<html><head><meta http-equiv="Content-Type" content="text/html; charset=utf-8"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class=""><div dir="auto" style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class=""><br class=""><div><blockquote type="cite" class=""><div class="">On Apr 17, 2020, at 8:53 AM, Donald Tillman <<a href="mailto:don@till.com" class="">don@till.com</a>> wrote:</div><div class=""><div class=""><br class=""><blockquote type="cite" class="">On Apr 17, 2020, at 1:56 AM, René Schmitz <<a href="mailto:synth@schmitzbits.de" class="">synth@schmitzbits.de</a>> wrote:<br class=""><br class="">Interesting circuit, and a great article.<br class="">I'm pretty sure I have seen a similar technique before, because I have used it. (cusp canceling)<br class=""></blockquote><br class="">I am very familiar with cusp cancellation. I've used it also. And it's mentioned in the article.<br class="">This is not cusp cancellation.</div></div></blockquote><br class=""></div><div><br class=""></div><div>I'd like to expand on this for a moment...</div><div><br class=""></div><div>I think the phrase "cusp cancellation" has, accidentally, been misused a lot. And that's caused confusion.</div><div><br class=""></div><div>"Cusp cancellation" should mean that we've already got a pretty good approximation going, but the cusps of the triangle are still coming through a little bit. And we can cancel those by subtracting a small amount of the original triangle wave. Sweet!</div><div><br class=""></div><div>This would be because the transfer curve of the diff amp pair isn't completely flat at the top and bottom. The tanh() curve is asymptotic, so there will always be a little slope on the peaks.</div><div><br class=""></div><div>The most common next step is to apply negative feedback around the diff amp pair. This could be in the form of a feedback resistor, or by adding small emitter resistors. The negative feedback plumps up the curve and flattens the slope at the peaks for a better overall fit. Nice!</div><div><br class=""></div><div>But here, with the Colin/Henry/Guest/Tillman (Have I got everybody? In order?) approach, the output of the diff amp pair isn't remotely close to a sine wave. Not even trying. And none of us are using negative feedback to plump out the curve. We're not in the cusp cancelling business, we're doing something else.</div><div><br class=""></div><div>I got here by applying actual cusp cancellation to an actual diff amp pair with negative feedback and a pretty good sine approximation. Then I refined it with thousands of simulations, which lead me away from cusp cancelling, and toward considering a compound curve of tanh(x) - βx, expressly for the bumps and the sine shape in between. And the rest as I described.</div><div><br class=""></div><div>So I guess Dennis Colin (ARP, Aries) got to the circuit first.</div><div><br class=""></div><div>So, I'll claim that if a small amount of the original triangle wave is subtracted from a wave that's roughly sinusoidal, then it's actual cusp cancellation.</div><div><br class=""></div><div>But if the diff amp pair contribution doesn't look like a sine wave, and there's no negative feedback, and the transfer function can be put into the form tanh(x) - βx, then it's this other approach that Dennis Colin pioneered.</div><div><br class=""></div><div> -- Don</div><div class=""><div style="color: rgb(0, 0, 0); font-family: Menlo; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px;">--<br class="">Donald Tillman, Palo Alto, California<br class=""><a href="http://www.till.com" class="">http://www.till.com</a></div>
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