Modelica.ComplexMath

Information

This package contains basic mathematical functions operating on complex numbers (such as sin(..)), as well as functions operating on vectors of complex numbers.

Extends from Modelica.Icons.Package (Icon for standard packages).

Package Content

Name	Description
j=Complex(0, 1)	Imaginary unit
Vectors	Library of functions operating on complex vectors
sin	Sine of complex number
cos	Cosine of complex number
tan	Tangent of complex number
asin	Arc-sine of complex number
acos	Arc-cosine of complex number
atan	Arc-tangent of complex number
sinh	Hyperbolic-sine of complex number
cosh	Hyperbolic-cosine of complex number
tanh	Hyperbolic-tangent of complex number
asinh	Area-hyperbolic-sine of complex number
acosh	Area-hyperbolic-cosine of complex number
atanh	Area-hyperbolic-tangent of complex number
exp	Exponential of complex number
log	Logarithm of complex number
'abs'	Absolute value of complex number
arg	Phase angle of complex number
conj	Conjugate of complex number
real	Real part of complex number
imag	Imaginary part of complex number
fromPolar	Complex from polar representation
'sqrt'	Square root of complex number
'max'	Return maximum element of complex vector
'min'	Return minium element of complex vector
'sum'	Return sum of complex vector
'product'	Return product of complex vector

Types and constants

  final constant Complex j = Complex(0,1) "Imaginary unit";

Modelica.ComplexMath.sin

Information

This function returns the Complex sine of the Complex input.

Inputs

Type	Name	Default	Description
Complex	c1		Complex number

Outputs

Type	Name	Description
Complex	c2	sin(c1)

Modelica definition

function sin "Sine of complex number"
  input Complex c1 "Complex number";
  output Complex c2 "sin(c1)";
algorithm 
   c2 := (exp(Complex(-c1.im, +c1.re)) - exp(Complex(+c1.im, -c1.re)))/Complex(0, 2);
end sin;

Modelica.ComplexMath.cos

Information

This function returns the Complex cosine of the Complex input.

Inputs

Type	Name	Default	Description
Complex	c1		Complex number

Outputs

Type	Name	Description
Complex	c2	= cos(c1)

Modelica definition

function cos "Cosine of complex number"
  input Complex c1 "Complex number";
  output Complex c2 "= cos(c1)";
algorithm 
  c2 := (exp(Complex(-c1.im, +c1.re)) + exp(Complex(+c1.im, -c1.re)))/2;
end cos;

Modelica.ComplexMath.tan

Information

This function returns the Complex tangent of the Complex input.

Inputs

Type	Name	Default	Description
Complex	c1		Complex number

Outputs

Type	Name	Description
Complex	c2	= tan(c1)

Modelica definition

function tan "Tangent of complex number"
  input Complex c1 "Complex number";
  output Complex c2 "= tan(c1)";
algorithm 
  c2 := sin(c1)/cos(c1);
end tan;

Modelica.ComplexMath.asin

Information

This function returns the inverse Complex sine of the Complex input.

Inputs

Type	Name	Default	Description
Complex	c1		Complex number

Outputs

Type	Name	Description
Complex	c2	arc_sin(c1)

Modelica definition

function asin "Arc-sine of complex number"
  input Complex c1 "Complex number";
  output Complex c2 "arc_sin(c1)";
algorithm 
  c2 := -j*log(j*c1 + 'sqrt'(1 - c1*c1));
end asin;

Modelica.ComplexMath.acos

Information

This function returns the inverse Complex cosine of the Complex input.

Inputs

Type	Name	Default	Description
Complex	c1		Complex number

Outputs

Type	Name	Description
Complex	c2	= arc_cos(c1)

Modelica definition

function acos "Arc-cosine of complex number"
  input Complex c1 "Complex number";
  output Complex c2 "= arc_cos(c1)";
algorithm 
  c2 := -j*log(c1 + j*'sqrt'(1 - c1*c1));
end acos;

Modelica.ComplexMath.atan

Information

This function returns the inverse Complex tangent of the Complex input.

Inputs

Type	Name	Default	Description
Complex	c1		Complex number

Outputs

Type	Name	Description
Complex	c2	= arc_tan(c1)

Modelica definition

function atan "Arc-tangent of complex number"
  input Complex c1 "Complex number";
  output Complex c2 "= arc_tan(c1)";
algorithm 
  c2 := 0.5*j*log((j + c1)/(j - c1));
end atan;

Modelica.ComplexMath.sinh

Information

This function returns the Complex hyperbolic sine of the Complex input.

Inputs

Type	Name	Default	Description
Complex	c1		Complex number

Outputs

Type	Name	Description
Complex	c2	sinh(c1)

Modelica definition

function sinh "Hyperbolic-sine of complex number"
  input Complex c1 "Complex number";
  output Complex c2 "sinh(c1)";
algorithm 
  c2 := Complex(Math.sinh(c1.re)*Math.cos(c1.im), Math.cosh(c1.re)*Math.sin(c1.im));
end sinh;

Modelica.ComplexMath.cosh

Information

This function returns the Complex hyperbolic cosine of the Complex input.

Inputs

Type	Name	Default	Description
Complex	c1		Complex number

Outputs

Type	Name	Description
Complex	c2	= cosh(c1)

Modelica definition

function cosh "Hyperbolic-cosine of complex number"
  input Complex c1 "Complex number";
  output Complex c2 "= cosh(c1)";
algorithm 
  c2 := Complex(Math.cosh(c1.re)*Math.cos(c1.im), Math.sinh(c1.re)*Math.sin(c1.im));
end cosh;

Modelica.ComplexMath.tanh

Information

This function returns the Complex hyperbolic tangent of the Complex input.

Inputs

Type	Name	Default	Description
Complex	c1		Complex number

Outputs

Type	Name	Description
Complex	c2	= tanh(c1)

Modelica definition

function tanh "Hyperbolic-tangent of complex number"
  input Complex c1 "Complex number";
  output Complex c2 "= tanh(c1)";
algorithm 
  c2 := sinh(c1)/cosh(c1);
end tanh;

Modelica.ComplexMath.asinh

Information

This function returns the inverse Complex hyperbolic sine of the Complex input.

Inputs

Type	Name	Default	Description
Complex	c1		Complex number

Outputs

Type	Name	Description
Complex	c2	ar_sinh(c1)

Modelica definition

function asinh "Area-hyperbolic-sine of complex number"
  input Complex c1 "Complex number";
  output Complex c2 "ar_sinh(c1)";
algorithm 
  c2 := log(c1 + 'sqrt'(c1*c1 + 1));
end asinh;

Modelica.ComplexMath.acosh

Information

This function returns the inverse Complex hyperbolic cosine of the Complex input.

Inputs

Type	Name	Default	Description
Complex	c1		Complex number

Outputs

Type	Name	Description
Complex	c2	= ar_cosh(c1)

Modelica definition

function acosh "Area-hyperbolic-cosine of complex number"
  input Complex c1 "Complex number";
  output Complex c2 "= ar_cosh(c1)";
algorithm 
  c2 := log(c1 + (c1 + 1)*'sqrt'((c1 - 1)/(c1 + 1)));
end acosh;

Modelica.ComplexMath.atanh

Information

This function returns the inverse Complex hyperbolic tangent of the Complex input.

Inputs

Type	Name	Default	Description
Complex	c1		Complex number

Outputs

Type	Name	Description
Complex	c2	= ar_tanh(c1)

Modelica definition

function atanh "Area-hyperbolic-tangent of complex number"
  input Complex c1 "Complex number";
  output Complex c2 "= ar_tanh(c1)";
algorithm 
  c2 := 0.5*log((1 + c1)/(1 - c1));
end atanh;

Modelica.ComplexMath.exp

Information

This function returns the Complex natural exponential of the Complex input.

Inputs

Type	Name	Default	Description
Complex	c1		Complex number

Outputs

Type	Name	Description
Complex	c2	= exp(c1)

Modelica definition

function exp "Exponential of complex number"
  input Complex c1 "Complex number";
  output Complex c2 "= exp(c1)";
algorithm 
  c2 := Complex(Math.exp(c1.re)*Math.cos(c1.im), Math.exp(c1.re)*Math.sin(c1.im));
end exp;

Modelica.ComplexMath.log

Information

This function returns the Complex natural logarithm of the Complex input.

Inputs

Type	Name	Default	Description
Complex	c1		Complex number

Outputs

Type	Name	Description
Complex	c2	= log(c1)

Modelica definition

function log "Logarithm of complex number"
  input Complex c1 "Complex number";
  output Complex c2 "= log(c1)";
algorithm 
  c2 := Complex(Modelica.Math.log('abs'(c1)), arg(c1));
end log;

Modelica.ComplexMath.'abs'

Information

This function returns the Real absolute of the Complex input, i.e., it's length.

Inputs

Type	Name	Default	Description
Complex	c		Complex number

Outputs

Type	Name	Description
Real	result	= abs(c)

Modelica definition

function 'abs' "Absolute value of complex number"
  input Complex c "Complex number";
  output Real result "= abs(c)";
algorithm 
  result := (c.re^2 + c.im^2)^0.5; //changed from sqrt
end 'abs';

Modelica.ComplexMath.arg

Information

This function returns the Real argument of the Complex input, i.e., it's angle.

Inputs

Type	Name	Default	Description
Complex	c		Complex number
Angle	phi0	0	Phase angle phi shall be in the range: -pi < phi-phi0 < pi [rad]

Outputs

Type	Name	Description
Angle	phi	= phase angle of c [rad]

Modelica definition

function arg "Phase angle of complex number"
  input Complex c "Complex number";
  input Modelica.SIunits.Angle phi0=0 
    "Phase angle phi shall be in the range: -pi < phi-phi0 < pi";
  output Modelica.SIunits.Angle phi "= phase angle of c";
algorithm 
  phi := Modelica.Math.atan3(
      c.im,
      c.re,
      phi0);
end arg;

Modelica.ComplexMath.conj

Information

This function returns the Complex conjugate of the Complex input.

Inputs

Type	Name	Default	Description
Complex	c1		Complex number

Outputs

Type	Name	Description
Complex	c2	= c1.re - j*c1.im

Modelica definition

function conj "Conjugate of complex number"
  input Complex c1 "Complex number";
  output Complex c2 "= c1.re - j*c1.im";
algorithm 
  c2 := Complex(c1.re, -c1.im);
end conj;

Modelica.ComplexMath.real

Information

This function returns the real part of the Complex input.

Inputs

Type	Name	Default	Description
Complex	c		Complex number

Outputs

Type	Name	Description
Real	r	= c.re

Modelica definition

function real "Real part of complex number"
  input Complex c "Complex number";
  output Real r "= c.re ";
algorithm 
  r := c.re;
end real;

Modelica.ComplexMath.imag

Information

This function returns the imaginary part of the Complex input.

Inputs

Type	Name	Default	Description
Complex	c		Complex number

Outputs

Type	Name	Description
Real	r	= c.im

Modelica definition

function imag "Imaginary part of complex number"
  input Complex c "Complex number";
  output Real r "= c.im ";
algorithm 
  r := c.im;
end imag;

Modelica.ComplexMath.fromPolar

Information

This function constructs a Complex number from it's length (absolute) and angle (argument).

Inputs

Type	Name	Default	Description
Real	len		abs of complex
Angle	phi		arg of complex [rad]

Outputs

Type	Name	Description
Complex	c	= len*cos(phi) + j*len*sin(phi)

Modelica definition

function fromPolar "Complex from polar representation"
  input Real len "abs of complex";
  input Modelica.SIunits.Angle phi "arg of complex";
  output Complex c "= len*cos(phi) + j*len*sin(phi)";
algorithm 
  c := Complex(len*Modelica.Math.cos(phi), len*Modelica.Math.sin(phi));
end fromPolar;

Modelica.ComplexMath.'sqrt'

Information

This function returns the Complex square root of the Complex input.

Inputs

Type	Name	Default	Description
Complex	c1		Complex number

Outputs

Type	Name	Description
Complex	c2	= sqrt(c1)

Modelica definition

function 'sqrt' "Square root of complex number"
  input Complex c1 "Complex number";
  output Complex c2 "= sqrt(c1)";
algorithm 
  c2 := Complex(sqrt('abs'(c1))*Math.cos(arg(c1)/2), sqrt('abs'(c1))*Math.sin(arg(c1)/2));
end 'sqrt';

Modelica.ComplexMath.'max'

Information

This function returns the largest element of the Complex input vector, defined by the Complex absolute.

Inputs

Type	Name	Default	Description
Complex	v[:]		Vector

Outputs

Type	Name	Description
Complex	result	Element of v with largest absolute value
Integer	index	v[index] has the largest absolute value

Modelica definition

function 'max' "Return maximum element of complex vector"
  input Complex v[:] "Vector";
  output Complex result "Element of v with largest absolute value";
  output Integer index "v[index] has the largest absolute value";
protected 
  Real absv_i;
  Real absres;
algorithm 
  if size(v,1) > 0 then
    absres := 'abs'(v[1]);
    index  := 1;
    for i in 2:size(v,1) loop
      absv_i := 'abs'(v[i]);
      if absv_i > absres then
        absres := absv_i;
        index := i;
      end if;
    end for;
    result :=v[index];
  else
    result := Complex(0);
    index  := 0;
  end if;
end 'max';

Modelica.ComplexMath.'min'

Information

This function returns the smallest element of the Complex input vector, defined by the Complex absolute.

Inputs

Type	Name	Default	Description
Complex	v[:]		Vector

Outputs

Type	Name	Description
Complex	result	Element of v with smallest absolute value
Integer	index	v[index] has the smallest absolute value

Modelica definition

function 'min' "Return minium element of complex vector"
  input Complex v[:] "Vector";
  output Complex result "Element of v with smallest absolute value";
  output Integer index "v[index] has the smallest absolute value";
protected 
  Real absv_i;
  Real absres;
algorithm 
  if size(v,1) > 0 then
    absres := 'abs'(v[1]);
    index  := 1;
    for i in 2:size(v,1) loop
      absv_i := 'abs'(v[i]);
      if absv_i < absres then
        absres := absv_i;
        index := i;
      end if;
    end for;
    result :=v[index];
  else
    result := Complex(0);
    index  := 0;
  end if;
end 'min';

Modelica.ComplexMath.'sum'

Information

This function returns the Complex sum of the Complex input vector

Inputs

Type	Name	Default	Description
Complex	v[:]		Vector

Outputs

Type	Name	Description
Complex	result	Complex sum of vector elements

Modelica definition

function 'sum' "Return sum of complex vector"
  input Complex v[:] "Vector";
  output Complex result "Complex sum of vector elements";
algorithm 
  result:=Complex(0);
  for i in 1:size(v,1) loop
    result:=result + v[i];
  end for;
end 'sum';

Modelica.ComplexMath.'product'

Information

This function returns the Complex product of the Complex input vector

Inputs

Type	Name	Default	Description
Complex	v[:]		Vector

Outputs

Type	Name	Description
Complex	result	Complex product of vector elements

Modelica definition

function 'product' "Return product of complex vector"
  input Complex v[:] "Vector";
  output Complex result "Complex product of vector elements";
algorithm 
  result:=Complex(1);
  for i in 1:size(v,1) loop
    result:=result * v[i];
  end for;
end 'product';