Digital Signal Processing Reference
In-Depth Information
Envisaging applications
The main application of complex calculus in the technological field is electro-technology
and in the field of natural sciences it is physics. Of course, electro-technology is founded
on the physical basis of electromagnetism as well as on phenomena of charge flow in
solid, liquid and gaseous materials and so-called plasma (gas partially or completely
consisting of free charge carriers such as ions or electrons).
That is why there are only three basic components in electro- technology: OHM's
resistance R, inductance of a coil L and the capacitance of a capacitor C.
The resistance of Ohm here stands for the heat effect of the electric current or the charge
flow in materials, i.e. the transformation of electromagnetic energy into “mechanical”
thermal energy. Strictly speaking, simultaneously occurring heat radiation is also
electromagnetic energy, though in a very high frequency range (infrared).
Direct current flowing through a coil creates a magnetic field. In electro- technology, this
is represented by the inductance L as a component. Between the plates of a capacitor fed
with direct current voltage, an electrical field manifests itself; the capacitance C - a
component dedicated to the electric field - describes the proportion of the charge
quantity supplied and the voltage level.
This simplified explanation is not intended to conceal the fact that any temporally
alterable current always creates an electromagnetic field, and that any temporally
alterable magnetic field creates a temporally alterable electrical field and vice versa.
The effects of the electromagnetic field are measured by voltages and currents. There are
5 fundamental equations operating this electromagnetic interplay. Illustration 308
demonstrates these.
Now that all the relevant mathematical (and signalling) processes have been dealt with
within the framework of complex calculation, an application of both great practical and
theoretical relevance will be calculated: the oscillating circuit. Here this stands for any
frequency selective or frequency dependent processes and systems. In Illustration 308 the
series resonant circuit, the series connection of R, L and C, is selected.
Because the correlation of voltage and current at the coil and the capacitor is described by
means of the differential coefficient or integral calculus, the process of the practical cal-
culation of circuits and networks result in differential equations. This is where EULER's
relation (Illustration 303) comes in. The e-function is very easy to differentiate and inte-
grate. A relatively straightforward algebraic equation (Illustration 309) results from the
the differential equation.
In electro technology, for the imaginary unit the letter j (j² = -1) has been chosen, because
i had already been assigned to electric current.
