diff --git a/content/Dirak δ-function (lec 21).md b/content/Dirak δ-function (lec 21).md index 8916318..203819e 100644 --- a/content/Dirak δ-function (lec 21).md +++ b/content/Dirak δ-function (lec 21).md @@ -37,12 +37,14 @@ $w(t)=\frac{1}{4}\mathcal{L}^{-1}\left\{ \frac{1}{s+1} - \frac{1}{s+5} \right\}+ $=\underbrace{ \frac{1}{4}(e^{-t}-e^{ -5t }) }_{ \text{this came from initial conditions} }+\dots$ $\mathcal{L}^{-1}\{e^{as}F(s)\}=f(t-a)u(t-a)$ $$y(t)= \frac{1}{4}(e^{-t}-e^{ -5t }) +\frac{e}{4}u(t-1)(e^{ -(t-1) }-e^{ -5(t-1) })$$ -notice that this RHS came from the impulse delta and the effect it has on the system. +notice that the right most term came from the impulse and the effect it had on the system. side note: delta functions are useful for quantum physics. Lets start modelling some electric circuits again: ![[Drawing 2023-10-25 13.43.26.excalidraw]] -$0.2I_{1}+0.1I_{3}'+2I_{1}=g(t)$ +the circuit is starts switched on and is then switched off at $t=1$ +Applying KVL: +$0.2I_{1}+0.1I_{3}'+2I_{1}=g(t) \qquad \text{where } g(t)=\begin{cases}6, & 0\leq t\leq 1 \\0, & 1