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Why screw with the integrator and the diode mess when you can use a single chip and a few resistors to get a pretty good approximation to a sine wave. A simple filter will get rid of the harmonics. The more steps you use, t he more attenuation you get with the same filter. A 14 pin, 8 FF shift reg ister (74xx164) and anything that inverts plus 8 resistors and a single cap acitor gets you a pretty simple and effective circuit.
Fewer chips than the integrator/diode thing. Simpler. Easy to see if you have something wrong.
BTW, the 2 FF ring counter doesn't even give you as good an approximation a s the other circuit since the timing is equal and the steps have to be equa l. The original circuit has unequal timing on the two amplitude bits and u nequal amplitude to give a better approximation. The OP's simulation is no t correct. It appears his weighting resistors are approximately equal so t he amplitudes are the same and it ends up looking like a triangle more than a sine.