Glossaire des fonctions mathématiques disponibles¶
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 | max(x) |
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 | module |
x<y | idem abs(x) |
x<=y | carré du module |
x>y | conjugué |
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) | arg cosinus hyperbolique |
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) | conversion paramètres s vers Y |
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 |