UNIT B.2 — MATHEMATICS

Calculus for circuits: rates and accumulation

BRIDGE · EE Foundations Bridge · ~12 min

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Calculus has two moves. The derivative measures how fast something changes: velocity is the derivative of position; on a graph it's the slope at a point. The integral accumulates: total distance from a velocity curve; on a graph it's the area underneath.

Electricity runs on both. A capacitor's current is proportional to how FAST the voltage across it changes: i = C·(dv/dt). Constant voltage, zero current — that's why capacitors block DC. An inductor is the mirror image: v = L·(di/dt), which is why interrupting an inductive load (a motor, a coil) produces a voltage spike and an arc across the opening contacts.

You've already met an integral: energy = the accumulation of power over time. kWh (Unit 1.7) is literally the area under the power curve. Calculus just gives you the machinery to do that with curves instead of rectangles.

Go deeper: Gilbert Strang's free Calculus textbook plus MIT's Calculus 1A/1B/1C — Semester 1 of the Go Deeper pathway.

Derivative
Rate of change · slope
Integral
Accumulation · area
Capacitor
i = C · dv/dt
Knowledge check

Quiz — 3 questions

1. The derivative of a quantity tells you its:

2. A capacitor with constant DC voltage across it carries:

3. Breaking an inductive circuit causes an arc because:

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