How to get coordinate of point from plots?

Question

I used BodePlot function to plot the magnitude and phase for a transfer function as below.

BodePlot[(1+s/10^6)/((1+s/10^2) (1+s/10^12)),{0.1,10^20}]

Now I want to do the following things:

1. Find the magnitude of the transfer function from the magnitude plot at 2*10^7 Hz.
2. Find the phase of the transfer function from the phase plot at 2*10^7 Hz.

Is there a way to get exact these numbers? I tried to use Getcoordinate () function from drawing tool and then move cursor to that point but I only get the approximated result.

Answer 1

Update (answer simplified).

Defining the function you're using as

t[s_] := (1 + s/10^6) / ( (1 + s/10^2) (1 + s/10^12) )

you can get the exact magnitude and phase at 2*10^7 Hertz from AbsArg:

AbsArg[t[I 2 10^7]]
(* {50000 Sqrt[401/100000000042500000001], -ArcTan[10003000001/199999850020]} *)

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Comments (based on old answer).

While not necessary for the present question, you can be interested in defining a FunctionTransferModel for the function t:

tfm = TransferFunctionModel[t[s], s]

Different properties for the model can be given at construction time by specifying particular values for the options of TransferFunctionModel. Also, other symbols that act on TransferFunctionModel expressions can be used to access in a simple and direct way properties of the model, such as TransferFunctionZeros and TransferFunctionPoles.

With tfm, the exact magnitude and phase are obtained in a similar way:

AbsArg[tfm[I 2 10^7][[1, 1]]]
(* {50000 Sqrt[401/100000000042500000001], -ArcTan[10003000001/199999850020]} *)

% === AbsArg[t[I 2 10^7]]
(* True *)

Answer 2

If you have the plots, you can simply use Cases to extract the datapoints being plotted, and Interpolation, to create a functional form.

{magnitude, phase} = Interpolation /@ Cases[
   BodePlot[(1 + s/10^6)/((1 + s/10^2) (1 + s/10^12)), {0.1, 10^20}],
   Line[pts_] :> pts, Infinity]

Notice that the domain is from {-1, 20}, which means that the inputs to these functions are actually the base-10 logarithm, so to get the values you just use

magnitude[Log10[2 10^7]]
phase[Log10[2 10^7]]
(* -79.9892 *)
(* -2.86327 *)

These correspond to the values shown in the plot.