55 lines
2.6 KiB
Markdown
55 lines
2.6 KiB
Markdown
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#start of lec 19
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This lecture we will learn about periodic functions, specifically, non-sinusoidal periodic functions.
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# Periodic function
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Definition:
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$f$ is periodic with period $T \quad (T>0)$ if:
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$$f(t)=f(t+T), \quad \forall\ t\in \mathbb{R}$$
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![[Drawing 2023-10-20 13.06.35.excalidraw]]
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We will now compute laplace transforms of these periodic functions. Computing DE's containing these periodic functions using something like #voparam would not be easy.
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If we take the windowed version of the function (one period, where everywhere else is 0, ie:)
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$f_{T}(t)=\begin{cases}f(t)\ ,\ & 0\leq t\leq T \\0\ ,\ & \text{otherwise}\end{cases}$
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we can "glue together" many of these windows together to rebuild our $f(t)$, like this:
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$f(t)=f_{T}(t)+f_{T}(t-T)u(t-T)+f_{T}(t-2T)u(t-2T)+\dots$
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$\mathcal{L}\{f\}=\mathcal{L}\{f_{T}\}+\mathcal{L}\{f(t-T)u(t-T)\}+\dots$
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recall the formula from last lec: $\mathcal{L}\{u(t-a)f(t-a)\}=e^{-as}F(s)$
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then:
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$\mathcal{L}\{f\}=\mathcal{L}\{f_{T}\}(1+e^{-TS}+e^{-2TS}+e^{-3TS}+\dots)$
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$\mathcal{L}\{f\}=\mathcal{L}\{f_{T}\}(1+e^{-TS}+(e^{-TS})^{2}+(e^{-TS})^{3}+\dots)$
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This is a geometric series! $1+r+r^2+\dots$
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Geometric series are convergent when $|r|<1$
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and equal to: $\frac{1}{1-r}$
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in this case, $r=e^{-Ts}$
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so:
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$$\mathcal{L}\{f\}=\mathcal{L}\{f_{T}\} \frac{1}{1-e^{-Ts}}$$
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handy formula! ^ will be used again.
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#ex
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imagine another function: (image is of a square wave with a period of 2a, oscillates between 1 and 0, starts at 1 when t=0.)
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![[Drawing 2023-10-20 13.27.58.excalidraw]]
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$\mathcal{L}\{f\}=\mathcal{L}\{f_{2a}\} \frac{1}{1-e^{-2as}}$
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$f_{2a}=u(t)-u(t-a)$ (this is the first period piece)
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$\implies \mathcal{L}\{f_{2a}\}=\mathcal{L}\{u(t)\}-\mathcal{L}\{u(t-a)\}=\frac{1}{s}- \frac{e^{-as}}{s}$
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plug back in:
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$\mathcal{L}\{f\}=\mathcal{L}\{f_{2a}\} \frac{1}{1-e^{-2as}}=\frac{1}{s}\cancel{ (1-e^{-as}) } \frac{1}{\cancel{ (1-e^{-as}) }(1+e^{-as})}$
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$$\mathcal{L}\{f\}=\frac{1}{s(1+e^{-as})}$$
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#ex
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$y''+3y'+2y=f(t)$ where $f(t)$ is from the previous example
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$y(0)=y'(0)=0,\ a=1$ (a is width of 1/2 period in the function f(t))
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$s^2Y+3sY+2Y=\mathcal{L}\{f\}= \frac{1}{s(1+e^{-s})}$
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$Y(s)=\frac{1}{s(s+1)(s+2)} \frac{1}{1+e^{-s}}$
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$=F(s) \frac{1}{1+e^{-s}}$
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$\mathcal{L}^{-1}\{F\}=\mathcal{L}^{-1}\{\frac{1}{2} \frac{1}{s}+\frac{1}{2} \frac{1}{s+2}-\frac{1}{s+1}\}$
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$=\frac{1}{2}+\frac{1}{2}e^{-2t}-e^{-t}$
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$y(t)= \dots$
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we want to use formula from earlier
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so we need to change $F(s) \frac{1}{1+e^{-s}}$
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to: $F(s) \frac{1e^{-s}}{1-e^{-2s}}$
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$y(t)=p(t)$, periodic with period of 2 ($T=2$)
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$$f_{2a}(t)=\mathcal{L}^{-1}\{F(s)-F(s)e^{-s}\}=\frac{1}{2}+\frac{1}{2}e^{-2t}-e^{-t}-\left( \frac{1}{2}+\frac{1}{2} e^{-2(t-1)}-e^{-(t-1)})u(t-1 \right)$$
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