Package Modelica.​Blocks.​Continuous.​Internal.​Filter.​Utilities
Utility functions for filter computations

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Extends from Modelica.​Icons.​InternalPackage (Icon for an internal package (indicating that the package should not be directly utilized by user)).

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Function Modelica.​Blocks.​Continuous.​Internal.​Filter.​Utilities.​BesselBaseCoefficients
Return coefficients of normalized low pass Bessel filter (= gain at cut-off frequency 1 rad/s is decreased 3dB)

Information

The transfer function H(p) of a n 'th order Bessel filter is given by

        Bn(0)
H(p) = -------
        Bn(p)
 

with the denominator polynomial

         n             n  (2n - k)!       p^k
Bn(p) = sum c_k*p^k = sum ----------- * -------   (1)
        k=0           k=0 (n - k)!k!    2^(n-k)

and the numerator

               (2n)!     1
Bn(0) = c_0 = ------- * ---- .                    (2)
                n!      2^n
 

Although the coefficients c_k are integer numbers, it is not advisable to use the polynomials in an unfactorized form because the coefficients are fast growing with order n (c_0 is approximately 0.3e24 and 0.8e59 for order n=20 and order n=40 respectively).

Therefore, the polynomial Bn(p) is factorized to first and second order polynomials with real coefficients corresponding to zeros and poles representation that is used in this library.

The function returns the coefficients which resulted from factorization of the normalized transfer function

H'(p') = H(p),  p' = p/w0

as well as

alpha = 1/w0

the reciprocal of the cut of frequency w0 where the gain of the transfer function is decreased 3dB.

Both, coefficients and cut off frequency were calculated symbolically and were eventually evaluated with high precision calculation. The results were stored in this function as real numbers.

Calculation of normalized Bessel filter coefficients

Equation

abs(H(j*w0)) = abs(Bn(0)/Bn(j*w0)) = 10^(-3/20)

which must be fulfilled for cut off frequency w = w0 leads to

[Re(Bn(j*w0))]^2 + [Im(Bn(j*w0))]^2 - (Bn(0)^2)*10^(3/10) = 0

which has exactly one real solution w0 for each order n. This solutions of w0 are calculated symbolically first and evaluated by using high precise values of the coefficients c_k calculated by following (1) and (2).

With w0, the coefficients of the factorized polynomial can be computed by calculating the zeros of the denominator polynomial

        n
Bn(p) = sum w0^k*c_k*(p/w0)^k
        k=0

of the normalized transfer function H'(p'). There exist n/2 of conjugate complex pairs of zeros (beta +-j*gamma) if n is even and one additional real zero (alpha) if n is odd. Finally, the coefficients a, b1_k, b2_k of the polynomials

 a*p + 1,  n is odd 

and

b2_k*p^2 + b1_k*p + 1,   k = 1,... div(n,2)

results from

a = -1/alpha

and

b2_k = 1/(beta_k^2 + gamma_k^2) b1_k = -2*beta_k/(beta_k^2 + gamma_k^2)

Extends from Modelica.​Icons.​Function (Icon for functions).

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Function Modelica.​Blocks.​Continuous.​Internal.​Filter.​Utilities.​toHighestPowerOne
Transform filter to form with highest power of s equal 1

Information

This icon indicates Modelica functions.

Extends from Modelica.​Icons.​Function (Icon for functions).

Inputs

Outputs

Function Modelica.​Blocks.​Continuous.​Internal.​Filter.​Utilities.​normalizationFactor
Compute correction factor of low pass filter such that amplitude at cut-off frequency is -3db (=10^(-3/20) = 0.70794...)

Information

This icon indicates Modelica functions.

Extends from Modelica.​Icons.​Function (Icon for functions).

Inputs

Outputs

Function Modelica.​Blocks.​Continuous.​Internal.​Filter.​Utilities.​bandPassAlpha
Return alpha for band pass

Information

A band pass with bandwidth "w" is determined from a low pass

  1/(p^2 + a*p + b)

with the transformation

  new(p) = (p + 1/p)/w

This results in the following derivation:

  1/(p^2 + a*p + b) -> 1/( (p+1/p)^2/w^2 + a*(p + 1/p)/w + b )
                     = 1 /( ( p^2 + 1/p^2 + 2)/w^2 + (p + 1/p)*a/w + b )
                     = w^2*p^2 / (p^4 + 2*p^2 + 1 + (p^3 + p)a*w + b*w^2*p^2)
                     = w^2*p^2 / (p^4 + a*w*p^3 + (2+b*w^2)*p^2 + a*w*p + 1)

This 4th order transfer function shall be split in to two transfer functions of order 2 each for numerical reasons. With the following formulation, the fourth order polynomial can be represented (with the unknowns "c" and "alpha"):

  g(p) = w^2*p^2 / ( (p*alpha)^2 + c*(p*alpha) + 1) * ( (p/alpha)^2 + c*(p/alpha) + 1)
       = w^2*p^2 / ( p^4 + c*(alpha + 1/alpha)*p^3 + (alpha^2 + 1/alpha^2 + c^2)*p^2
                                                   + c*(alpha + 1/alpha)*p + 1 )

Comparison of coefficients:

  c*(alpha + 1/alpha) = a*w           -> c = a*w / (alpha + 1/alpha)
  alpha^2 + 1/alpha^2 + c^2 = 2+b*w^2 -> equation to determine alpha

  alpha^4 + 1 + a^2*w^2*alpha^4/(1+alpha^2)^2 = (2+b*w^2)*alpha^2
    or z = alpha^2
  z^2 + a^2*w^2*z^2/(1+z)^2 - (2+b*w^2)*z + 1 = 0

Therefore the last equation has to be solved for "z" (basically, this means to compute a real zero of a fourth order polynomial):

   solve: 0 = f(z)  = z^2 + a^2*w^2*z^2/(1+z)^2 - (2+b*w^2)*z + 1  for "z"
              f(0)  = 1  > 0
              f(1)  = 1 + a^2*w^2/4 - (2+b*w^2) + 1
                    = (a^2/4 - b)*w^2  < 0
                    // since b - a^2/4 > 0 requirement for complex conjugate poles
   -> 0 < z < 1

This function computes the solution of this equation and returns "alpha = sqrt(z)";

Extends from Modelica.​Icons.​Function (Icon for functions).

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