[IEEE 2014 IEEE International Conference on Intelligent Energy and Power Systems (IEPS) - Kyiv,...

2014 IEEE International Conference on Intelligent Energy and Power Systems (IEPS)

266

Estimation of Electromechanical Systems Power Controllability According to Instantaneous Power Components

Mykhaylo Zagirnyak Electric Machines and Devices department

Kremenchuk Mykhailo Ostrohradskyi National University Kremenchuk, Ukraine [email protected]

Tetyana Korenkova, Viktoriya Kovalchuk Systems of Automatic Control and Electric Drive department

Kremenchuk Mykhailo Ostrohradskyi National University Kremenchuk, Ukraine

[email protected]

Abstract — It has been shown that the analysis of power conversion processes in all the power channel elements, based on instantaneous power harmonic components, underlies the estimation of electromechanical systems power controllability. Presence of additional power variable component reflecting energy exchange processes in the system results in the decrease of the object power controllability. It has been established that effective power is a quality measure for power processes in the system. An index of system power controllability has been offered. It has been proved that decomposition of instantaneous power signal into orthogonal components reflects real power condition of the electromechanical system.

Keywords — electromechanical systems, instantaneous power, effective power, power controllability, energy conversion, energy mode

I. INTRODUCTION Behavior of an electromechanical complex including an

electric drive, a production mechanism with a transfer device, is completely determined by power processes taking place in separate elements of the complex. On the one hand, these processes are conditioned by the actions on the part of the power supply, and on the other hand, they are determined by electrical, electromagnetic and mechanical parameters of separate elements of the system and the complex.

Considerable attention was paid to the problem of the analysis of power processes, improvement of the power efficiency of the energy converter – electric motor system [1–6]. Papers [1–3] deal with processes and quality indices of power conversion in power converter installations. The problems of development and research of the systems of optimization of electric drives power conditions are described in [4, 5]. However, as applied to electromechanical complexes, presenting a complicated channel for power transmission, conversion and consumption, this problem is less investigated.

The process of power transfer and conversion in an electromechanical system (EMS) is accompanied by partial loss of energy at each of the elements and is characterized by an ability to accumulate energy to a certain degree depending on the type of the analyzed element. The above said can be

proved, for example, by losses of energy released in the form of heat, kinetic energy accumulated in the rotating masses of electric drive. In this case power components connected with the increase or decrease of kinetic energy of masses moving in a rotary or linear way, can be both positive (during acceleration) and negative (during breaking) as to their direction.

The matter of estimation of the system power controllability (PC) taking into consideration varied energy conditions of the power channel, energy flux non-unidirectionality, presence of various energy storage devices is rather important for solution of problems of technological process control, improvement of EMS functioning quality.

A common parameter characterizing a mode of EMS, however complex it may be, is its power expressed in the same metric units independently of the physical nature of components included in the expressions for its determination (voltage and current, moment and speed, liquid pressure and discharge). This is why EMS PC is directly connected with the processes of energy consumption or generation.

Improvement of EMS PC is a rather money-consuming operation requiring installation connected with acquisition of expensive equipment (converters, filters, protective devices) and directed to rational power consumption conditions, provision of the system reliability and economy. Decrease of PC results from manifestation of specific electromechanical equipment properties, influence of elements with nonlinear characteristics, presence of energy storage systems, which is accompanied by certain conditions of energy nodal transfer in all the EMS elements.

To analyze power processes in EMS one usually uses integral estimation of power [6, 7] based on averaging of physical quantities in the assigned time interval and connected with information loss due to integration. In the mentioned papers the theory of power is formed on the basis of notions about components of complete power and power coefficients in the systems with valve inverters of various types and supplied by an ideal AC voltage three-phase circuit. As a rule, process repetition cycle is assumed to be equal or multiple of the main supply voltage cycle.

978-1-4799-2266-6/14/$31.00 ©2014 IEEE


267

Further development of methods of power processes analysis and calculation resulted in creation of modern power theories operating with instantaneous values of voltages, currents and power [8–12]. The use of the function of instantaneous power equal to the product of instantaneous values of current and voltage makes it possible to most completely characterize the change of electromagnetic energy in the time domain. Knowing the instantaneous values it is easy to determine their mean, mean-square estimations at any time interval including the signal repetition cycle.

At present p-q, cross-vector and p-q-r instantaneous power theories are successfully applied during the synthesis of systems of power active filters automated control and active controlled rectifiers [8–11]. The use of such approaches allows one to solve the problems of compensation of reactive power component, attenuation of higher harmonics of current consumed by the conversion system from the mains, etc. Efficiency of the application of a particular power theory depends on the type of the conversion system, considered operation conditions, load nonlinear characteristics, automated control device requirements, etc.

Obviously, a number of harmonic power components can be determined on the basis of the instantaneous power function. Their analysis enables the estimation of electromechanical complexes power channel energy conversion processes characterized by different physical nature of the considered signals, their form, cyclicity, presence of storage devices, nonlinear elements reflecting the special character of the technological mechanism electric drive operation, etc. In this case, equations of balance of harmonic components of power supply instantaneous power and EMS elements or complex [12] should underlie the theoretical basis of instantaneous power method.

II. RESEARCH METHOD Power conversion processes in EMS are complicated and

various. One of instantaneous power components, a component variable (inactive) in time, characterizes the power exchanging process between the mains supply and the consumer, between the technological mechanism and the motor. Presence of additional variable power components reduces the efficiency of energy conversion process, results in occurrence of higher harmonics in power spectra, is accompanied by the growth of power effective values and leads to decrease of the system PC on the whole. The above said is conditioned by the complicated structure of the energy conversion power channel in technological complexes EMS (nonsinusoidal form of the supply voltage, occurrence of nonlinear processes in technological circuit, etc.)

As power processes taking place in EMS are of the character of periodical (stationary) vibrations of power, at the analyzed period of time it is admissible to synthesize power time function by a harmonic series that is formed on the basis of orthogonal harmonic components of voltages and currents.

Thus, the purpose of paper consists in a substantiation of the approach to EMS PC estimation by means of the analysis of the power conversion processes in all the power channel elements on the basis of instantaneous power components.

For the simplest system – a circuit with ideal controlling properties, where resistance is connected to alternating-current mains with regulated amplitude and frequency, the voltage and current signals are of the form:

( ) ( )tcosUtu a Ω= ; (1)

( ) ( ) ( )tcosIRtuti a Ω== (2)

where aa I,U – amplitude values of voltage and current, correspondingly; fπ=Ω 2 – circular frequency of the signal of voltage or current; f – signal change frequency; t – time of signal change; R – resistance.

Then, instantaneous power:

( ) ( ) ( ) ( )( ) ( ).tcosPPtcosIUIU

tcosIUtitutpaaaaa

aaΩ+=Ω+=

=Ω==2222 0

2 (3)

Obviously, the amplitude àP of the power variable component does not depend on frequency and equals to the power constant 0P component.

Effective power eP is estimation of quality of energy conversion processes determined as a root-mean-square value of voltage and current instantaneous values product.

For the above considered case effective power value equals to square root of sum of power constant value squared and half of the power variable component squared:

( ) 2321 022

00

2 PPPdttpTP a

T

e =+== ∫ (4)

where T – period of change of the power signal.

Under real conditions when current and voltage at EMS energy converter output present periodical nonsinusoidal time functions with the cycle equal to the cycle of the supply nonsinusoidal voltage, they can be presented by trigonometric series of the form [13]:

( ) ( )

( ) ( )0

0 0

cos

cos sin

N

n nn

N N

na nbn n

u t U n t

U n t U n t

φ=

= =

= Ω − =

= Ω + Ω

∑

∑ ∑

(5)

( ) ( )

( ) ( )∑∑

∑

==

=

Ω+Ω=

=ψ−Ω=M

mmb

M

mma

M

mmm

tmsinItmcosI

tmcosIti

00

0 (6)

where n, m – are voltage and current harmonic numbers, correspondingly; N, M – number of voltage and current


268

components; ψϕ, – phase angles; mn ,ΩΩ – circular frequencies of the change of voltage and current signals, respectively; ;cosUU nnna ϕ= nnnb sinUU ϕ= – orthogonal cosine and sine components of voltage signal;

;cosII mmma ψ= mmmb sinII ψ= – orthogonal cosine and sine components of current signal.

Harmonic analysis of time power function allowed one to single out the following components in it:

( ) ( ) ( )( ) ( )

( ) ( )∑∑∑

∑∑

===

==

Ω+Ω+=

=ψ−Ωϕ−Ω=

==

K

kkb

K

kka

K

kk

M

mmm

N

nnn

tksinPtkcosPP

tmcosItncosU

titutp

1110

11 (7)

where ∑=

K

kkP

10 – is instantaneous power total constant

component; kaP – amplitude value of the k-th instantaneous power cosine component; kbP – amplitude value of the k-th instantaneous power sine component; k – power harmonic number ( )mnk ±= ; K – power components number.

Further analysis of (7) shows that instantaneous power includes the sum of the following components groups.

The first group is determined by multiplication of single-frequency ( )mn = voltage and current components and forms constant and canonic power components.

Instantaneous power constant component:

( ) ( )

.IU

tmcosItncosUPN

n

M

mmn

M

mmm

N

nnn

21 1

110

∑ ∑

∑∑

= =

==

=

=ψ−Ωϕ−Ω= (8)

Instantaneous power canonic order cosine component:

( ) ( ) ( )

( ) ( )

( ) 21 1

11

11

∑ ∑

∑∑

∑∑

= =

==

==

Ω=

=ΩΩ+

+ΩΩ=

N

n

M

mmn

M

mm

N

nn

M

mm

N

nnac

tkcosIU

tmsinItnsinU

tmcosItncosUtP

(9)

where mnk += . Instantaneous power canonic order sine component:

( ) ( ) ( )

( ) ( )

( ) 21 1

11

11

∑ ∑

∑∑

∑∑

= =

==

==

Ω=

=ΩΩ+

+ΩΩ=

N

n

M

mmn

M

mm

N

nn

M

mm

N

nnbc

tksinIU

tmcosItnsinU

tmsinItncosUtP

(10)

where mnk += .

The second group of components is determined by the product of various-frequency ( mn ≠ ) voltage and current components and presents non-canonic power components.

Instantaneous power non-canonic order cosine component:

( ) ( ) ( )

( ) ( )

( ) 21 1

11

11

∑ ∑

∑∑

∑∑

= =

==

==

Ω=

=ΩΩ+

+ΩΩ=

N

n

M

mmn

M

mm

N

nn

M

mm

N

nnas

tkcosIU

tmsinItnsinU

tmcosItncosUtP

(11)

where mnk ±= .

Instantaneous power non-canonic order sine component:

( ) ( ) ( )

( ) ( )

( ) 21 1

11

11

∑ ∑

∑∑

∑∑

= =

==

==

Ω=

=ΩΩ+

+ΩΩ=

N

n

M

mmn

M

mm

N

nn

M

mm

N

nnbs

tksinIU

tmcosItnsinU

tmsinItncosUtP

(12)

where mnk ±= .

Thus, instantaneous power consists of the sum of constant component and canonic and non-canonic harmonic power components:

( ) ( ) ( ) ( ) ( )tPtPtPtPPtp bsbcasac ++++= 0 . (13)

Power effective value ( )∫=T

e dttpTP0

21 is a quality

measure for power processes in the system when control signals (e.g. action frequency) change, nonlinearities occur in energy conversion circuit or energy storages of various types are introduced. Instantaneous power components (13) present initial parameters for such estimation. Expression for determination of the effective power of voltage and current polyharmonic signals is given in Table 1.

Knowing effective power in an ideal system eiP (when most typical nonlinearities are absent) and in a system with nonlinearities efP reflecting the special character of technological mechanism electric drive operation, it is possible to determine EMS PC (electromechanical system power controllability) index: efeic PPk = . In this case the master controls in the ideal and nonlinear systems are assumed equal as to the amplitude of the constant component, as well as to the amplitude and frequency of the variable component.

Table 1 contains an analytic expression for determination of PC index in EMS with a polyharmonic voltage source. This expression takes in to account the power losses in EMS power


269

channel, influence of factors decreasing power efficiency (current phase shift with respect to voltage due to the presence of reactive elements). Decrease of PC is often accompanied by power processes with high harmonic in the power spectrum and increase of the value of the system effective power. Obviously, if the system is completely controllable, the controllability coefficient ck =1; when the system is uncontrollable, ck tends to zero.

III. MODELING RESULTS A structural scheme of a model of a thyristor transducer –

motor (TT-M) with negative velocity feedback (Fig. 1) was considered for the analysis of power processes in an EMS. The motor is described by a linearized mathematical model, where

ΣΣ= RLTe – electromagnetic time constant; ΣR – motor total resistance; ΣL – total inductance; Σ= Rkd 1 – transfer coefficient; J – motor inertia moment. Operation of a unidirectional TT is presented by nonlinearity of a “saturation” type. Master control fed to the system input is of

the form: ( ) ( )tcosUUtu a Ω+= 0 , where 0U – voltage constant component; ( )tcosU a Ω – input voltage variable component. Electric current that can be presented in the form of series (6) flows through the power channel of the system. Using (7), (14) let us determine instantaneous power and effective power in the EMS, correspondingly, in the absence of a nonlinear element and in its presence. The modeled system parameters: 75=nP kW; 220=nU V; 9340.I n = А;

15000 =n rpm; 0320.R =Σ Ohm; 000820.L =Σ H; 2531.kd = ; 331.k f = ; 8962.J = kgm2; 0260.Te = s;

100=rM Nm; 22=tk ; 8921 .kr = ; 7752 .kr = ; 0640.kos = ; 0190.km = .

The analysis of instantaneous power amplitude values spectra (Fig. 2) for the considered case showed that higher order harmonics – 3, 4, 5, etc. exist in the nonlinear system instantaneous power spectrum. The obtained power spectrum confirms the thesis about the system elements nonlinear properties influence on power processes taking place in it.

TABLE I. ANALYTIC EXPRESSION FOR POWER PROCESSES ESTIMATION

Power effective value

22222

1

2

1

2

1

2

1

2

10 ⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= ∑∑∑∑∑=====

K

kkbs

K

kkbc

K

kkas

K

kkac

K

kke PPPPPP ;

EMS PC index

222222

1

2

1

2

1

2

1

2

10

2

1

2

10 ⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= ∑∑∑∑∑∑∑=======

K

kkbs

K

kkbc

K

kkas

K

kkac

K

kk

K

kkac

K

kkc PPPPPPPk .

0U

)tcos(Ua Ω

2rk

1rk

p1

tk1+pT

k

e

dfk

Jp1

×

rM

osk

fk

−+++

++ +

−

−+

)t(Ud )t(I

)t(P

)t(ω)t(E

)t(U

)t(Uos

cM 2ωmk

Fig. 1. An electromechanical system model structural scheme

a


270

b

Fig. 2. Waveforms of change of instantaneous values of voltage ( )tu , current ( )ti , power ( )tp , torque ( )tM , resistance moment ( )tMr , angular velocity ( )tω

and instantaneous power amplitude spectra of the electric motor in linear (a) and nonlinear (b) systems when ( ) ( )tcostu Ω+= 25 , 2=f Hz

As a result of the research, it was pointed out that when the amplitude of voltage variable component decreases, the spectrum of EMS instantaneous power approaches the type of spectrum in an ideal system – the amplitude values of higher harmonics considerably reduce.

Figs. 3 and 4 present curves of effective power change in an ideal eiP and factual efP systems, where constM r = and EMS PC coefficient ck when the amplitude aU and frequency f of the fed voltage variable component are changed, correspondingly. It can be seen from the analysis of the obtained curves that the value of ck does not practically change in the voltage signal variable component low-value area – an equality of effective power values in a linear and a nonlinear system can be observed. In this case EMS is insensitive to an introduced nonlinear element. When the amplitude of assignment signal variable component increases, the PC index starts to decrease sharply – this is the area where the influence of the object nonlinear characteristics is most noticeable. When the frequency of the input signal grows, the values of effective power, both in linear and nonlinear systems, increase and coefficient ck decreases. For the system with parameters

( ) ( )tcostu Ω+= 35 and f=2 Hz the value of coefficient ck changes within the range of 1 to 0.77.

Fig. 3. Curves of effective power change in ideal and factual systems depending on the voltage variable component amplitude at different values of the input signal frequency

Thus, EMS PC is an index characterizing object power reaction to change of controlling and disturbing effects in the analyzed system.

The offered approach to estimation of EMS PC with the application of harmonic analysis of instantaneous power makes it possible to research power processes when they are changed in real time with preservation of the complete information of the initial power forming signals, take into account manifestation of electromechanical equipment specific properties, appearance of power variable components loading power channel with additional components. To estimate EMS power processes it is expedient to use PC index based on determination of effective powers in ideal and real systems taking into consideration constant and variable components, where the former determines active power in the considered scheme, and the latter component determines instantaneous power in an alternating current circuit, as well as power of an interchange character.

Fig. 4. Dependence of EMS PC coefficient on the voltage variable component amplitude at different values of the input signal frequency

The considered matter acquires priority significance in the problems of technological process or complex control, improvement of EMS operation quality, which provide economical conditions of power consumption and required technological reliability.


271

IV. EXPERIMENTAL VERIFICATION To confirm the obtained results an experimental electric

hydraulic complex (EHC) has been created. It includes two rotary pumps, equal as to their parameters, with induction motors on one shaft; a manifold pipe network with installed shut-off-and-regulating fittings and receiving tanks; a regulated rotary shutter with an electric drive; frequency transformers for changing rotation frequency of pump electric motors; a control instrumentation (current, voltage, rotation frequency, pressure and discharge sensors). Electrical performance of the used equipment is shown in Table 2.

TABLE II. ELECTRICAL EQUIPMENT PERFORMANCE

Parameter name Parameter valueRated power, W 830 Rated voltage, V 380 Rated current, A 1.7

Mains frequency, Hz 50 Rotation frequency , rpm 2900

Maximum productivity, m3/hour 8 Maximum head, m 22

Rated power, W 550

The developed experimental complex embraces a whole spectrum of research and scientific applied problems. One of the main items comprises research of processes of energy transformation in all the links of hydrotransport unit presenting a complicated interconnected complex of different, as to its physical nature, equipment: electromechanical and hydraulic. Characteristics of pumping units, hydrodynamic network, pipe fittings present nonlinear dependences.

The developed EHC makes it possible to research the processes of energy conversion in all the links of a hydrotransport plant presenting a complicated system of equipment different as to its nature: electromechanical and hydraulic. Operation of hydrotransport systems is accompanied by a complicated character of change in time of output technological parameters (water head and productivity), which is conditioned by a liquid flow turbulence, cavitation phenomena in the pump and pipeline, pressure pulsations during pipeline armature handling, etc.

During the experimental research the pump cavitational processes influence on the power controllability of EHC was considered. Cavitation is accompanied by pressure periodic self-oscillations in the hydrosystem with the frequency of 0.1–0.3 Hz. Fig. 5 contains instantaneous electric ( )tpel and hydraulic

( )tphc power variation waveforms in linear (without cavitation) and nonlinear (with cavitation) systems, respectively, when the required pressure value is supported in

the pipeline. Development of cavitation processes is accompanied by occurrence of harmonic components of low-frequency order in the instantaneous electric power spectrum (Fig. 6), results in the growth of effective power values at each EHC element, reduction of the coefficient of EHC power controllability. The above said is confirmed by the results given in Table 3 for different degrees of cavitation development

0VVn ikav = , where 0V,Vi – current and initial volumes of cavitation cavern, respectively.

TABLE III. NUMERICAL VALUES OF EFFECTIVE POWERS AND EHC POWER CONTROLLABILITY COEFFICIENT

Parameter name

Linear system

System with cavitation

1=kavn 1061.nkav = 5022.nkav =

eleP 277.125 451.687 511.32

hceP 34.368 34.526 34.898

ck 1 0.614 0.542

Thus, presence of instantaneous power signals variable component reflecting the specific character of hydrodynamic processes behavior enables their presentation as a trigonometric series including two components – constant and variable. The variable component amplitude grows in cavitational operating modes. It is for this reason that the research of EHC processes can be performed with the use of the proposed approach taking into consideration the properties of the systems of the electric drive, pump and hydraulic network.

Increase of PC in the case under consideration is important in control of pressure and discharge, in optimization of power operating conditions of turbomechanism electric drives operation, in control of processes taking place in development of consequences of pumps emergency shut-offs, sharp valve closing, etc.

V. CONCLUSION It has been proved that electromechanical systems power

controllability estimation is based on the analysis of power conversion processes in all the elements of electromechanical complex power channel, using harmonic instantaneous power components. It has been established that power variable component reflects the real pattern of energy exchanging processes in the system, conditioned by presence of non-linear elements, power storage devices and results in reduction of the object power controllability. Mathematical estimation of power controllability of electromechanical systems has been proposed. It is based on the determination of effective power in an ideal and a real system taking into account instantaneous power constant and variable components.


272

а b

Fig. 5. Instantaneous electric and hydraulic power variation waveforms in linear (a) and nonlinear (b) systemsV

Fig. 6. Instantaneous power spectra in linear and nonlinear systems

Obviously, the power approach to the estimation of power controllability can be used in the analysis of more complicated conditions of technological complexes electromechanical systems. For example, in the problems of the analysis of head and consumption signals at a pumping installation output and in a hydrodynamic network with the aim of determination of restrictions by allowed values when emergency operating conditions connected with sudden power cutting-off, hydraulic impacts, etc. appear. In control system of pumping complex electric drive such protective functions can be realized by formation of the technological parameters (head and consumption) rise rate feedback.

REFERENCES [1] Herrera, R. S., Salmeryn, P., Kim, H. (2008) Application of instantaneous

power theory to the problems of compensation with the help of active filters: various approaches, calculations and experimental results. IEEE Transactions on Industrial Electronics (Vol. 55, no. 1, pp. 184–196).

[2] Herrera R. S., Salmeryn, P. (2009) Instantaneous reactive power theory: a reference in the nonlinear loads compensation. IEEE Transactions on Industrial Electronics (Vol. 56, no. 6, pp. 2015–2022).

[3] Mikhalskii, V. M. (2013) Means of improving quality of electric energy at the input and output of frequency converters and voltage transducers with pulse-duration modulation. – Kiev. The Institute of Electrodynamics of the National Academy of Sciences of Ukraine, 340 p. (in Ukraine).

[4] Ghozzi, S., Jelassi, K., Roboam, X. (2004) Energy optimization of induction motor drives,” in Proc. IEEE Conf. Industrial Technology (ICIT), pp. 602–610.

[5] Gnacinski, P. (2007) Energy saving work of frequency controlled induction cage machine. Energy Conversion and Management (Vol. 48, pp. 919–926).

[6] Tonkal, V. E., Novoseltsev, A.V., Denisiuk, S. P. et al. (1992) Energy Balance in Power Circuits, Kiev: Naukova Dumka, 312 p. (in Russian).

[7] Zinoviev, G.S. (2003) Fundamentals of power electronics, NSTU, Novosibirsk, 220 p. (in Russian).

[8] Akagi, H., Watanabe, E. H., Aredes, M. (2007) Instantaneous Power Theory and Applications to Power Conditioning. New York: Wiley. – 379 p.

[9] Soares, V., Verdelho, P., Marques, G. D. (2000) An instantaneous active and reactive current component method for active filters. IEEE Transactions Power Electronics (Vol. 15, pp. 660–669).

[10] Peng, Z., Ott, G. W., Adams, D. J. (1998) Harmonics and reactive power compensation based on the Generaized instantaneous reactive power theory for three-phaze fourwire systems. IEEE Transactions Power Electronics (Vol. 13, no. 6, pp. 1174–1181).

[11] Kim, Hyosung, Blaabjerg, F., Bak-Jensen, B., Choi, Jaeho (2002) Instantaneous power compensation in three-phase systems by using p-q-r theory. IEEE Transactions Power Electronics (Vol. 12, no. 5, pp. 701–710).

[12] Rodkin, D. I. (2003) Polyharmonic Signals Instantaneous Power Components. Elektrotekhnika, Moscow (Vol. 3, pp. 38–42) (in Russian).

[13] Zharskyi B.K., Novskyi V.O., Golubev V.V. (2013) Transformation of electromegnatic energy parameters by valve commutators. – Kiev. The Institute of Electrodynamics of the National Academy of Sciences of Ukraine, 323 p. (in Ukraine).

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