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<span style="font-size: 12pt;">David - what you show as the Landen transform (in the box) is EXACTLY called iteration............</span>! - Bernie</div>
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<div id="divRplyFwdMsg" dir="ltr"><font face="Calibri, sans-serif" style="font-size:11pt" color="#000000"><b>From:</b> David G Dixon <dixon@mail.ubc.ca><br>
<b>Sent:</b> Saturday, January 9, 2021 8:16 PM<br>
<b>To:</b> Bernard Arthur Hutchins, Jr <bah13@cornell.edu>; synth-diy@synth-diy.org <synth-diy@synth-diy.org><br>
<b>Subject:</b> RE: [sdiy] 90-degree phase displacement network calculations </font>
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<div dir="ltr" align="left"><span class="x_661413100-10012021"><font color="#0000ff" size="2" face="Arial">Hello Bernie,</font></span></div>
<div dir="ltr" align="left"><span class="x_661413100-10012021"><font color="#0000ff" size="2" face="Arial"></font></span> </div>
<div dir="ltr" align="left"><span class="x_661413100-10012021"><font color="#0000ff" size="2" face="Arial">No, it is not iterative. It is closed form. The step-wise procedure is an implementation of the Landen transformation to enable calculation of elliptic
sine from circular sine. That is all. If Excel had a function call for elliptic sine, I would use that instead. Once the elliptic sine is calculated the poles are obtained directly. Once half the poles are obtained, then you just take the logarithms, set
them negative, and take the antilogs to find the other half.</font></span></div>
<div dir="ltr" align="left"><span class="x_661413100-10012021"><font color="#0000ff" size="2" face="Arial"></font></span> </div>
<div dir="ltr" align="left"><span class="x_661413100-10012021"><font color="#0000ff" size="2" face="Arial">The reason for the steps is that you must transform the k and k' values using a simple formula until you obtain k = 0 and k' = 1. At that point, sn =
sin, so you can simply find the sines of the angles (2n-1)*(pi/2)/4N and take those as the elliptic sines. Then, you go back up the chain of k and k' values, calculating the elliptic sines from those below (of which the first one, on the bottom of the table,
is a circular sine). When you get to the top of the table, you have an elliptic sine at the desired modulus. This is then used to calculate the pole directly.</font></span></div>
<div dir="ltr" align="left"><span class="x_661413100-10012021"><font color="#0000ff" size="2" face="Arial"></font></span> </div>
<div dir="ltr" align="left"><span class="x_661413100-10012021"><font color="#0000ff" size="2" face="Arial">To make this more plain, here is a table showing the calculations in Excel.</font></span></div>
<div dir="ltr" align="left"><span class="x_661413100-10012021"><font color="#0000ff" size="2" face="Arial"></font></span> </div>
<div dir="ltr" align="left"><span class="x_661413100-10012021"><font color="#0000ff" size="2" face="Arial"><span class="x_661413100-10012021"><font color="#0000ff" size="2" face="Arial"><img border="0" hspace="0" alt="" align="baseline" data-outlook-trace="F:1|T:1" src="cid:661413100@10012021-3393">
<img border="0" hspace="0" alt="" align="baseline" data-outlook-trace="F:1|T:1" src="cid:661413100@10012021-339A"></font></span></font></span></div>
<div dir="ltr" align="left"><span class="x_661413100-10012021"><font color="#0000ff" size="2" face="Arial"><span class="x_661413100-10012021"></span></font></span> </div>
<span class="x_661413100-10012021"><font color="#0000ff" size="2"><span class="x_661413100-10012021">
<div dir="ltr" align="left">
<table border="0" cellspacing="0" cellpadding="0" width="1090" style="width:818pt; border-collapse:collapse">
<colgroup><font face="Arial"><col span="10" width="100" style="width:75pt"><col width="90" style="width:68pt"></font></colgroup>
<tbody>
<tr height="21" style="height:15.75pt">
<td height="21" width="100" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; width:75pt; height:15.75pt; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial"></font></td>
<td class="x_xl35" width="100" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; width:75pt; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">4N</font></td>
<td class="x_xl24" width="100" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; width:75pt; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">32</font></td>
<td class="x_xl24" width="100" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; width:75pt; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">1</font></td>
<td class="x_xl24" width="100" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; width:75pt; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">3</font></td>
<td class="x_xl24" width="100" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; width:75pt; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">5</font></td>
<td class="x_xl24" width="100" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; width:75pt; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">7</font></td>
<td class="x_xl24" width="100" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; width:75pt; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">9</font></td>
<td class="x_xl24" width="100" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; width:75pt; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">11</font></td>
<td class="x_xl24" width="100" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; width:75pt; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">13</font></td>
<td class="x_xl24" width="90" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; width:68pt; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">15</font></td>
</tr>
<tr height="21" style="height:15.75pt">
<td class="x_xl36" height="21" style="border-bottom:windowtext 0.5pt solid; border-left:#ece9d8; background-color:transparent; height:15.75pt; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">i</font></td>
<td class="x_xl36" style="border-bottom:windowtext 0.5pt solid; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">k</font></td>
<td class="x_xl36" style="border-bottom:windowtext 0.5pt solid; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">k'</font></td>
<td class="x_xl36" style="border-bottom:windowtext 0.5pt solid; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">sn(π/64)</font></td>
<td class="x_xl36" style="border-bottom:windowtext 0.5pt solid; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">sn(3π/64)</font></td>
<td class="x_xl36" style="border-bottom:windowtext 0.5pt solid; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">sn(5π/64)</font></td>
<td class="x_xl36" style="border-bottom:windowtext 0.5pt solid; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">sn(7π/64)</font></td>
<td class="x_xl36" style="border-bottom:windowtext 0.5pt solid; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">sn(9π/64)</font></td>
<td class="x_xl36" style="border-bottom:windowtext 0.5pt solid; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">sn(11π/64)</font></td>
<td class="x_xl36" style="border-bottom:windowtext 0.5pt solid; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">sn(13π/64)</font></td>
<td class="x_xl36" style="border-bottom:windowtext 0.5pt solid; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">sn(15π/64)</font></td>
</tr>
<tr height="21" style="height:15.75pt">
<td class="x_xl24" height="21" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; height:15.75pt; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">—</font></td>
<td class="x_xl29" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font color="#ff0000" face="Arial"><strong>0.000100</strong></font></td>
<td class="x_xl30" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font color="#3366ff" face="Arial">0.319549</font></strong></td>
<td class="x_xl30" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font color="#3366ff" face="Arial">0.758823</font></strong></td>
<td class="x_xl30" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font color="#3366ff" face="Arial">0.929639</font></strong></td>
<td class="x_xl30" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font color="#3366ff" face="Arial">0.980794</font></strong></td>
<td class="x_xl30" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font color="#3366ff" face="Arial">0.994857</font></strong></td>
<td class="x_xl30" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font color="#3366ff" face="Arial">0.998630</font></strong></td>
<td class="x_xl30" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font color="#3366ff" face="Arial">0.999635</font></strong></td>
<td class="x_xl30" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font color="#3366ff" face="Arial">0.999903</font></strong></td>
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<tr height="21" style="height:15.75pt">
<td class="x_xl24" height="21" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; height:15.75pt; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">1</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.999800</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.019998</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.164092</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.459577</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.679394</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.820798</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.903443</font></td>
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<font face="Arial">0.949062</font></td>
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<font face="Arial">0.973452</font></td>
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<font face="Arial">2</font></td>
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<font face="Arial">0.960788</font></td>
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<font face="Arial">3</font></td>
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<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.427556</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.514103</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.595700</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.671559</font></td>
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<tr height="21" style="height:15.75pt">
<td class="x_xl24" height="21" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; height:15.75pt; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">6</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.000001</font></td>
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<font face="Arial">1.000000</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.049068</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.146730</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.242980</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.336890</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.427555</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.514103</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.595699</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.671559</font></td>
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<tr height="21" style="height:15.75pt">
<td class="x_xl24" height="21" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; height:15.75pt; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">7</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.000000</font></td>
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<font face="Arial">1.000000</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.049068</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.146730</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.242980</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.336890</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.427555</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.514103</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.595699</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.671559</font></td>
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<tr height="21" style="height:15.75pt">
<td class="x_xl24" height="21" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; height:15.75pt; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">8</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.000000</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">1.000000</font></td>
<td class="x_xl31" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font face="Arial">0.049068</font></strong></td>
<td class="x_xl31" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font face="Arial">0.146730</font></strong></td>
<td class="x_xl31" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font face="Arial">0.242980</font></strong></td>
<td class="x_xl31" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font face="Arial">0.336890</font></strong></td>
<td class="x_xl31" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font face="Arial">0.427555</font></strong></td>
<td class="x_xl31" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font face="Arial">0.514103</font></strong></td>
<td class="x_xl31" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font face="Arial">0.595699</font></strong></td>
<td class="x_xl31" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font face="Arial">0.671559</font></strong></td>
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<tr height="22" style="height:16.5pt">
<td class="x_xl24" height="22" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; height:16.5pt; border-top:#ece9d8; border-right:#ece9d8">
<strong><font face="Arial"></font></strong></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font face="Arial"></font></strong></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font face="Arial"></font></strong></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font face="Arial"></font></strong></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font face="Arial"></font></strong></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font face="Arial"></font></strong></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font face="Arial"></font></strong></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font face="Arial"></font></strong></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font face="Arial"></font></strong></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font face="Arial"></font></strong></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font face="Arial"></font></strong></td>
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<tr height="21" style="height:15.75pt">
<td class="x_xl24" height="21" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; height:15.75pt; border-top:#ece9d8; border-right:#ece9d8">
<strong><font face="Arial"></font></strong></td>
<td class="x_xl26" style="border-bottom:windowtext 0.5pt solid; border-left:windowtext 1pt solid; background-color:transparent; border-top:windowtext 1pt solid; border-right:windowtext 1pt solid">
<font face="Arial">omega</font></td>
<td class="x_xl28" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font face="Arial">sn</font></strong></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.319549</font></td>
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<font face="Arial">0.758823</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.929639</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.980794</font></td>
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<font face="Arial">0.994857</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.998630</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.999635</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.999903</font></td>
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<tr height="22" style="height:16.5pt">
<td class="x_xl24" height="22" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; height:16.5pt; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial"></font></td>
<td class="x_xl27" style="border-bottom:windowtext 1pt solid; border-left:windowtext 1pt solid; background-color:transparent; border-top:#ece9d8; border-right:windowtext 1pt solid">
<font face="Arial">100.000000</font></td>
<td class="x_xl28" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font face="Arial">cn</font></strong></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.947570</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.651297</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.368472</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.195047</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.101294</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.052333</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.027000</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.013925</font></td>
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<tr height="22" style="height:16.5pt">
<td class="x_xl24" height="22" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; height:16.5pt; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial"></font></td>
<td class="x_xl24" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial"></font></td>
<td class="x_xl28" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font face="Arial">pole</font></strong></td>
<td class="x_xl32" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">296.533</font></td>
<td class="x_xl32" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">85.830</font></td>
<td class="x_xl32" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">39.636</font></td>
<td class="x_xl32" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">19.887</font></td>
<td class="x_xl32" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">10.182</font></td>
<td class="x_xl32" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">5.240</font></td>
<td class="x_xl32" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">2.701</font></td>
<td class="x_xl32" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">1.393</font></td>
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<tr height="21" style="height:15.75pt">
<td class="x_xl24" height="21" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; height:15.75pt; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial"></font></td>
<td class="x_xl26" style="border-bottom:windowtext 0.5pt solid; border-left:windowtext 1pt solid; background-color:transparent; border-top:windowtext 1pt solid; border-right:windowtext 1pt solid">
<font face="Arial">f</font></td>
<td class="x_xl28" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font face="Arial">log pole</font></strong></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">2.472074</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">1.933639</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">1.598090</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">1.298561</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">1.007822</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.719369</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.431521</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.143830</font></td>
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<tr height="22" style="height:16.5pt">
<td class="x_xl24" height="22" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; height:16.5pt; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial"></font></td>
<td class="x_xl27" style="border-bottom:windowtext 1pt solid; border-left:windowtext 1pt solid; background-color:transparent; border-top:#ece9d8; border-right:windowtext 1pt solid">
<font face="Arial">628.318531</font></td>
<td class="x_xl28" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
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<font face="Arial">-2.472074</font></td>
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<font face="Arial">-1.933639</font></td>
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<font face="Arial">-1.598090</font></td>
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<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">-0.719369</font></td>
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<font face="Arial">-0.431521</font></td>
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<tr height="22" style="height:16.5pt">
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<font face="Arial"></font></td>
<td class="x_xl28" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<strong><font face="Arial">−pole</font></strong></td>
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<font face="Arial">0.003372</font></td>
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<font face="Arial">0.011651</font></td>
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<font face="Arial">0.025230</font></td>
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<font face="Arial">0.050285</font></td>
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<font face="Arial">0.098215</font></td>
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<font face="Arial">0.190823</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.370236</font></td>
<td class="x_xl25" style="border-bottom:#ece9d8; border-left:#ece9d8; background-color:transparent; border-top:#ece9d8; border-right:#ece9d8">
<font face="Arial">0.718076</font></td>
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<strong><font face="Arial">RC</font></strong></td>
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<strong><font face="Arial">0.471948</font></strong></td>
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<strong><font face="Arial">0.136603</font></strong></td>
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<strong><font face="Arial">0.063083</font></strong></td>
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<strong><font face="Arial">0.031651</font></strong></td>
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<strong><font face="Arial">0.016205</font></strong></td>
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<strong><font face="Arial">0.008340</font></strong></td>
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<strong><font face="Arial">0.004299</font></strong></td>
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<strong><font face="Arial">0.002216</font></strong></td>
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<font face="Arial"> </font></td>
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<strong><font face="Arial">0.000005</font></strong></td>
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<strong><font face="Arial">0.000019</font></strong></td>
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<strong><font face="Arial">0.000040</font></strong></td>
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<strong><font face="Arial">0.000080</font></strong></td>
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<strong><font face="Arial">0.000156</font></strong></td>
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<strong><font face="Arial">0.000304</font></strong></td>
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<strong><font face="Arial">0.000589</font></strong></td>
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<div dir="ltr" align="left"><font face="Arial"></font> </div>
<div dir="ltr" align="left"><span class="x_661413100-10012021"><font face="Arial">So, you see, there is no iteration. You simply have to transform k' until it equals 1. I have determined that this always occurs within 8 steps, as shown. Here it actually
occurred in 6 steps, and the calculated sn values were equal to the sine value until i = 4 to six decimal places.</font></span></div>
<div dir="ltr" align="left"><span class="x_661413100-10012021"><font face="Arial"></font></span> </div>
<div dir="ltr" align="left"><span class="x_661413100-10012021"><font face="Arial">The calculation for "p" is closed-form, but there is no more convenient way to calculate elliptic sines.</font></span></div>
</span></font></span>
<div dir="ltr" align="left"><span class="x_661413100-10012021"><font color="#0000ff" size="2" face="Arial"></font></span> </div>
<div dir="ltr" lang="en-us" class="x_OutlookMessageHeader" align="left">
<hr tabindex="-1">
<font size="2" face="Tahoma"><b>From:</b> Bernard Arthur Hutchins, Jr [mailto:bah13@cornell.edu]
<br>
<b>Sent:</b> Saturday, January 09, 2021 1:35 PM<br>
<b>To:</b> synth-diy@synth-diy.org<br>
<b>Cc:</b> David G Dixon<br>
<b>Subject:</b> [sdiy] 90-degree phase displacement network calculations <br>
</font><br>
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<div></div>
<span style="background-color:#ffecb3; color:#000000; font-size:12px"><span style="background-color:#ffecb3; color:#000000; font-size:12px"><span style="padding-bottom:3px; line-height:1.6; background-color:#ffecb3; font-style:normal; padding-left:3px; padding-right:3px; color:#000000; font-size:12px; font-weight:normal; padding-top:3px">[<strong>CAUTION:</strong>
Non-UBC Email]</span></span></span>
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<p class="x_MsoNormal" style="line-height:107%; margin:0in 0in 8pt"><font color="#ff0000" face="Arial, sans-serif"><span style="background-color:rgb(250,250,250)"></span></font> </p>
<p class="x_MsoNormal" style="line-height:107%; margin:0in 0in 8pt; font-family:Calibri,sans-serif; color:rgb(0,0,0); font-size:11pt">
<span style="font-family:'Arial',sans-serif; font-size:12pt">Thanks again David – </span></p>
<p class="x_MsoNormal" style="line-height:107%; margin:0in 0in 8pt; font-family:Calibri,sans-serif; color:rgb(0,0,0); font-size:11pt">
<span style="font-family:'Arial',sans-serif; font-size:12pt"></span></p>
<p class="x_MsoNormal" style="line-height:107%; margin:0in 0in 8pt; font-family:Calibri,sans-serif; color:rgb(0,0,0); font-size:11pt">
<span style="font-family:'Arial',sans-serif; font-size:12pt">I see. Your function is NOT closed form – it is iterative; self-terminating at 8 “for loops” – absolutely NOTHING wrong with that.<span style="">
</span> </span></p>
<p class="x_MsoNormal" style="line-height:107%; margin:0in 0in 8pt; font-family:Calibri,sans-serif; color:rgb(0,0,0); font-size:11pt">
<span style="font-family:'Arial',sans-serif; font-size:12pt"></span></p>
<p class="x_MsoNormal" style="line-height:107%; margin:0in 0in 8pt; font-family:Calibri,sans-serif; color:rgb(0,0,0); font-size:11pt">
<span style="font-family:'Arial',sans-serif; font-size:12pt">In equiripple designs we have the powerful “alternation theorem” to tell us WHEN we have “found” an acceptable ER solution (at least - close enough).
<span style=""> </span>At the same time, the mathematicians assure us (prove!) that we will never find closed form!<span style="">
</span>(Don’t you hate it when you are shown that they are right?) </span></p>
<p class="x_MsoNormal" style="line-height:107%; margin:0in 0in 8pt; font-family:Calibri,sans-serif; color:rgb(0,0,0); font-size:11pt">
<span style="font-family:'Arial',sans-serif; font-size:12pt"></span></p>
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<span style="font-family:'Arial',sans-serif; font-size:12pt">What happens (if you know) if you terminate after just a couple of iterations?<span style=""> That is: </span>How does convergence progress? </span></p>
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<span style="font-family:'Arial',sans-serif; font-size:12pt"></span></p>
<p class="x_MsoNormal" style="line-height:107%; margin:0in 0in 8pt; font-family:Calibri,sans-serif; color:rgb(0,0,0); font-size:11pt">
<span style="font-family:'Arial',sans-serif; font-size:12pt">-Bernie </span></p>
<br>
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