# Circuit Theory/Convolution Integral/Examples/example49/Vc

series LRC circuit ... find voltage across the resistor

Given that the source voltage is (2t-3t2), find voltage across the resistor.

This is the Vc solution.

Outline:

## Contents

### Transfer Function

${\displaystyle H(s)={\frac {V_{C}}{V_{S}}}={\frac {\frac {1}{0.25s}}{4+s+{\frac {1}{0.25s}}}}}$
simplify((1/(s*0.25))/(4 + s + 1/(0.25*s)))
${\displaystyle H(s)={\frac {4}{s^{2}+4s+4}}}$

### Homogeneous Solution

solve(s^2 + 4.0*s + 4.0,s)

There are two equal roots at s = -2, so the solution has the form:

${\displaystyle V_{C_{h}}(t)=Ae^{-2t}+Bte^{-2t}+C_{1}}$

### Particular Solution

After a long time attached to a unit step function source, the inductor has shorted and the capacitor has opened. All the drop is across the capacitor. The current is zero.

${\displaystyle V_{C_{p}}=1}$

This also means that C1 is still unknown.

### Initial Conditions

So far the full equation is:

${\displaystyle V_{C}(t)=1+Ae^{-2t}+Bte^{-2t}+C_{1}}$

At t = ∞ what is the B term's value?

limit(B*t*exp(-t),t = infinity)

Mupad says 0. So This means that at t = ∞:

${\displaystyle 1=1+C_{1}}$
${\displaystyle C_{1}=0}$

Initial voltage across the capacitor is 0 so:

${\displaystyle 0=1+A+C_{1}}$
${\displaystyle A=-1}$

Initial current through the series leg is zero because of the assumed initial conditions of the inductor. This means i(0) = 0, so:

${\displaystyle i(t)=C*{dV_{C}(t) \over dt}={\frac {1}{4}}((-2A+B)e^{-2t}-2Bte^{-2t})}$
${\displaystyle 0={\frac {1}{4}}(-2A+B)}$
${\displaystyle B=2A}$
${\displaystyle B=-2}$

This means that i(t) =:

${\displaystyle i(t)={\frac {1}{4}}((-2(-1)+(-2))e^{-2t}-2(-2)te^{-2t})=te^{-2t}}$
${\displaystyle V_{R}(t)=4*i(t)=4te^{-2t}}$

### Impulse Solution

Taking the derivative of the above get:

${\displaystyle V_{R}\delta (t)=4e^{-2t}-8te^{-2t}}$

### Convolution Integral

${\displaystyle V_{R}(t)=\int _{0}^{t}(4e^{-2(t-x)}-8(t-x)e^{-2(t-x)})(2x-3x^{2})dx}$
f := (4*exp(-2*(t-x)) - 8*(t-x)exp(-2*(t-x)))*(2*x-3*x^2);
S :=int(f,x=0..t)

${\displaystyle V_{R}(t)=8-8e^{-2t}-10te^{-2t}-6t}$

There will not be any constant since again, V_R(t) = 0 after a long time ... and the capacitor opens.