference should reverse its signed voltage and current representation consistently without changing physical power. Changing the reference node should shift node potentials but leave branch results unchanged.

For each calculation, verify:

  1. inputs, units, and topology are valid;
  2. the numerical solver reports success;
  3. residuals meet scale-aware tolerances;
  4. power or energy balances close;
  5. limiting cases agree with simpler physics;
  6. time-step refinement converges for transient results.

These checks test different failure modes. Passing one does not make the others optional.

Model boundaries

The lumped model assumes that components are localized and signals propagate across the circuit much faster than the behavior of interest changes. At sufficiently high frequency or over long conductors, transmission-line delay, electromagnetic coupling, radiation, skin effect, and distributed capacitance and inductance matter. Real components also have tolerance, temperature dependence, noise, parasitic behavior, and finite operating limits.

A numerical result is only as physical as its component models and assumptions. Circuit simulation is most valuable when it makes those assumptions testable and reproducible, not when it gives extra decimal places to an oversimplified model.

A worked Python notebook

Hold the upper divider resistor at one kilohm and the supply at five volts while varying the lower resistor. The question for this calculation is precise: how does output voltage change when we change second divider resistance, 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.

The output is unloaded; an attached load changes the effective lower resistance. 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 100 to 5000 ohm. 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 second divider resistance, expressed in ohm. The output is output voltage, expressed in V. 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.

\[f(q) = \frac{5 q}{q + 1000}\]

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 2000 ohm, the expression gives 3.33333 V. 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("5*q/(1000+q)", locals={"q": q})
reference_input = 2000.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "ohm")
print("Predicted output voltage:", reference_output, "V")

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 0.454545 V. The midpoint, at 2550 ohm, gives 3.59155 V. The upper endpoint gives 4.16667 V. Read these together with the assumptions: they describe different choices of second divider resistance 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([100.0, 2550.0, 5000.0])
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} ohm -> {output_value:.6g} V")
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 second divider resistance; the vertical axis shows output voltage. 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(100.0, 5000.0, 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("Second divider resistance (ohm)")
ax.set_ylabel("Output voltage (V)")
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 output voltage change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 2000 ohm is 0.000555556 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.

\[f'(q) = - \frac{5 q}{\left(q + 1000\right)^{2}} + \frac{5}{q + 1000}\]

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 = 2024.5
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, "V")
print("Full expression:", full_prediction, "V")
print("Approximation difference:", full_prediction - linear_prediction, "V")

Here the changed input is 2024.5 ohm. The tangent estimate is 3.34694 V, while the full expression gives 3.34683 V. 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 = 4900.0
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 second divider resistance inside the stated interval, predict whether output voltage 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: The output is unloaded; an attached load changes the effective lower resistance. 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 V: 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(5000.0) - evaluate(100.0))
for points in [101, 201, 401]:
    grid = np.linspace(100.0, 5000.0, 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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