# 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 xy Larger than x>=y Larger than or equal

## 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

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