ODE-05

RC charging

Step response.

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ODERC

Governing equation

V=V_s(1-e^{-t/RC})

$t \in \left[0, 2000\right]$

where

V_s: Vs (V)
R: R (Ω)
C: C (µF)
t: t (ms)
V: V (V)

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

V=V_s(1-e^{-t/RC})

$t \in \left[0, 2000\right]$

$t = 0$ $V = 0$

Along the curve

②

Domain of the variable — blue

t_{\min} = 0,\quad t_{\max} = 2000

$t \in \left[0, 2000\right]$

$t = 0$ $t = 0$

Along the curve

③

Response curve — green

V=V_s(1-e^{-t/RC})

$t \in \left[0, 2000\right]$

$t = 0$ $V = 0$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}V}{\mathrm{d}t}

$t \in \left[0, 2000\right]$

$t = 12.58$ $V^{\prime} = 0.044$

Along the curve

Lecture brief

Historical brief

Exponential decay, the harmonic oscillator, SIR epidemics, Lotka–Volterra and Bessel are the first solvable ODEs of nature and populations. The lab is rate, period and phase portrait in closed form. This sheet (ODE-05 — RC charging) is the form associated with RC. Working symbols: $V_s$ , $R$ , $C$ , $t$ $\rightarrow$ $V$ . Step response.

Purpose

Purpose: compute

V

from

V_s

R

C

t

in Differential equations via

V=V_s(1-e^{-t/RC})

Step response. Use it when a real differential equations question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

V_s = 5.000\,\mathrm{V}

R = 1000.000\,\mathrm{Ω}

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

t = 50.000\,\mathrm{ms}

, the governing relation

V=V_s(1-e^{-t/RC})

yields

V = 1.967\,\mathrm{V}

. Step response. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Vs V_s5 VR R1000 ΩC C100 µFt t50 ms

Outputs

V V1.967 V

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Narration of this film

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