How to solve this simple non-linear equation numerically with Abs[]?

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1

Hi,

I was just trying to get into Mathematica a little more but I’ve been stuck when trying to solve an equation. The H function models a low pass filter and I want to find out the cut off frequency.

Z[\[Omega]_] := 1 /(\[ImaginaryJ]*\[Omega]*c)
H[\[Omega]_] := Z[\[Omega]]/(r + Z[\[Omega]])

c = 1*^-6; r = 1000;
LogLogPlot[Abs[H[2 \[Pi]*f]], {f, 1, 1000000}, ImageSize -> Large, AxesOrigin -> {1, 1*^-3}, GridLines -> {{160}, {}}]

NSolve[Abs[H[2 \[Pi]*f]] == 1/Sqrt[2], f, Reals]
NSolve[Abs[H[2 \[Pi]*f]] <= 1/Sqrt[2], f, Reals]

f = 160; Abs[H[2 \[Pi]*f]] < 1/Sqrt[2]

Plotting the simple diagram worked. Now, when I try to numerically solve this (in)equation, I get this error:

NSolve::nddc: "The system ... contains a nonreal constant -500000\ I. With the domain Reals specified, all constants should be real."

Even though the Abs[] around it should get rid of the i. However, when I “manually” test the inequation, I get back true. What am I doing wrong?

Thanks for any help

2

From Documentation on NSolve (see Details section) you can find out that

NSolve deals primarily with linear and polynomial equations.

You obviously deal with non-polynomial equation due to Abs[] function. In this cases use FindRoot function. From plot we see that the solution is located around 160:

Z[\[Omega]_] := 1/(\[ImaginaryJ]*\[Omega]*c)
H[\[Omega]_] := Z[\[Omega]]/(r + Z[\[Omega]])

c = 1*^-6; r = 1000;
LogLogPlot[{1/Sqrt[2], Abs[H[2 \[Pi]*f]]}, {f, 1, 1000000}, 
 ImageSize -> Large, AxesOrigin -> {1, 1*^-3}, GridLines -> {{160}, {}}]

as23-LK320.png
as23-LK320.png404×239 6.18 KB

And this simple line solves your problem:

FindRoot[Abs[H[2 \[Pi]*f]] == 1/Sqrt[2], {f, 100}]
(* {f -> 159.155} *)
3 
In[1]:= Z[?_] := 1/(\[ImaginaryJ]*?*c);
H[?_] := Z[?]/(r + Z[?]);
c = 1*^-6; r = 1000;
Reduce[Abs[H[2 ?*f]] == 1/Sqrt[2] && f > 0, f]

Out[4]= f == 500/?

if you want the result in terms of radians instead of a decimal approximation, just as long as you don’t use a decimal point anywhere