ISSN 1068-3712, Russian Electrical Engineering, 2017, Vol. 88, No. 12, pp. 842–844. © Allerton Press, Inc., 2017.
Original Russian Text © A.V. Gorelik, V.Yu. Gorelik, V.I. Apattsev, A.P. Baturin, V.A. Kobzev, I.A. Zhuravlev, 2017, published in Elektrotekhnika, 2017, No. 12, pp. 76–79.
A Method of Investigating Electrical Systems with Periodically
A. V. Gorelik*, V. Yu. Gorelik, V. I. Apattsev, A. P. Baturin, V. A. Kobzev, and I. A. Zhuravlev
Russian University of Transport, Moscow, 127994 Russia
Received November 14, 2017
Abstract⎯A method is presented of investigating the impulse-response-function stability of electrical sys-
tems with periodically time-varying parameters. The method uses the Laplace transformation and Hill’s
determinants. The method is of general character and imposes minimum restrictions on the form of the dif-
ferential equation. Expressions for the characteristic equation of the impulse-response function are derived.
An example of calculation for the second-order differential equation is given.
Keywords: control systems, parametric amplifiers, variables, Hill determinants, transient processes
Interest in the development of methods of analysis
and synthesis of periodically nonstationary systems
has been unabating for several decades due to the
importance of the problem, as well as the considerable
difficulties arising in investigations of similar systems.
In electrical engineering, radio-engineering and
telecommunications devices are used that are based on
resonance and other properties of electric circuits with
variable parameters. This is because basic transforma-
tions, such as generation of high-frequency oscillations,
modulation, detection, frequency conversion, and many
others, can be only realized using nonlinear systems or
linear systems with variable parameters [1–3].
For example, a synchronous electrical machine is
of the most widely used technical systems with period-
ically changing parameters.
In a synchronous machine with protruding poles,
periodic changes of the self-inductance coefficients of
stator phases and coefficients of mutual inductance
between phase windings take place.
In the general case, the electromagnetic processes
in a synchronous machine are described by a system of
differential equations with periodic coefficients [3, 4].
Low-power-capacitor asynchronous motors
(CAMs) are the most common motors in modern
electric drives powered by one-phase alternating-cur-
rent networks. The main problem in CAM analysis is
the fact that these motors possess asymmetry, and,
therefore, transient electromechanical processes in
the natural (phase) coordinate system also can be
described by the system of nonlinear differential equa-
tions with periodic coefficients .
Modern radiotechnical, especially radio-receiving,
equipment includes amplifiers with automatic gain
control (AGC) designed to ensure small changes of the
output signal level at large changes of the input signal.
An AGC system is a system with variable parameters,
and its behavior is described by a linear differential
equation with varying coefficients (in the case of peri-
odic signals with periodic coefficients) .
Solving differential equations with periodic coeffi-
cients is required in the cases of both absolute and
periodic stability of nonlinear systems .
Investigations of stability of the systems of this kind
are especially interesting now, because the mathemat-
ical packages widely used in practice cannot be
Let the system under consideration be described
with the following differential equation:
where x is an adjustable coordinate, δ(t – ξ) is the
Dirac delta function, and ξ is the external-impact
−≥ = =
2, 1, const,
rl c c
fn t d