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Use and properties of the Series command.

JohnArcher
7-Bedrock

Use and properties of the Series command.

Good morning, all.

I need some guidance on the use of the Series command.

I uinderstand the stricture of not expanding a Talyors Series too far, but I have a quotient with numerator and denonminator having many terms involving exp(x), cosh(x) etc, all based upon the exponential property. These should be solvable using expansions of exp(x) and the binomial theorem.

Does Mathcad 15 or Prime have such built-in functionality built into Series to avoid Taylor?

My equations contain data spanning dc to nanoseconds and I need more than a three term expansion.

Practically the quotient expands into a ratio of terms involving root(s), the coomplex variable. After rationaliziing, I find thr LHP roots of the denominator and use a contour integral solution.

I really need some help, here. Still pushing boundaries and learning at age 79!

Thanks, John Archer.

1 ACCEPTED SOLUTION

Accepted Solutions

Have you tried using the symbolic rewrite keyword to express your initial result in terms of exp?

Is there a simpler way of deriving the expression (are you using Laplace, for example)?

Stuart

View solution in original post

6 REPLIES 6
RichardJ
19-Tanzanite
(To:JohnArcher)

I'm not sure what you mean by "avoiding Taylor". The symbolic series keyword expands a function as a Taylor series. If you need more than three terms use Taylor,N, where N is the power of the highest term. Does that solve your problem?

series.png

Thanks ror tour prompt reply, Richard.

Because of the circuit properties (high speed transitions on mis-terminated tramsmission lines for arc fault suppression) the denominator and numerator of my quotient each produce a series in mutiples of root(s), which cleans up after Simplification. A 15th order Series command finally produces an 8th order quotient and a result that makes sense. I know that I will later need at least twice this resolution for the second half of the process. Mathcad 15 is very slow for very high order series involving cosh or sinh and complex arguements.

Mathcad Help and my books warn of errors if a series expansion is pushed too far. How do I determine "too far"?

Exp(x) is a fundamental series that does not depend on anyone's series approximation method. By inference, are sine and cosh, with complex arguements, also immune to expansion errors using Mathcad if expanded correctly? Would 64bit Prime give a better or faster result than 32bit Mathcad 15?

I already have a provisional patent on this device and will file for the full patent shortly. Before i spend several thousand dollars of my retirement money on this application, I want to make sure that I am doing things correctly.

Sorry to be a pain, but I am a creative magnetics engineer, not a math wizaerd.

John Archer

Have you tried using the symbolic rewrite keyword to express your initial result in terms of exp?

Is there a simpler way of deriving the expression (are you using Laplace, for example)?

Stuart

Thank you Stuart. We have not spoken for quite a while.

Yes, I used Rewrite, exp when Mathcad 15 took forever producing an s^30 series. I have a new, very fast, HP desktop machine with heat pipe board cooling and an added 40GB solid state hard drive. It was made for ultimate gaming but my long series take an awful time, probably because of the cosh/sinh arguements. There were sign changes and I was afraid of getting into some non- converging region. I rationalised the awfully long quotient and used Solve to pull-out one root at a time. it was tedious but steered clear of convergence problems, real or imagined! Hopefully I can ignore RHP poles because they only exist before the fault happens. There is no feedback, just standing waves.

Richard ( thankyou) has assured me that my fears were groundless and I do not need 17 digit accuracy.

I have invented an inexpensive way to control and prevent large surge currents and high voltages in a shorted power feed. Carelessness or a cannon shell may trip a circuit breaker, but there will be no arcing damage or fuel ignition.

This calculation was eased by a greatly simplified solution of the transmission line equations, for this application, which I produced when I retired from EMC work in the JSF Program Office at Pax.

RichardJ
19-Tanzanite
(To:JohnArcher)

Mathcad Help and my books warn of errors if a series expansion is pushed too far. How do I determine "too far"?

The symbolic processor will give you an exact expansion for as many terms as you want, as long as the function you are expanding does not contain floating point numbers. If it does contain floating point numbers then the accuracy is good to 20 decimal places. You can increase that accuracy by using the float keyword. The accuracy of the numeric processor is only about 15 decimal places. Where in the Mathcad help does it say there will be errors?

Exp(x) is a fundamental series that does not depend on anyone's series approximation method. By inference, are sine and cosh, with complex arguements, also immune to expansion errors using Mathcad if expanded correctly? Would 64bit Prime give a better or faster result than 32bit Mathcad 15?

What I said above about expansions using the symbolic processor are true regardless of the function you expand, and it makes no difference which version of Mathcad you use. They use the exact same symbolic processor.

Thankyou again, Richard, you have always been very helpful. I guess that I can push the series until my key design current at t = 0 does not change much. I aim to prevent supply current cut-off as the arc fault develops as this could be an EMC problem.

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