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Section 9. Technical sciences
Amirov Sultan Fayzullaevich, doctor of technical sciences, professor E-mail: sulton.amirov@bk.ru Rustamov Dilshod Shavkatovich, doctor of philosophy (PhD), assistant professor E-mail: rustamov_d1976@mail.ru Babanazarova Nargisa Kamilovna, senior lecturer Tashkent Institute of Railway Engineers E-mail: nargisa2003@list.ru
RESEARCH OF DYNAMIC CHARACTERISTICS OF ELECTROMAGNETIC CURRENT TRANSDUCER
Abstract. An analytical equation for the transient characteristic of the developed electromagnetic current transducer is obtained. By the analysis of the transient response equation and their curves at different values of the ratio of the modulating voltage ®M to the frequency of the converted current ax have been shown that at harmonic input quantity, the transient characteristic of the developed current transducer is a harmonic function with amplitude, decreasing exponentially and consisting of three components, varying with frequencies respectively ax, (ax -2mM) and (®x + 2mM). It has been defined that transient time decreases with increasing of (®M / ax) relationship.
Keywords: electromagnetic current transducer, parametric block diagram, dynamic characteristic, complex sensitivity, physical-technical effect, modulation.
In practice, in many cases controlled (convertible) current Ix changes in time quite quickly and in a large range. In these cases, the electromagnetic current transducer (EMCT) operates in a dynamic mode. If the law of change of the converted current Ix (t) is specified, then substituting it into a differential equation describing the dynamic state of EMCT, and solving this equation allows us to find the type of output voltage Usux (t). Knowing the form of the ^ (t) and Uma () functions, we can find specific values of the output voltage for any moment of time.
Analytical expression of the dynamic characteristics of the new EMCT can be find using the method of parametric structural schemes (PSS) [1; 2], compiled for its dynamic mode (Fig. 1).
In the PSS involved the following (PTE), parameters and values:
1) the effect of ampere-turns - interchain PTE between the converted electric current I3x and magnetic voltage U Mx with a conversion factor K, U = wx, [-]; wx - the
number of turns of the winding (bus) with convertible current; 2) Interchain PTE between the magnetic capacitance CMM and the magnetic voltage QMM in the magnetic
circuit of modulation with a conversion factor
M - differential relative magnet-
K dV
a^ I dB
^m
Wb
ic permeability of the material of magnetic conductor, [-]; lMM, [m] - average length of magnetic flux in the magnetic
modulation circuit; 3) C^M = , [H] - magnetic capaci-
U ^m
tance of the modulation magnetic circuit in the path of magnetic flux; 4) U^s = Q^xW^s, [A] - magnetic voltage drop in the working gap from magnetic flux QMx, generated
by convertible current I3x; W^s =—— and S^s - respectively, the magnetic rigidity (magnetic resistance by the classical analogy of the chains) of the working gap Sp and the area of this gap in the path of the magnetic flux QMx ;
m=mM, [rad/s] - angular frequency of supply voltage UM;
Qnz, [Wb] - resulting magnetic flux created by convertible
and modulating currents; I^ = -Q-, [V] - resulting magnetic current;
Figure 1. Parametric block diagram of the developed
5) interchain PFE of electromagnetic induction between the resulting magnetic current I^ and output voltage U3ebix with conversion factor K^ = wu, [-]; wu - I^ and output voltage U3eblx with conversion factor KI U = wu, [-]; wu - the number of turns of the measuring winding; 6) interchain PTE of ampere-turns between electric current I3M and magnetic voltage in a modulation circuit with a conversion factor
Kt U = wu, [-]; 7) GM, [S] - electrical conductivity of modulating winding; U , [V] - supply voltage of modulating winding.
We will write the system of equations for the PSS, describing the developed EMPT in a dynamic mode. To facilitate the generation of equations for the PSS sensor, we divide it into four (I-IV) sections. In order to simplify the calculation of the dynamic characteristics of EMPT in the first approximation, we can neglect the magnetic inductances (electric capacitances in the path of eddy currents in the magnetic conductor) LMM and Lx of magnetic circuits of modulation and
convertible current, as well as electric capacitance C3M
EMPT to determination of its dynamic characteristics
of electrical circuit modulation due to the smallness of their values (in the PSS branches with these parameters are indicated by dotted lines).
For the I section of the MSS we have the following:
U,Bra (p) = 10 (p) = pK^uQ^ (p), (1)
p - complex variable (operator).
The following equations can be written for the II section of the PSS:
Qx (P) = Cx U^ (p) = C^ [U^ (p) - U(p)] = = Cx[Ux (p)-U,xr (p)-U,xS(p)] = =C x [k.„uJx (p) - pRxQx (p) - WA (p)] From here we find Q^ (p):
KI„U C»Jx (p)
QAp ) = i+(+w)
" 1 + ( + W)cqm (p)
(2)
The magnetic flux expression Q^M (p) is found from the PSS II section as:
U m (p )C,M (p)
U(P) =
k,mug*m umm (p)
(4)
Q„m (P ) = :
1 + pL C
r MM MM
(3) Sequentially substituting (4) to (3), (3) to (2), and (2)
1 + pR^M (p )' to (l), we finally obtain an analytical equation for the output
For the IV section of the PSS we have: voltage of the developed EMPT in the operator form, i.e.:
pk^u, kqmckmum ^.ugimuim (p cm (p )l3x (p)
u 3.8ux (p )
(1+p^C^)( + p^gm) + (( + pR,x ^q^Ktm^UG
From (5) we find one of the dynamic characteristics of the EMPT - the transfer function, which has the following form:
u
3 2 0 -1 -2
3. (¡mx 100
50
0
-50
-100
N u3. ebixh U T 3.6bix2, ^3.ebix3.
h 4
• * ■ » a * ' / '<l
• / | / , * \ $ < (0M/t «v=50
u
0 0.1 0.2 0.3 0.4 0.5
a)
2 1 0 -1
-2
3.6b!x 20
10 0 -10 -20
if |\ bixl, U3 3.6MX3 .
I ** f\v 3 2 1
» \ 1 ' t • v. 1 ( ■ 1 x' ~
, 1 ,1 < )x =10
0 0.1 0.2 0.3
C)
0.4 0.5
K1
K (p )= (p)-= —-,
[C,M (t)] F (p)
(5)
(6)
4
3
2 0 -2 -3
U3.ebix 60
40 20
0 -20
-40 -60
» r/' TJ ',ix2 y 3. tbix3 .
iV
4 3
1 / ;/
OK/i Ox =25
U,
0 0.1 0.2 0.3 b)
0.4 0.5
-2
3. kmx 10
-5
-10
Us.ebL 17* TJ' c3.
yYf 2 (~ 1 Yx>
A A /
1 iyM/i( >x = 5
0.1
0.2 0.3 d)
0.4 0.5
Figure 2. Curves of dependence U*bhx = f (t) and their components at different values of the ratio
(H 1 1 -UIb*K.I'>2 ^..MLT^4 -UMBMX.
where K1 = ki„u, kqmcmki,mu„m Ki,uG mum [-] - proportion- f2{p) = r,mc,m0l3mc3mpl + (rmc ,m0 + lmmcmm + k q^ i^mmgmmc ,m0 x
ality factor; L{— [CM (t)lx (t)] - Laplace image of function xR,xp + Kq^c,iKimumUmmGmmC^moW^s +1 - denominator ofproper
— c )i ~ (t)] ;
fraction K(p); C^o - average value Cm (t) .
In expression F2 (p) we will introduce following designation:
Ti = KfiM Cm<> , [s] ; T2 = hu^u, H ;
эм эм '
: = T3> [s];T4 = ^ [s].
Subject to above mentioned designations for F2 (p ) we
have:
F2 (p) = TT p2+(T + t2+T3 )p+Tf+1. (7)
T4
In consideration of (7), operator equation (6) will have following expression:
K (p )=-K-. (8)
TiT2 p2 +(Ti + T2 + T3 )p + +1 T4
Transfer function K (p) in calculating measuring technology often calls as complex sensitivity of measuring converter [3]. It is a complex value and is the main function of the transducer, fully defining dynamic characteristics.
Find the original (8) using the decomposition theorem [4]:
K(t) = ^( -e^), (9)
+ T2 + T3 )±
where pl 2 =-
roots of the
(
(T +T2 + T3)2 - 4TT2
Tl+1
V T4
2tt
12
characteristic equation
F2 (p) = O ;
T =4 (Ti + T2 + T3)2 - 4TT2
T
± +1 T
V 4
In view of (9), we write the equation for the output voltage U_ (t) depending on the change in the converted current I3x (t) in the following form:
U,„ (t) = f(-ep2t)-~C, (t)I„ (t)]. (10)
The law of change of variable parameter of magnetic ca-
pacitance (t) in magnetic modulation sensors at has the following expression [5]:
C„M (t) = C,,MOsm2mt = . -C,iM0(1 + cos2rnt), (ll)
^M \ f ftmû m v 2 ftmo ^ m ' '
where ®m - angular frequency of power supply voltage of modulation winding.
We assume that in the dynamic mode convertible current I ax (t ) varies according to the following sinusoidal law:
lax (t)= lavish, (12)
where rnx - angular frequency of current change Iax n the dynamic mode.
Taking into account (ll) and (12), the transition equation of the developed EMPT (10) ta .(< )=K-C '
U
loao1 axpmx (epit _ ep2t )
4T
ces the following form:
t
cos® t +
'1 _ 2
xcos(cax -2ahi)t +
1 + 2
ffl
x
Turning to dimensionless quantities, we have:
cos(cax -2ahi)t
(13)
U* =(epi' - ep2t )
a-BBix \ J
cos m t +
' m ^ 1 - 2^ m
cos(mx -2mlA)t-
( \ 1 + 2—
cos (— + 2—м)t
= u*
-u*
-u*
(14)
In (fig. 2) are shown the graphics U*BHX = f (t) and their components at different values of the ratio (®M / ).
Thus, the analysis of the obtained equation of the transient response (13) and their curves at different values of the ratio of the frequency of the modulating voltage coM to the frequency of the converted current cox shows that with a harmonic input value, the transient response of the developed EMPT is a harmonic function with amplitude, decreasing by exponential law and consists of three components, varying with frequencies, respectively cox, (ax -2®M) and (mx + 2®M) . It have been established that transient time decreases with increasing of ratio (mM / ax)
References:
1. Patent of Republic of Uzbekistan. No. 04217. A device for current conversion / Amirov S. F., Safarov A. M., Turdy-bekov K. Kh., Rustamov D. Sh., Khushbokov B. Kh. // Official bulletin, 2010.- No. 8.
2. Zaripov M. F., Zaynullin N. R., Petrova I. Yu. Energy-information method of scientific and technical creativity.- Moscow: VNIIPI SCST, 1988.- 124 p.
3. Atamalyan E. G. Instruments and methods of measuring of electrical quantities: Tutorial.- M., Drofa, 2005.- 415 p.
4. Demirchyan K. S., Neyman L. R., Korovkin N. V., Chechurin V L. Theoretical basis of electrical engineering: In 3 vol. Textbook for universities. Ed. 4th - St. Petersburg: Piter, 2006.- Vol. 1 (p. 464).- Vol. 2 (p. 576).- Vol. 3 (p. 384).
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