Answer3

THE MAGAZINE FOR COMPUTER APPLICATIONS Circuit Cellar Online offers articles illustrating creative solutions and unique applications through complete projects, practical tutorials, and useful design techniques.

	WHAT'S YOUR ENGINEERING QUOTIENT?
Problem 3—If you have four flip-flops and three resistors, how would you calculate the resistor values to get the best approximation to a sinewave? What is the advantage of synthesizing sinewaves this way? Answer: Start by drawing the phasor diagram for this case. We now have three waveforms, A, B and C, separated by 45° as shown below. The fundamentals have the relationship shown below. The B signal is aligned with the 0° axis, but the A and C waveforms are at –45° and +45°, respectively. The net sum will be B + 1.414×(A or C). The third harmonics a A and C have 3× the phase shift of the fundamentals, placing them at –135° and +135°, respectively, as shown below. It becomes clear that the sum of A and C can be used to cancel B if the amplitudes of A and C are equal to each other, and equal to sqrt(2) = 0.707 that of B. Going back to the fundamental diagram, this means that the net total of that component will be 2× the level of B alone. Similarly, the fifth harmonics a A and C have 5× the phase shift of the fundamentals, placing them at –225° and +225°, respectively, as shown below. Although they have switched positions, the A and C components will cancel the B component exactly as in the third-harmonic case shown above. This technique can be generalized to even more stages. Each added stage cancels another set of harmonics if the resistor values are set correctly. By cancelling low-order harmonics in this way, only the higher-order ones need to be filtered out, making it easy to synthesize high-quality sinewaves with a simple combination of digital and analog components. Contributor: Dave Tweed 12-01 — NEXT Q&A For questions or comments about Test Your EQ, e-mail eq@circuitcellar.com.