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Engineering
Electrical Engineering
Power Series Equation for Amplifier and Harmonics
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[QUOTE="FactChecker, post: 5916491, member: 500115"] I am not sure if this is what you are looking for, but here are my two cents: For harmonics, the best expansion would be in terms of trigonometric functions, not a power series. That being said, here is a brief description of the power series. a[SUB]0[/SUB] + a[SUB]1[/SUB]x + a[SUB]2[/SUB]x[SUP]2[/SUP] + a[SUB]3[/SUB]x[SUP]3[/SUP] + ... + a[SUB]n[/SUB]x[SUP]n[/SUP] is a Taylor series approximation. Since it has powers of (x-x[SUB]0[/SUB]), where x[SUB]0[/SUB]=0, it is expanded around x[SUB]0[/SUB]=0 and is called a Maclaurin series. The first term, a[SUB]0[/SUB], is the function value at x=0. The second term, a[SUB]1[/SUB]x, adjusts for the slope (first derivative) of the function at x=0. The third term, a[SUB]2[/SUB]x[SUP]3[/SUP], adjusts for the curvature (second derivative) of the function at x=0. For well behaved functions, more terms give better approximations of the function farther away from the central point, x[SUB]0[/SUB] = 0. Since you are interested in harmonics, the expansion of sin(x) and cos(x) will be of special interest. Here is a figure showing Taylor series expansions of f(x)=cos(x) at x[SUB]0[/SUB] = 0 with more and more terms. The function cos(x) is an even function, so the coefficients of the odd powers of x are all 0. g(x)=1-x[SUP]2[/SUP]/2 looks ok very near x=0, but the errors get large away from x=0. As more and more terms are added, the functions, g(x), h(x), p(x), q(x), r(x), and s(x) get more and more accurate farther from x=0. They follow the higher derivatives of f(x) better. The final function, t(x), shows the error between s(x) and f(x). You can see that the error is fairly small out to about x=5 and then grows rapidly. [ATTACH=full]218010[/ATTACH] [/QUOTE]
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Power Series Equation for Amplifier and Harmonics
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