Yugra State University BulletinYugra State University Bulletin1816-92282078-9114Yugra State University791010.17816/byusu201814138-45Research ArticleTransient and Steady-State Analysis of Single Switched Capacitor ConverterKushnerovAlexander<p>Department of Electrical and Computer Engineering</p>kushnero@ee.bgu.ac.ilBen-Gurion University of the Negev09032018141384513022018Copyright © 2018, Kushnerov A.2018<p>The paper derives closed form expressions for transient and steady-state operation of a DC-DC converter with single switched capacitor. To this end, the result of each switching is considered as a point in the iterative process, and the function between the points is reconstructed. As opposed to the commonly accepted approach, when each of the topologies is approximated by a first order circuit, the proposed analysis is carried out for second order circuits. This allows obtaining the waveform of output voltage ripple and paves the way to more accurate calculation of equivalent resistance. The obtained analytical expressions were verified by simulations and an excellent agreement between the results was found.</p>
<p></p>charge pumpdiscrete-time systemsequivalent resistanceswitched capacitor converter<h3>Introduction</h3>
<p>Switched capacitor converters (SCCs) are favored in some applications due to low EMI and compatibility with IC technology. Over the past few decades, ongoing research has shifted towards sophisticated SCCs with large number of capacitors and advanced control circuits. However, only a few studies use analytical methods. The analysis presented in this paper is based on the method of difference equations [1],[2],[3], which allows finding the solution for the transient and steady-state operation. Although this method is unified, it was applied previously only to the switched inductor converters [4],[5],[6]. However, some analytical methods for analysis of switched capacitor circuits that bear resemblance of the proposed one, were presented in [7],[8],[9],[10].</p>
<p>Let us consider the SCC shown schematically in Fig. 1(a). It comprises four switches <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mn>1</mn></msub><mo></mo><msub><mi>S</mi><mn>4</mn></msub></math>withon-resistances <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>1</mn></msub><mo></mo><msub><mi>r</mi><mn>4</mn></msub></math>. The corresponding pair of switches is turned on/off by two non-overlapping clocks <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi></mi><mn>1</mn></msub></math>and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi></mi><mn>2</mn></msub></math>shown in Fig. 1(b). Thus, during <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub></math>the capacitor <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mn>1</mn></msub></math>is charged by <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>i</mi></msub><mi>n</mi></math>through <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mn>1</mn></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mn>3</mn></msub></math>and then, during <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>2</mn></msub></math>, is discharged to the load through <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mn>2</mn></msub><mo>,</mo><mo></mo><msub><mi>S</mi><mn>4</mn></msub><mo>.</mo></math>Thus, we have two topologies, which are considered separately in Sections II. Applying to each topology the Kirchhoffs voltage and current laws (KVL and KCL), we write a system of two first-order differential equations. These equations are then solved using the Laplace transform. The solution for the first topology defines the initial conditions for the second one. This enables us to compose a system of two first-order difference equations, which is solved using the Z-transform. Thus, for a given period, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>, we know the initial values of state variables and functions according to which these variables change. This is the solution is closed form.</p>
<p> </p>
<center>
<div class="preview fancybox" style="text-align: center;"><a title="Fig. 1: Considered SCC (a) and two non-overlapping clocks and (b)." href="/files/journals/17/articles/7910/supp/7910-17102-1-SP.png" rel="simplebox"><img style="max-height: 300px; max-width: 300px;" src="/files/journals/17/articles/7910/supp/7910-17102-1-SP.png" /></a></div>
</center>
<p><strong>Fig. 1: Considered SCC (a) and two non-overlapping clocks<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi></mi><mn>1</mn></msub></math> and<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi></mi><mn>2</mn></msub></math> (b).</strong></p>
<p></p>
<h3>Differential equations</h3>
<ol>
<li><em>First topology</em></li>
</ol>
<p>The switches <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mn>1</mn></msub></math>and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mn>3</mn></msub></math>in Fig. 1(a) are turned on during <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mi>T</mi><mo></mo><mi>t</mi><mo></mo><mi>n</mi><mi>T</mi><mo>+</mo><msub><mi>t</mi><mn>1</mn></msub></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>is the number of period. Thus, in the first topology (Fig. 2), <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>1</mn></msub><mo>=</mo><msub><mi>r</mi><mn>1</mn></msub><mo>+</mo><msub><mi>r</mi><mn>3</mn></msub></math>.</p>
<p></p>
<center>
<div class="preview fancybox" style="text-align: center;"><a title="Fig. 2: First topology of the considered SCC." href="/files/journals/17/articles/7910/supp/7910-17103-1-SP.png" rel="simplebox"><img style="max-height: 300px; max-width: 300px;" src="/files/journals/17/articles/7910/supp/7910-17103-1-SP.png" /></a></div>
</center>
<p><strong>Fig. 2: First topology of the considered SCC.</strong></p>
<p></p>
<p>The KVL and KCL equations for this circuit are:</p>
<table width="100%">
<tbody>
<tr>
<td width="39%"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><msub><mi>V</mi><mi>i</mi></msub><mi>n</mi><mo>=</mo><msub><mi>i</mi><mn>1</mn></msub><msub><mi>R</mi><mn>1</mn></msub><mo>+</mo><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>+</mo><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi></mtd></mtr><mtr><mtd><msub><mi>i</mi><mn>1</mn></msub><mo>=</mo><msub><mi>i</mi><mn>2</mn></msub><mo>+</mo><msub><mi>i</mi><mn>3</mn></msub></mtd></mtr></mtable></mfenced></math></td>
<td width="8%">
<p>or</p>
</td>
<td width="47%"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><msub><mi>V</mi><mi>i</mi></msub><mi>n</mi><mo>=</mo><msub><mi>C</mi><mn>1</mn></msub><mfrac><mrow><mi>d</mi><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><msub><mi>R</mi><mn>1</mn></msub><mo>+</mo><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>+</mo><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi></mtd></mtr><mtr><mtd><msub><mi>C</mi><mn>1</mn></msub><mfrac><mrow><mi>d</mi><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mi>C</mi><mi>o</mi></msub><mfrac><mrow><mi>d</mi><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi></mrow><msub><mi>R</mi><mi>o</mi></msub></mfrac></mtd></mtr></mtable></mfenced></math></td>
<td width="4%">
<p>(1)</p>
</td>
</tr>
</tbody>
</table>
<p>Let us write (1) in the Laplace domain using</p>
<table width="100%">
<tbody>
<tr>
<td width="35%"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mfenced open="{" close="}"><mi>K</mi></mfenced><mo>=</mo><mfrac><mi>K</mi><mi>S</mi></mfrac></math></td>
<td width="8%">
<p>and</p>
</td>
<td width="49%"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mfenced open="{" close="}"><mfrac><mrow><mi>d</mi><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac></mfenced><mo>=</mo><mi>s</mi><mi>F</mi><mo>(</mo><mi>s</mi><mo>)</mo><mo>-</mo><mi>f</mi><mo>(</mo><mn>0</mn><mo>)</mo></math></td>
<td width="6%">
<p>(2)</p>
</td>
</tr>
</tbody>
</table>
<p>such that</p>
<table width="100%">
<tbody>
<tr>
<td width="92%"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mo>(</mo><mi>s</mi><msub><mi>R</mi><mn>1</mn></msub><msub><mi>C</mi><mn>1</mn></msub><mo>+</mo><mn>1</mn><mo>)</mo><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>+</mo><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>=</mo><msub><mi>R</mi><mn>1</mn></msub><msub><mi>C</mi><mn>1</mn></msub><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>n</mi><mo>)</mo><mo>+</mo><mfrac><mrow><msub><mi>V</mi><mi>i</mi></msub><mi>n</mi></mrow><mi>s</mi></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mi>s</mi><msub><mi>C</mi><mn>1</mn></msub><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>+</mo><mo>(</mo><mi>s</mi><msub><mi>C</mi><mi>o</mi></msub><mo>+</mo><mfrac><mn>1</mn><msub><mi>R</mi><mi>o</mi></msub></mfrac><mo>)</mo><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>=</mo><msub><mi>C</mi><mi>o</mi></msub><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>-</mo><msub><mi>C</mi><mn>1</mn></msub><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>n</mi><mo>)</mo></mtd></mtr></mtable></mfenced></math></td>
<td width="7%">
<p>(3)</p>
</td>
</tr>
</tbody>
</table>
<p>The solution of (3) can be written as:</p>
<table width="100%">
<tbody>
<tr>
<td width="92%"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>=</mo><mfrac><mrow><mo>(</mo><mi>s</mi><mo>-</mo><msub><mi>r</mi><mn>1</mn></msub><mo>)</mo><mo>(</mo><mi>s</mi><mo>-</mo><msub><mi>r</mi><mn>2</mn></msub><mo>)</mo></mrow><mrow><mi>s</mi><mo>(</mo><mi>s</mi><mo>-</mo><msub><mi>s</mi><mn>1</mn></msub><mo>)</mo><mo>(</mo><mi>s</mi><mo>-</mo><msub><mi>s</mi><mn>2</mn></msub><mo>)</mo></mrow></mfrac><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>n</mi><mo>)</mo><mo>;</mo><mo></mo><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>=</mo><mfrac><mrow><mo>(</mo><mi>s</mi><mo>-</mo><mi>p</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>s</mi><mo>-</mo><msub><mi>s</mi><mn>1</mn></msub><mo>)</mo><mo>(</mo><mi>s</mi><mo>-</mo><msub><mi>s</mi><mn>2</mn></msub><mo>)</mo></mrow></mfrac><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></td>
<td width="7%">
<p>(4)</p>
</td>
</tr>
</tbody>
</table>
<p>where</p>
<table width="100%">
<tbody>
<tr>
<td width="92%"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>1</mn></msub><mo>,</mo><mn>2</mn><mo>=</mo><mo>-</mo><mi></mi><mo></mo><mo></mo><msup><mi></mi><mn>2</mn></msup><mo>-</mo><mfrac><mrow><msubsup><mi></mi><mn>0</mn><mn>2</mn></msubsup><msub><mi>V</mi><mi>i</mi></msub><mi>n</mi></mrow><mrow><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mfrac><mo>;</mo><mo></mo><mi></mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced open="[" close="]"><mrow><mfenced><mrow><mfrac><mn>1</mn><msub><mi>R</mi><mn>1</mn></msub></mfrac><mo>+</mo><mfrac><mn>1</mn><msub><mi>R</mi><mi>o</mi></msub></mfrac></mrow></mfenced><mfrac><mn>1</mn><msub><mi>C</mi><mi>o</mi></msub></mfrac><mo>+</mo><mfrac><mrow><msub><mi>V</mi><mi>i</mi></msub><mi>n</mi><mo>-</mo><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mrow><msub><mi>R</mi><mn>1</mn></msub><msub><mi>C</mi><mn>1</mn></msub><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mfrac></mrow></mfenced><mo>;</mo></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mo>-</mo><mfenced><mrow><mfrac><mrow><msub><mi>V</mi><mi>i</mi></msub><mi>n</mi><mo>-</mo><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mrow><msub><mi>R</mi><mn>1</mn></msub><msub><mi>C</mi><mi>o</mi></msub><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mn>1</mn></msub><msub><mi>C</mi><mn>1</mn></msub></mrow></mfrac></mrow></mfenced></math></td>
<td width="7%">
<p>(5)</p>
</td>
</tr>
</tbody>
</table>
<p>and</p>
<table width="100%">
<tbody>
<tr>
<td width="92%"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>s</mi><mn>1</mn></msub><mo>,</mo><mn>2</mn><mo>=</mo><mo>-</mo><mi></mi><mo></mo><mo></mo><mo>(</mo><msup><mi></mi><mn>2</mn></msup><mo>-</mo><msubsup><mi></mi><mn>0</mn><mn>2</mn></msubsup><mo>)</mo><mo>;</mo><mo></mo><mi></mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced open="[" close="]"><mrow><mfenced><mrow><mfrac><mn>1</mn><msub><mi>C</mi><mn>1</mn></msub></mfrac><mo>+</mo><mfrac><mn>1</mn><msub><mi>C</mi><mi>o</mi></msub></mfrac></mrow></mfenced><mfrac><mn>1</mn><msub><mi>R</mi><mn>1</mn></msub></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mi>o</mi></msub><msub><mi>C</mi><mi>o</mi></msub></mrow></mfrac></mrow></mfenced><mo>;</mo><mo></mo><msubsup><mi></mi><mn>0</mn><mn>2</mn></msubsup><mo>=</mo><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mn>1</mn></msub><msub><mi>C</mi><mn>1</mn></msub><msub><mi>R</mi><mi>o</mi></msub><msub><mi>C</mi><mi>o</mi></msub></mrow></mfrac></math></td>
<td width="7%">
<p>(6)</p>
</td>
</tr>
</tbody>
</table>
<p>The inverse Laplace transform of (4) is:</p>
<table width="100%">
<tbody>
<tr>
<td width="92%">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>L</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced open="{" close="}"><mrow><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn></mrow></mfenced><mo>=</mo><mfenced><mrow><mfrac><mrow><mo>(</mo><msub><mi>r</mi><mn>1</mn></msub><mo>-</mo><msub><mi>s</mi><mn>1</mn></msub><mo>)</mo><mo>(</mo><msub><mi>r</mi><mn>2</mn></msub><mo>-</mo><msub><mi>s</mi><mn>1</mn></msub><mo>)</mo></mrow><mrow><msub><mi>s</mi><mn>1</mn></msub><mo>(</mo><msub><mi>s</mi><mn>1</mn></msub><mo>-</mo><msub><mi>s</mi><mn>2</mn></msub><mo>)</mo></mrow></mfrac><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo>-</mo><mfrac><mrow><mo>(</mo><msub><mi>r</mi><mn>1</mn></msub><mo>-</mo><msub><mi>s</mi><mn>2</mn></msub><mo>)</mo><mo>(</mo><msub><mi>r</mi><mn>2</mn></msub><mo>-</mo><msub><mi>s</mi><mn>2</mn></msub><mo>)</mo></mrow><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>(</mo><msub><mi>s</mi><mn>1</mn></msub><mo>-</mo><msub><mi>s</mi><mn>2</mn></msub><mo>)</mo></mrow></mfrac><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>2</mn></msub><mi>t</mi></mrow></msup><mo>+</mo><mfrac><mrow><msub><mi>r</mi><mn>1</mn></msub><msub><mi>r</mi><mn>2</mn></msub></mrow><mrow><msub><mi>s</mi><mn>1</mn></msub><msub><mi>s</mi><mn>2</mn></msub></mrow></mfrac></mrow></mfenced><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>n</mi><mo>)</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>L</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced open="{" close="}"><mrow><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mo>(</mo><msub><mi>s</mi><mn>2</mn></msub><mo>-</mo><mi>p</mi><mo>)</mo><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>2</mn></msub><mi>t</mi></mrow></msup><mo>-</mo><mo>(</mo><msub><mi>s</mi><mn>1</mn></msub><mo>-</mo><mi>p</mi><mo>)</mo><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>1</mn></msub><mi>t</mi></mrow></msup></mrow><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>-</mo><msub><mi>s</mi><mn>1</mn></msub></mrow></mfrac><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></p>
</td>
<td width="7%">
<p>(7)</p>
</td>
</tr>
</tbody>
</table>
<p>Using and</p>
<table width="100%">
<tbody>
<tr>
<td width="7%">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><msub><mi>t</mi><mn>1</mn></msub></math></p>
</td>
<td width="83%"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>1</mn></msub><msub><mi>r</mi><mn>2</mn></msub><mo>=</mo><mfrac><mrow><msubsup><mi></mi><mn>0</mn><mn>2</mn></msubsup><msub><mi>V</mi><mi>i</mi></msub><mi>n</mi></mrow><mrow><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mfrac><mo>;</mo><mo></mo><msub><mi>s</mi><mn>1</mn></msub><msub><mi>s</mi><mn>2</mn></msub><mo>=</mo><msubsup><mi></mi><mn>0</mn><mn>2</mn></msubsup><mo>;</mo><mo></mo><msub><mi>r</mi><mn>1</mn></msub><mo>+</mo><msub><mi>r</mi><mn>2</mn></msub><mo>=</mo><mo>-</mo><mn>2</mn><mi></mi></math></td>
<td width="9%">
<p>(8)</p>
</td>
</tr>
</tbody>
</table>
<p>we can rewrite (7) as:</p>
<table width="100%">
<tbody>
<tr>
<td width="7%">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><msub><mi>t</mi><mn>1</mn></msub><mo>)</mo><mo>=</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>(</mo><msub><mi>V</mi><mi>i</mi></msub><mi>n</mi><mo>-</mo><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo><mo>+</mo><mi>c</mi><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>n</mi><mo>)</mo></math></p>
</td>
<td width="83%"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><msub><mi>t</mi><mn>1</mn></msub><mo>)</mo><mo>=</mo><mi>d</mi><mo>(</mo><msub><mi>V</mi><mi>i</mi></msub><mi>n</mi><mo>-</mo><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo><mo>+</mo><mi>e</mi><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></td>
<td width="9%">
<p>(9)</p>
</td>
</tr>
</tbody>
</table>
<p>where</p>
<table width="100%">
<tbody>
<tr>
<td width="90%">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><msub><mi>s</mi><mn>1</mn></msub><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>2</mn></msub><msub><mi>t</mi><mn>1</mn></msub></mrow></msup><mo>-</mo><msub><mi>s</mi><mn>2</mn></msub><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>1</mn></msub><msub><mi>t</mi><mn>1</mn></msub></mrow></msup></mrow><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>-</mo><msub><mi>s</mi><mn>1</mn></msub></mrow></mfrac></mrow></mfenced><msub><mi>V</mi><mi>i</mi></msub><mi>n</mi><mo>;</mo><mo></mo><mi>b</mi><mo>=</mo><mfrac><mrow><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>2</mn></msub><msub><mi>t</mi><mn>1</mn></msub></mrow></msup><mo>-</mo><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>1</mn></msub><msub><mi>t</mi><mn>1</mn></msub></mrow></msup></mrow><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>-</mo><msub><mi>s</mi><mn>1</mn></msub></mrow></mfrac><mfenced><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mn>1</mn></msub><msub><mi>C</mi><mn>1</mn></msub></mrow></mfrac></mfenced><mo>;</mo><mo></mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mfrac><mrow><mfenced open="[" close="]"><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>+</mo><mfenced><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><msub><mi>R</mi><mn>1</mn></msub></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msub><mi>R</mi><mi>o</mi></msub></mfrac></mstyle></mrow></mfenced><mstyle displaystyle="true"><mfrac><mn>1</mn><msub><mi>C</mi><mi>o</mi></msub></mfrac></mstyle></mrow></mfenced><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>2</mn></msub><msub><mi>t</mi><mn>1</mn></msub></mrow></msup><mo>-</mo><mfenced open="[" close="]"><mrow><msub><mi>s</mi><mn>1</mn></msub><mo>+</mo><mfenced><mrow><mfrac><mn>1</mn><msub><mi>R</mi><mn>1</mn></msub></mfrac><mo>+</mo><mfrac><mn>1</mn><msub><mi>R</mi><mi>o</mi></msub></mfrac></mrow></mfenced><mfrac><mn>1</mn><msub><mi>C</mi><mi>o</mi></msub></mfrac></mrow></mfenced><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>1</mn></msub><msub><mi>t</mi><mn>1</mn></msub></mrow></msup></mrow><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>-</mo><msub><mi>s</mi><mn>1</mn></msub></mrow></mfrac><mo>;</mo><mo></mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mfrac><mrow><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>2</mn></msub><msub><mi>t</mi><mn>1</mn></msub></mrow></msup><mo>-</mo><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>1</mn></msub><msub><mi>t</mi><mn>1</mn></msub></mrow></msup></mrow><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>-</mo><msub><mi>s</mi><mn>1</mn></msub></mrow></mfrac><mfenced><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mn>1</mn></msub><msub><mi>C</mi><mi>o</mi></msub></mrow></mfrac></mfenced><mo>;</mo><mo></mo><mi>e</mi><mo>=</mo><mfrac><mrow><mfenced><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>+</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mn>1</mn></msub><msub><mi>C</mi><mn>1</mn></msub></mrow></mfrac></mstyle></mrow></mfenced><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>2</mn></msub><msub><mi>t</mi><mn>1</mn></msub></mrow></msup><mo>-</mo><mfenced><mrow><msub><mi>s</mi><mn>1</mn></msub><mo>+</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mn>1</mn></msub><msub><mi>C</mi><mn>1</mn></msub></mrow></mfrac></mstyle></mrow></mfenced><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>1</mn></msub><msub><mi>t</mi><mn>1</mn></msub></mrow></msup></mrow><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>-</mo><msub><mi>s</mi><mn>1</mn></msub></mrow></mfrac></math></p>
</td>
<td width="9%">
<p>(10)</p>
</td>
</tr>
</tbody>
</table>
<p></p>
<ol>
<li><em>Second topology</em></li>
</ol>
<p>The switches <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mn>2</mn></msub></math>and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mn>4</mn></msub></math>in Fig. 1(a) are turned on during <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mi>T</mi><mo>+</mo><msub><mi>t</mi><mn>1</mn></msub><mo></mo><mi>t</mi><mo></mo><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mi>T</mi></math>, such that in the second topology (Fig. 3) <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>2</mn></msub><mo>=</mo><msub><mi>r</mi><mn>2</mn></msub><mo>+</mo><msub><mi>r</mi><mn>4</mn></msub></math>.</p>
<p></p>
<center>
<div class="preview fancybox" style="text-align: center;"><a title="Fig. 3: Second topology of the considered SCC." href="/files/journals/17/articles/7910/supp/7910-17104-1-SP.png" rel="simplebox"><img style="max-height: 300px; max-width: 300px;" src="/files/journals/17/articles/7910/supp/7910-17104-1-SP.png" /></a></div>
</center>
<p><strong>Fig. 3: Second topology of the considered SCC.</strong></p>
<p></p>
<p>The KVL and KCL equations for this circuit are:</p>
<table width="100%">
<tbody>
<tr>
<td width="41%"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>=</mo><mo>-</mo><msub><mi>i</mi><mn>1</mn></msub><msub><mi>R</mi><mn>2</mn></msub><mo>+</mo><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi></mtd></mtr><mtr><mtd><mo>-</mo><msub><mi>i</mi><mn>1</mn></msub><mo>=</mo><msub><mi>i</mi><mn>2</mn></msub><mo>+</mo><msub><mi>i</mi><mn>3</mn></msub></mtd></mtr></mtable></mfenced></math></td>
<td width="6%">
<p>or</p>
</td>
<td width="48%"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>=</mo><mo>-</mo><msub><mi>C</mi><mn>1</mn></msub><mfrac><mrow><mi>d</mi><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><msub><mi>R</mi><mn>2</mn></msub><mo>+</mo><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi></mtd></mtr><mtr><mtd><mo>-</mo><msub><mi>C</mi><mn>1</mn></msub><mfrac><mrow><mi>d</mi><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mi>C</mi><mi>o</mi></msub><mfrac><mrow><mi>d</mi><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi></mrow><msub><mi>R</mi><mi>o</mi></msub></mfrac></mtd></mtr></mtable></mfenced></math></td>
<td width="4%">
<p>(11)</p>
</td>
</tr>
</tbody>
</table>
<p>Using (2), we write (11) in the Laplace domain:</p>
<table width="100%">
<tbody>
<tr>
<td width="92%"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mo>(</mo><mi>s</mi><msub><mi>R</mi><mn>2</mn></msub><msub><mi>C</mi><mn>1</mn></msub><mo>+</mo><mn>1</mn><mo>)</mo><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>-</mo><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>=</mo><msub><mi>R</mi><mn>2</mn></msub><msub><mi>C</mi><mn>1</mn></msub><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>n</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>s</mi><msub><mi>C</mi><mn>1</mn></msub><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>+</mo><mfenced><mrow><mi>s</mi><msub><mi>C</mi><mi>o</mi></msub><mo>+</mo><mfrac><mn>1</mn><msub><mi>R</mi><mi>o</mi></msub></mfrac></mrow></mfenced><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>=</mo><msub><mi>C</mi><mn>1</mn></msub><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>n</mi><mo>)</mo><mo>+</mo><msub><mi>C</mi><mi>o</mi></msub><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo></mtd></mtr></mtable></mfenced></math></td>
<td width="7%">
<p>(12)</p>
</td>
</tr>
</tbody>
</table>
<p>The solution of (12) is:</p>
<table width="100%">
<tbody>
<tr>
<td width="92%"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>=</mo><mfrac><mrow><mo>(</mo><mi>s</mi><mo>-</mo><msub><mi>p</mi><mn>1</mn></msub><mo>)</mo></mrow><mrow><mo>(</mo><mi>s</mi><mo>-</mo><msub><mi>s</mi><mn>1</mn></msub><mo>)</mo><mo>(</mo><mi>s</mi><mo>-</mo><msub><mi>s</mi><mn>2</mn></msub><mo>)</mo></mrow></mfrac><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>n</mi><mo>)</mo><mo>;</mo><mo></mo><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>=</mo><mfrac><mrow><mo>(</mo><mi>s</mi><mo>-</mo><msub><mi>p</mi><mn>2</mn></msub><mo>)</mo></mrow><mrow><mo>(</mo><mi>s</mi><mo>-</mo><msub><mi>s</mi><mn>1</mn></msub><mo>)</mo><mo>(</mo><mi>s</mi><mo>-</mo><msub><mi>s</mi><mn>2</mn></msub><mo>)</mo></mrow></mfrac><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></td>
<td width="7%">
<p>(13)</p>
</td>
</tr>
</tbody>
</table>
<p>where</p>
<table width="100%">
<tbody>
<tr>
<td width="4%">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mn>1</mn></msub><mo>=</mo><mo>-</mo><mfenced><mrow><mfrac><mrow><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mrow><msub><mi>C</mi><mi>o</mi></msub><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><msub><mi>C</mi><mn>1</mn></msub></mfrac></mrow></mfenced><mfrac><mn>1</mn><msub><mi>R</mi><mn>2</mn></msub></mfrac><mo>;</mo><mo></mo></math></p>
</td>
<td width="87%"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mn>2</mn></msub><mo>=</mo><mo>-</mo><mfenced><mrow><mfrac><mn>1</mn><msub><mi>R</mi><mi>o</mi></msub></mfrac><mo>+</mo><mfrac><mn>1</mn><msub><mi>R</mi><mn>2</mn></msub></mfrac></mrow></mfenced><mfrac><mn>1</mn><msub><mi>C</mi><mi>o</mi></msub></mfrac><mo>-</mo><mfenced><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mn>2</mn></msub><msub><mi>C</mi><mn>1</mn></msub></mrow></mfrac></mfenced><mfrac><mrow><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mrow><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mfrac></math></td>
<td width="7%">
<p>(14)</p>
</td>
</tr>
</tbody>
</table>
<p>and</p>
<table width="100%">
<tbody>
<tr>
<td width="92%"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>s</mi><mn>1</mn></msub><mo>,</mo><mn>2</mn><mo>=</mo><mo>-</mo><mi></mi><mo></mo><mo></mo><mo>(</mo><msup><mi></mi><mn>2</mn></msup><mo>-</mo><msubsup><mi></mi><mn>0</mn><mn>2</mn></msubsup><mo>)</mo><mo>;</mo><mo></mo><mi></mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced open="[" close="]"><mrow><mfenced><mrow><mfrac><mn>1</mn><msub><mi>C</mi><mn>1</mn></msub></mfrac><mo>+</mo><mfrac><mn>1</mn><msub><mi>C</mi><mi>o</mi></msub></mfrac></mrow></mfenced><mfrac><mn>1</mn><msub><mi>R</mi><mn>2</mn></msub></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mi>o</mi></msub><msub><mi>C</mi><mi>o</mi></msub></mrow></mfrac></mrow></mfenced><mo>;</mo><mo></mo><msubsup><mi></mi><mn>0</mn><mn>2</mn></msubsup><mo>=</mo><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mn>2</mn></msub><msub><mi>C</mi><mn>1</mn></msub><msub><mi>R</mi><mi>o</mi></msub><msub><mi>C</mi><mi>o</mi></msub></mrow></mfrac></math></td>
<td width="7%">
<p>(15)</p>
</td>
</tr>
</tbody>
</table>
<p>The inverse Laplace transform of (13) is:</p>
<table width="100%">
<tbody>
<tr>
<td width="92%">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>L</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced open="{" close="}"><mrow><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn></mrow></mfenced><mo>=</mo><mfrac><mrow><mo>(</mo><msub><mi>s</mi><mn>2</mn></msub><mo>-</mo><msub><mi>p</mi><mn>1</mn></msub><mo>)</mo><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>2</mn></msub><mi>t</mi></mrow></msup><mo>-</mo><mo>(</mo><msub><mi>s</mi><mn>1</mn></msub><mo>-</mo><msub><mi>p</mi><mn>1</mn></msub><mo>)</mo><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>1</mn></msub><mi>t</mi></mrow></msup></mrow><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>-</mo><msub><mi>s</mi><mn>1</mn></msub></mrow></mfrac><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>n</mi><mo>)</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>L</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced open="{" close="}"><mrow><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mo>(</mo><msub><mi>s</mi><mn>2</mn></msub><mo>-</mo><msub><mi>p</mi><mn>2</mn></msub><mo>)</mo><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>2</mn></msub><mi>t</mi></mrow></msup><mo>-</mo><mo>(</mo><msub><mi>s</mi><mn>1</mn></msub><mo>-</mo><msub><mi>p</mi><mn>2</mn></msub><mo>)</mo><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>1</mn></msub><mi>t</mi></mrow></msup></mrow><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>-</mo><msub><mi>s</mi><mn>1</mn></msub></mrow></mfrac><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></p>
</td>
<td width="7%">
<p>(16)</p>
</td>
</tr>
</tbody>
</table>
<p>Since for the second topology <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo></mo><mi>t</mi><mo></mo><msub><mi>t</mi><mn>2</mn></msub></math>, the initial conditions will be <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><msub><mi>t</mi><mn>1</mn></msub><mo>)</mo></math>and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><msub><mi>t</mi><mn>1</mn></msub><mo>)</mo></math>, whereas <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><msub><mi>t</mi><mn>2</mn></msub><mo>-</mo><msub><mi>t</mi><mn>1</mn></msub></math>.</p>
<p></p>
<h3>Difference equations</h3>
<p>Substituting (14) and (15) into (16), we obtain the recurrent equations:</p>
<table width="100%">
<tbody>
<tr>
<td style="width: 11.1807%;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>=</mo><mi>f</mi><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><msub><mi>t</mi><mn>1</mn></msub><mo>)</mo><mo>+</mo><mi>g</mi><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><msub><mi>t</mi><mn>1</mn></msub><mo>)</mo></math></p>
</td>
<td style="width: 78.8193%;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>=</mo><mi>h</mi><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><msub><mi>t</mi><mn>1</mn></msub><mo>)</mo><mo>+</mo><mi>k</mi><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><msub><mi>t</mi><mn>1</mn></msub><mo>)</mo></math></td>
<td style="width: 9%;">
<p>(17)</p>
</td>
</tr>
</tbody>
</table>
<p>where</p>
<table width="100%">
<tbody>
<tr>
<td width="90%">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>=</mo><mfrac><mrow><mfenced><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>+</mo><mfenced><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><msub><mi>R</mi><mi>o</mi></msub></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msub><mi>R</mi><mn>2</mn></msub></mfrac></mstyle></mrow></mfenced><mstyle displaystyle="true"><mfrac><mn>1</mn><msub><mi>C</mi><mi>o</mi></msub></mfrac></mstyle></mrow></mfenced><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>(</mo><msub><mi>t</mi><mn>2</mn></msub><mo>-</mo><msub><mi>t</mi><mn>1</mn></msub><mo>)</mo></mrow></msup><mo>-</mo><mfenced><mrow><msub><mi>s</mi><mn>1</mn></msub><mo>+</mo><mfenced><mrow><mfrac><mn>1</mn><msub><mi>R</mi><mi>o</mi></msub></mfrac><mo>+</mo><mfrac><mn>1</mn><msub><mi>R</mi><mn>2</mn></msub></mfrac></mrow></mfenced><mfrac><mn>1</mn><msub><mi>C</mi><mi>o</mi></msub></mfrac></mrow></mfenced><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>1</mn></msub><mo>(</mo><msub><mi>t</mi><mn>2</mn></msub><mo>-</mo><msub><mi>t</mi><mn>1</mn></msub><mo>)</mo></mrow></msup></mrow><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>-</mo><msub><mi>s</mi><mn>1</mn></msub></mrow></mfrac></math>;</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mfrac><mrow><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>(</mo><msub><mi>t</mi><mn>2</mn></msub><mo>-</mo><msub><mi>t</mi><mn>1</mn></msub><mo>)</mo></mrow></msup><mo>-</mo><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>1</mn></msub><mo>(</mo><msub><mi>t</mi><mn>2</mn></msub><mo>-</mo><msub><mi>t</mi><mn>1</mn></msub><mo>)</mo></mrow></msup></mrow><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>-</mo><msub><mi>s</mi><mn>1</mn></msub></mrow></mfrac><mfenced><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mn>2</mn></msub><msub><mi>C</mi><mn>1</mn></msub></mrow></mfrac></mfenced><mo>;</mo><mo></mo><mi>h</mi><mo>=</mo><mfrac><mrow><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>(</mo><msub><mi>t</mi><mn>2</mn></msub><mo>-</mo><msub><mi>t</mi><mn>1</mn></msub><mo>)</mo></mrow></msup><mo>-</mo><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>1</mn></msub><mo>(</mo><msub><mi>t</mi><mn>2</mn></msub><mo>-</mo><msub><mi>t</mi><mn>1</mn></msub><mo>)</mo></mrow></msup></mrow><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>-</mo><msub><mi>s</mi><mn>1</mn></msub></mrow></mfrac><mfenced><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mn>2</mn></msub><msub><mi>C</mi><mi>o</mi></msub></mrow></mfrac></mfenced></math>;</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mfrac><mrow><mfenced><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>+</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mn>2</mn></msub><msub><mi>C</mi><mn>1</mn></msub></mrow></mfrac></mstyle></mrow></mfenced><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>(</mo><msub><mi>t</mi><mn>2</mn></msub><mo>-</mo><msub><mi>t</mi><mn>1</mn></msub><mo>)</mo></mrow></msup><mo>-</mo><mfenced><mrow><msub><mi>s</mi><mn>1</mn></msub><mo>+</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mn>2</mn></msub><msub><mi>C</mi><mn>1</mn></msub></mrow></mfrac></mstyle></mrow></mfenced><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>1</mn></msub><mo>(</mo><msub><mi>t</mi><mn>2</mn></msub><mo>-</mo><msub><mi>t</mi><mn>1</mn></msub><mo>)</mo></mrow></msup></mrow><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>-</mo><msub><mi>s</mi><mn>1</mn></msub></mrow></mfrac></math></p>
</td>
<td width="9%">
<p>(18)</p>
</td>
</tr>
</tbody>
</table>
<p>Now, substituting (9) into (17), we obtain the difference equations:</p>
<table width="100%">
<tbody>
<tr>
<td width="7%">
<p></p>
</td>
<td width="83%"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>=</mo><mi>A</mi><mo>+</mo><mi>B</mi><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>+</mo><mi>G</mi><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>n</mi><mo>)</mo></mtd></mtr><mtr><mtd><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>=</mo><mi>E</mi><mo>+</mo><mi>F</mi><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>+</mo><mi>H</mi><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>n</mi><mo>)</mo></mtd></mtr></mtable></mfenced></math></td>
<td width="9%">
<p>(19)</p>
</td>
</tr>
</tbody>
</table>
<p>where</p>
<table width="100%">
<tbody>
<tr>
<td width="90%"><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><mi>A</mi><mo>=</mo><mo>(</mo><mi>f</mi><mi>b</mi><mo>+</mo><mi>g</mi><mi>d</mi><mo>)</mo><msub><mi>V</mi><mi>i</mi></msub><mi>n</mi><mo>+</mo><mi>f</mi><mi>a</mi><mo>;</mo></mtd><mtd><mi>B</mi><mo>=</mo><mi>g</mi><mi>e</mi><mo>-</mo><mi>f</mi><mi>b</mi><mo>;</mo></mtd><mtd><mi>G</mi><mo>=</mo><mi>f</mi><mi>c</mi><mo>-</mo><mi>g</mi><mi>d</mi></mtd></mtr><mtr><mtd><mi>E</mi><mo>=</mo><mo>(</mo><mi>h</mi><mi>b</mi><mo>+</mo><mi>k</mi><mi>d</mi><mo>)</mo><msub><mi>V</mi><mi>i</mi></msub><mi>n</mi><mo>+</mo><mi>h</mi><mi>a</mi><mo>;</mo></mtd><mtd><mi>F</mi><mo>=</mo><mi>k</mi><mi>e</mi><mo>-</mo><mi>h</mi><mi>b</mi><mo>;</mo></mtd><mtd><mi>H</mi><mo>=</mo><mi>h</mi><mi>c</mi><mo>-</mo><mi>k</mi><mi>d</mi></mtd></mtr></mtable></math></td>
<td width="9%">
<p>(20)</p>
</td>
</tr>
</tbody>
</table>
<p>Before we proceed to the solution of (19), let us consider the steady-state operation, where</p>
<table width="100%">
<tbody>
<tr>
<td width="7%">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>=</mo><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>n</mi><mo>)</mo></math></p>
</td>
<td width="83%"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>=</mo><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></td>
<td width="9%">
<p>(21)</p>
</td>
</tr>
</tbody>
</table>
<p>substituting (21) into (19), we have</p>
<table width="100%">
<tbody>
<tr>
<td width="7%">
<p></p>
</td>
<td width="83%"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mo>(</mo><mi>G</mi><mo>-</mo><mn>1</mn><mo>)</mo><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>n</mi><mo>)</mo><mo>+</mo><mi>B</mi><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mo>-</mo><mi>A</mi></mtd></mtr><mtr><mtd><mi>H</mi><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>n</mi><mo>)</mo><mo>+</mo><mo>(</mo><mi>F</mi><mo>-</mo><mn>1</mn><mo>)</mo><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mo>-</mo><mi>E</mi></mtd></mtr></mtable></mfenced></math></td>
<td width="9%">
<p>(22)</p>
</td>
</tr>
</tbody>
</table>
<p>The solution of (22) is:</p>
<table width="100%">
<tbody>
<tr>
<td width="7%">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>B</mi><mi>E</mi><mo>-</mo><mi>A</mi><mo>(</mo><mi>F</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>F</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>G</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>-</mo><mi>B</mi><mi>H</mi></mrow></mfrac></math></p>
</td>
<td width="83%"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>H</mi><mi>A</mi><mo>-</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>F</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>G</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>-</mo><mi>B</mi><mi>H</mi></mrow></mfrac></math></td>
<td width="9%">
<p>(23)</p>
</td>
</tr>
</tbody>
</table>
<p>Let us write (19) in the Z-domain using</p>
<table width="100%">
<tbody>
<tr>
<td width="36%"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi><mfenced open="{" close="}"><mi>K</mi></mfenced><mo>=</mo><mi>K</mi><mfrac><mi>z</mi><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></mfrac></math></td>
<td width="7%">
<p>and</p>
</td>
<td width="49%"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi><mfenced open="{" close="}"><mrow><mi>f</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><mo>=</mo><mi>z</mi><mi>F</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>-</mo><mi>f</mi><mo>(</mo><mn>0</mn><mo>)</mo></math></td>
<td width="6%">
<p>(24)</p>
</td>
</tr>
</tbody>
</table>
<p>such that</p>
<table width="100%">
<tbody>
<tr>
<td width="92%"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mo>(</mo><mi>z</mi><mo>-</mo><mi>G</mi><mo>)</mo><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>z</mi><mo>)</mo><mo>-</mo><mi>B</mi><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>=</mo><mi>A</mi><mfrac><mi>z</mi><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mi>z</mi><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mn>0</mn><mo>)</mo></mtd></mtr><mtr><mtd><mo>-</mo><mi>H</mi><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>z</mi><mo>)</mo><mo>+</mo><mo>(</mo><mi>z</mi><mo>-</mo><mi>F</mi><mo>)</mo><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>=</mo><mi>E</mi><mfrac><mi>z</mi><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mi>z</mi><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mn>0</mn><mo>)</mo></mtd></mtr></mtable></mfenced></math></td>
<td width="7%">
<p>(25)</p>
</td>
</tr>
</tbody>
</table>
<p>The solution of (25) is:</p>
<table width="100%">
<tbody>
<tr>
<td width="92%"><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>z</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mo>(</mo><mi>z</mi><mo>-</mo><mi>F</mi><mo>)</mo><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mn>0</mn><mo>)</mo><mo>+</mo><mi>B</mi><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>1</mn></msub><mo>)</mo><mo>(</mo><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>2</mn></msub><mo>)</mo></mrow></mfrac><mi>z</mi><mo>+</mo><mfrac><mrow><mo>(</mo><mi>z</mi><mo>-</mo><mi>F</mi><mo>)</mo><mi>A</mi><mo>+</mo><mi>B</mi><mi>E</mi></mrow><mrow><mo>(</mo><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>1</mn></msub><mo>)</mo><mo>(</mo><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>2</mn></msub><mo>)</mo></mrow></mfrac><mfenced><mfrac><mi>z</mi><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></mfrac></mfenced></mtd></mtr><mtr><mtd><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>H</mi><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mn>0</mn><mo>)</mo><mo>+</mo><mo>(</mo><mi>z</mi><mo>-</mo><mi>G</mi><mo>)</mo><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>1</mn></msub><mo>)</mo><mo>(</mo><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>2</mn></msub><mo>)</mo></mrow></mfrac><mi>z</mi><mo>+</mo><mfrac><mrow><mi>H</mi><mi>A</mi><mo>+</mo><mo>(</mo><mi>z</mi><mo>-</mo><mi>G</mi><mo>)</mo><mi>E</mi></mrow><mrow><mo>(</mo><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>1</mn></msub><mo>)</mo><mo>(</mo><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>2</mn></msub><mo>)</mo></mrow></mfrac><mfenced><mfrac><mi>z</mi><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></mfrac></mfenced></mtd></mtr></mtable></math></td>
<td width="7%">
<p>(26)</p>
</td>
</tr>
</tbody>
</table>
<p>where</p>
<table width="100%">
<tbody>
<tr>
<td width="7%">
<p></p>
</td>
<td width="84%"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>,</mo><mn>2</mn><mo>=</mo><mfrac><mrow><mi>G</mi><mo>+</mo><mi>F</mi><mo></mo><mo></mo><mo>(</mo><mo>(</mo><mi>G</mi><mo>+</mo><mi>F</mi><msup><mo>)</mo><mn>2</mn></msup><mo>-</mo><mn>4</mn><mo>(</mo><mi>G</mi><mi>F</mi><mo>-</mo><mi>B</mi><mi>H</mi><mo>)</mo><mo>)</mo></mrow><mn>2</mn></mfrac></math></td>
<td width="7%">
<p>(27)</p>
</td>
</tr>
</tbody>
</table>
<p>Since (26) is represented in the form of partial fractions, we can apply the inverse Z-transform to each term separately:</p>
<p></p>
<table width="100%">
<tbody>
<tr>
<td width="92%">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>A</mi><mo>(</mo><msub><mi>z</mi><mn>2</mn></msub><mo>-</mo><mi>F</mi><mo>)</mo><mo>+</mo><mi>B</mi><mi>E</mi></mrow><mrow><msub><mi>z</mi><mn>2</mn></msub><mo>-</mo><msub><mi>z</mi><mn>1</mn></msub></mrow></mfrac><mfenced><mfrac><mrow><msubsup><mi>z</mi><mn>2</mn><mi>n</mi></msubsup><mo>-</mo><mn>1</mn></mrow><mrow><msub><mi>z</mi><mn>2</mn></msub><mo>-</mo><mn>1</mn></mrow></mfrac></mfenced><mo>-</mo><mfrac><mrow><mi>A</mi><mo>(</mo><msub><mi>z</mi><mn>1</mn></msub><mo>-</mo><mi>F</mi><mo>)</mo><mo>+</mo><mi>B</mi><mi>E</mi></mrow><mrow><msub><mi>z</mi><mn>2</mn></msub><mo>-</mo><msub><mi>z</mi><mn>1</mn></msub></mrow></mfrac><mfenced><mfrac><mrow><msubsup><mi>z</mi><mn>1</mn><mi>n</mi></msubsup><mo>-</mo><mn>1</mn></mrow><mrow><msub><mi>z</mi><mn>1</mn></msub><mo>-</mo><mn>1</mn></mrow></mfrac></mfenced><mo>+</mo><mfrac><mrow><mo>(</mo><msub><mi>z</mi><mn>2</mn></msub><mo>-</mo><mi>F</mi><mo>)</mo><msubsup><mi>z</mi><mn>2</mn><mi>n</mi></msubsup><mo>-</mo><mo>(</mo><msub><mi>z</mi><mn>1</mn></msub><mo>-</mo><mi>F</mi><mo>)</mo><msubsup><mi>z</mi><mn>1</mn><mi>n</mi></msubsup></mrow><mrow><msub><mi>z</mi><mn>2</mn></msub><mo>-</mo><msub><mi>z</mi><mn>1</mn></msub></mrow></mfrac><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mn>0</mn><mo>)</mo><mo>+</mo><mfrac><mrow><msubsup><mi>z</mi><mn>2</mn><mi>n</mi></msubsup><mo>-</mo><msubsup><mi>z</mi><mn>1</mn><mi>n</mi></msubsup></mrow><mrow><msub><mi>z</mi><mn>2</mn></msub><mo>-</mo><msub><mi>z</mi><mn>1</mn></msub></mrow></mfrac><mi>B</mi><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mn>0</mn><mo>)</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>E</mi><mo>(</mo><msub><mi>z</mi><mn>2</mn></msub><mo>-</mo><mi>G</mi><mo>)</mo><mo>+</mo><mi>A</mi><mi>H</mi></mrow><mrow><msub><mi>z</mi><mn>2</mn></msub><mo>-</mo><msub><mi>z</mi><mn>1</mn></msub></mrow></mfrac><mfenced><mfrac><mrow><msubsup><mi>z</mi><mn>2</mn><mi>n</mi></msubsup><mo>-</mo><mn>1</mn></mrow><mrow><msub><mi>z</mi><mn>2</mn></msub><mo>-</mo><mn>1</mn></mrow></mfrac></mfenced><mo>-</mo><mfrac><mrow><mi>E</mi><mo>(</mo><msub><mi>z</mi><mn>1</mn></msub><mo>-</mo><mi>G</mi><mo>)</mo><mo>+</mo><mi>A</mi><mi>H</mi></mrow><mrow><msub><mi>z</mi><mn>2</mn></msub><mo>-</mo><msub><mi>z</mi><mn>1</mn></msub></mrow></mfrac><mfenced><mfrac><mrow><msubsup><mi>z</mi><mn>1</mn><mi>n</mi></msubsup><mo>-</mo><mn>1</mn></mrow><mrow><msub><mi>z</mi><mn>1</mn></msub><mo>-</mo><mn>1</mn></mrow></mfrac></mfenced><mo>+</mo><mfrac><mrow><mo>(</mo><msub><mi>z</mi><mn>2</mn></msub><mo>-</mo><mi>G</mi><mo>)</mo><msubsup><mi>z</mi><mn>2</mn><mi>n</mi></msubsup><mo>-</mo><mo>(</mo><msub><mi>z</mi><mn>1</mn></msub><mo>-</mo><mi>G</mi><mo>)</mo><msubsup><mi>z</mi><mn>1</mn><mi>n</mi></msubsup></mrow><mrow><msub><mi>z</mi><mn>2</mn></msub><mo>-</mo><msub><mi>z</mi><mn>1</mn></msub></mrow></mfrac><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>+</mo><mfrac><mrow><msubsup><mi>z</mi><mn>2</mn><mi>n</mi></msubsup><mo>-</mo><msubsup><mi>z</mi><mn>1</mn><mi>n</mi></msubsup></mrow><mrow><msub><mi>z</mi><mn>2</mn></msub><mo>-</mo><msub><mi>z</mi><mn>1</mn></msub></mrow></mfrac><mi>H</mi><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mn>0</mn><mo>)</mo></math></p>
</td>
<td width="7%">
<p>(28)</p>
</td>
</tr>
</tbody>
</table>
<p></p>
<h3>Simulation results</h3>
<p>To verify the obtained analytical expressions (9), (17) and (28), the circuit shown in Fig. 4 was simulated in PSIM 9.0. Since the PSIM bidirectional switches have zero on-resistance, two external resistors corresponding to <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>1</mn></msub></math>and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>2</mn></msub></math>in Fig. 2 and Fig. 3 were added. The parameters of the circuit in Fig. 4 are as follows: <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>i</mi></msub><mi>n</mi><mo>=</mo><mn>10</mn><mi>V</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>C</mi></msub><mn>1</mn><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><msub><mi>V</mi><mi>C</mi></msub><mi>o</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn><mi>V</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo>=</mo><msub><mi>t</mi><mn>2</mn></msub><mo>=</mo><mn>5</mn><mi></mi><mi>s</mi><mo>,</mo></math><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>1</mn></msub><mo>=</mo><msub><mi>R</mi><mn>2</mn></msub><mo>=</mo><mn>1</mn><mi></mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mn>1</mn></msub><mo>=</mo><mn>10</mn><mi></mi><mi>F</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mi>o</mi></msub><mo>=</mo><mn>100</mn><mi></mi></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mi>o</mi></msub><mo>=</mo><mn>100</mn><mi></mi><mi>F</mi></math>.</p>
<p></p>
<center>
<div class="preview fancybox" style="text-align: center;"><a title="Fig. 4: Simulation circuit for the voltage-halving SCC." href="/files/journals/17/articles/7910/supp/7910-17105-1-SP.png" rel="simplebox"><img style="max-height: 300px; max-width: 300px;" src="/files/journals/17/articles/7910/supp/7910-17105-1-SP.png" /></a></div>
</center>
<p><strong>Fig. 4: Simulation circuit for the voltage-halving SCC.</strong></p>
<p></p>
<p>Since <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>1</mn></msub><mo>=</mo><msub><mi>R</mi><mn>2</mn></msub></math>and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo>=</mo><msub><mi>t</mi><mn>2</mn></msub><mo>,</mo></math>, the expressions (10) and (18) are reduced to:</p>
<table width="100%">
<tbody>
<tr>
<td width="92%">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><msub><mi>s</mi><mn>1</mn></msub><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>2</mn></msub><msub><mi>t</mi><mn>1</mn></msub></mrow></msup><mo>-</mo><msub><mi>s</mi><mn>2</mn></msub><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>1</mn></msub><msub><mi>t</mi><mn>1</mn></msub></mrow></msup></mrow><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>-</mo><msub><mi>s</mi><mn>1</mn></msub></mrow></mfrac></mrow></mfenced><msub><mi>V</mi><mi>i</mi></msub><mi>n</mi><mo>;</mo><mo></mo><mi>b</mi><mo>=</mo><mi>g</mi><mo>=</mo><mfrac><mrow><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>2</mn></msub><msub><mi>t</mi><mn>1</mn></msub></mrow></msup><mo>-</mo><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>1</mn></msub><msub><mi>t</mi><mn>1</mn></msub></mrow></msup></mrow><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>-</mo><msub><mi>s</mi><mn>1</mn></msub></mrow></mfrac><mfenced><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mn>1</mn></msub><msub><mi>C</mi><mn>1</mn></msub></mrow></mfrac></mfenced><mo>;</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mi>f</mi><mo>=</mo><mfrac><mrow><mfenced open="[" close="]"><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>+</mo><mfenced><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><msub><mi>R</mi><mn>1</mn></msub></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msub><mi>R</mi><mi>o</mi></msub></mfrac></mstyle></mrow></mfenced><mstyle displaystyle="true"><mfrac><mn>1</mn><msub><mi>C</mi><mi>o</mi></msub></mfrac></mstyle></mrow></mfenced><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>2</mn></msub><msub><mi>t</mi><mn>1</mn></msub></mrow></msup><mo>-</mo><mfenced open="[" close="]"><mrow><msub><mi>s</mi><mn>1</mn></msub><mo>+</mo><mfenced><mrow><mfrac><mn>1</mn><msub><mi>R</mi><mn>1</mn></msub></mfrac><mo>+</mo><mfrac><mn>1</mn><msub><mi>R</mi><mi>o</mi></msub></mfrac></mrow></mfenced><mfrac><mn>1</mn><msub><mi>C</mi><mi>o</mi></msub></mfrac></mrow></mfenced><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>1</mn></msub><msub><mi>t</mi><mn>1</mn></msub></mrow></msup></mrow><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>-</mo><msub><mi>s</mi><mn>1</mn></msub></mrow></mfrac><mo>;</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo>=</mo><mi>k</mi><mo>=</mo><mfrac><mrow><mfenced><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>+</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mn>1</mn></msub><msub><mi>C</mi><mn>1</mn></msub></mrow></mfrac></mstyle></mrow></mfenced><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>2</mn></msub><msub><mi>t</mi><mn>1</mn></msub></mrow></msup><mo>-</mo><mfenced><mrow><msub><mi>s</mi><mn>1</mn></msub><mo>+</mo><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mn>1</mn></msub><msub><mi>C</mi><mn>1</mn></msub></mrow></mfrac></mrow></mfenced><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>1</mn></msub><msub><mi>t</mi><mn>1</mn></msub></mrow></msup></mrow><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>-</mo><msub><mi>s</mi><mn>1</mn></msub></mrow></mfrac><mo>;</mo><mo></mo><mi>d</mi><mo>=</mo><mi>h</mi><mo>=</mo><mi>b</mi><mfrac><msub><mi>C</mi><mn>1</mn></msub><msub><mi>C</mi><mi>o</mi></msub></mfrac></math></p>
</td>
<td width="7%">
<p>(29)</p>
</td>
</tr>
</tbody>
</table>
<p>These constants are substituted into (20) and then into (28), which sets the initial conditions for (9) and (17). The voltages across <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mn>1</mn></msub></math>and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mi>o</mi></msub></math>during the first ten periods ( <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo></mo><mi>t</mi><mo></mo><mn>10</mn><mi>T</mi></math>) is shown Fig. 5, which compares the MathCAD calculation and PSIM simulation.</p>
<p></p>
<center>
<div class="preview fancybox" style="text-align: center;"><a title="Fig. 5: Voltages across and for , MathCAD (a) and PSIM (b)." href="/files/journals/17/articles/7910/supp/7910-17106-1-SP.png" rel="simplebox"><img style="max-height: 300px; max-width: 300px;" src="/files/journals/17/articles/7910/supp/7910-17106-1-SP.png" /></a></div>
</center>
<p><strong>Fig. 5: Voltages across<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mn>1</mn></msub></math> and<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mi>o</mi></msub></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo></mo><mi>t</mi><mo></mo><mn>10</mn><mi>T</mi><mn>0</mn><mo></mo><mi>t</mi><mo></mo><mn>10</mn><mi>T</mi></math>, MathCAD (a) and PSIM (b).The horizontal scale is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mi></mi><mi>s</mi><mo></mo><mi>d</mi><mi>i</mi><mi>v</mi></math>, i.e. each division is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>.</mo></math></strong></p>
<p></p>
<p>The voltages in Fig. 5 were measured at the points and are given in Table I along with the relative error, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi></mi></math>.</p>
<p></p>
<p><strong>Table I: Measured values of the voltages in Fig. 5.</strong></p>
<table>
<tbody>
<tr>
<td rowspan="2"></td>
<td colspan="2" width="120"></td>
<td rowspan="2" width="50">
<p>[%]</p>
</td>
<td colspan="2"></td>
<td rowspan="2">
<p>[%]</p>
</td>
</tr>
<tr>
<td>
<p>Calc.</p>
</td>
<td width="64">
<p>Simul.</p>
</td>
<td>
<p>Calc.</p>
</td>
<td>
<p>Simul.</p>
</td>
</tr>
<tr>
<td>
<p>1</p>
</td>
<td>
<p>2.5146</p>
</td>
<td width="64">
<p>2.5046</p>
</td>
<td width="50">
<p>0.398</p>
</td>
<td>
<p>0.5174</p>
</td>
<td>
<p>0.5119</p>
</td>
<td>
<p>1.074</p>
</td>
</tr>
<tr>
<td>
<p>2</p>
</td>
<td>
<p>3.4986</p>
</td>
<td width="64">
<p>3.4937</p>
</td>
<td width="50">
<p>0.140</p>
</td>
<td>
<p>0.9542</p>
</td>
<td>
<p>0.9449</p>
</td>
<td>
<p>0.984</p>
</td>
</tr>
<tr>
<td>
<p>3</p>
</td>
<td>
<p>3.9147</p>
</td>
<td width="64">
<p>3.9144</p>
</td>
<td width="50">
<p>0.008</p>
</td>
<td>
<p>1.3381</p>
</td>
<td>
<p>1.3259</p>
</td>
<td>
<p>0.920</p>
</td>
</tr>
<tr>
<td>
<p>4</p>
</td>
<td>
<p>4.1172</p>
</td>
<td width="64">
<p>4.1193</p>
</td>
<td width="50">
<p>0.051</p>
</td>
<td>
<p>1.6815</p>
</td>
<td>
<p>1.6670</p>
</td>
<td>
<p>0.870</p>
</td>
</tr>
<tr>
<td>
<p>5</p>
</td>
<td>
<p>4.2366</p>
</td>
<td width="64">
<p>4.2397</p>
</td>
<td width="50">
<p>0.073</p>
</td>
<td>
<p>1.9909</p>
</td>
<td>
<p>1.9747</p>
</td>
<td>
<p>0.820</p>
</td>
</tr>
<tr>
<td>
<p>6</p>
</td>
<td>
<p>4.3211</p>
</td>
<td width="64">
<p>4.3254</p>
</td>
<td width="50">
<p>0.010</p>
</td>
<td>
<p>2.2706</p>
</td>
<td>
<p>2.2531</p>
</td>
<td>
<p>0.777</p>
</td>
</tr>
<tr>
<td>
<p>7</p>
</td>
<td>
<p>4.3890</p>
</td>
<td width="64">
<p>4.3943</p>
</td>
<td width="50">
<p>0.121</p>
</td>
<td>
<p>2.5236</p>
</td>
<td>
<p>2.5054</p>
</td>
<td>
<p>0.726</p>
</td>
</tr>
<tr>
<td>
<p>8</p>
</td>
<td>
<p>4.4474</p>
</td>
<td width="64">
<p>4.4534</p>
</td>
<td width="50">
<p>0.135</p>
</td>
<td>
<p>2.7526</p>
</td>
<td>
<p>2.7339</p>
</td>
<td>
<p>0.684</p>
</td>
</tr>
<tr>
<td>
<p>9</p>
</td>
<td>
<p>4.4991</p>
</td>
<td width="64">
<p>4.5055</p>
</td>
<td width="50">
<p>0.142</p>
</td>
<td>
<p>2.9601</p>
</td>
<td>
<p>2.9410</p>
</td>
<td>
<p>0.649</p>
</td>
</tr>
<tr>
<td>
<p>10</p>
</td>
<td>
<p>4.5454</p>
</td>
<td width="64">
<p>4.5455</p>
</td>
<td width="50">
<p>0.002</p>
</td>
<td>
<p>7.1478</p>
</td>
<td>
<p>7.1282</p>
</td>
<td>
<p>0.627</p>
</td>
</tr>
</tbody>
</table>
<p></p>
<p>Fig. 6 compares the MathCAD calculation and PSIM simulation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo></mo><mi>t</mi><mo></mo><mn>100</mn><mi>T</mi></math>. Note that at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo></mo><mn>60</mn><mi>T</mi></math>the SCC reaches the steady-state.</p>
<p></p>
<center>
<div class="preview fancybox" style="text-align: center;"><a title="Fig. 6: Voltages across and for , MathCAD (a) and PSIM (b)." href="/files/journals/17/articles/7910/supp/7910-17107-1-SP.png" rel="simplebox"><img style="max-height: 300px; max-width: 300px;" src="/files/journals/17/articles/7910/supp/7910-17107-1-SP.png" /></a></div>
</center>
<p><strong>Fig. 6: Voltages across<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mn>1</mn></msub></math> and<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mi>o</mi></msub></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo></mo><mi>t</mi><mo></mo><mn>100</mn><mi>T</mi></math>, MathCAD (a) and PSIM (b).The horizontal scale is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mi></mi><mi>s</mi><mo></mo><mi>d</mi><mi>i</mi><mi>v</mi></math>, i.e. each division is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mi>T</mi></math>.</strong></p>
<p></p>
<p>The steady-state voltages are shown Fig. 7. Their discrete values were calculated by (23) and then substituted as the initial conditions into (9) and (17).</p>
<p></p>
<center>
<div class="preview fancybox" style="text-align: center;"><a title="Fig. 7: Steady-state voltages across and for ." href="/files/journals/17/articles/7910/supp/7910-17108-1-SP.png" rel="simplebox"><img style="max-height: 300px; max-width: 300px;" src="/files/journals/17/articles/7910/supp/7910-17108-1-SP.png" /></a></div>
</center>
<p><strong>Fig. 7: Steady-state voltages across<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mn>1</mn></msub></math> and<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mi>o</mi></msub></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>95</mn><mi>T</mi><mo></mo><mi>t</mi><mo></mo><mn>100</mn><mi>T</mi></math>.</strong></p>
<p></p>
<h3>Conclusion</h3>
<p>Based on the method of difference equations the closed form expressions for the voltages across the capacitors in the voltage-halving SCC were derived. The solution of these equations allows us to predict the SCC behavior in both the transient and steady-state operation. That is for a given period, , we know the initial values of the voltages and functions according to which these voltages change. The obtained expressions were verified by simulations. As evident from Table I, the deviation between the theoretical and simulation results does not exceed 1.1%. The used method however, is very complex even in the case of the two-phase SCC and its extension to the multi-phase SCC will apparently require some special assumptions.</p>[M. F. Gardner and J. L. Barnes, Transients in Linear Systems, Studied by the Laplace Transformation, vol. I, Wiley, 1942.][Y. Z. Tsypkin, Sampling Systems Theory and its Application, vol. I, Macmillan, 1964.][P. M. Derusso, R. J. Roy and C. M. Close, "§6.4 The Concept of State," in State Variables for Engineers, New York: Wiley, 1965, pp. 413-415.][L. R. Nerone, "Analytical solutions of the class D inverter," in IEEE International Symposium on Circuits and Systems (ISCAS), 2008.][A. Shoihet and M. A. Slonim, "Analysis of buck converter processes using difference equations method," in IEEE Convention of Electrical and Electronics Engineers in Israel (IEEEI), 2010.][N. Krihely, M. A. Slonim and S. Ben-Yaakov, "Transient and steady-state analysis of three-phase PWM buck rectifier," IET Power Electronics, vol. 5, no. 9, pp. 1764-1775, 2012.][A. Lamantia, P. Maranesi and L. Radrizzani, "Small-signal model of the Cockcroft-Walton voltage multiplier," IEEE Transactions on Power Electronics, vol. 9, no. 1, pp. 18-25, 1994.][V. Vitchev, "Calculating essential charge-pump parameters," Power Electronics Technology, vol. 7, pp. 30-45, 2006.][M. E. Karagozler, S. C. Goldstein and D. S. Ricketts, "Analysis and modeling of capacitive power transfer in microsystems," IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 59, no. 7, pp. 1557-1566, 2012.][R. Ramezani and A. Yakovlev, "Capacitor discharging through asynchronous circuit switching," in IEEE Int. Symposium on Asynchronous Circuits and Systems (ASYNC), 2013.]