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The Advanced symbolic method of alternating current technology is a generalization of the complex AC circuit analysis on exponentially swelling and decaying sinusoidal signals. Wherein the transition from the imaginary frequency to the complex frequency. This formal extension has several advantages for the theoretical treatment of AC networks, especially for the circuit synthesis. At the same time this representation in harmony with the results of the Laplace transform and the operational calculus by Mikusinski.

Requirements

To understand the following versions knowledge of complex numbers, electric networks and the complex AC circuit analysis are necessary.

Just as for the established practice in the complex AC circuit analysis applies also for their expansion:

  • Both methods are applicable only to linear time-invariant systems.
  • It may just be the so-called steady state can be determined.

Signals

The extended symbolic method of alternating current technology is based on exponentially rising and decaying sinusoidal input signals. In the steady state a linear time invariant system then occur only provide those signals with the same frequency and the same Hüllkurvenkonstante within the system. Practically, however, such signals little importance, but their observation brings several mathematical benefits. Substituting 0, we immediately obtain the usual sinusoidal signals. Below is an example always considered the voltage, although all statements of course also apply to the current and other physical quantities.

Mathematical basis

The starting point is derivable from the Euler formula relations

And

These allow the representation of the trigonometric functions as a superposition of two exponential functions with imaginary argument. This results, for example for a generalized characterized by exponentially increasing or decreasing sinusoidal AC voltage

Or

A real signal is thus composed of two complex signals. Here, the right term exactly the complex conjugate left- Term is due to the force superposition theorem, it is sufficient to perform all calculations with only the left term, and to use the real and imaginary parts of the result at the end.

Complex voltage and complex current

Therefore Transfer the complex voltage (or the complex current):

As is known from the complex AC statement can not be with such complex signals, the problems of the (linear) alternating current circuits dissolved considerably easier than with the (real) trigonometric functions.

Complex amplitudes

With the already used in the complex AC circuit analysis time-independent complex amplitude

One can write

Complex frequency

As a shortcut, you eventually leads the complex frequency ( in the literature are also the symbols or used ), and then receives for the complex voltage

With this signal representation now can be the calculation of the desired complex signals.

Inverse transformation

To the desired real power (or the real current ) to obtain, after it takes the calculation of the desired complex signal only to add the conjugate complex signal ( the cosine ) or subtracted ( during sinus ) and 2 or by 2j to share. The same can be reached easily by the real part or Imaginärteilbildung:

Or

It has been shown that this inverse transformation is not even in practice, but necessary because of the complex amplitude of the result are zero magnitude and phase immediately legible.

Differential operator

While employed in the complex AC statement differential operator of the purely imaginary term (which is why the complex AC circuit analysis is often called bill also ), now is the complex frequency s occurs as a differential operator, for it is true, for example:

Impedance and admittance function

As in the complex AC circuit analysis we define the impedance function of a dipole as

Admittance function is referred to as the reciprocal of the impedance function.

This yields the following elementary impedance functions:

  • Ohmic resistance R:
  • Inductance L:
  • Capacitance C:

The impedance or Admittanzfunktionen complex circuits are provided "as usual " is calculated (and often just read ):

  • Series resonant circuit:
  • Parallel resonant circuit:

Any complex impedance or Admittanzfunktionen called Zweipolfunktionen. They can be represented as a rational function in s and are the basis for the network synthesis. In particular, these functions can be displayed clearly in the pole-zero diagram.

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