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<h2 class="hd hd-2 unit-title">5.1. Rules and applications.</h2>
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<h3 class="hd hd-3 problem-header">Objectives.</h3><p>
After completing this lecture you will be able to </p><ol class="enumerate"><li value="1"><p>
Use the <b class="bfseries"><span style="color:#0000FF">partial fraction decomposition</span></b> to find the inverse Laplace transform of any rational function [mathjaxinline]Q(s)/P(s)[/mathjaxinline] with [mathjaxinline]\deg Q<\deg P[/mathjaxinline]. </p></li><li value="2"><p>
Use the pole diagram of [mathjaxinline]\mathcal{L}(f(t);s)[/mathjaxinline] to describe the <b class="bfseries"><span style="color:#0000FF">long-term behavior</span></b> of a function [mathjaxinline]f(t)[/mathjaxinline]. </p></li><li value="3"><p>
Analyze the <b class="bfseries"><span style="color:#0000FF">stability</span></b> of an LTI system by means of the pole diagram of its transfer function. </p></li><li value="4"><p>
Find Laplace transforms using the <b class="bfseries"><span style="color:#0000FF">[mathjaxinline]s[/mathjaxinline]-derivative rule</span></b> and the <b class="bfseries"><span style="color:#0000FF">exponential-shift formula.</span></b> </p></li></ol>
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<h2 class="hd hd-2 unit-title">5.2. Review.</h2>
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The Laplace transform carries functions of time, [mathjaxinline]f(t)[/mathjaxinline] for [mathjaxinline]t\geq 0[/mathjaxinline], to complex functions of a complex variable [mathjaxinline]s[/mathjaxinline]. The integral definition is </p><table id="a0000000547" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]L(f(t);s)=\int _0^\infty f(t)e^{-st}\, dt\, .[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Laplace transform technique depends upon developing two lists: A list of <b class="bf">rules</b> and a list of <b class="bf">computations</b>. We will add a lot more entries today. So far, these two lists look like this. </p><h3 class="hd hd-3 problem-header">Rules.</h3><ol class="enumerate"><li value="1"><p><b class="bfseries"><span style="color:#0000FF">Linearity.</span></b></p><table id="a0000000548" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]af(t)+bg(t)\rightsquigarrow aF(s)+bG(s)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table></li><li value="2"><p><b class="bfseries"><span style="color:#0000FF">Inverse transform.</span></b> [mathjaxinline]F(s)[/mathjaxinline] essentially determines [mathjaxinline]f(t)[/mathjaxinline]. </p></li><li value="3"><p><b class="bfseries"><span style="color:#0000FF"> [mathjaxinline]t[/mathjaxinline]-derivative rules.</span></b> For a function [mathjaxinline]f(t)[/mathjaxinline] of exponential type, </p><table id="a0000000549" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]f'(t)\rightsquigarrow sF(s)-f(0)\, .[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
and </p><table id="a0000000550" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]f^{(n)}(t)\rightsquigarrow s^ nF(s)-\left(f(0)s^{n-1}+f'(0)s^{n-2}+\cdots +f^{(n-1)}(0)\right)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table></li></ol><h3 class="hd hd-3 problem-header"> Calculations.</h3><ul class="itemize"><li><p>
[mathjaxinline]\displaystyle {1\rightsquigarrow \frac{1}{s}}[/mathjaxinline], [mathjaxinline]\mathrm{Re}\, (s)>0[/mathjaxinline]. </p></li><li><p>
[mathjaxinline]\displaystyle {e^{rt}\rightsquigarrow \frac{1}{s-r}}[/mathjaxinline], [mathjaxinline]\mathrm{Re}\, (s)>\mathrm{Re}\, (r)[/mathjaxinline]. </p></li><li><p>
[mathjaxinline]\displaystyle {\cos (\omega t)\rightsquigarrow \frac{s}{s^2+\omega ^2}}[/mathjaxinline], [mathjaxinline]\mathrm{Re}\, (s)>0[/mathjaxinline]. </p></li><li><p>
[mathjaxinline]\displaystyle {\sin (\omega t)\rightsquigarrow \frac{\omega }{s^2+\omega ^2}}[/mathjaxinline], [mathjaxinline]\mathrm{Re}\, (s)>0[/mathjaxinline]. </p></li></ul>
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<h2 class="hd hd-2 unit-title">5.3. s-derivative rule.</h2>
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We definitely need to expand our table of calculations: at this point we don't even know how to solve [mathjaxinline]\ddot x=1[/mathjaxinline] with rest initial conditions! It leads to [mathjaxinline]s^2X=1/s[/mathjaxinline], and we don't have [mathjaxinline]1/s^3[/mathjaxinline] anywhere in our table. It's not too different from [mathjaxinline]1/s^2[/mathjaxinline], though; in fact </p><table id="a0000000551" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\frac{1}{s^3}=-\frac12\frac{d}{ds}\frac{1}{s^2}\, .[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
So let's go back to the integral definition and try to recognize what [mathjaxinline]F'(s)[/mathjaxinline] is in general: </p><table id="a0000000552" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000553"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle F'(s)=[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{d}{ds}\int _0^\infty f(t)e^{-st}\, dt[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000554"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \int _0^\infty f(t)\frac{d}{ds}e^{-st}\, dt[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000555"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \int _0^\infty f(t)(-t)e^{-st}\, dt=-\mathcal{L}(tf(t);s)\, .[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><p>
This gives us a new entry in our table of rules: </p><p><b class="bfseries"><span style="color:#0000FF">The [mathjaxinline]s[/mathjaxinline]-derivative rule.</span></b></p><table id="a0000000556" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]tf(t)\rightsquigarrow -F'(s)\, .[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
In general, this rule becomes </p><table id="a0000000557" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]t^ nf(t) \rightsquigarrow (-1)^ nF^{(n)}(s)\, .[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
For example, starting with [mathjaxinline]1\rightsquigarrow s^{-1}[/mathjaxinline] we find, in sequence, </p><table id="a0000000558" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000559"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle 1[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow s^{-1}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000560"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle t=t\cdot 1[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow -\frac{d}{ds}s^{-1}=s^{-2}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000561"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle t^2=t\cdot t[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow -\frac{d}{ds}s^{-2}=2s^{-3}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000562"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle t^3=t\cdot t^2[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow -\frac{d}{ds}(2s^{-3})=3\cdot 2\, s^{-4}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><p>
and in general </p><table id="a0000000563" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]t^ n\rightsquigarrow \frac{n!}{s^{n+1}}\, ,\quad n=0,1,2,\ldots \, .[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Note that these all converge for [mathjaxinline]\mathrm{Re}\, s >0[/mathjaxinline]. </p><p>
So, to return to [mathjaxinline]\ddot x=1[/mathjaxinline] with rest initial conditions, we find </p><table id="a0000000564" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000565"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle s^2X=[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \, \frac{1}{s},[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000566"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle X=[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \, \frac{1}{s^3},[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000567"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle x=[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \, \frac{1}{2}t^2,[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><p>
which is indeed the solution to [mathjaxinline]\ddot x=1[/mathjaxinline] with rest initial conditions [mathjaxinline]x(0)=0[/mathjaxinline], [mathjaxinline]\dot x(0) = 0[/mathjaxinline]. </p>
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Use the [mathjaxinline]s[/mathjaxinline]-derivative rule to find [mathjaxinline]\mathcal{L}\left(te^{at}\right)[/mathjaxinline]. </p>
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Practice problem 2
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Use the [mathjaxinline]s[/mathjaxinline]-derivative rule to find [mathjaxinline]\mathcal{L}\left(t^{5}e^{at}\right)[/mathjaxinline]. </p>
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<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
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<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
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<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
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<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
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Inverse Laplace practice
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Find the inverse Laplace transform [mathjaxinline]\displaystyle \mathcal{L}^{-1}\left(\frac{4}{s^3}\right)[/mathjaxinline] </p>
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(Enter answer as a function of time [mathjaxinline]t[/mathjaxinline].) </p>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
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<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
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<h2 class="hd hd-2 unit-title">5.4. Resonance.</h2>
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<p><b class="bfseries"><span style="color:#0000FF">Resonance</span></b> occurs when the input frequency is close to a natural frequency of the system. In that case the periodic system response will have exceptionally large amplitude. In the complete absence of damping (a mathematical possibility) there is no periodic response if the two frequencies coincide. </p><p><p><b class="bfseries">Example 4.1 </b> Suppose we are driving a spring system through the spring, and there is negligible damping: say </p><table id="a0000000581" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\ddot x+9x=9\cos (3t)\, .[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Because the input frequency coincides with the natural frequency of the harmonic oscillator, we are in resonance. There are still solutions, but none of them are periodic. One can be found using complex replacement and the generalized ERF. But we can also find solutions using Laplace transform. As usual with the Laplace transform method, we need to specify initial conditions; let's say that [mathjaxinline]x(0)=2,\dot x(0)=0[/mathjaxinline]. </p><p>
Apply [mathjaxinline]\mathcal{L}[/mathjaxinline] to both sides, and remembering the terms arising from the initial condition: </p><table id="a0000000582" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000583"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle 9\cos (3t)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{9s}{s^2+9}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.9)</td></tr><tr id="a0000000584"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle x[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle X[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.10)</td></tr><tr id="a0000000585"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \ddot x[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle s^2X - sx(0) -x'(0) = s^2X-2s[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.11)</td></tr></table><p>
we get the algebraic equation in [mathjaxinline]X[/mathjaxinline]: </p><table id="a0000000586" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax](s^2+9)X-2s=\frac{9s}{s^2+9}\, ,[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
or </p><table id="a0000000587" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]X=\frac{9s}{(s^2+9)^2}+\frac{2s}{s^2+9}\, .[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
This is already in standard partial fractions form! We need to find the inverse Laplace transform of this expression. The second term is easy, but the first is not in our table. But it very close to the derivative (with respect to [mathjaxinline]s[/mathjaxinline]) of [mathjaxinline](s^2+9)^{-1}[/mathjaxinline], which is a known Laplace transform. So let's use the [mathjaxinline]s[/mathjaxinline]-derivative rule: Since </p><table id="a0000000588" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\sin (\omega t)\rightsquigarrow \frac{\omega }{s^2+\omega ^2}\, ,[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
we have </p><table id="a0000000589" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000590"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle t\sin (\omega t)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle -\frac{d}{ds}\frac{\omega }{s^2+\omega ^2}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000591"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{2\omega s}{(s^2+\omega ^2)^{2}}\, .[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><p>
Returning to our original problem, with [mathjaxinline]\omega =3[/mathjaxinline], </p><table id="a0000000592" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\frac{3}{2}t\sin (3t)\rightsquigarrow \frac{9s}{(s^2+9)^2}\, ,[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
and we find </p><table id="a0000000593" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]x=\frac{3}{2}t\sin (3t)+2\cos (3t)\, .[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Just as a check, compute that [mathjaxinline]x(0)=2[/mathjaxinline] and [mathjaxinline]\dot x(0)=0[/mathjaxinline]. </p><p>
The presence of a repeated factor in the denominator of the Laplace transform [mathjaxinline]X(s)[/mathjaxinline] is the mark of resonance in the [mathjaxinline]s[/mathjaxinline]-domain. As you see, in this example one factor came from a characteristic polynomial of the system, the other from the Laplace transform of the input signal. Repeated factors produce repeated roots, and when they occur in the denominator they produce poles of “higher multiplicity". </p><h3 class="hd hd-3 problem-header">The [mathjaxinline]s[/mathjaxinline]-derivative rule and the multiplicity of poles:</h3><table id="a0000000594" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]tf(t) \rightsquigarrow -F'(s).[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
By iterating this, we find </p><table id="a0000000595" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]t^ kf(t) \rightsquigarrow (-1)^ kF^{(k)}(s).[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
For example, since [mathjaxinline]e^{rt} \rightsquigarrow (s-r)^{-1}[/mathjaxinline], </p><table id="a0000000596" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]t^{k-1}e^{rt} \rightsquigarrow \frac{(k-1)!}{(s-r)^ k}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
The [mathjaxinline]s[/mathjaxinline]-derivative rule associates a pole at [mathjaxinline]s=r[/mathjaxinline] of multiplicity [mathjaxinline]k[/mathjaxinline] with a polynomial of degree [mathjaxinline]k-1[/mathjaxinline] times [mathjaxinline]e^{rt}[/mathjaxinline]. </p></p>
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<h2 class="hd hd-2 unit-title">5.5. More computations with quadratic denominators.</h2>
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The previous example used the [mathjaxinline]s[/mathjaxinline]-derivative to compute [mathjaxinline]\mathcal{L}(t\sin (\omega t);s)[/mathjaxinline]. The [mathjaxinline]s[/mathjaxinline]-derivative rule also leads to a computation of [mathjaxinline]\mathcal{L}(t\cos (\omega t);s)[/mathjaxinline], and we have two new calculations. </p><ul class="itemize"><li><p>
[mathjaxinline]\displaystyle {t\sin (\omega t)\rightsquigarrow \frac{2\omega s}{(s^2+\omega ^2)^2}}[/mathjaxinline] </p></li><li><p>
[mathjaxinline]\displaystyle {t\cos (\omega t)\rightsquigarrow \frac{s^2-\omega ^2}{(s^2+\omega ^2)^2}}[/mathjaxinline] </p></li></ul><p>
Because we are often using our Laplace tables to go backwards, that is to compute [mathjaxinline]\mathcal{L}^{-1}[/mathjaxinline], it would be useful to have [mathjaxinline]\displaystyle {\mathcal{L}^{-1}\left(\frac{1}{(s^2+\omega ^2)^2};t\right)}[/mathjaxinline] on our table as well. </p>
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Find the expression for [mathjaxinline]\displaystyle {\mathcal{L}^{-1}\left(\frac{1}{(s^2+\omega ^2)^2};t\right)}[/mathjaxinline] by combining the following two transforms: </p>
<table id="a0000000597" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000598">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle t\cos (\omega t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{s^2-\omega ^2}{(s^2+\omega ^2)^2}[/mathjaxinline]
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<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(4.12)</td>
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<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \sin (\omega t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{\omega }{s^2+\omega ^2} =\frac{(s^2+\omega ^2)\omega }{(s^2+\omega ^2)^2}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(4.13)</td>
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<h2 class="hd hd-2 unit-title">5.6. The s-shift rule.</h2>
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We have found the Laplace transform of sinusoidal functions. Damped sinusoids are just as important and occur as transients of an LTI system response, so we should work that out too. </p>
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Find the Laplace transform of [mathjaxinline]e^{at}\cos (\omega t)[/mathjaxinline] by writing [mathjaxinline]e^{at}\cos (\omega t)[/mathjaxinline] as the real part of a complex exponentials, and using this expression to determine its Laplace transform as a rational function with real coefficients. </p>
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(Enter the Laplace transform as a function of [mathjaxinline]s[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], and [mathjaxinline]\omega[/mathjaxinline].) </p>
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<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
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<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
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<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
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<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
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<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
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<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
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<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
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A similar calculation to the previous problem, using </p><table id="a0000000609" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\sin (\omega t) = \mathrm{Im}\, \left(e^{i\omega t} \right)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
gives us the Laplace transform </p><table id="a0000000610" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]e^{at}\sin (\omega t) \rightsquigarrow \frac{\omega }{(s-a)^2 + \omega ^2}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
By forming linear combinations, we can find the Laplace transform of any damped sinusoid. But the form of the answer here gives us pause; it's very close to the Laplace transform of the <b class="bf">undamped</b> sinusoid [mathjaxinline]\cos (\omega t)[/mathjaxinline]. Let's see if this is something general. So suppose we know [mathjaxinline]f(t)\rightsquigarrow F(s)[/mathjaxinline], and use the integral expression for the Laplace transform to compute </p><table id="a0000000611" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000612"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle L(e^{at}f(t);s)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\, \int _0^\infty (e^{at}f(t))e^{-st}\, dt[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000613"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\, \int _0^\infty f(t)e^{-(s-a)t}\, dt[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000614"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\, F(s-a)\, .[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><p>
That is, </p><p><b class="bfseries"><span style="color:#0000FF">The [mathjaxinline]s[/mathjaxinline]-shift rule:</span></b></p><table id="a0000000615" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]e^{at}f(t)\rightsquigarrow F(s-a)\, .[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
This is consistent with the calculation you just did, and shows also that </p><table id="a0000000616" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]e^{at}\sin (\omega t)\rightsquigarrow \frac{\omega }{(s-a)^2+\omega ^2}\, .[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p><h3 class="hd hd-3 problem-header">Consistency.</h3><br/></p><p>
It is a good exercise to check for consistency among our various formulas: </p><ul class="itemize"><li><p>
We have [mathjaxinline]\mathcal{L}(1) = 1/s[/mathjaxinline], so the [mathjaxinline]s[/mathjaxinline]-shift formula gives [mathjaxinline]\mathcal{L}(e^{at}\cdot 1) = 1/(s-a)[/mathjaxinline]. This matches our formula for [mathjaxinline]\mathcal{L}(e^{at})[/mathjaxinline]. </p></li></ul>
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Find the Laplace transform of [mathjaxinline]e^{-t}\cos (3t)[/mathjaxinline]. </p>
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<td class="formulainput">Integers</td>
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<font color="#0078b0">2520</font>
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<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
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<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
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<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
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Practice 2
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Find the Laplace transform of [mathjaxinline]e^{2t}t\cos (3t)[/mathjaxinline]. </p>
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<font color="#0078b0">2520</font>
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<p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">Laplace table<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent"><p><h3>Calculations</h3></p><table id="a0000000620" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000621"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle 1[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac1{s}, \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s>0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.16)</td></tr><tr id="a0000000622"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle e^{rt}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac1{s-r}, \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s>\mathrm{Re}\, r[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.17)</td></tr><tr id="a0000000623"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \cos \omega t[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{s}{s^2+\omega ^2}, \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s>0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.18)</td></tr><tr id="a0000000624"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \sin \omega t[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{\omega }{s^2+\omega ^2}, \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s>0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.19)</td></tr><tr id="a0000000625"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle t[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac1{s^2}, \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s > 0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.20)</td></tr><tr id="a0000000626"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle t^ n[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{n!}{s^{n+1}}, \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s > 0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.21)</td></tr><tr id="a0000000627"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle t\sin (\omega t)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{2\omega s}{(s^2+\omega ^2)^2}, \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s>0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.22)</td></tr><tr id="a0000000628"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle t\cos (\omega t)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{s^2-\omega ^2}{(s^2+\omega ^2)^2}, \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s>0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.23)</td></tr><tr id="a0000000629"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \frac{1}{2\omega }t\sin (\omega t)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{s}{(s^2+\omega ^2)^2}, \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s>0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.24)</td></tr><tr id="a0000000630"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \frac{1}{2\omega ^2}\left(\frac{1}{\omega }\sin (\omega t)-t\cos (\omega t)\right)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{1}{(s^2+\omega ^2)^2} , \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s>0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.25)</td></tr></table><p><h3>Rules</h3></p><table id="a0000000631" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000632"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle f'(t)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle sF(s) - f(0), \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle t\text {-derivative rule}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.26)</td></tr><tr id="a0000000633"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle tf(t)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle -F'(s), \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
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[mathjaxinline]\displaystyle s\text {-derivative rule}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.27)</td></tr><tr id="a0000000634"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle e^{at}f(t)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle F(s-a), \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle s\text {-shift rule}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.28)</td></tr></table></div><p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px"><a href="javascript: {return false;}">Show</a></p></div></p><SCRIPT src="/assets/courseware/v1/631e447105fca1b243137b21b9ed6f90/asset-v1:OCW+18.031+2019_Spring+type@asset+block/latex2edx.js" type="text/javascript"/><LINK href="/assets/courseware/v1/daf81af0af57b85a105e0ed27b7873a0/asset-v1:OCW+18.031+2019_Spring+type@asset+block/latex2edx.css" rel="stylesheet" type="text/css"/>
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Inverse Laplace practice 1
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Find the inverse Laplace transform [mathjaxinline]\displaystyle \mathcal{L}^{-1}\left(\frac{4}{(s-5)^3}\right)[/mathjaxinline] </p>
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(Enter answer as a function as time [mathjaxinline]t[/mathjaxinline].) </p>
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<p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">Laplace table<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent"><p><h3>Calculations</h3></p><table id="a0000000640" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000641"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle 1[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac1{s}, \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s>0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.32)</td></tr><tr id="a0000000642"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle e^{rt}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac1{s-r}, \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s>\mathrm{Re}\, r[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.33)</td></tr><tr id="a0000000643"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \cos \omega t[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{s}{s^2+\omega ^2}, \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s>0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.34)</td></tr><tr id="a0000000644"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \sin \omega t[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{\omega }{s^2+\omega ^2}, \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s>0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.35)</td></tr><tr id="a0000000645"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle t[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac1{s^2}, \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s > 0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.36)</td></tr><tr id="a0000000646"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle t^ n[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{n!}{s^{n+1}}, \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s > 0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.37)</td></tr><tr id="a0000000647"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle t\sin (\omega t)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{2\omega s}{(s^2+\omega ^2)^2}, \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s>0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.38)</td></tr><tr id="a0000000648"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle t\cos (\omega t)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{s^2-\omega ^2}{(s^2+\omega ^2)^2}, \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s>0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.39)</td></tr><tr id="a0000000649"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \frac{1}{2\omega }t\sin (\omega t)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{s}{(s^2+\omega ^2)^2}, \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s>0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.40)</td></tr><tr id="a0000000650"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \frac{1}{2\omega ^2}\left(\frac{1}{\omega }\sin (\omega t)-t\cos (\omega t)\right)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{1}{(s^2+\omega ^2)^2} , \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s>0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.41)</td></tr></table><p><h3>Rules</h3></p><table id="a0000000651" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000652"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle f'(t)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle sF(s) - f(0), \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle t\text {-derivative rule}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.42)</td></tr><tr id="a0000000653"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle tf(t)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle -F'(s), \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle s\text {-derivative rule}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.43)</td></tr><tr id="a0000000654"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle e^{at}f(t)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle F(s-a), \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle s\text {-shift rule}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.44)</td></tr></table></div><p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px"><a href="javascript: {return false;}">Show</a></p></div></p><SCRIPT src="/assets/courseware/v1/631e447105fca1b243137b21b9ed6f90/asset-v1:OCW+18.031+2019_Spring+type@asset+block/latex2edx.js" type="text/javascript"/><LINK href="/assets/courseware/v1/daf81af0af57b85a105e0ed27b7873a0/asset-v1:OCW+18.031+2019_Spring+type@asset+block/latex2edx.css" rel="stylesheet" type="text/css"/>
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Inverse Laplace practice 2
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Find the inverse Laplace transform [mathjaxinline]\displaystyle \mathcal{L}^{-1}\left(\frac{1}{s^2+4s+13}\right)[/mathjaxinline]. <br/>(This will look better if you complete the square. Use the Laplace table above, linearity, and the [mathjaxinline]s[/mathjaxinline]-shift rule.) </p>
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(Enter answer as a function as time [mathjaxinline]t[/mathjaxinline].) </p>
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<h2 class="hd hd-2 unit-title">5.7. The Pole Diagram and Homogeneous Solutions.</h2>
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<p>In <a href="/courses/course-v1:OCW+18.031+2019_Spring/courseware/week1/lec2/1" target="_blank">Lecture 2</a>, we learned how to read stability information of an LTI system from the pole diagram of its transfer function. In <a href="/courses/course-v1:OCW+18.031+2019_Spring/courseware/week2/lec3/1" target="_blank">Lecture 3</a>, we learned how to take the Laplace Transform of a function [mathjaxinline]\, f(t) \,[/mathjaxinline] in the time domain, and express it instead in the frequency domain, [mathjaxinline]\, F(s)[/mathjaxinline]. This gave us a <a href="/courses/course-v1:OCW+18.031+2019_Spring/courseware/week2/lec4/1" target="_blank">new method</a> to find a system response of an LTI system to a wide variety of input signals using algebra, partial fractions, and inverse Laplace Transforms read off of a table.</p>
<p>In practice, the real power of Laplace transform is that it gives you very useful information about the solutions <i class="itshape">without</i> requiring you to solve it all out explicitly. In particular, <b class="bf">the pole diagram of the transfer function <b class="bfseries"><span style="color: #ff7800;">is</span></b> the pole diagram of a generic zero input response (ZIR)</b>. (Recall that the zero input response is the solution when the input is [mathjaxinline]0[/mathjaxinline], or the solution to the associated homogeneous equation.) This means that important information about the homogeneous solutions can be read off directly from the pole diagram of the transfer function.</p>
<ul class="itemize">
<li>
<p>a real pole of [mathjaxinline]F(s)[/mathjaxinline] at [mathjaxinline]s=a[/mathjaxinline] comes from a term in [mathjaxinline]f(t)[/mathjaxinline] of the form [mathjaxinline]e^{at}[/mathjaxinline].</p>
</li>
<li>
<p>a conjugate pair [mathjaxinline]a\pm i\omega[/mathjaxinline] comes from a term in [mathjaxinline]f(t)[/mathjaxinline] of the form [mathjaxinline]e^{at}\cos (\omega t-\phi )[/mathjaxinline].</p>
</li>
</ul>
<p><b class="bf">Examples</b></p>
<ol class="enumerate">
<li value="1">
<p>An LTI system has the following transfer function:</p>
<table id="a0000000662" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout: auto;">
<tbody>
<tr>
<td class="equation" style="width: 80%; border: none;">[mathjax]H(s) = \frac{Q(s)}{P(s)} = \frac{1}{(s^2+2s+2)(s+2)} = \frac{As+B}{s^2+2s+2} + \frac{C}{s+2}.[/mathjax]</td>
<td class="eqnnum" style="width: 20%; border: none;"> </td>
</tr>
</tbody>
</table>
<p>This system is stable since the poles [mathjaxinline]s=-1\pm i, -2[/mathjaxinline] have negative real part.</p>
<center><img src="/assets/courseware/v1/b9804e1c46fa6d583b4dbe963a6b10e8/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_c4-polesexample1.svg" width="300px" style="margin: 0px 10px 10px 10px;" /></center>
<p>The pole diagram implies that the general zero input response is</p>
<table id="a0000000663" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout: auto;">
<tbody>
<tr>
<td class="equation" style="width: 80%; border: none;">[mathjax]Ae^{-t}\cos (t) + Be^{-t}\sin (t) + Ce^{-2t} = De^{-t}\cos (t-\phi ) + Ce^{-2t}.[/mathjax]</td>
<td class="eqnnum" style="width: 20%; border: none;"> </td>
</tr>
</tbody>
</table>
<p>We can read this homogeneous solution immediately off of the pole diagram, which is the pole diagram for the Laplace transform of a generic homogeneous solution.</p>
<p>The poles at [mathjaxinline]-1\pm i[/mathjaxinline] contribute to a transient term [mathjaxinline]De^{-t}\cos (t-\phi )[/mathjaxinline]. The pole at [mathjaxinline]s=-2[/mathjaxinline] contributes to a transient term [mathjaxinline]Ce^{-2t}[/mathjaxinline].</p>
</li>
<li value="2">
<p>A certain LTI system has the following pole diagram.</p>
<center><img src="/assets/courseware/v1/0d093acb5742f8337be6d53a1e5a123b/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_c4-polesexample2.svg" width="300px" style="margin: 0px 10px 10px 10px;" /></center>
<p>From the pole diagram, we see that a generic homogeneous solution for this system takes the form</p>
<table id="a0000000664" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout: auto;">
<tbody>
<tr>
<td class="equation" style="width: 80%; border: none;">[mathjax]A_1e^{-t} + Be^{-2t} + C\cos (2t) + D\sin (2t).[/mathjax]</td>
<td class="eqnnum" style="width: 20%; border: none;"> </td>
</tr>
</tbody>
</table>
<p>The first two terms coming from poles in the left half plane are transient and tend to zero as [mathjaxinline]t[/mathjaxinline] tends to [mathjaxinline]\infty[/mathjaxinline]. The poles along the imaginary axis correspond to</p>
<table id="a0000000665" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout: auto;">
<tbody>
<tr>
<td class="equation" style="width: 80%; border: none;">[mathjax]C\cos (2t) + D\sin (2t).[/mathjax]</td>
<td class="eqnnum" style="width: 20%; border: none;"> </td>
</tr>
</tbody>
</table>
<p>This system is not stable since the poles do not all have negative real part.</p>
</li>
</ol>
<div class="hideshowbox">
<h4 onclick="hideshow(this);" style="margin: 0px;">Summary of new terminology as it relates to old terminology<span class="icon-caret-down toggleimage"></span></h4>
<div class="hideshowcontent">
<p>Given an LTI system [mathjaxinline]P(D)x = Q(D)y[/mathjaxinline], where [mathjaxinline]P[/mathjaxinline] and [mathjaxinline]Q[/mathjaxinline] have no common factors. Then the transfer function is [mathjaxinline]\displaystyle H(s)= \frac{Q(s)}{P(s)}[/mathjaxinline].</p>
<table class="tabular" cellspacing="0" style="table-layout: auto;">
<tbody>
<tr>
<td style="text-align: left; border: none;"><b class="bf">New terminology</b></td>
<td style="text-align: left; border: none;"> </td>
<td style="text-align: left; border: none;"><b class="bf">Old terminology</b></td>
</tr>
<tr>
<td style="text-align: left; border: none;">Poles of [mathjaxinline]H(s)[/mathjaxinline]</td>
<td style="text-align: left; border: none;">=</td>
<td style="text-align: left; border: none;">The zeros of [mathjaxinline]P(D)[/mathjaxinline]</td>
</tr>
<tr>
<td style="text-align: left; border: none;">ZIRs (zero input responses)</td>
<td style="text-align: left; border: none;">=</td>
<td style="text-align: left; border: none;">Homogeneous solutions (solutions to [mathjaxinline]P(D)x = 0[/mathjaxinline])</td>
</tr>
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Determine long-term behavior from pole diagram
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A system has the following pole diagram. </p>
<center>
<img src="/assets/courseware/v1/67112c2f26262a98cb9f945120f72d0b/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_c4-polesexample3.svg" width="300px" style="margin: 0px 10px 10px 10px"/>
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<p>
Determine which terms exist as summands of <b class="bf">some</b> zero input response (ZIR). (Choose all that apply.) <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lec5-tab7-problem1_2_1">
<fieldset aria-describedby="status_lec5-tab7-problem1_2_1">
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<input type="checkbox" name="input_lec5-tab7-problem1_2_1[]" id="input_lec5-tab7-problem1_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="lec5-tab7-problem1_2_1-choice_0-label" for="input_lec5-tab7-problem1_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_lec5-tab7-problem1_2_1"> <text>[mathjaxinline]e^{-t}[/mathjaxinline]</text>
</label>
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<input type="checkbox" name="input_lec5-tab7-problem1_2_1[]" id="input_lec5-tab7-problem1_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="lec5-tab7-problem1_2_1-choice_1-label" for="input_lec5-tab7-problem1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lec5-tab7-problem1_2_1"> <text>[mathjaxinline]e^{-2t}[/mathjaxinline]</text>
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<input type="checkbox" name="input_lec5-tab7-problem1_2_1[]" id="input_lec5-tab7-problem1_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="lec5-tab7-problem1_2_1-choice_2-label" for="input_lec5-tab7-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lec5-tab7-problem1_2_1"> <text>[mathjaxinline]\cos (t-\phi )[/mathjaxinline]</text>
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<input type="checkbox" name="input_lec5-tab7-problem1_2_1[]" id="input_lec5-tab7-problem1_2_1_choice_6" class="field-input input-checkbox" value="choice_6"/><label id="lec5-tab7-problem1_2_1-choice_6-label" for="input_lec5-tab7-problem1_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_lec5-tab7-problem1_2_1"> <text>[mathjaxinline]e^{-2t}\cos (t-\phi )[/mathjaxinline]</text>
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<input type="checkbox" name="input_lec5-tab7-problem1_2_1[]" id="input_lec5-tab7-problem1_2_1_choice_7" class="field-input input-checkbox" value="choice_7"/><label id="lec5-tab7-problem1_2_1-choice_7-label" for="input_lec5-tab7-problem1_2_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_lec5-tab7-problem1_2_1"> <text>[mathjaxinline]e^{-t}\cos (2t-\phi )[/mathjaxinline]</text>
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Long term behavior of ZIRs
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Recall that there is an infinite family of ZIR's, with each pole in the pole diagram determining one parameter. </p>
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True or false: all ZIRs of the system with the given pole diagram tend to zero as time grows large. </p>
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<h2 class="hd hd-2 unit-title">5.8. Region of convergence again.</h2>
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<p>
We defined the Laplace transform [mathjaxinline]F(s)=\mathcal{L}(f(t);s)[/mathjaxinline] using an improper integral, and made the point that this integral will generally only converge for [mathjaxinline]s[/mathjaxinline] to the right of some vertical line in the complex plane. Now we can understand better why the convergence fails: it fails when you hit a pole. Generally, the region of convergence is the half-plane to the right of the rightmost pole. </p><p>
On the other hand, we have just been discussing the pole diagram of functions such as [mathjaxinline]F(s)[/mathjaxinline], apparently making sense of it outside its region of convergence. What's going on? </p><p>
The fact is that knowledge of functions such as [mathjaxinline]F(s)[/mathjaxinline] in some right half plane suffices to determine a function defined everywhere in the complex plane (except for a scattering of points, the poles). You will learn about this if you take a course in complex variables; the process is known as <b class="bfseries"><span style="color:#0000FF">analytic continuation.</span></b> </p><p>
Most of the functions [mathjaxinline]F(s)[/mathjaxinline] that arise in this course are actually rational functions. Their expression, as a quotient of one polynomial by another, allows computation of the values of the function for almost every complex number. It provides us with the analytic continuation of the function given by the integral expression for the Laplace transform. </p>
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