Using an FFT for power computing
A complex number is a number consisting of a real and an imaginary part. This number can be represented
as a point or position vector in a two-dimensional Cartesian coordinate system called the complex plane.
The numbers are conventionally plotted using the real part as the horizontal component, and imaginary
part as the vertical
Another way of encoding points in the complex plane, other than using the x- and y-coordinates, is to use
the distance of a point z to O, the point whose coordinates are (0,0), and the angle of the line through z and
O. This idea leads to the polar form of complex numbers. The absolute value (or magnitude) of a complex
number z = x + iy is
The argument or phase of z is defined as:
Together, r and Phi show another way of representing complex numbers, the polar form, as the combination
of modulus and argument fully specify the position of a point on the plane.
Root Mean Square Computing
In electrical engineering, the Root Mean Square (RMS) is a fundamental measurement of the magnitude
of an AC signal. The RMS value assigned to an AC signal is the amount of DC required to produce an
equivalent amount of heat in the same load. In a complex plane, the RMS value of the current (IRMS) and
the voltage (URMS) is the same as the summation of their magnitudes associated
for each harmonic. Tthe total RMS values of current and voltage in the frequency
domain are defined as:
Complex power computing
AC power flow has three components: real or true power (P) measured in watts (W), apparent power (S)measured in volt-amperes (VA), and reactive power (Q) measured in reactive volt-amperes (VAr). These
three types of power—true, reactive, and apparent—relate to one another in a trigonometric form. This is
called a power triangle
Angle Phi in this picture is the phase of voltage relative to current. A complex power is then defined as:
Where U is a voltage vector (U = URe + jUIm) and I is a current vector (I = IRe + jIIm). Note, that I* is a
complex conjugate current vector.
The length of a complex power (|S|) is the apparent power (VA) actually. In terms
of current and voltage phasors (FFT outputs), and in terms of next Equation the complex power in Cartesian
form can be finally expressed as:
Where:
IRE(k), URE(k) are real parts of kth harmonics of current and voltage.
IIM(k), UIM(k) are imaginary parts of kth harmonics of current and voltage.
final...
As seen in an application:
