im
Journal of	JET v°lume 8 (2015) p.p. 11-16
Issue 2, October 2015 Typology of article 1.01
Technology	www.fe.um.si/en/jet.html
DETERMINATION OF THE CONDITIONS FOR THE EXISTENCE OF HIGHER-ORDER DIFFERENTIAL ELECTROMAGNETIC INVARIANTS
DOLOČITEV POGOJEV ZA OBSTOJ DIFERENCIALNIH ELEKTROMAGNETNIH INVARIANT VIŠJEGA REDA
Boris Nevzlin1, Valentina ZagirnyakR, Veronika Zahorulko2
Keywords: four-element dipole, electromagnetic invariants, differential transformations, conditions for the existence of higher-order invariants
Abstract
A four-element dipole representation by first-order electromagnetic invariants according to differential transformation and increments is well known. The paper deals with a most general description of the conditions of existence of an electromagnetic invariant for a four-element dipole with active-reactive components in a differential form and as increments of any order. It is shown analytically that invariants exist at mutual transformations of increments into differentials and differentials into increments.
R Corresponding author: Valentina Zagirnyak, (Eng.), Tel.: +38 05366 36218, Fax: +38 05366 36000, Mailing address: Kremenchuk Mykhailo Ostrohradskyi National University, Manufacturing Engineering Department, Vul. Per-shotravneva, 20, 39600, Kremenchuk, Ukraine, E-mail address: mzagirn@kdu.edu.ua
1	Kremenchuk Mykhailo Ostrohradskyi National University, Electric machines and Apparatus Department, Vul. Per-shotravneva, 20, 39600, Kremenchuk, Ukraine
2	Kremenchuk Mykhailo Ostrohradskyi National University Electric machines and Apparatus Department, Vul. Per-shotravneva, 20, 39600, Kremenchuk, Ukraine
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Povzetek
Predstavitev dipola, sestavljenega iz štirih elementov, z elektromagnetno invarianto prvega reda je z vidika diferencialne transformacije in inkrementov že dobro poznana. Članek obravnava splošen opis pogojev za obstoj elektromagnetnih invariant dipola, sestavljenega iz štirih elementov, z delovno in jalovo komponento v diferencialni obliki in kot inkrementi poljubnega reda. Analitično je dokazano, da pri medsebojni transformaciji inkrementov v diference in diferenc v inkremente invariante obstajajo.
1
INTRODUCTION
A four-element dipole with active-reactive components (Fig. 1, a, b, c) is known [1, 2] to represent electromagnetic invariants according to differential transformation and increments that are of the form:
dC dro dg dro
d£_ dg
C - C
2 M
C1 + C2 ro2 - ro1
gi + g2 ' g2 - g1
ro2	-roi
R2	- Ri
Aa AR
a
-a2
R1 + R2
1
8L_ dro dR
dro
AC
"Ag"
dL
dR
c1 +c2
gi + g 2 OK
3a
dro dR dro
da
dR
a1 +a2 R1 + R2
l + l2
R1 + R2
L2 A
R2 - R1
AL AR
L +l2 R1 + R2
(1)
where ro -arbitrary circular frequency; ro2, roj -circular frequencies and ro2 > roj;
C2, a2, L2, g-2, R2-values C, a, L, g, R at frequency ro2; Cj, aj,L, gi, Ri -values C, a, L, g, R at frequency roi .
Conditions for invariants existence consist of, respectively:
Cig2 -C2gi # 0, aiR2 -a2Ri # 0, LiR2 -L2Ri # 0.	(2)
The same papers [1, 2] state that (1) there exist invariants not only according to frequency ro, but also according to order n of derivatives and increments:
d "N d "g"
on dg
N +N
2
gi + g2
3"a 3 "R
da OR
a1 +a R1 + R
2
2
3 "L 3"R
OL
3R
Li +L
2
R + R
(3)
2
but conditions for their existence are not given.
The purpose of this paper: determination of conditions for the existence of higher order invariants.
2 MATERIAL AND RESULTS OF RESEARCH
Second derivatives with respect to C and g:
Q2$ -2 (Cj + C2 ) (Cjg2 - C2gj )2 [(gj + g2 )2 - 3ro2 (Cj + C2 )2 ]
dro2
[(gi +g2)2 +ro2 (Ci +C2)2 ]
m
a
CO o - CO
ro
ro - ro
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Determination of the conditionsforthe existence of higher-order differential electromagnetic invariants
d 2 g 2 (g! + g 2 )( g 2 - C2 g! )2 [(g! + g 2 )2 - 3®2 ( + C2 )21
— =-[-1.	(4)
^	[(g! +g2 )2 +®2 (C +C2 )2 ]3
The second condition for the existence of a second-order differential invariant follows from (4):
g! + g2	C + C2),	(5)
which is supplementary to (2).
Analogously, for a third-order differential invariant, the second condition for existence (the first one, as before, is (2)) is of the form:
g! + g2 *ro(Q + C2),	(6)
and for the fourth one:
(g! + g2 )4 - 5® 2 (( + C2 )2 [2 (g! + g2 )2 - ® 2 (C! + C2 )2 ] * 0.	(7)
Condition (7), obviously, always exists at
n/2 (g! + g 2 )=®(C + C2 ) .	(8)
C	C2
g!
g2
Q-
-tf C,
a)
R
a!
a
2
R
2
a, R 5)
L
R

^2 J^YV
R
2
-* L, R *-
B)
Figure 1: Circuits hf fhut-slsmssh diphlss: e) Ci, C2 esd C- chsdssssts cepecihiss esd hstmisel cepecihescs hf hhs circuit eh etbihtety ftsqusscy, g!, g2 esd g - hhs sems chscstsisg chsduchiohy; b) a!, a2 esd a - phhsshiel chsfficisshs hf chsdssssts esd hostell phhsshiel
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JET Vol.8 (2015)
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cnrfficirnt nf tOr circuit, Rj, R2 and R - tOr snmr concerning resistances; c) Li, L2, and L -brnncOrs inductances and norrnll inductance nf tOr circuit, Ri, R2, and R - tOr samr concerning resistances
Let us determine a derivative of the (n+i)-th order by a mathematical induction method, [3], in accordance with which a formula is considered true for any transformation, if it is proved that it, being true for n-th transformation, is also true for (n+i)-th transformation. As invariant of the n-th order is of the form:
\2/,~r v> \2
(( + N2)(( + N2)2 ((g2 -N2g! )2 fn ( C2, g1; g2, ®)
(g! +g2 )2 + ®2 (Ni +N2)
9nN
d"N 9®n	_
d"g	(g! + g2 )(Ni + N2 )2 (%2 - N2gi )2 fn (Ci, C2, gi, g2, ® )
9®"
(9)
[(gi +g2)2 + ®2 (Ni +N2)2
where fn (Ci, C2, gi, g2, ®) -the determined function for the derivative of the n-th order, e.g.
for n = 3 fn = -24® [(gi + g2 )2 -®2 (Ci + C2 )2 ] .
Invariant of the (n+i)-th order is obtained:
(( +N2)(( +N2)2 (Ng -Ng)2 { (gi + g2)2 + ®2 (( +N2)2
an+iN
9n+iN =
rjn+i	sjn+i
9 g 9_g
9®n+i
	(gi + g2 )2 +®2 (Ni +N2)2		n+2	
(gi + g2) (( +N2)2 (%2 -N2gi)2{		(gi + g2)2 +®2 (( +N2)2		X -->
[(gi + g2 )2 + ®2 (Ni +N2 )2 <{[(gi + g2 )2 +®2 (Ni +N2 )2 ]9®[fn (Ci, C2, gi, g2, ® )] -
(gi + g 2 )2 + ®2 (Ni + N2 )2 ] 9- [fn (Ci, C2, gi, g 2, ®)] -
-2® (n + i)(Ni + N2 )2 fn (Ci, C2, gi, g2, ® )}
Ni + N2
-2® (n + i)(Ni + N2 )2 fn (Ci, C2, gi, g2, ® )} gi + g2
[(gi +g2)2 +®2 (Ni +N2)2]|®[fn (Ci, C2, gi, g2, ®)] =
(10)
In this case, the condition for existence of (n+1)-th invariant, except (2), is of the form:
X2 ]_9_
j9®L'«v	------- /J	(11)
* 2(n + i)®(Ni + N2)2 fn (Ci, C2, gi, g2, ®). Thus, it can be stated that as a relation of any order derivatives of C and g with respect to frequency represents an invariant with respect to frequency, it is also an invariant with respect to the order of differential transformation with the condition for existence (2) for all invariants and (5) or (6), or (7)-(8) - for invariants of the determined orders (in a general form (11)).
>
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Determination of the conditionsforthe existence of higher-order differential electromagnetic invariants
Analogously to the circuits shown in Fig. 1, b, c, higher-order invariants are of the form (3) and, accordingly, conditions for their existence are of the form (except (2):
ro(Rj +R2 (aj +a2 ),	(12)
Rj + R2 ^-\/3ro(L + L2) - second-order invariant,
ra(Rj +R2 )^aj +a2,	(13)
Rj + R2 ^ -\/3ro(L + L2) - third-order invariant,
V2ro(Rj + R2)^aj +a2, 42(Rj + R2)*ro(Lj + L2) or
(14)
ro2 (Rj +R2 )4-5 (aj +a2 )2 [2 (Rj +R2 )2 ro2-(aj +a2 )2 0,
-order invariant.
[ro2 (Rj + R2 )2 + (aj + a2 )2 ] ^TO[.fn (( > a2 ' Rj> R2 ' ro )] :
(Rj + R2 )4 - 5ro2 (L + L2 )2 [2 (Rj +R2 )2-ro2 (( +L2 )2 ] * 0 - fourth-c
In the general form, the condition for existence:
d_
J 3ro 1
#	2ro (n + j )(a j +a2)2 fn (a j, a2, Rj, R2, ro);
[(Rj +R2 )2 +ro2 (L +L2 )2	(L, L2, Rj, R2, ro)
[	] oro	(15)
*	2ro(n + j )(L + L2)2 fn (L,L2, Rj,R2, ro).
Let us determine conditions for the existence of invariants according to increments. Second-order increment (otherwise -finite differences of the second order [4]) can be determined as:
^Vro2 - N N3 -N2 - ((2 - Nj ) (Nj + N2 )(Nj S2 -N2Sj )2 (ro3 j )
v2
A2N _ Aro2 -Aro j _ ro3 -ro2 -(ro2 -ro j) _	(ro3 +ro j-2ro2)
A2g Agro2 -Agro j S3 - g2 - (S2 - Sj ) (g + S2 ) (N g2 -N2g )2 ( -TOj ) > Aro2-Aro j ro3-ro2-(ro2-ro j)	(ro3 +ro j-2ro2 )x
\2	\2 r
—■ -->
x{(gj + S2 ) -(Nj + N2 ) [ro2 (ro3 +ro j) + ro3ro j]} 3
xn[(Sj + S2)2-ro2 (N +N)2
n + n2
(16)
X{(g1 + g2 )2 -(^1 + ^2 )2 [W2 (3 Q + Wj® 1]} g + g2
^	3
Xn[(^1 + g 2 )2 -™2 ( +n2 )2
1=1
where Q, C2, C3, g1; g2, &3 -values C and g, respectively, at frequencies ffl2' W . The condition for existence in this case:
(1 +g2)2 +#2)2 [®2 (3 + ®1 )+®3®1 ]■	(17)
Analogously, for other invariants:
A2a _ aj + a2 A2L _ L + L2
A2 R Rj + R2 A2 R Rj + R2
(18)
>
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conditions for existence:
(«1 +«2 )	+R2 ) [«2 (CO3 + «i) + «3«i];
(R1 +R2 )2 *(L +L2 )2 [«2 («3 + «1 ) + «3®1 ] ■
Obviously, value «2 («3 +«1 )+«3«i at approximation Ara^-0, where
ra3 = «2 +Ara =ra1 + 2Ara tends to value 3«2 then conditions (17) and (19) turn into (5) and (12), which confirms the correctness of the performed transformations.
For increments of any n-th order, the condition for the existence of an invariant is of the form, e.g. with respect to C and g,
fn C C2, gl, gl, «1 -®n+i)* 0 .	(20)
It should be noted that invariants also exist in mutual transformations of increments into differentials and differentials into increments, i.e. (omitting lengthy intermediate transformations):
A(dC) = 5 (AC )= C1 + C2;	A(5a) = 5(Aa)= a1 + a2;
A(g) = 5(Ag) "" gi + g 2'	A(5R) = s(aR) "" Ri + R,'
A(5L) 5(AL) =_ L + L2
A(CR ) = 5(AR )= R + R2
In this case, the condition for existence of an invariant e.g. with respect to C and g in transformations of increments into derivatives:
(gi +g2)4 * 2®i«2 ((1 +N)2 [(gi +g2)2 + (« i2 + « 1 «2 + «2)(( +N)2],	(22)
and transformation of derivatives into increments provide an invariant existing at any positive real values Q, C2, gi, g2 (meeting (2).
Analogous results also take place for circuits 6,R, and L, R.
3 CONCLUSIONS
Conditions for the existence of invariants of a four-element dipole with active-reactive components at differential transformations and in increments of any order and also for mutual differential-difference transformations have been determined.
References
[1]	B. I. Nevzlin: Identification and application of an electromagnetic invariant of a mathematical model of the controlled environment with active-reactive components, Herald of East-Ukrainian State. Univ., Iss. 2, p.p. 155-161, 1997
[2]	B. I. Nevzlin: About extension of the scope of existence of an electromagnetic invariant, Herald of East-Ukrainian State. Univ., Iss. 4., p.p. 12-14, 1997
[3]	Mathematical encyclopedia. - M. : Sov. Encyclopedia, Vol. 3, p.p. 563-564, 1982
[4]	Mathematical encyclopedia. - M. : Sov. Encyclopedia, Vol. 2, p.p. 1026, 1979
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