
The study the response of high pass RC circuits to a square wave input; numerical examples with graphs of voltages are presented.
Problem with SolutionFind and graph the voltages across the capacitor \( R \) as function of time in the high pass \( RC \) ciruit below given that the input voltage is \( v_i(t) \) is a square wave as shown in the graph below. Solution to the ProblemIn the study of the low pass RC circuit response to a square wave, it was found that the voltage across the capacitor is given by \( \displaystyle v_C(t) = \displaystyle V_0 \sum_{n=0}^{n=\infty} \left \{ u(tnT) \; \left(1  e^{ \dfrac{t  n \; T}{R \;C} } \right) \\\\ \quad \quad \quad \quad  u(t(n+1/2)T) \; \left(1  e^{\dfrac{ t  (n + 1/2) T}{\; R \; C} } \right) \right\} \) when the input \( v_i(t) \) voltage is a square wave modelled by a sum of positive and negative step unit functions of the form \( \displaystyle v_i(t) = V_0 \sum_{n=0}^{n=\infty} \left\{ u(t  n\;T) u (t(n+1/2)\;T) \right\} \) In this study we need to find the voltage \( v_R(t) \) cross the resistor which is given by \( v_R(t) = v_i(t)  v_C(t)\) When \( v_i(t) \) and \( v_C(t) \) are substituted by their expression given above, we can simplify \( v_R(t) \) to \( \displaystyle v_R(t) = \displaystyle V_0 \sum_{n=0}^{n=\infty} \left \{ u(tnT) \; \left(e^{ \dfrac{t  n \; T}{R \;C} } \right) \\\\ \quad \quad \quad  u(t(n+1/2)T) \; \left(e^{\dfrac{ t  (n + 1/2) T}{\; R \; C} } \right) \right\} \)
Numerical Applications
b) \( T = 10 RC = 10 \) s c) \( T = 5 RC = 5 \) s d) \( T = 2 RC = 2 \) s More References and LinksRC Circuit Response to a Step VoltageSolve Differential Equations Using Laplace Transform Laplace transforms Engineering Mathematics with Examples and Solutions 