Operator calculation according to Heaviside

The operator calculation according to Heaviside describes an empirical operator calculation named after Oliver Heaviside , which was published in 1887 in his famous work "Electromagnetic Theory" . ( English operator calculus or operational calculus )

General

When solving differential equations, the purpose of the operator calculation is to replace the operation of differentiation with the algebraic operation of multiplying with an operator and thus to "transform" a differential equation that is usually relatively difficult to solve into an easier to solve algebraic equation . Based on preliminary work by Leibniz and Cauchy , Heaviside was the first to develop the operator calculation into a calculus and thus solved many of the theoretical problems of electrical engineering that were pending at the time .

To do this, he generalized the complex alternating current calculus , in which differentiation replaces, by introducing the differential operator and using it as a multiplicative factor without any deeper justification: ${\displaystyle \mathrm {j} \omega ={\frac {d}{dt}}}$ ${\displaystyle p={\frac {d}{dt}}}$

${\displaystyle p\cdot f(t)={\frac {d}{dt}}f(t)={\frac {df(t)}{dt}}=f'(t)}$

Finally, it separates from the time functions and gives the resulting expressions their “own existence” as an operator. Intuitively, he concludes that the reciprocal logically represents the operator for the integration. ${\displaystyle p}$ ${\displaystyle {\frac {1}{p}}}$

The Heaviside operator calculation (also known as Heaviside calculus) represents a generalization of the extended symbolic method to non-stationary signals and is thus a forerunner of modern operator calculations , such as the Laplace transformation and the operator calculation according to Mikusiński .

example

RC element

The course of the voltage of u _a (t) on the capacitor of an RC element is to be calculated if the direct voltage U _{e is} "switched on" at the input of the capacitor at time t = 0. With Heaviside's notation for the unit jump as “bold one”, u _e (t) = U _e · 1 .

The relationship is obtained from the circuit

${\displaystyle i_{C}=C\cdot {\frac {du_{a}}{dt}}={\frac {u_{e}-u_{a}}{R}}}$

From this follows the inhomogeneous differential equation for t ≥ 0:

${\displaystyle RC\cdot {\frac {du_{a}}{dt}}+u_{a}=U_{e}}$.

Heaviside now uses the differential operator p:

${\displaystyle RC\cdot p\cdot u_{a}+u_{a}=U_{e}}$

He leaves out

${\displaystyle (RC\cdot p+1)\cdot u_{a}=U_{e}}$

... and resolves according to the size you are looking for

${\displaystyle u_{a}={\frac {U_{e}}{1+pCR}}}$.

The “operator for the result” is found, but what does this expression mean? Heaviside tries the solution through series expansion:

${\displaystyle u_{a}={\frac {U_{e}}{pCR}}\cdot {\frac {1}{1+{\frac {1}{pCR}}}}={\frac {U_{e}}{pCR}}\cdot \left(1-{\frac {1}{pCR}}+{\frac {1}{(pCR)^{2}}}-{\frac {1}{(pCR)^{3}}}+...\right)=U_{e}\cdot \left({\frac {1}{pCR}}-{\frac {1}{(pCR)^{2}}}+{\frac {1}{(pCR)^{3}}}-{\frac {1}{(pCR)^{4}}}+...\right)}$

By interpreting as an integration operator (applied to the unit jump), one obtains the series members (which, however, vanish for t <0): ${\displaystyle {\frac {1}{p}}}$

${\displaystyle {\frac {1}{pCR}}\cdot 1={\frac {1}{RC}}\int _{0}^{t}1\cdot d\tau =t/RC}$

${\displaystyle {\frac {1}{(pCR)^{2}}}\cdot 1={\frac {1}{pCR}}\cdot \left({\frac {1}{pCR}}\cdot 1\right)={\frac {1}{(RC)^{2}}}\int _{0}^{t}\tau \cdot d\tau ={\frac {(t/RC)^{2}}{2}}}$

and general

${\displaystyle {\frac {1}{(pCR)^{n}}}\cdot 1={\frac {(t/RC)^{n}}{n!}}}$.

So

${\displaystyle u_{a}=U_{e}\cdot \left(t/RC-{\frac {(t/RC)^{2}}{2}}+{\frac {(t/RC)^{3}}{6}}-{\frac {(t/RC)^{4}}{24}}+...\right)=U_{e}\cdot \left(1-\left(1-t/RC+{\frac {(t/RC)^{2}}{2}}-{\frac {(t/RC)^{3}}{6}}+{\frac {(t/RC)^{4}}{24}}-...\right)\right)}$

Here you have to recognize - like Heaviside - "with a trained eye" that the exponential function (with negative argument) is hidden behind this series and thus receives the self-contained solution (for t ≥ 0):

${\displaystyle u_{a}=U_{e}\cdot \left(1-e^{-t/RC}\right)}$.

The transfer function

Heaviside now defines as a characteristic of the system a transfer function independent of the signals as an operator. For the above Example one gets:

${\displaystyle H={\frac {u_{a}}{u_{e}}}={\frac {1}{1+pCR}}}$.

This definition is identical to the transfer function in the sense of the extended symbolic method of alternating current technology and other operator calculations and is still of paramount importance today.

Interpretation by decomposition

Heaviside interpreted the results obtained in operator form by partial fraction decomposition or series expansion (as in the example above). Heaviside developed a reliable method for the partial fraction decomposition, which is still used as Heaviside's expansion theorem in modern operator calculations. In practice, however, there are problems with determining the roots if the degree of the denominator polynomial to be decomposed is greater than 4, if a root is 0 or occurs several times.

In contrast to this, the decomposition by series expansion is in principle quite difficult and, depending on the approach, possible in different ways. The result becomes ambiguous and so this method is "not well suited for practical engineering work". Disassembled in the above Example convert the operator into a power series as follows

${\displaystyle u_{a}=U_{e}\cdot {\frac {1}{1+pCR}}=U_{e}\cdot \left(1-pCR+(pCR)^{2}-(pCR)^{3}+...\right)}$

and interprets p as a differential operator, then one gets a wrong or meaningless result. To be on the safe side, extensive mathematical studies on the convergence or divergence of the power series would be required.

criticism

Heaviside viewed mathematics as an experimental science and believed that success justified the process. He made no distinction between the operators and the objects on which he applied them. A mathematical theory of the body would have been necessary for this, but it had not yet been worked out at the time. Heaviside always assumed (implicitly) vanishing initial conditions of the differential equations, ie “discharged energy stores” at time 0. Although Heaviside solved many of the then current problems with his operator calculation, it could not prevail and was exposed to many “attacks” by mathematicians. Only through the interpretation of the operators with the help of the Laplace transformation could the operator calculation on the basis of the integral transformation and the function theory become established in theory and practice. Finally, in 1950 the mathematician Jan Mikusiński justified an operator calculation “without Laplace transformation” with algebraic methods in a mathematically exact manner.

Web links

• Heaviside's Operator Calculus (English)

literature

• Oliver Heaviside: Electromagnetic Theory, Volume 1 (Classic Reprint) . Forgotten Books, 2010, ISBN 978-1-4400-8252-8 ( online ).
• Oliver Heaviside: Electromagnetic Theory, Volume 2 (Classic Reprint) . Forgotten Books, 2010, ISBN 978-1-4400-8877-3 ( online ).
• Oliver Heaviside: Electromagnetic Theory, Volume 3 (Classic Reprint) . Forgotten Books, 2010, ISBN 978-1-4400-8253-5 ( online ).
• FH Lange: Signals and Systems - Volume 1: Spectral Representation . Verlag Technik, Berlin 1965.
• Gerhard Wunsch : History of Systems Theory . Akademie-Verlag., Leipzig 1985.