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<span style="font-family: Arial, sans-serif;"><o:p style=""><span style="font-size: 18px; background-color: rgb(250, 250, 250);"> </span><span style="font-size: 12pt;"> </span></o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif"><span style="mso-spacerun:yes"> </span>The “design” of 90-Degree Phase-Differencing Networks (90-degreePDNs) can involve some deeper theoretical discussions as well as practical ”plug and chug” equation
summaries.<span style="mso-spacerun:yes"> </span>The earliest offerings in EN involved references to academic presentations from which practical equations were largely just cribbed {D. Weaver, "Design of RC Wide-Band 90-Degree Phase-Difference Network," Proc.
IRE, Vol. 42, pp 671-676 (1954); W. Albershein & F. Shirley, "Computation Methods for Broad-Band 90° Phase-Difference Networks," IEEE Trans. Ckt. Theory, CT-16, pg 189 (1969) }.<span style="mso-spacerun:yes">
</span>See a summary at:<o:p> </o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif"><a href="http://electronotes.netfirms.com/MEHCh6aPart.PDF">http://electronotes.netfirms.com/MEHCh6aPart.PDF</a><o:p> </o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif">with details at:<span style="mso-spacerun:yes">
</span><a href="http://electronotes.netfirms.com/AN167-AN170.PDF">http://electronotes.netfirms.com/AN167-AN170.PDF</a><o:p> </o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif">and a quit important contribution to practical implementations at: “Design of Wideband Analog 90° Phase Differencing Networks Without a Large Spread of Capacitor Values”;
<a href="http://electronotes.netfirms.com/EN168-90degreePDN.PDF">http://electronotes.netfirms.com/EN168-90degreePDN.PDF</a><o:p> </o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif">The offering here from David seems to be (like our own!) deficient in theoretical explanations.
<span style="mso-spacerun:yes"> </span>Is there a link/reference?<span style="mso-spacerun:yes">
</span>Perhaps no one cares? <span style="mso-spacerun:yes"> </span><span style="mso-spacerun:yes"> </span><span style="mso-spacerun:yes"> </span>But some notion of a “UNIQUE-BEST ” (UB) (can’t do better) solution requires theory – in this case it seems to
be the “alternation theorem” (AT) as applied to the phase error.<o:p> </o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif">The AT allows us to RECOGNIZE the UB equi-ripple solution in terms of the finding of a correct
<span style="mso-spacerun:yes"> </span>SET of “extremal frequencies” as found by ITERATING an “exchange algorithm” (Remez or Parks-McClellan).
<span style="mso-spacerun:yes"> </span>No closed form is possible, but convergence is very rapid.<br>
<span style="mso-spacerun:yes"> </span><span style="mso-spacerun:yes"> </span><span style="mso-spacerun:yes"> </span><span style="mso-spacerun:yes"> </span><o:p> </o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif">I have not attempted this.<o:p> </o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif"><o:p>-Bernie</o:p></span></p>
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