How to Implement x**2-y**2=1
How to? create a circuit that solves the algebraic equation. X**2-y**2=1?? There are many ways. I found a simple one by meditation.?
I wrote the equation as (x+y)(x-y)=1. “When the product of two things becomes unity” is a powerful way of computing functional inverse. What are these things? Meditate. Consider them as gains. The current- mirror provides exponential current-gain e**n when we connect a PTAT v- source in series with the diode connected device. If V= nvT where vT = kT/q. Similarly, if we move the v-source to? the other side, we get an attenuation e**(-n). If we feed the output of the first to the input of the second mirror? We get product of two gains!? When that becomes unity (1). Some? visible magic happens. What happens when a loop gain becomes > unity in a closed loop +ve feedback system?? It explodes to infinity! If it becomes less than unity, it decays to zero. This is a comparator with positive feedback. Now we realize that the job is simple.? I will draw a picture of a cicuit that dances hyperbolically.? Note that I connected this circuit to the pull up and pull down I designed for my friend Itfan. Please simulate? it ? You can see that I made one v-source variable so that Iwill attain the correct value at a known speed by integraing a current. Here the starting current is e**n. Iin. When the magic happens you reset the integrating capacitor. I do it by making the current go to zero. This will make go? zero when the output transistor of the mirror acting as a? switch opens!? Remember that it makes the circuit adiabatic. Furthermore, the voltage gain tends to infinity when go goes to zero.
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Now you see the elegance of Siva’s dance. The cicuit puts out an impulse discharge current. The integrated charge is C. Delta Vth. Please simulate it. Put an initial condition zero on the cap. This way the circuit will start with a loop gain e **n charging the cap almost instantaneously. As the cap charges the gain will reduce. When it goes below unity the transistor will shut off. . My V- source (deta Vth) is not PTAT. I chose it for convenience. You can make it PTAT by the Xi- Chen circuit (his MS thesis at UT).?
University of Texas at Austin Texas
1 年Barrie Hilbert and George Wilson were my friends. Wilson hired my student Ken Schlotzhauer from Waterloo and I remember visiting Tektronix around 1980.
I may recall TRV teaching us the Gilbert Cell Analog Multiplier in late 1970s. This is a modern nonlinear analog version using CMOS devices! It is fun to design nonlinear function analog circuits.
University of Texas at Austin Texas
1 年I have a simpler faster circuit. Simulate it The same idea but uses complementary mirrors to simplify the topology. Otherwise it works the same way. Simulate it. https://share.icloud.com/photos/06c99UUKQ1WjraOtiDgocJBOA
University of Texas at Austin Texas
1 年For those who have not seen this https://docs.google.com/document/d/1c7BB1pEhOU548mSyHbHgxO6ng_RII0MBoQUNquBbNoU/edit
University of Texas at Austin Texas
1 年There other ways of getting hyperbolic movement. I will present it in another article. Those who are new comers you can see all my articles in Linkedin.