Switching a Circuit On and Off

Flip a switch at one moment and off again later, and the input is a pulse. Step functions write that switching as math, and the Laplace transform turns each switch into a simple factor. The capacitor charges while the pulse is on, then drains.

How it works

Step functions

In one sentence

The Heaviside function $u(t - c)$ is 0 for $t < c$ and 1 for $t \ge c$ , and $\mathcal{L}\{u(t - c)f(t - c)\} = e^{-cs}F(s)$ , so switched inputs become exponential factors in $s$ .

General formula

{\htmlData{sym=3}{R}}C\,{\htmlData{sym=4}{V}}' + {\htmlData{sym=4}{V}} = {\htmlData{sym=2}{V_s}}\left[{\htmlData{sym=0}{u(t - a)}} - {\htmlData{sym=1}{u(t - b)}}\right]

Each switch flip starts a fresh copy of the circuit's step response at the moment it happens, and the copies add.

What the symbols mean

$u(t - a)$
Switch on
0 before time a, 1 after: the switch closing.
$u(t - b)$
Switch off
Subtracting it turns the input back off at time b.
$V_s$
Supply voltage
The voltage applied while the switch is on.
$R$
Resistance
With C, sets how fast the capacitor responds.
$V$
Capacitor voltage
The unknown function of time.

The unit step $u(t - a)$ is 0 before $t = a$ and 1 after: a switch closing at time $a$ .
$u(t - a) - u(t - b)$ is a pulse, on from $a$ to $b$ .
With Laplace transforms each step becomes $e^{-as}/s$ , so the response is the circuit's step response started at $a$ , minus another started at $b$ .
The capacitor charges toward $V_s$ while the pulse is on, then discharges once it ends.

Learn the math

Inverse Laplace TransformsDiff Eq · Lesson 24, Laplace Transforms Step FunctionsDiff Eq · Lesson 25, Laplace Transforms Solving IVPs with Laplace TransformsDiff Eq · Lesson 26, Laplace Transforms IVPs with Step FunctionsDiff Eq · Lesson 27, Laplace Transforms RC CircuitsE&M · Lesson 15, Current and DC Circuits

Learn the math

Inverse Laplace TransformsDiff Eq · Lesson 24, Laplace Transforms

Step FunctionsDiff Eq · Lesson 25, Laplace Transforms

Solving IVPs with Laplace TransformsDiff Eq · Lesson 26, Laplace Transforms

IVPs with Step FunctionsDiff Eq · Lesson 27, Laplace Transforms

RC CircuitsE&M · Lesson 15, Current and DC Circuits