- Qucs -
Quite Universal Circuit Simulator

Short Description of Mathematical Functions

The following operations and functions can be applied in Qucs equations. For detailed description please refer to the "Measurement Expressions Reference Manual". Parameters in brackets "[]" are optional.

Operators

Arithmetic Operators

`+x`	Unary plus
`-x`	Unary minus
`x+y`	Addition
`x-y`	Subtraction
`x*y`	Multiplication
`x/y`	Division
`x%y`	Modulo (remainder of division)
`x^y`	Power

Logical Operators

`!x`	Negation
`x&&y`	And
`x\|\|y`	Or
`x^^y`	Exclusive or
`x?y:z`	Abbreviation for conditional expression "if `x` then `y` else `z`"
`x==y`	Equal
`x!=y`	Not equal
`x<y`	Less than
`x<=y`	Less than or equal
`x>y`	Larger than
`x>=y`	Larger than or equal

Alphabetical List of Functions

abs adjoint angle arccos arccosec arccot arcosech arcosh arcoth arcsec arcsin arctan arg arsech arsinh artanh avg besseli0 besselj bessely ceil conj cos cosec cosech cosh cot coth cumavg cumprod cumsum dB dbm dbm2w ddx deg2rad det dft diff erf erfc erfcinv erfinv exp eye fft fftshift fix floor Freq2Time GaCircle GpCircle hypot idft ifft imag integrate interpolate inverse kbd length limexp linspace ln log10 log2 logspace mag max min Mu Mu2 NoiseCircle norm phase PlotVs polar prod rad2deg random real rms Rollet round rtoswr rtoy rtoz runavg sec sech sign sin sinc sinh sqr sqrt srandom StabCircleL StabCircleS StabFactor StabMeasure stddev step stos stoy stoz sum tan tanh Time2Freq transpose twoport unwrap variance vt w2dbm xvalue ytor ytos ytoz yvalue ztor ztos ztoy

Math Functions

Vectors and Matrices: Creation

`eye(n)`	Creates `n` x `n` identity matrix
`length(y)`	Returns the length of the given vector
`linspace(from,to,n)`	Creates a real vector with `n` linearly spaced components between `from` and `to`, both inclusively
`logspace(from,to,n)`	Creates a real vector with `n` logarithmically spaced components between `from` and `to`, both inclusively

Vectors and Matrices: Basic Matrix Functions

`adjoint(x)`	Adjoint matrix of `x` (transposed and conjugate complex)
`det(x)`	Determinant of a matrix `x`
`inverse(x)`	Inverse matrix of `x`
`transpose(x)`	Transposed matrix of `x` (rows and columns exchanged)

Elementary Mathematical Functions: Basic Real and Complex Functions

`abs(x)`	Absolute value, magnitude of complex number
`angle(x)`	Phase angle in radians of a complex number. Synonym for `arg()`
`arg(x)`	Phase angle in radians of a complex number
`conj(x)`	Conjugate of a complex number
`deg2rad(x)`	Converts phase from degrees into radians
`hypot(x,y)`	Euclidean distance function
`imag(x)`	Imaginary part of a complex number
`mag(x)`	Magnitude of a complex number
`norm(x)`	Square of the absolute value of a vector
`phase(x)`	Phase angle in degrees of a complex number
`polar(m,p)`	Transform from polar coordinates (magnitude `m`, phase `p`) into complex number
`rad2deg(x)`	Converts phase from radians into degrees
`real(x)`	Real part of a complex number
`sign(x)`	Signum function
`sqr(x)`	Square (power of two) of a number
`sqrt(x)`	Square root
`unwrap(p[,tol[,step]])`	Unwraps the angle `p` (in radians -- default `step` is 2*pi) using the optional tolerance value `tol` (default is pi)

Elementary Mathematical Functions: Exponential and Logarithmic Functions

`exp(x)`	Exponential function to basis e
`limexp(x)`	Limited exponential function
`log10(x)`	Decimal logarithm
`log2(x)`	Binary logarithm
`ln(x)`	Natural logarithm (base e )

Elementary Mathematical Functions: Trigonometry

`cos(x)`	Cosine function
`cosec(x)`	Cosecant
`cot(x)`	Cotangent function
`sec(x)`	Secant
`sin(x)`	Sine function
`tan(x)`	Tangent function

Elementary Mathematical Functions: Inverse Trigonometric Functions

`arccos(x)`	Arc cosine (also known as "inverse cosine")
`arccosec(x)`	Arc cosecant
`arccot(x)`	Arc cotangent
`arcsec(x)`	Arc secant
`arcsin(x)`	Arc sine (also known as "inverse sine")
`arctan(x[,y])`	Arc tangent (also known as "inverse tangent")

Elementary Mathematical Functions: Hyperbolic Functions

`cosh(x)`	Hyperbolic cosine
`cosech(x)`	Hyperbolic cosecant
`coth(x)`	Hyperbolic cotangent
`sech(x)`	Hyperbolic secant
`sinh(x)`	Hyperbolic sine
`tanh(x)`	Hyperbolic tangent

Elementary Mathematical Functions: Inverse Hyperbolic Functions

`arcosh(x)`	Hyperbolic area cosine
`arcosech(x)`	Hyperbolic area cosecant
`arcoth(x)`	Hyperbolic area cotangent
`arsech(x)`	Hyperbolic area secant
`arsinh(x)`	Hyperbolic area sine
`artanh(x)`	Hyperbolic area tangent

Elementary Mathematical Functions: Rounding

`ceil(x)`	Round to the next higher integer
`fix(x)`	Truncate decimal places from real number
`floor(x)`	Round to the next lower integer
`round(x)`	Round to nearest integer

Elementary Mathematical Functions: Special Mathematical Functions

`besseli0(x)`	Modified Bessel function of order zero
`besselj(n,x)`	Bessel function of first kind and `n`-th order
`bessely(n,x)`	Bessel function of second kind and `n`-th order
`erf(x)`	Error function
`erfc(x)`	Complementary error function
`erfinv(x)`	Inverse error function
`erfcinv(x)`	Inverse complementary error function
`sinc(x)`	Sinc function (sin(`x`)/`x` or 1 at `x` = 0)
`step(x)`	Step function

Data Analysis: Basic Statistics

`avg(x[,range])`	Arithmetic average of vector elements; if a range is given then `x` must have a single data dependency
`cumavg(x)`	Cumulative average of vector elements
`max(x,y)`	Returns the greater of the values `x` and `y`
`max(x[,range])`	Maximum value in vector; if a range is given then `x` must have a single data dependency
`min(x,y)`	Returns the lesser of the values `x` and `y`
`min(x[,range])`	Minimum value in vector; if a range is given then `x` must have a single data dependency
`rms(x)`	Root Mean Square of vector elements
`runavg(x)`	Running average of vector elements
`stddev(x)`	Standard deviation of vector elements
`variance(x)`	Variance of vector elements
`random()`	Random number between 0.0 and 1.0
`srandom(x)`	Give random seed

Data Analysis: Basic Operation

`cumprod(x)`	Cumulative product of vector elements
`cumsum(x)`	Cumulative sum of vector elements
`interpolate(f,x[,n])`	Equidistant spline interpolation of real function vector `f(x)` using `n` equidistant datapoints; the latter can be omitted and defaults to a reasonable value
`prod(x)`	Product of vector elements
`sum(x)`	Sum of vector elements
`xvalue(f,yval)`	Returns the x-value which is associated with the y-value nearest to a specified y-value `yval` in a given vector `f`; therefore the vector `f` must have a single data dependency
`yvalue(f,xval)`	Returns the y-value of the given vector `f` which is located nearest to the x-value `xval`; therefore the vector `f` must have a single data dependency

Data Analysis: Differentiation and Integration

`ddx(expr,var)`	Derives mathematical expression `expr` with respect to the variable `var`
`diff(y,x[,n])`	Differentiate vector `y` with respect to vector `x` `n` times. If `n` is omitted it defaults to `n` = 1
`integrate(x,h)`	Integrate vector `x` numerically assuming a constant step-size `h`

Data Analysis: Signal Processing

`dft(x)`	Discrete Fourier Transform of vector `x`
`fft(x)`	Fast Fourier Transform of vector `x`
`fftshift(x)`	Shuffles the FFT values of vector `x` in order to move the frequency 0 to the center of the vector
`Freq2Time(V,f)`	Inverse Discrete Fourier Transform of function `V(f)` interpreting it physically
`idft(x)`	Inverse Discrete Fourier Transform of vector `x`
`ifft(x)`	Inverse Fast Fourier Transform of vector `x`
`kbd(x[,n])`	Kaiser-Bessel derived window
`Time2Freq(v,t)`	Discrete Fourier Transform of function `v(t)` interpreting it physically

Electronics Functions

Unit Conversion

`dB(x)`	dB value
`dbm(x)`	Convert voltage to power in dBm
`dbm2w(x)`	Convert power in dBm to power in Watts
`w2dbm(x)`	Convert power in Watts to power in dBm
`vt(t)`	Thermal voltage for a given temperature `t` in Kelvin

Reflection Coefficients and VSWR

`rtoswr(x)`	Converts reflection coefficient to voltage standing wave ratio (VSWR)
`rtoy(x[,zref])`	Converts reflection coefficient to admittance; by default reference `zref` is 50 ohms
`rtoz(x[,zref])`	Converts reflection coefficient to impedance; by default reference `zref` is 50 ohms
`ytor(x[,zref])`	Converts admittance to reflection coefficient; by default reference `zref` is 50 ohms
`ztor(x[,zref])`	Converts impedance to reflection coefficient; by default reference `zref` is 50 ohms

N-Port Matrix Conversions

`stos(s,zref[,z0])`	Converts S-parameter matrix to S-parameter matrix with different reference impedance(s)
`stoy(s[,zref])`	Converts S-parameter matrix to Y-parameter matrix
`stoz(s[,zref])`	Converts S-parameter matrix to Z-parameter matrix
`twoport(m,from,to)`	Converts a two-port matrix from one representation into another, possible values for `from` and `to` are 'Y', 'Z', 'H', 'G', 'A', 'S' and 'T'.
`ytos(y[,z0])`	Converts Y-parameter matrix to S-parameter matrix
`ytoz(y)`	Converts Y-parameter matrix to Z-parameter matrix
`ztos(z[,z0])`	Converts Z-parameter matrix to S-parameter matrix
`ztoy(z)`	Converts Z-parameter matrix to Y-parameter matrix

Amplifiers

`GaCircle(s,Ga[,arcs])`	Circle(s) with constant available power gain `Ga` in the source plane
`GpCircle(s,Gp[,arcs])`	Circle(s) with constant operating power gain `Gp` in the load plane
`Mu(s)`	Mu stability factor of a two-port S-parameter matrix
`Mu2(s)`	Mu' stability factor of a two-port S-parameter matrix
`NoiseCircle(Sopt, Fmin,Rn,F[,Arcs])`	Generates circle(s) with constant Noise Figure(s) `F`. `Arcs` specifies the angles in degrees created by e.g. `linspace(0,360,100)`. If Arcs is a number it specifies the number of equally spaced circle segments, if it is omitted this number defaults to a reasonable value
`PlotVs(data,dep)`	Returns a data item based upon vector or matrix vector `data` with dependency on a given vector `dep`, e.g. `PlotVs(Gain,frequency/1e9)`
`Rollet(s)`	Rollet stability factor of a two-port S-parameter matrix
`StabCircleL(s[,arcs])`	Stability circle in the load plane
`StabCircleS(s[,arcs])`	Stability circle in the source plane
`StabFactor(s)`	Stability factor of a two-port S-parameter matrix. Synonym for `Rollet()`
`StabMeasure(s)`	Stability measure B1 of a two-port S-parameter matrix

Nomenclature

Ranges

`LO:HI`	Range from `LO` to `HI`
`:HI`	Up to `HI`
`LO:`	From `LO`
`:`	No range limitations

Matrices and Matrix Elements

`M`	The whole matrix `M`
`M[2,3]`	Element being in 2nd row and 3rd column of matrix `M`
`M[:,3]`	Vector consisting of 3rd column of matrix `M`

Immediate

`2.5`	Real number
`1.4+j5.1`	Complex number
`[1,3,5,7]`	Vector
`[11,12;21,22]`	Matrix

Number suffixes

`E`	exa, * 1e+18
`P`	peta, * 1e+15
`T`	tera, * 1e+12
`G`	giga, * 1e+9
`M`	mega, * 1e+6
`k`	kilo, * 1e+3
`m`	milli, * 1e-3
`u`	micro, * 1e-6
`n`	nano, * 1e-9
`p`	pico, * 1e-12
`f`	femto, * 1e-15
`a`	atto, * 1e-18

Name of Values

`S[1,1]`	S-parameter value
nodename.`V`	DC voltage at node nodename
name.`I`	DC current through component name
nodename.`v`	AC voltage at node nodename
name.`i`	AC current through component name
nodename.`vn`	AC noise voltage at node nodename
name.`in`	AC noise current through component name
nodename.`Vt`	Transient voltage at node nodename
name.`It`	Transient current through component name

Note: All voltages and currents are peak values.
Note: Noise voltages are RMS values at 1 Hz bandwidth.

Constants

`i, j`	Imaginary unit ("square root of -1")
`pi`	4*arctan(1) = 3.14159...
`e`	Euler = 2.71828...
`kB`	Boltzmann constant = 1.38065e-23 J/K
`q`	Elementary charge = 1.6021765e-19 C

back to the top

- Qucs -Quite Universal Circuit Simulator