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2022-06-07 03:18:09 +08:00
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<h1 dir="auto" id="chapter-1">Chapter 1</h1>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mrow><mi>d</mi><mi>q</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>⇔</mo><mi>q</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo>∫</mo><mi>i</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>d</mi><mi>t</mi></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>P</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>v</mi><mo>×</mo><mi>i</mi><mo>=</mo><mi>i</mi><mo>×</mo><mfrac><mi>W</mi><mi>q</mi></mfrac></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>v</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mi>W</mi><mi>q</mi></mfrac><mo>⇔</mo><mi>W</mi><mo>=</mo><mi>v</mi><mo>×</mo><mi>q</mi><mo>=</mo><mo>∫</mo><mi>v</mi><mo>×</mo><mi>i</mi><mo>⋅</mo><mi>d</mi><mi>t</mi></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">
\begin{align}
i(t)&amp;=\frac{dq}{dt} \Leftrightarrow q(t)=\int i(t)\cdot dt\\
P&amp;=v\times i=i\times \frac{W}{q}\\
v&amp;=\frac{W}{q} \Leftrightarrow W=v\times q = \int v\times i \cdot dt\\
\end{align}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:7.6152em;vertical-align:-3.5576em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.0576em;"><span style="top:-6.0576em;"><span class="pstrut" style="height:3.3714em;"></span><span class="mord"><span class="mord mathnormal">i</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span><span style="top:-3.535em;"><span class="pstrut" style="height:3.3714em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span><span style="top:-0.9943em;"><span class="pstrut" style="height:3.3714em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.5576em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.0576em;"><span style="top:-6.0576em;"><span class="pstrut" style="height:3.3714em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">i</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.535em;"><span class="pstrut" style="height:3.3714em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span
</math>
<h1 dir="auto" id="chapter-4">Chapter 4</h1>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Load line: </mtext><msub><mi>i</mi><mi>x</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>−</mo><mfrac><msub><mi>v</mi><mi>x</mi></msub><msub><mi>R</mi><mi>T</mi></msub></mfrac><mo>+</mo><mfrac><msub><mi>v</mi><mi>t</mi></msub><msub><mi>R</mi><mi>T</mi></msub></mfrac></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>General: </mtext><msub><mi>A</mi><mi>v</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><msub><mi>v</mi><mrow><mi>o</mi><mi>u</mi><mi>t</mi></mrow></msub><msub><mi>v</mi><mrow><mi>i</mi><mi>n</mi></mrow></msub></mfrac></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Inverting: </mtext><msub><mi>A</mi><mi>v</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>−</mo><mfrac><msub><mi>R</mi><mi>f</mi></msub><msub><mi>R</mi><mrow><mi>i</mi><mi>n</mi></mrow></msub></mfrac></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Non Inverting: </mtext><msub><mi>A</mi><mi>v</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><msub><mi>R</mi><mi>f</mi></msub><msub><mi>R</mi><mrow><mi>i</mi><mi>n</mi></mrow></msub></mfrac></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">
\begin{align}
\text{Load line: } i_x &amp;= -\frac{v_x}{R_T}+ \frac{v_t}{R_T}\\
\text{General: } A_v &amp;= \frac{v_{out}}{v_{in}}\\
\text{Inverting: } A_v &amp;= -\frac{R_f}{R_{in}}\\
\text{Non Inverting: } A_v &amp;= 1+\frac{R_f}{R_{in}}\\
\end{align}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:9.4798em;vertical-align:-4.4899em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.9899em;"><span style="top:-7.2427em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord text"><span class="mord">Load line: </span></span><span class="mord"><span class="mord mathnormal">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.9991em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord text"><span class="mord">General: </span></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.5028em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord text"><span class="mord">Inverting: </span></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-0.0064em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord text"><span class="mord">Non Inverting: </span></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.4899em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.9899em;"><span style="top:-7.2427em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span c
</math>
<h2 dir="auto" id="inverting">Inverting</h2>
<div dir="auto" ></div>
<!-- ![](2022-05-31-15-52-51.png) -->
<h2 dir="auto" id="non-inverting">Non-inverting</h2>
<div dir="auto" ></div>
<!-- ![](2022-05-31-15-53-03.png) -->
<h1 dir="auto" id="chapter-5">Chapter 5</h1>
<h2 dir="auto" id="capacitor">Capacitor</h2>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>C</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mi>q</mi><mi>v</mi></mfrac></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>C</mi><mfrac><mrow><mi>d</mi><mi>v</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇔</mo><mi>v</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>C</mi></mfrac><msubsup><mo>∫</mo><mn>0</mn><mi>t</mi></msubsup><mi>i</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>d</mi><mi>t</mi></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Series: </mtext><mfrac><mn>1</mn><msub><mi>C</mi><mi>T</mi></msub></mfrac></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mi>N</mi></munderover><mfrac><mn>1</mn><msub><mi>C</mi><mi>i</mi></msub></mfrac></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Parallel: </mtext><msub><mi>C</mi><mi>T</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mi>N</mi></munderover><msub><mi>C</mi><mi>i</mi></msub></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Energy: </mtext><mi>E</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>C</mi><msup><mi>v</mi><mn>2</mn></msup></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">
\begin{align}
C &amp;= \frac{q}{v}\\
i(t) = C\frac{dv}{dt} &amp;\Leftrightarrow v(t) = \frac{1}{C}\int_0^t i(t)\cdot dt\\
\text{Series: }\frac{1}{C_T} &amp;= \sum^N_{i=0}\frac{1}{C_i}\\
\text{Parallel: }C_T &amp;= \sum^N_{i=0}C_i\\
\text{Energy: }E &amp;= \frac{1}{2}Cv^2
\end{align}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:13.9684em;vertical-align:-6.7342em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:7.2342em;"><span style="top:-9.955em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span><span style="top:-7.4255em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord mathnormal">i</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-4.3852em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord text"><span class="mord">Series: </span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-0.9792em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord text"><span class="mord">Parallel: </span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T
</math>
<h3 dir="auto" id="differential-equation-solution">Differential equation solution</h3>
<p dir="auto">Where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mi>s</mi></msub><mo>=</mo><msub><mi>v</mi><mi mathvariant="normal">∞</mi></msub></mrow><annotation encoding="application/x-tex">v_s=v_\infty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">∞</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>τ</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>R</mi><mo>×</mo><mi>C</mi></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>v</mi><mi>C</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtable rowspacing="0.16em" columnalign="left right" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>v</mi><mn>0</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>t</mi><mo>≤</mo><mn>0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>v</mi><mi mathvariant="normal">∞</mi></msub><mo>+</mo><mo stretchy="false">(</mo><msub><mi>v</mi><mn>0</mn></msub><mo>−</mo><msub><mi>v</mi><mi mathvariant="normal">∞</mi></msub><mo stretchy="false">)</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>t</mi><mi mathvariant="normal">/</mi><mi>τ</mi></mrow></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>t</mi><mo>&gt;</mo><mn>0</mn></mrow></mstyle></mtd></mtr></mtable></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi>v</mi><mn>0</mn></msub><msup><mi>e</mi><mrow><mo>−</mo><mi>t</mi><mi mathvariant="normal">/</mi><mi>τ</mi></mrow></msup><mtext> (Natural response, no input)</mtext></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi>v</mi><mi mathvariant="normal">∞</mi></msub><mrow><mo fence="true">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>t</mi><mi mathvariant="normal">/</mi><mi>τ</mi></mrow></msup><mo fence="true">)</mo></mrow><mtext> (Forced response, input)</mtext></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>i</mi><mi>C</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mrow><msub><mi>v</mi><mi>s</mi></msub><mo>−</mo><msub><mi>v</mi><mi>C</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mi>R</mi></mfrac><mo>=</mo><mfrac><mrow><mo>−</mo><mo stretchy="false">(</mo><msub><mi>v</mi><mn>0</mn></msub><mo>−</mo><msub><mi>v</mi><mi mathvariant="normal">∞</mi></msub><mo stretchy="false">)</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>t</mi><mi mathvariant="normal">/</mi><mi>τ</mi></mrow></msup></mrow><mi>R</mi></mfrac></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">
\begin{align}
\tau &amp;= R\times C\\
v_C(t) &amp;=
\begin{cases}
\begin{array}{lr}
v_0 &amp; t\leq 0\\
v_\infty+(v_0-v_\infty)e^{-t/\tau} &amp; t &gt; 0
\end{array}
\end{cases}\\
&amp; v_0 e^{-t/\tau} \text{ (Natural response, no input)}\\
&amp; v_\infty\left(1-e^{-t/\tau}\right) \text{ (Forced response, input)}\\
i_C(t) &amp;= \frac{v_s-v_C(t)}{R}=\frac{-(v_0-v_\infty)e^{-t/\tau}}{R}
\end{align}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:10.497em;vertical-align:-4.9985em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.4985em;"><span style="top:-8.2235em;"><span class="pstrut" style="height:3.565em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span></span></span><span style="top:-6.0895em;"><span class="pstrut" style="height:3.565em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span><span style="top:-3.8775em;"><span class="pstrut" style="height:3.565em;"></span><span class="mord"></span></span><span style="top:-2.0675em;"><span class="pstrut" style="height:3.565em;"></span><span class="mord"></span></span><span style="top:0.4475em;"><span class="pstrut" style="height:3.565em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.9985em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.4985em;"><span style="top:-8.2235em;"><span class="pstrut" style="height:3.565em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span><span style="top:-6.0895em;"><span class="pstrut" style="height:3.565em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.474em;"><span style="top:-3.474em;"><span class="pstrut" style="height:3.474em;"></span><span class="mord"><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.474em;"><span style="top
</math>
<ol dir="auto">
<li dir="auto">Remove all independent sources, find equivalent resistance and capacitance, find <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span></span></span></span>.</li>
<li dir="auto">Set C as open circuit, find initial capacitor voltage <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">v_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> at <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">t=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></li>
<li dir="auto">Set C as open circuit, find final capacitor voltage <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mi mathvariant="normal">∞</mi></msub></mrow><annotation encoding="application/x-tex">v_\infty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">∞</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> at <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">t\to\infty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord">∞</span></span></span></span></li>
</ol>
<h2 dir="auto" id="inductor">Inductor</h2>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>L</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mi>λ</mi><mi>i</mi></mfrac></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>v</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>L</mi><mfrac><mrow><mi>d</mi><mi>i</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇔</mo><mi>i</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>L</mi></mfrac><msubsup><mo>∫</mo><mn>0</mn><mi>t</mi></msubsup><mi>v</mi><mo>⋅</mo><mi>d</mi><mi>t</mi></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Series: </mtext><msub><mi>L</mi><mi>T</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mi>N</mi></munderover><msub><mi>L</mi><mi>i</mi></msub></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Parallel: </mtext><mfrac><mn>1</mn><msub><mi>L</mi><mi>T</mi></msub></mfrac></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mi>N</mi></munderover><mfrac><mn>1</mn><msub><mi>L</mi><mi>i</mi></msub></mfrac></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Energy: </mtext><mi>E</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>L</mi><msup><mi>i</mi><mn>2</mn></msup></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">
\begin{align}
L &amp;= \frac{\lambda}{i}\\
v(t)=L\frac{di}{dt} &amp;\Leftrightarrow i(t)=\frac{1}{L}\int_0^tv\cdot dt\\
\text{Series: }L_T &amp;= \sum^N_{i=0}L_i\\
\text{Parallel: }\frac{1}{L_T} &amp;= \sum^N_{i=0}\frac{1}{L_i}\\
\text{Energy: }E &amp;= \frac{1}{2}Li^2
\end{align}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:14.2323em;vertical-align:-6.8661em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:7.3661em;"><span style="top:-9.823em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord mathnormal">L</span></span></span><span style="top:-7.2936em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">L</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-4.2533em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord text"><span class="mord">Series: </span></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-0.8473em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord text"><span class="mord">Parallel: </span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span cla
</math>
<h3 dir="auto" id="differential-equation-solution-1">Differential equation solution</h3>
<p dir="auto">Where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mi>s</mi></msub><mi mathvariant="normal">/</mi><mi>R</mi><mo>=</mo><msub><mi>i</mi><mi mathvariant="normal">∞</mi></msub></mrow><annotation encoding="application/x-tex">v_s/R=i_\infty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">∞</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>τ</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mi>L</mi><mi>R</mi></mfrac></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtable rowspacing="0.16em" columnalign="left right" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>i</mi><mn>0</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>t</mi><mo>≤</mo><mn>0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>i</mi><mi mathvariant="normal">∞</mi></msub><mo>+</mo><mo stretchy="false">(</mo><msub><mi>i</mi><mn>0</mn></msub><mo>−</mo><msub><mi>i</mi><mi mathvariant="normal">∞</mi></msub><mo stretchy="false">)</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>t</mi><mi mathvariant="normal">/</mi><mi>τ</mi></mrow></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>t</mi><mo>&gt;</mo><mn>0</mn></mrow></mstyle></mtd></mtr></mtable></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi>i</mi><mn>0</mn></msub><msup><mi>e</mi><mrow><mo>−</mo><mi>t</mi><mi mathvariant="normal">/</mi><mi>τ</mi></mrow></msup><mtext> (Natural response, no input)</mtext></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi>i</mi><mi mathvariant="normal">∞</mi></msub><mrow><mo fence="true">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>t</mi><mi mathvariant="normal">/</mi><mi>τ</mi></mrow></msup><mo fence="true">)</mo></mrow><mtext> (Forced response, input)</mtext></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">
\begin{align}
\tau &amp;= \frac{L}{R}\\
i(t) &amp;=
\begin{cases}
\begin{array}{lr}
i_0 &amp; t\leq 0\\
i_\infty+(i_0-i_\infty)e^{-t/\tau} &amp; t &gt; 0
\end{array}
\end{cases}\\
&amp; i_0 e^{-t/\tau} \text{ (Natural response, no input)}\\
&amp; i_\infty\left(1-e^{-t/\tau}\right) \text{ (Forced response, input)}\\
\end{align}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:8.7924em;vertical-align:-4.1462em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.6462em;"><span style="top:-6.7598em;"><span class="pstrut" style="height:3.474em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span></span></span><span style="top:-4.2998em;"><span class="pstrut" style="height:3.474em;"></span><span class="mord"><span class="mord mathnormal">i</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span><span style="top:-2.0878em;"><span class="pstrut" style="height:3.474em;"></span><span class="mord"></span></span><span style="top:-0.2778em;"><span class="pstrut" style="height:3.474em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.1462em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.6462em;"><span style="top:-6.7598em;"><span class="pstrut" style="height:3.474em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-4.2998em;"><span class="pstrut" style="height:3.474em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.474em;"><span style="top:-3.474em;"><span class="pstrut" style="height:3.474em;"></span><span class="mord"><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.474em;"><span style="top:-3.634em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.386em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">i</span><span class="msupsub"><
</math>
<h2 dir="auto" id="voltage-drop-in-dc-for-capacitor-and-inductor-at-steady-state">Voltage drop in DC for capacitor and inductor at steady state</h2>
<pre><code dir="auto"><code><div>CAPACITOR: INDUCTOR:
v_T _ v_T _
| &lt;- V_1 | &lt;- V_1
C1 = ) &lt;- V_D1 L1 3 ) &lt;- V_D1
| |
C2 = C2 3
| |
... ...
| |
CN = LN 3
| |
GND * GND *
</div></code></code></pre>
<h2 dir="auto" id="capacitor-1">Capacitor</h2>
<p dir="auto">Current through capacitors in series is the same, so all capacitors have same charge stored <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span></span></span></span>.</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow><mtext>Voltage drop over capacitor </mtext><mstyle scriptlevel="0" displaystyle="false"><mi>i</mi></mstyle><mtext>: </mtext></mrow><msub><mi>v</mi><mrow><mi>D</mi><mi>i</mi></mrow></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi>v</mi><mi>T</mi></msub><mfrac><msub><mi>C</mi><mi>T</mi></msub><msub><mi>C</mi><mi>i</mi></msub></mfrac></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Voltage divider: </mtext><msub><mi>v</mi><mi>i</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi>v</mi><mi>T</mi></msub><mfrac><msub><mi>C</mi><mi>T</mi></msub><mrow><mfrac><mn>1</mn><msub><mi>C</mi><mi>i</mi></msub></mfrac><mo>+</mo><mfrac><mn>1</mn><msub><mi>C</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mfrac><mo>+</mo><mo>⋯</mo><mo>+</mo><mfrac><mn>1</mn><msub><mi>C</mi><mi>N</mi></msub></mfrac></mrow></mfrac></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align}
\text{Voltage drop over capacitor $i$: } v_{Di} &amp;= v_T\frac{C_T}{C_i}\\
\text{Voltage divider: } v_i &amp;= v_T\frac{C_T}{\frac{1}{C_i}+\frac{1}{C_{i+1}}+\dots+\frac{1}{C_N}}\\
\end{align}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.3785em;vertical-align:-2.4393em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9393em;"><span style="top:-4.9393em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord text"><span class="mord">Voltage drop over capacitor </span><span class="mord mathnormal">i</span><span class="mord">: </span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">D</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.4429em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord text"><span class="mord">Voltage divider: </span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.4393em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9393em;"><span style="top:-4.9393em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></sp
<h2 dir="auto" id="inductor-1">Inductor</h2>
<p dir="auto">No voltage drop in steady state (Inductor is a short circuit)</p>
<h1 dir="auto" id="chapter-7">Chapter 7</h1>
<h2 dir="auto" id="maximum-power-transfer-in-ac">Maximum power transfer in AC</h2>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mtext>Condition: </mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mover accent="true"><msub><mi>Z</mi><mi>L</mi></msub><mo stretchy="true">‾</mo></mover><mo>=</mo><mover accent="true"><msubsup><mi>Z</mi><mi>S</mi><mo>∗</mo></msubsup><mo stretchy="true">‾</mo></mover></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mtext>Maximum power to load (50%): </mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mn>2</mn><msub><mi>P</mi><mtext>avg</mtext></msub><mo>=</mo><msub><mi>P</mi><mtext>rms</mtext></msub><mo>⋅</mo><msqrt><mn>2</mn></msqrt><mo>=</mo><msub><mi>P</mi><mtext>max</mtext></msub><mo>=</mo><mfrac><msup><mrow><mi mathvariant="normal">∣</mi><msub><mi>V</mi><mi>S</mi></msub><mi mathvariant="normal">∣</mi></mrow><mn>2</mn></msup><mrow><mn>4</mn><msub><mi>R</mi><mi>S</mi></msub></mrow></mfrac></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><msup><mrow><mi mathvariant="normal">∣</mi><msub><mi>V</mi><mi>L</mi></msub><mi mathvariant="normal">∣</mi></mrow><mn>2</mn></msup><mrow><mn>4</mn><msub><mi>R</mi><mi>L</mi></msub></mrow></mfrac></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mtext>Total maximum power: </mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mn>2</mn><msub><mi>P</mi><mtext>avg</mtext></msub><mo>=</mo><msub><mi>P</mi><mtext>rms</mtext></msub><mo>⋅</mo><msqrt><mn>2</mn></msqrt><mo>=</mo><msub><mi>P</mi><mtext>max</mtext></msub><mo>=</mo><mfrac><msup><mrow><mi mathvariant="normal">∣</mi><msub><mi>V</mi><mi>S</mi></msub><mi mathvariant="normal">∣</mi></mrow><mn>2</mn></msup><mrow><mn>2</mn><msub><mi>R</mi><mi>S</mi></msub></mrow></mfrac></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">
\begin{align}
\text{Condition: } &amp;\overline{Z_L} = \overline{Z_S^*}\\
\text{Maximum power to load (50\%): } &amp;2P_\text{avg}=P_\text{rms}\cdot\sqrt{2} = P_\text{max} = \frac{{|V_S|}^2}{4R_S}\\
&amp; = \frac{{|V_L|}^2}{4R_L}\\
\text{Total maximum power: } &amp;2P_\text{avg}=P_\text{rms}\cdot\sqrt{2} = P_\text{max} = \frac{{|V_S|}^2}{2R_S}
\end{align}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:9.8444em;vertical-align:-4.6722em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.1722em;"><span style="top:-7.9199em;"><span class="pstrut" style="height:3.631em;"></span><span class="mord"><span class="mord text"><span class="mord">Condition: </span></span></span></span><span style="top:-5.6288em;"><span class="pstrut" style="height:3.631em;"></span><span class="mord"><span class="mord text"><span class="mord">Maximum power to load (50%): </span></span></span></span><span style="top:-2.8618em;"><span class="pstrut" style="height:3.631em;"></span><span class="mord"></span></span><span style="top:-0.0948em;"><span class="pstrut" style="height:3.631em;"></span><span class="mord"><span class="mord text"><span class="mord">Total maximum power: </span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.6722em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.1722em;"><span style="top:-7.9199em;"><span class="pstrut" style="height:3.631em;"></span><span class="mord"><span class="mord"></span><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6705em;"><span style="top:-2.4065em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">S</span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2935em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2935em;"><span></span></span></span></span></span></span></span><span style="top:-
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54
c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
c69,-144,104.5,-217.7,106.5,-221
l0 -0
c5.3,-9.3,12,-14,20,-14
H400000v40H845.2724
s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
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c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54
c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
c69,-144,104.5,-217.7,106.5,-221
l0 -0
c5.3,-9.3,12,-14,20,-14
H400000v40H845.2724
s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
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</math>
<p dir="auto"><img src="2022-06-01-16-41-13.png" alt="" class="loading" id="image-hash-1999406576" data-src="2022-06-01-16-41-13.png"></p>
<h2 dir="auto" id="complex-power">Complex Power</h2>
<p dir="auto">Where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>V</mi><mo>ˉ</mo></mover><mo>=</mo><mi>V</mi><mi mathvariant="normal">∠</mi><mi>θ</mi></mrow><annotation encoding="application/x-tex">\bar{V}=V\angle\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8201em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8201em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mord">∠</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>I</mi><mo>ˉ</mo></mover><mo>=</mo><mi>I</mi><mi mathvariant="normal">∠</mi><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\bar{I}=I\angle\phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8201em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8201em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1389em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mord">∠</span><span class="mord mathnormal">ϕ</span></span></span></span>:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Complex [VA]: </mtext><mover accent="true"><mi>S</mi><mo>ˉ</mo></mover></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mover accent="true"><mi>V</mi><mo>ˉ</mo></mover><mtext>rms</mtext></msub><mo>×</mo><msubsup><mover accent="true"><mi>I</mi><mo>ˉ</mo></mover><mtext>rms</mtext><mo>∗</mo></msubsup><mo>=</mo><mfrac><mrow><mover accent="true"><mi>V</mi><mo>ˉ</mo></mover><mo>×</mo><msup><mover accent="true"><mi>I</mi><mo>ˉ</mo></mover><mo>∗</mo></msup></mrow><mn>2</mn></mfrac><mo>=</mo><mfrac><mrow><mi>V</mi><mi>I</mi></mrow><mn>2</mn></mfrac><mi mathvariant="normal">∠</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>−</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Apparent [VA]: </mtext><mi mathvariant="normal">∣</mi><mi>S</mi><mi mathvariant="normal">∣</mi></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Real [W]: </mtext><mi>P</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi mathvariant="normal">∣</mi><mi>S</mi><mi mathvariant="normal">∣</mi><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>θ</mi><mo>−</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo>=</mo><mtext>Re</mtext><mo stretchy="false">(</mo><mover accent="true"><mi>S</mi><mo>ˉ</mo></mover><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Reactive [VAR]: </mtext><mi>Q</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi mathvariant="normal">∣</mi><mi>S</mi><mi mathvariant="normal">∣</mi><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>θ</mi><mo>−</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo>=</mo><mtext>Im</mtext><mo stretchy="false">(</mo><mover accent="true"><mi>S</mi><mo>ˉ</mo></mover><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>Q</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>P</mi><mi>tan</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>arccos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mtext>Power factor</mtext><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mtext>Power Factor</mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mi>P</mi><mrow><mi mathvariant="normal">∣</mi><mi>S</mi><mi mathvariant="normal">∣</mi></mrow></mfrac><mo>=</mo><mi>cos</mi><mo>⁡</mo><mrow><mo fence="true">(</mo><mi>arctan</mi><mo>⁡</mo><mrow><mo fence="true">(</mo><mfrac><mi>Q</mi><mi>P</mi></mfrac><mo fence="true">)</mo></mrow><mo fence="true">)</mo></mrow></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">
\begin{align}
\text{Complex [VA]: }\bar{S} &amp;= \bar{V}_\text{rms}\times \bar{I}_\text{rms}^* = \frac{\bar{V}\times \bar{I}^*}{2} = \frac{VI}{2}\angle(\theta-\phi)\\
\text{Apparent [VA]: } |S|\\
\text{Real [W]: } P &amp;= |S| \cos(\theta-\phi) = \text{Re}(\bar{S})\\
\text{Reactive [VAR]: } Q &amp;= |S| \cos(\theta-\phi) = \text{Im}(\bar{S})\\
Q &amp;= P\tan(\arccos(\text{Power factor}))\\
\text{Power Factor} &amp;= \frac{P}{|S|} = \cos\left(\arctan\left(\frac{Q}{P}\right)\right)
\end{align}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:11.1831em;vertical-align:-5.3416em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.8416em;"><span style="top:-7.8416em;"><span class="pstrut" style="height:3.4971em;"></span><span class="mord"><span class="mord text"><span class="mord">Complex [VA]: </span></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8201em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">ˉ</span></span></span></span></span></span></span></span></span><span style="top:-6.0156em;"><span class="pstrut" style="height:3.4971em;"></span><span class="mord"><span class="mord text"><span class="mord">Apparent [VA]: </span></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mord">∣</span></span></span><span style="top:-4.5156em;"><span class="pstrut" style="height:3.4971em;"></span><span class="mord"><span class="mord text"><span class="mord">Real [W]: </span></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span><span style="top:-3.0156em;"><span class="pstrut" style="height:3.4971em;"></span><span class="mord"><span class="mord text"><span class="mord">Reactive [VAR]: </span></span><span class="mord mathnormal">Q</span></span></span><span style="top:-1.5156em;"><span class="pstrut" style="height:3.4971em;"></span><span class="mord"><span class="mord mathnormal">Q</span></span></span><span style="top:0.5944em;"><span class="pstrut" style="height:3.4971em;"></span><span class="mord"><span class="mord text"><span class="mord">Power Factor</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.3416em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.8416em;"><span style="top:-7.8416em;"><span class="pstrut" style="height:3.4971em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8201em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">rms</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8201em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span></s
</math>
<p dir="auto">Where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>S</mi><mo>ˉ</mo></mover><mo>=</mo><mi mathvariant="normal">∣</mi><mi>S</mi><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∠</mi><mi>φ</mi></mrow><annotation encoding="application/x-tex">\bar{S}=|S|\angle\varphi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8201em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8201em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mord">∣∠</span><span class="mord mathnormal">φ</span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>φ</mi><mo>=</mo><mi>arctan</mi><mo>⁡</mo><mrow><mo fence="true">(</mo><mfrac><mi>Q</mi><mi>P</mi></mfrac><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex"> \varphi = \arctan\left(\frac{Q}{P}\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">φ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mop">arctan</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">Q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p>
</math>
<table dir="auto">
<thead dir="auto">
<tr dir="auto">
<th></th>
<th>Lagging</th>
<th>Leading</th>
</tr>
</thead>
<tbody dir="auto">
<tr dir="auto">
<td>Voltage</td>
<td>Current behind</td>
<td>Current ahead</td>
</tr>
<tr dir="auto">
<td>Load type</td>
<td>Inductive</td>
<td>Capacitive</td>
</tr>
<tr dir="auto">
<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span></span></span></span></td>
<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">Q&gt;0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></td>
<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">Q&lt;0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></td>
</tr>
<tr dir="auto">
<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>φ</mi></mrow><annotation encoding="application/x-tex">\varphi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">φ</span></span></span></span></td>
<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>φ</mi><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\varphi&gt;0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">φ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></td>
<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>φ</mi><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\varphi&lt;0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">φ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></td>
</tr>
</tbody>
</table>
<h1 dir="auto" id="chapter-8">Chapter 8</h1>
<p dir="auto">Constants</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>μ</mi><mn>0</mn></msub><mo>=</mo><mn>4</mn><mi>π</mi><mo>×</mo><mn>1</mn><msup><mn>0</mn><mrow><mo>−</mo><mn>7</mn></mrow></msup></mrow><annotation encoding="application/x-tex">
\mu_0=4\pi \times 10^{-7}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">4</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span></span></span></p>
</math>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Faraday’s law: </mtext><mi>ε</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>−</mo><mi>N</mi><mfrac><mrow><mi>d</mi><mi>Φ</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Ampere’s law: </mtext><mi>B</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>π</mi><mi>r</mi></mrow></mfrac></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">
\begin{align}
\text{Faraday&#x27;s law: }\varepsilon &amp;= -N\frac{d\varPhi}{dt}\\
\text{Ampere&#x27;s law: }B &amp;= \frac{\mu_0 I}{2\pi r}\\
\end{align}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.7038em;vertical-align:-2.1019em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6019em;"><span style="top:-4.6019em;"><span class="pstrut" style="height:3.3714em;"></span><span class="mord"><span class="mord text"><span class="mord">Faraday’s law: </span></span><span class="mord mathnormal">ε</span></span></span><span style="top:-2.2556em;"><span class="pstrut" style="height:3.3714em;"></span><span class="mord"><span class="mord text"><span class="mord">Ampere’s law: </span></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1019em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6019em;"><span style="top:-4.6019em;"><span class="pstrut" style="height:3.3714em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathit">Φ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.2556em;"><span class="pstrut" style="height:3.3714em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></spa
</math>
<h2 dir="auto" id="transformer">Transformer</h2>
<p dir="auto">Step up: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>&gt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n&gt;1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span><br>
Step down: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>&lt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n&lt;1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mi>V</mi><mi>s</mi></msub><msub><mi>V</mi><mi>p</mi></msub></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><msub><mi>N</mi><mi>s</mi></msub><msub><mi>N</mi><mi>p</mi></msub></mfrac><mo>=</mo><mfrac><msub><mi>i</mi><mi>p</mi></msub><msub><mi>i</mi><mi>s</mi></msub></mfrac><mo>=</mo><mi>n</mi></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mover accent="true"><mi>Z</mi><mo>ˉ</mo></mover><mrow><mi>i</mi><mi>n</mi></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mn>1</mn><msup><mi>n</mi><mn>2</mn></msup></mfrac><msub><mover accent="true"><mi>Z</mi><mo>ˉ</mo></mover><mi>L</mi></msub></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">
\begin{align}
\frac{V_s}{V_p} &amp;= \frac{N_s}{N_p}=\frac{i_p}{i_s}=n\\
\bar{Z}_{in} &amp;= \frac{1}{n^2}\bar{Z}_L
\end{align}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.9399em;vertical-align:-2.2199em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.7199em;"><span style="top:-4.7199em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9721em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.1264em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8201em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">in</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.2199em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.7199em;"><span style="top:-4.7199em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nu
</math>
<h2 dir="auto" id="motor">Motor</h2>
<p dir="auto">For permanent motors, define permanent torque constant <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mrow><mi>T</mi><mi>P</mi></mrow></msub><mo>=</mo><msub><mi>k</mi><mi>T</mi></msub><mi>Φ</mi></mrow><annotation encoding="application/x-tex">k_{TP}=k_T\varPhi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">TP</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathit">Φ</span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>T</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi>k</mi><mi>T</mi></msub><mo>×</mo><mi>Φ</mi><mo>×</mo><msub><mi>i</mi><mi>a</mi></msub><mo>=</mo><msub><mi>k</mi><mrow><mi>T</mi><mi>P</mi></mrow></msub><mo>×</mo><msub><mi>i</mi><mi>a</mi></msub></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>P</mi><mtext>mech</mtext></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi>ω</mi><mtext>mech</mtext></msub><mi>T</mi><mo>=</mo><msub><mi>ω</mi><mtext>mech</mtext></msub><mo>×</mo><msub><mi>k</mi><mrow><mi>T</mi><mi>P</mi></mrow></msub><mo>×</mo><msub><mi>i</mi><mi>a</mi></msub></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">
\begin{align}
T &amp;= k_T\times\varPhi \times i_a = k_{TP} \times i_a\\
P_\text{mech} &amp;= \omega_\text{mech} T = \omega_\text{mech} \times k_{TP}\times i_a
\end{align}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">mech</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathit">Φ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">TP</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" styl
</math>
<h3 dir="auto" id="back-emf">Back emf</h3>
<p dir="auto">Define for permanent motors, define permanent armature constant <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mrow><mi>a</mi><mi>P</mi></mrow></msub><mo>=</mo><msub><mi>k</mi><mi>a</mi></msub><mi>Φ</mi></mrow><annotation encoding="application/x-tex">k_{aP} = k_a\varPhi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">P</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathit">Φ</span></span></span></span><br>
Note, back emf should oppose <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mi>a</mi></msub></mrow><annotation encoding="application/x-tex">v_a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>i</mi><mi>a</mi></msub></mrow><annotation encoding="application/x-tex">i_a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>e</mi><mi>b</mi></msub><mo>=</mo><msub><mi>k</mi><mi>a</mi></msub><mo>×</mo><mi>Φ</mi><mo>×</mo><msub><mi>ω</mi><mtext>mech</mtext></msub><mo>=</mo><msub><mi>k</mi><mrow><mi>a</mi><mi>P</mi></mrow></msub><mo>×</mo><msub><mi>ω</mi><mtext>mech</mtext></msub></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">
\begin{align}
e_b=k_a\times \varPhi\times \omega_\text{mech} = k_{aP} \times\omega_\text{mech}\\
\end{align}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5em;vertical-align:-0.5em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1em;"><span style="top:-3.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathit">Φ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">mech</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">P</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">mech</span></span></sp
</math>
<h3 dir="auto" id="summary">Summary</h3>
<p dir="auto">For ideal motor, torque and armature constants are the same: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mi>a</mi></msub><mo>=</mo><msub><mi>k</mi><mi>T</mi></msub></mrow><annotation encoding="application/x-tex">k_a=k_T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><br>
Define <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span> as number of magnetic poles and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span></span></span></span> as the number of parallel paths in armature winding.</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Power dissipated: </mtext><msub><mi>P</mi><mi>e</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi>e</mi><mi>b</mi></msub><mo>×</mo><msub><mi>i</mi><mi>a</mi></msub><mo>=</mo><msub><mi>k</mi><mrow><mi>a</mi><mi>P</mi></mrow></msub><mo>×</mo><msub><mi>ω</mi><mtext>mech</mtext></msub><mo>×</mo><msub><mi>i</mi><mi>a</mi></msub></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Constants for ideal motor: </mtext><msub><mi>k</mi><mi>a</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi>k</mi><mi>T</mi></msub><mo>=</mo><mfrac><mrow><mi>p</mi><mi>N</mi></mrow><mrow><mn>2</mn><mi>π</mi><mi>M</mi></mrow></mfrac></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">
\begin{align}
\text{Power dissipated: } P_e &amp;= e_b\times i_a = k_{aP} \times \omega_\text{mech} \times i_a\\
\text {Constants for ideal motor: } k_a &amp;= k_T = \frac{pN}{2\pi M}
\end{align}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.8463em;vertical-align:-1.6732em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.1732em;"><span style="top:-4.6935em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord text"><span class="mord">Power dissipated: </span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.6732em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord text"><span class="mord">Constants for ideal motor: </span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.6732em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.1732em;"><span style="top:-4.6935em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-le
</math>
<p dir="auto">For permanent magnet DC motor in DC steady state:<br>
Define viscous frictional damping coefficient <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span> and load torque <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mi>L</mi></msub></mrow><annotation encoding="application/x-tex">T_L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>=</mo><msub><mi>v</mi><mi>a</mi></msub><mo>−</mo><msub><mi>i</mi><mi>a</mi></msub><msub><mi>R</mi><mi>a</mi></msub><mo>−</mo><msub><mi>k</mi><mrow><mi>a</mi><mi>P</mi></mrow></msub><msub><mi>ω</mi><mtext>mech</mtext></msub><mo>=</mo><msub><mi>v</mi><mi>a</mi></msub><mo>−</mo><msub><mi>i</mi><mi>a</mi></msub><msub><mi>R</mi><mi>a</mi></msub><mo>−</mo><msub><mi>e</mi><mi>b</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>k</mi><mrow><mi>T</mi><mi>P</mi></mrow></msub><msub><mi>i</mi><mi>a</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>=</mo><msub><mi>T</mi><mi>L</mi></msub><mo>+</mo><mi>b</mi><mo>×</mo><msub><mi>ω</mi><mtext>mech</mtext></msub></mrow></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtext>Analog speed control (Voltage): </mtext><mi>T</mi><mo>=</mo><mfrac><msub><mi>k</mi><mrow><mi>T</mi><mi>P</mi></mrow></msub><mi>R</mi></mfrac><msub><mi>v</mi><mi>s</mi></msub><mo>−</mo><mfrac><mrow><msub><mi>k</mi><mrow><mi>T</mi><mi>P</mi></mrow></msub><msub><mi>k</mi><mrow><mi>a</mi><mi>P</mi></mrow></msub></mrow><mi>R</mi></mfrac><msub><mi>ω</mi><mtext>mech</mtext></msub></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtext>Analog speed control (Current): </mtext><mi>T</mi><mo>=</mo><mfrac><mrow><msub><mi>k</mi><mrow><mi>T</mi><mi>P</mi></mrow></msub><msub><mi>R</mi><mi>S</mi></msub></mrow><mi>R</mi></mfrac><msub><mi>i</mi><mi>s</mi></msub><mo>−</mo><mfrac><mrow><msub><mi>k</mi><mrow><mi>T</mi><mi>P</mi></mrow></msub><msub><mi>k</mi><mrow><mi>a</mi><mi>P</mi></mrow></msub></mrow><mi>R</mi></mfrac><msub><mi>ω</mi><mtext>mech</mtext></msub></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">
\begin{align}
&amp;\begin{cases}
0 &amp;= v_a - i_a R_a - k_{aP} \omega_\text{mech} = v_a - i_a R_a - e_b \\
k_{TP} i_a &amp;= T_L + b\times \omega_\text{mech}
\end{cases}\\
&amp;\text{Analog speed control (Voltage): } T = \frac{k_{TP}}{R}v_s - \frac{k_{TP}k_{aP}}{R} \omega_\text{mech}\\
&amp;\text{Analog speed control (Current): } T = \frac{k_{TP}R_S}{R}i_s - \frac{k_{TP}k_{aP}}{R} \omega_\text{mech}
\end{align}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:8.0149em;vertical-align:-3.7575em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.2575em;"><span style="top:-6.2575em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"></span></span><span style="top:-3.336em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"></span></span><span style="top:-0.9785em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.7575em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.2575em;"><span style="top:-6.2575em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">TP</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-righ
</math>
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