Descripción breve de las Funciones matemáticas¶
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 |
Math Functions¶
Vectors and Matrices: Creation¶
eye(n) |
Creates n x n identity matrix |
length(y) |
Returns the length of the y vector |
linspace(from,to,n) |
Real vector with n lin spaced components between from and to |
logspace(from,to,n) |
Real vector with n log spaced components between from and to |
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 polar coordinates m and p into a 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]]) |
Unwrap angle p (radians) – defaults step = 2pi, tol = 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]) |
Average of vector x . If range given 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 of vector x . If range given x must have a single data dependency |
min(x,y) |
Returns the lesser of the values x and y |
min(x[,range]) |
Minimum of vector x . If range is given 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]) |
Spline interpolation of vector f using n equidistant points of x |
prod(x) |
Product of vector elements |
sum(x) |
Sum of vector elements |
xvalue(f,yval) |
Returns x-value nearest to yval in single dependency vector f |
yvalue(f,xval) |
Returns y-value nearest to xval in single dependency vector f |
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. 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 to move DC 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; default zref = 50 ohms |
rtoz(x[,zref]) |
Converts reflection coefficient to impedance; default zref = 50 ohms |
ytor(x[,zref]) |
Converts admittance to reflection coefficient; default zref = 50 ohms |
ztor(x[,zref]) |
Converts impedance to reflection coefficient; default zref = 50 ohms |
N-Port Matrix Conversions¶
stos(s,zref[,z0]) |
Converts S-parameter matrix to S-parameter matrix with a different Z0 |
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 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]) |
Available power gain Ga circles (source plane ) |
GpCircle(s,Gp[,arcs]) |
Operating power gain Gp circles (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]) |
Noise Figure(s) F circles |
PlotVs(data,dep) |
Returns data selected from data : dependency dep |
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 |
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 |