Version 4 SHEET 1 1476 884 WIRE 272 -368 240 -368 WIRE 240 -368 240 -240 WIRE 240 -368 208 -368 WIRE 240 -240 272 -240 WIRE 240 -240 208 -240 WIRE 336 -368 400 -368 WIRE 400 -368 400 -224 WIRE 400 -224 336 -224 WIRE 400 -224 448 -224 WIRE 272 -208 240 -208 WIRE 240 -208 240 -192 WIRE 304 -256 304 -272 WIRE 304 -192 304 -176 WIRE 400 -224 400 -144 WIRE 400 -144 80 -144 WIRE 80 -144 80 -240 WIRE 80 -240 128 -240 WIRE 128 -368 80 -368 WIRE 272 80 240 80 WIRE 240 80 240 208 WIRE 240 80 208 80 WIRE 240 208 272 208 WIRE 240 208 208 208 WIRE 336 80 400 80 WIRE 400 80 400 224 WIRE 400 224 336 224 WIRE 400 224 448 224 WIRE 272 240 240 240 WIRE 240 240 240 256 WIRE 304 192 304 176 WIRE 304 256 304 272 WIRE 400 224 400 304 WIRE 400 304 80 304 WIRE 80 304 80 208 WIRE 80 208 128 208 WIRE 128 80 80 80 WIRE 240 -32 208 -32 WIRE 128 -32 80 -32 WIRE 240 80 240 -32 WIRE 272 496 240 496 WIRE 240 496 240 624 WIRE 240 496 208 496 WIRE 240 624 272 624 WIRE 240 624 208 624 WIRE 336 496 400 496 WIRE 400 496 400 640 WIRE 400 640 336 640 WIRE 400 640 448 640 WIRE 272 656 240 656 WIRE 240 656 240 672 WIRE 304 608 304 592 WIRE 304 672 304 688 WIRE 400 640 400 720 WIRE 400 720 80 720 WIRE 80 720 80 624 WIRE 80 624 128 624 WIRE 128 496 80 496 WIRE -304 -208 -304 -176 WIRE -304 160 -304 192 WIRE -304 80 -304 48 WIRE -304 -288 -304 -320 WIRE -304 432 -304 464 WIRE -176 432 -176 464 WIRE -304 352 -304 320 WIRE -176 352 -176 320 FLAG 240 -192 0 FLAG 304 -272 v+ FLAG 304 -176 v- FLAG 240 256 0 FLAG 304 176 v+ FLAG 304 272 v- FLAG 240 672 0 FLAG 304 592 v+ FLAG 304 688 v- FLAG 448 -224 x FLAG 448 224 y FLAG 448 640 z FLAG 80 -368 y FLAG 80 -32 x FLAG 80 80 xz FLAG 80 496 xy FLAG -304 -176 0 FLAG -304 192 0 FLAG -304 -320 xz FLAG -304 48 xy FLAG -304 464 0 FLAG -176 464 0 FLAG -304 320 v+ FLAG -176 320 v- SYMBOL Opamps\\LT1013 304 -288 R0 SYMATTR InstName U1 SYMBOL cap 272 -352 R270 WINDOW 0 32 32 VTop 0 WINDOW 3 0 32 VBottom 0 SYMATTR InstName C1 SYMATTR Value 0.1µ SYMATTR SpiceLine IC=0 SYMBOL res 112 -352 R270 WINDOW 0 32 56 VTop 0 WINDOW 3 0 56 VBottom 0 SYMATTR InstName R1 SYMATTR Value 100k SYMBOL res 112 -224 R270 WINDOW 0 32 56 VTop 0 WINDOW 3 0 56 VBottom 0 SYMATTR InstName R2 SYMATTR Value 100k SYMBOL Opamps\\LT1013 304 160 R0 SYMATTR InstName U2 SYMBOL cap 272 96 R270 WINDOW 0 32 32 VTop 0 WINDOW 3 0 32 VBottom 0 SYMATTR InstName C2 SYMATTR Value 0.1µ SYMATTR SpiceLine IC=0 SYMBOL res 112 96 R270 WINDOW 0 32 56 VTop 0 WINDOW 3 0 56 VBottom 0 SYMATTR InstName R3 SYMATTR Value 10k SYMBOL res 112 224 R270 WINDOW 0 32 56 VTop 0 WINDOW 3 0 56 VBottom 0 SYMATTR InstName R4 SYMATTR Value 1MEG SYMBOL res 112 -16 R270 WINDOW 0 32 56 VTop 0 WINDOW 3 0 56 VBottom 0 SYMATTR InstName R5 SYMATTR Value 37.5k SYMBOL Opamps\\LT1013 304 576 R0 SYMATTR InstName U3 SYMBOL cap 272 512 R270 WINDOW 0 32 32 VTop 0 WINDOW 3 0 32 VBottom 0 SYMATTR InstName C3 SYMATTR Value 0.1µ SYMATTR SpiceLine IC=0 SYMBOL res 112 512 R270 WINDOW 0 32 56 VTop 0 WINDOW 3 0 56 VBottom 0 SYMATTR InstName R6 SYMATTR Value 10k SYMBOL res 112 640 R270 WINDOW 0 32 56 VTop 0 WINDOW 3 0 56 VBottom 0 SYMATTR InstName R7 SYMATTR Value 374k SYMBOL bv -304 -304 R0 SYMATTR InstName B1 SYMATTR Value V=-V(x)*V(z)/10 SYMBOL bv -304 64 R0 SYMATTR InstName B2 SYMATTR Value V=V(x)*V(y)/10 SYMBOL voltage -304 336 R0 SYMATTR InstName V1 SYMATTR Value +15 SYMBOL voltage -176 336 R0 SYMATTR InstName V2 SYMATTR Value -15 TEXT 440 192 Left 0 ;-y TEXT -296 -496 Left 0 !.tran 0 10 0 1m TEXT -296 -464 Left 0 !.options plotwinsize=0 TEXT 544 -576 Left 0 ;http://frank.harvard.edu/~paulh/misc/lorenz.htm\n \n"Build a Lorenz Attractor"\nIn 1963 Edward Lorenz published his famous set of coupled nonlinear\nfirst-order ordinary differential equations; they are relatively simple,\nbut the resulting behavior is wonderfully complex. The equations are:\n \ndx/dt = s(y-x)\ndy/dt = rx-y-xz\ndz/dt = xy - bz\n \nwith suggested parameters s=10, r=28, and b=8/3. The solution executes\na trajectory, plotted in three dimensions, that winds around and around,\nneither predictable nor random, occupying a region known as its attractor.\nWith lots of computing power you can approximate the equations numerically,\nand many handsome plots can be found on the web. However, it's rather\neasy to implement these equations in an analog electronic circuit, with\njust 3 op-amps (each does both an integration and a sum) and two analog\nmultipliers (to form the products xy and xz).\n \nThe Circuit\nHere's the circuit: TEXT 536 184 Left 0 ;It's not hard to understand: the op-amps are wired as integrators, with\nthe various terms that make up each derivative summed at the inputs. The\nresistor values are scaled to 1 megohm, thus for example R3 weights the\nvariable x with a factor of 28 (1M/35.7k); this is combined with -y and\n-xz, each with unit weight. (note: the equations on the diagram are\nnormalized to 0.1V, hence the multiplier scale factor of 100.)\n \n\n \nThe Output\nThe circuit just sits there and produces three voltages x(t), y(t), and z(t);\nif you hook x and z into a `scope, you get a pattern like this... TEXT 536 656 Left 0 ;...the characteristic "owl's face" of the Lorenz attractor. The curve\nplays out in time, sometimes appearing to hesitate as it scales the\nboundary and decides which basin to drop back into. The value of C, the\nthree integrator capacitors, sets the time scale: at 0.47uF it does a\nleisurely wander; at 0.1uF it winds around like someone on a mission; and\nat 0.002uF it is fiercely busy solving its equations and delighting its\naudience. TEXT 96 -504 Left 0 ;Plot V(z) versus V(x)