Amplitude and Frequency Control: Stability of Limit Cycles in Phase-Shift and Twin-T Oscillators

<table>Waveforms of the Twin-T transient phenomenon with <svg height="14.6" id="M105" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 61.212502 14.6" width="61.212502" xmlns="http://www.w3.org/2000/svg">
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<text transform="matrix(1,0,0,-1,-65.7,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">1</tspan>
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<text transform="matrix(1,0,0,-1,-57.35,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">&lt;</tspan>
<tspan style="font-size: 12.50px; " x="12.027886" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="18.279387" y="0">/</tspan>
<tspan style="font-size: 12.50px; " x="21.75522" y="0">2</tspan>
<tspan style="font-size: 12.50px; " x="28.00672" y="0">9</tspan>
</text>
</g>
</g>
</svg>; <svg height="14.6" id="M106" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 61.212502 14.6" width="61.212502" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.6)">
<g transform="translate(72,-60.32)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝜂</tspan>
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<text transform="matrix(1,0,0,-1,-65.7,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">1</tspan>
</text>
<text transform="matrix(1,0,0,-1,-57.35,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.027886" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="18.279387" y="0">/</tspan>
<tspan style="font-size: 12.50px; " x="21.75522" y="0">2</tspan>
<tspan style="font-size: 12.50px; " x="28.00672" y="0">9</tspan>
</text>
</g>
</g>
</svg>;
<svg height="11.3125" id="M107" style="vertical-align:-3.2316pt" version="1.1" viewbox="0 0 12.0625 11.3125" width="12.0625" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,11.3125)">
<g transform="translate(72,-62.95)">
<text transform="matrix(1,0,0,-1,-71.95,66.22)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑣</tspan>
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<text transform="matrix(1,0,0,-1,-65.62,63.1)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑖</tspan>
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</svg>; <svg height="14.9" id="M108" style="vertical-align:-3.2316pt" version="1.1" viewbox="0 0 114.65 14.9" width="114.65" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.9)">
<g transform="translate(72,-60.08)">
<text transform="matrix(1,0,0,-1,-71.95,63.35)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑣</tspan>
</text>
<text transform="matrix(1,0,0,-1,-65.62,60.22)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑖</tspan>
</text>
<text transform="matrix(1,0,0,-1,-58.93,63.35)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.027886" y="0">−</tspan>
<tspan style="font-size: 12.50px; " x="20.592442" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="26.843941" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="29.96969" y="0">5</tspan>
<tspan style="font-size: 12.50px; " x="36.221191" y="0">3</tspan>
<tspan style="font-size: 12.50px; " x="42.472691" y="0">5</tspan>
<tspan style="font-size: 12.50px; " x="48.72419" y="0">7</tspan>
<tspan style="font-size: 12.50px; " x="54.975693" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="61.227192" y="0">4</tspan>
<tspan style="font-size: 12.50px; " x="69.566689" y="0">V</tspan>
</text>
</g>
</g>
</svg>; X-input: <svg height="14.9375" id="M109" style="vertical-align:-3.25793pt" version="1.1" viewbox="0 0 63.037498 14.9375" width="63.037498" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.9375)">
<g transform="translate(72,-60.05)">
<text transform="matrix(1,0,0,-1,-71.95,63.35)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑣</tspan>
</text>
<text transform="matrix(1,0,0,-1,-65.62,60.22)">
<tspan style="font-size: 8.75px; " x="0" y="0">0</tspan>
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<text transform="matrix(1,0,0,-1,-57.28,63.35)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.040389" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="18.291889" y="0">5</tspan>
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</g>
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</svg>; Y-input: <svg height="11.3125" id="M110" style="vertical-align:-3.2316pt" version="1.1" viewbox="0 0 12.0625 11.3125" width="12.0625" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,11.3125)">
<g transform="translate(72,-62.95)">
<text transform="matrix(1,0,0,-1,-71.95,66.22)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑣</tspan>
</text>
<text transform="matrix(1,0,0,-1,-65.62,63.1)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑖</tspan>
</text>
</g>
</g>
</svg>.</table>

Active and Passive Electronic Components

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Figure 3

Figure 3: Amplitude and Frequency Control: Stability of Limit Cycles in Phase-Shift and Twin-T Oscillators 