ELC-38

RC time constant

v(t) = V∞ + (V0−V∞) e^{−t/τ} with τ = R C. Charging or discharging.

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CircuitsTransient RC

Governing equation

v(t)=V_\infty+(V_0-V_\infty)e^{-t/RC}

$t \in \left[0.1, 5000\right]$

where

R: Resistance (kΩ)
C: Capacitance (µF)
V_0: Initial voltage (V)
V_\infty: Final voltage (V)
t: Time (ms)
\tau: Time constant (ms)
v: Voltage (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(t)=V_\infty+(V_0-V_\infty)e^{-t/RC}

$t \in \left[0.1, 5000\right]$

$t = 0.1$ $\tau = 0.003$

Along the curve

②

Domain of the variable — blue

t_{\min} = 0.1,\quad t_{\max} = 5000

$t \in \left[0.1, 5000\right]$

$t = 0.1$ $t = 0.1$

Along the curve

③

Response curve — green

v(t)=V_\infty+(V_0-V_\infty)e^{-t/RC}

$t \in \left[0.1, 5000\right]$

$t = 0.1$ $\tau = 0.003$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\tau}{\mathrm{d}t}

$t \in \left[0.1, 5000\right]$

$t = 31.55$ $\tau^{\prime} = 0.024$

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-38 — RC time constant) is the form associated with Transient RC. Working symbols: $R$ , $C$ , $V_0$ , $V_\infty$ , $t$ $\rightarrow$ $\tau$ , $v$ . KCL on the capacitor node is C dv/dt + v/R = Is. The homogeneous time is RC.

Purpose

Purpose: compute

\tau

v

from

R

C

V_0

V_\infty

t

in Electrical via

v(t)=V_\infty+(V_0-V_\infty)e^{-t/RC}

v(t) = V∞ + (V0−V∞) e^{−t/τ} with τ = R C. Charging or discharging. Use it when a real electrical question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

R = 10.000\,\mathrm{kΩ}

C = 47.000\,\mathrm{\mu F}

V_0 = 0.000\,\mathrm{V}

V_\infty = 12.000\,\mathrm{V}

t = 200.000\,\mathrm{ms}

, the governing relation

v(t)=V_\infty+(V_0-V_\infty)e^{-t/RC}

yields

\tau = 470.00\,\mathrm{ms}

v = 4.159\,\mathrm{V}

. A resistor, a capacitor, an exponential sag. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Resistance R10 kΩCapacitance C47 µFInitial voltage V_00 VFinal voltage V_\infty12 VTime t200 ms

Outputs

Time constant \tau470.00 ms
Voltage v4.159 V

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Narration of this film

A resistor, a capacitor, an exponential sag.

KCL on the capacitor node is C dv/dt + v/R = Is. The homogeneous time is RC.

Reading speed

Watch on YouTube