# Low Pass RC Circuit Response to a Square Wave - Graphing Calculator

Table of Contents

An online calculator to calculate and graph the voltage across a capacitor when the input \( v_i(t) \) is a square wave.

## Formulae for Voltage in a Series RC Circuit to Square Wave

We first give the formulas of the voltage across the capacitor when the input is a square wave.

The formulas developed in the study of low pass RC circuit response to a square wave is presented here its used in the calculator.

When a square wave of the form

\( \displaystyle v_i(t) = V_0 \sum_{n=0}^{n=\infty} \left\{ u(t - n\;T)- u (t-(n+1/2)\;T) \right\} \) were \( u(t) \) is the unit step function,

the voltage \( v_C(t) \) across the capacitor is given by

\( \displaystyle v_C(t) = \displaystyle V_0 \sum_{n=0}^{n=\infty} \left \{ u(t-nT) \; \left(1 - e^{- \dfrac{t - n \; T}{R \;C} } \right)
\\\\ \quad \quad \quad - u(t-(n+1/2)T) \; \left(1 - e^{-\dfrac{ t - (n + 1/2) T}{\; R \; C} } \right) \right\} \)

The voltage \( v_R(t) \) across the resistor is given by

\( v_R (t) = v_i(t) - v_C(t) \)

The current \( i(t) \) is given by

\( i(t) = \dfrac{v_R}{R} = \dfrac{v_i(t) - v_C(t)}{R} \)

\( \tau = R C \) is called the time constant of the circuit.

## Use of the calculator

Enter the the resistance \( R \), the capacitance \( C \), the period \( T \) of the square wave as a multiple of the time constant \( \tau = R \; C\) and the number of periods to be displayed then press "Calculate&Graph".

The time constant \( \tau \), the voltages \( v_C(t) \) (blue) and \( v_i(t) \) (red) are displayed.

The results shown are for \( V_0 = 1 \) V.
## Results

### More References and links

Low Pass RC Circuit Response to a Square Wave

series RLC circuit response to a step voltage

Engineering Mathematics with Examples and Solutions