Extended symbolic method of alternating current engineering

The extended symbolic method of alternating current technology is a generalization of the complex alternating current calculation to exponentially increasing and decreasing sinusoidal signals. This results in the transition from the imaginary frequency to the complex frequency . This formal extension has various advantages for the theoretical treatment of AC networks, in particular for circuit synthesis. At the same time, this representation harmonizes with the results of the Laplace transformation and the operator calculation according to Mikusiński . ${\displaystyle j\omega }$ ${\displaystyle s=\sigma +j\omega }$

requirements

Knowledge of complex numbers , electrical networks and the complex alternating current calculation is required to understand the following explanations .

As for the complex alternating current bill established in practice, the following also applies to its extension:

• Both methods can only be used for linear time-invariant systems .
• Only the so-called steady state can be determined.

Signals

The extended symbolic method of AC technology is based on exponentially increasing or decreasing sinusoidal input signals . In the steady state of a linear time-invariant system, only those signals with the same frequency and the same envelope curve constant occur within the system . In practice, however, such signals are of little importance, but their consideration has various mathematical advantages. If you set it to 0, you immediately get the usual sinusoidal signals. In the following, the voltage is always considered as an example, although all statements naturally also apply to the current and other physical quantities. ${\displaystyle \omega }$ ${\displaystyle \sigma }$ ${\displaystyle \sigma }$

Mathematical basis

The starting point are the relationships that can be derived from Euler's formula

${\displaystyle \cos \varphi ={\frac {e^{j\varphi }+e^{-j\varphi }}{2}}=\operatorname {Re} (e^{j\varphi })}$

and

${\displaystyle \sin \varphi ={\frac {e^{j\varphi }-e^{-j\varphi }}{2j}}=\operatorname {Im} (e^{j\varphi })}$.

These allow the representation of the trigonometric functions as a superposition of two exponential functions with imaginary arguments. This results, for example, for a generalized , characterized by exponentially increasing or decreasing sinusoidal alternating voltage ${\displaystyle \sigma }$

${\displaystyle u(t)={\hat {U}}e^{\sigma t}\cos {(\omega t+\varphi _{0})}={\hat {U}}\cdot {\frac {e^{\sigma t+j{(\omega t+\varphi _{0})}}+e^{\sigma t-j{(\omega t+\varphi _{0})}}}{2}}=\operatorname {Re} ({\hat {U}}{e^{\sigma t+j(\omega t+\varphi _{0})}})}$

or.

${\displaystyle u(t)={\hat {U}}e^{\sigma t}\sin {(\omega t+\varphi _{0})}={\hat {U}}\cdot {\frac {e^{\sigma t+j{(\omega t+\varphi _{0})}}-e^{\sigma t-j{(\omega t+\varphi _{0})}}}{2j}}=\operatorname {Im} ({\hat {U}}{e^{\sigma t+j(\omega t+\varphi _{0})}})}$.

${\displaystyle \omega }$- Radial frequency of the cosine oscillation

${\displaystyle \sigma }$- decay constant

${\displaystyle \varphi _{0}}$ - zero phase angle

A real signal is therefore composed of two complex signals. The right term is exactly the conjugate complex left term. Because of the applicable superposition theorem, it is sufficient to carry out all calculations only with the left term and to use the real or imaginary part of the result at the end.

Complex voltage and complex current

We therefore introduce the complex voltage (or the complex current ):

${\displaystyle {\underline {u}}(t)={\hat {U}}\cdot {e^{\sigma t+j{(\omega t+\varphi _{0})}}}}$.

As is known from complex AC calculations, the problems of (linear) AC circuits can be solved much more easily with such complex signals than with (real) trigonometric functions.

Complex amplitudes

With the time-independent complex amplitude already used in the complex AC calculation

${\displaystyle {\underline {U}}={\hat {U}}\cdot {e^{j\varphi _{0}}}}$

one can write

${\displaystyle {\underline {u}}(t)={\underline {U}}\cdot {e^{(\sigma +j{\omega )t}}}}$.

Complex frequency

Finally, as an abbreviation, one introduces the complex frequency (in the literature the symbols or are also used) and then receives for the complex voltage ${\displaystyle s=\sigma +j\omega }$ ${\displaystyle p}$ ${\displaystyle \lambda }$

${\displaystyle {\underline {u}}(t)={\underline {U}}\cdot {e^{st}}}$.

With this signal representation, the complex signals sought can now be calculated.

Inverse transformation

To get the real voltage (or the real current) you are looking for, after calculating the complex signal you only need to add its complex conjugate signal (for the cosine) or subtract it (for the sine) and add 2 or 2j share. The same can be achieved more easily by creating real or imaginary parts:

${\displaystyle u(t)={\frac {{\underline {u}}(t)+{\underline {u}}^{*}(t)}{2}}=\operatorname {Re} ({\underline {u}}(t))}$

or.

${\displaystyle u(t)={\frac {{\underline {u}}(t)-{\underline {u}}^{*}(t)}{2j}}=\operatorname {Im} ({\underline {u}}(t))}$

It has been shown that this inverse transformation is not even necessary in practice, because the amount and zero phase can be read immediately from the complex amplitude of the result.

Differential operator

While the purely imaginary expression is used as the differential operator in the complex AC calculus (which is why the complex AC calculus is often also called the calculation), the complex frequency s now appears as a differential operator , because z. B .: ${\displaystyle j\omega }$ ${\displaystyle j\omega }$

${\displaystyle {\frac {d{\underline {u}}(t)}{dt}}={\frac {d}{dt}}({\underline {U}}\cdot e^{st})={\underline {U}}\cdot {\frac {d}{dt}}(e^{st})={\underline {U}}\cdot s\cdot e^{st}=s\cdot ({\underline {U}}\cdot e^{st})=s\cdot {\underline {u}}(t)}$

Impedance and admittance function

As in the complex AC calculation, the impedance function of a two-terminal network is defined as

${\displaystyle {\underline {Z}}={\frac {{\underline {u}}(t)}{{\underline {i}}(t)}}={\frac {\underline {U}}{\underline {I}}}}$.

The reciprocal of the impedance function is called the admittance function.

This gives the following elementary impedance functions:

• ohmic resistance R:

  ${\displaystyle {\underline {Z_{R}}}={\frac {\underline {u}}{\underline {i}}}=R}$

• Inductance L:

  ${\displaystyle {\underline {Z_{L}}}={\frac {\underline {u}}{\underline {i}}}={\frac {L\cdot {\frac {d{\underline {i}}}{dt}}}{\underline {i}}}={\frac {sL\cdot {\underline {i}}}{\underline {i}}}=sL}$

• Capacity C:

  ${\displaystyle {\underline {Z_{C}}}={\frac {\underline {u}}{\underline {i}}}={\frac {\underline {u}}{C\cdot {\frac {d{\underline {u}}}{dt}}}}={\frac {\underline {u}}{sC\cdot {\underline {u}}}}={\frac {1}{sC}}}$

The impedance or admittance functions of complex circuits are calculated "as usual" (and often just read off):

• Series resonant circuit: ${\displaystyle {\underline {Z}}=R+sL+{\frac {1}{sC}}}$
• Parallel resonant circuit: ${\displaystyle {\underline {Y}}=G+sC+{\frac {1}{sL}}}$

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

See also

• Phasor

literature

• Hans Frühauf , Erich Trzeba: Synthesis and analysis of linear high frequency circuits . Academic publishing company Geest & Portig K.-G., Leipzig 1964.
• Eugen Philippow (editor): Taschenbuch Elektrotechnik, Volume 3 . Verlag Technik, Berlin 1969.
• Gerhard Wunsch : Elements of network synthesis . Verlag Technik, Berlin 1969.
• Gerhard Wunsch : History of Systems Theory . Akademie-Verlag, Leipzig 1985.