ELC-27

Capacitor current i=C dv/dt

i = C dv/dt. Current is capacitance times the voltage slope.

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CircuitsMaxwell

Governing equation

i=C\dfrac{\mathrm{d}v}{\mathrm{d}t}

$\mathrm{d}v/\mathrm{d}t \in \left[-500, 500\right]$

where

C: Capacitance (µF)
\mathrm{d}v/\mathrm{d}t: Voltage slope (V/s)
i: Current (mA)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

i=C\dfrac{\mathrm{d}v}{\mathrm{d}t}

$\mathrm{d}v/\mathrm{d}t \in \left[-500, 500\right]$

$\mathrm{d}v/\mathrm{d}t = -500$ $i = -50$

Along the curve

②

Domain of the variable — blue

\mathrm{d}v/\mathrm{d}t_{\min} = -500,\quad \mathrm{d}v/\mathrm{d}t_{\max} = 500

$\mathrm{d}v/\mathrm{d}t \in \left[-500, 500\right]$

$\mathrm{d}v/\mathrm{d}t = -500$ $\mathrm{d}v/\mathrm{d}t = -500$

Along the curve

③

Response curve — green

i=C\dfrac{\mathrm{d}v}{\mathrm{d}t}

$\mathrm{d}v/\mathrm{d}t \in \left[-500, 500\right]$

$\mathrm{d}v/\mathrm{d}t = -500$ $i = -50$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}i}{\mathrm{d}\mathrm{d}v/\mathrm{d}t}

$\mathrm{d}v/\mathrm{d}t \in \left[-500, 500\right]$

$\mathrm{d}v/\mathrm{d}t = -493.7$ $i^{\prime} = 0.1$

Along the curve

Lecture brief

Historical brief

Ohm (1827), Kirchhoff (1845) and Maxwell’s circuit reduction still run every board: RLC transients, transformers, skin effect and three-phase power. The sheets are those network laws, not a SPICE deck. This sheet (ELC-27 — Capacitor current i=C dv/dt) is the form associated with Maxwell. Working symbols: $C$ , $\mathrm{d}v/\mathrm{d}t$ $\rightarrow$ $i$ . Q = C V, so i = dQ/dt = C dv/dt for a linear capacitor. Dual of v = L di/dt.

Purpose

Purpose: compute

i

from

C

\mathrm{d}v/\mathrm{d}t

in Electrical via

i=C\dfrac{\mathrm{d}v}{\mathrm{d}t}

i = C dv/dt. Current is capacitance times the voltage slope. Use it when a real electrical question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

C = 100.000\,\mathrm{\mu F}

\mathrm{d}v/\mathrm{d}t = 50.000\,\mathrm{V/s}

, the governing relation

i=C\dfrac{\mathrm{d}v}{\mathrm{d}t}

yields

i = 5.000\,\mathrm{mA}

. A capacitor, a ramp voltage, a constant current. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Capacitance C100 µFVoltage slope \mathrm{d}v/\mathrm{d}t50 V/s

Outputs

Current i5.000 mA

Reading speed

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Narration of this film

A capacitor, a ramp voltage, a constant current.

Q = C V, so i = dQ/dt = C dv/dt for a linear capacitor. Dual of v = L di/dt.

Reading speed

Watch on YouTube