Equalization process

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A compensation process occurs in a physical or chemical system in which a stationary process is changed by an intervention - such as switching on, load change, disruption in the process - and changes into a new stationary process.

Compensation processes are of general physical importance and occur in many technical processes. They do not take place suddenly at the time of the intervention, but steadily and are described by the time behavior of certain state variables . For example, in electrical engineering when a capacitor is charging, the voltage can serve as this state variable.

In the general case, it is not a matter of course that an equalization process changes into a stable, stationary process. In the case of the linear passive networks, however, it can be shown that unstable states are not to be feared. This article is limited to the stable cases.

Cause, course

The physical cause of a compensation process is the energy or mass stored in components , which cannot be changed suddenly. In electrical engineering, for example, it is about electrical energy in capacities , magnetic energy in inductances and rotational energy in rotating machines. The result of the balancing process is a new continuous process. In systems that are capable of vibrations (e.g. alternating current network), the stationary process is characterized by the fact that the amplitude and frequency of all sinusoidal state variables are constant.

The equalization process can run aperiodically (sliding) or oscillating, so that the final state is also referred to as the steady state . In the equalization process, for example, in a water pipe due to a broken pipe (or a pump failure), in spite of the associated pressure drop, unacceptably high pressure surges can nevertheless occur, which trigger further pipe bursts.

In the case of an oscillating process, the extent of the equalization process cannot be foreseen, since its initial state is uncertain due to the randomness of the time of the intervention. In AC circuits in particular, considerable voltage or current surges depending on the point in time ( phase angle ) can occur at a higher frequency than the mains frequency , see, for example, frequency information on earth faults . The new stationary process, on the other hand, is independent of the initial state.

Mathematical treatment

The mathematical treatment of equalization processes leads to a linear ordinary differential equation with constant coefficients. In the case of storage elements, this is of the -th order. Alternatively, first order differential equations are possible. Your solution describes a superposition of the expected new steady process and the asymptotically decaying (volatile) equalization process. For example, in the case of an electric circuit, the current can be described as the superposition of a steady-state current and a compensating current, which mediates the continuous transition.

  • The stationary process follows from the particular solution of the inhomogeneous differential equation for .
In electrical engineering, this corresponds to a direct current or alternating current bill.
  • The fleeting process follows from the general solution of the homogeneous differential equation including the determination of constants.

Alternatively, the differential equations can be solved using the Laplace transform . Differentiating and integrating are reduced to multiplying and dividing through this transformation. A linear differential equation in the time domain becomes a linear algebraic equation in an image domain.

Individual evidence

  1. ↑ a b c Wilfried Weißgerber: Electrical engineering for engineers 3: compensation processes, Fourier analysis, four-pole theory. Springer Vieweg, 8th edition 2013, p. 1 ff
  2. ↑ a b Reinhold Paul, Steffen Paul: Repetitorium Elektrotechnik: Electromagnetic fields, networks, systems. Springer, 1996, p. 510
  3. ↑ a b c d Rolf Müller: Analyzing compensation processes in electro-mechanical systems with Maple. Vieweg + Teubner, 2011, p. 1
  4. ↑ Jörg Hugel: Electrical engineering: Basics and applications . Teubner, 1998, p. 368
  5. ↑ Amir M. Miri: Balancing processes in electrical energy systems: Mathematical introduction, electromagnetic and electromechanical processes . Springer, 2000, p. 2
  6. ↑ Amir M. Miri, p. 151
  7. ↑ Amir M. Miri, p. 150
  8. ↑ Alfred Fraenckel: Theory of alternating currents . Springer, 3rd edition 1930, p. 190
  9. ^ Karl Küpfmüller: Introduction to Theoretical Electrical Engineering . Springer, 10th ed. 1973, p. 491
  10. ↑ Reinhold Paul, Steffen Paul, p. 515
  11. ^ Alfred Fraenckel, p. 186
  12. ↑ Wilfried Weißgerber, p. 30 ff.
  13. ↑ Hans-Otto Seinsch: Compensation processes in electrical drives: Fundamentals of analytical and numerical calculation . Springer, 1991, p. 9 f