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<h2 class="hd hd-2 unit-title">2.1. Challenge question</h2>
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<h3 class="hd hd-3 problem-header">Challenge question 1</h3>
<p>A beautiful mathlet relating zero-pole diagrams with Bode and Nyquist plots is below.</p>
<p><iframe style="display: block; border-width: 0px; padding: 0px;" src="https://mathlets1803.surge.sh/bodeNyquistPlot.html" width="1100 px" height="640 px">
<p>
Use this mathlet to design a stable filter with 3 poles and 3 zeros that filters out frequencies in the range [0, 2] and has gain near one for [mathjaxinline]\omega > 2[/mathjaxinline]. (Click the Linear Bode button.) </p>
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<h4 onclick="hideshow(this);" style="margin: 0px;">Possible answer<span class="icon-caret-down toggleimage"></span></h4>
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One possible answer is shown below. Note that there are zeros at [mathjaxinline]0[/mathjaxinline], and [mathjaxinline]\pm 2[/mathjaxinline], a pole pair near [mathjaxinline]\pm 2[/mathjaxinline], and a real pole to help modulate the amplitudes in the Linear Bode plot (also pictured). <img src="/assets/courseware/v1/7e15e745dd301da4d4bd12e4fb9a7727/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_c2-challenge.png" width="800" /> </p>
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Note that to have a gain near [mathjaxinline]1[/mathjaxinline] for large [mathjaxinline]\omega[/mathjaxinline], we must have the same number of poles and zeros. </p>
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<p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px;"><a>Show</a></p>
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<h3 class="hd hd-3 problem-header">Challenge question 2</h3>
<p>
Use this mathlet to design a stable pass-band filter that suppresses frequencies less than about 2, having a gain close to 1 for frequencies between 2 and 4, and whose gain graph falls off to zero after that; using two pole pairs and some number of zeros. </p>
<iframe style="display: block; border-width: 0px; padding: 0px;" src="https://mathlets1803.surge.sh/bodeNyquistPlot1803.html" width="1100 px" height="640 px">
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<h4 onclick="hideshow(this);" style="margin: 0px;">Possible answer<span class="icon-caret-down toggleimage"></span></h4>
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We want to have a zero at [mathjaxinline]0[/mathjaxinline], which means we are going to have an odd number of zeros. Note that to have a gain near [mathjaxinline]0[/mathjaxinline] for large frequencies we must have the number of poles greater than the number of zeros. Since we have 4 poles, we can have [mathjaxinline]1[/mathjaxinline] or [mathjaxinline]3[/mathjaxinline] zeros. </p>
<p>
One possible answer is shown below. Note that there are zeros at [mathjaxinline]0[/mathjaxinline], and [mathjaxinline]\pm 1[/mathjaxinline], a pole pair near [mathjaxinline]\pm 2[/mathjaxinline], and a pole pair near [mathjaxinline]\pm 4[/mathjaxinline]. <img src="/assets/courseware/v1/e038c926f8f0defb0193aac21b776e6e/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_c2-challenge2.png" width="800" /> </p>
<p>
Optimizing the pole diagram to produce a given gain curve is an art; to achieve more control you have to use more poles (and zeros), which means a higher order system. </p>
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<p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px;"><a>Show</a></p>
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