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1ANALYSIS OF OPERATING MODES OF A FERRORESONANCE CURRENT STABILIZER WITH THREE-
PHASE INPUT
1Khalilov Nuritdin Abazovich, 2Nuraliev Almukhan Kalpakbaevich, 3Sheraliev Dostonbek Davlatmurod ugli, 4Eshmurodov Sobir Soat ugli
1Director of the Ulugbek Foundation, Doctor of Economics, Professor 2Candidate of technical sciences, associate professor, TSTU 3PhD doctoral student, TSTU 4PhD doctoral student, TSTU https://doi.org/10.5281/zenodo.10974961
Abstract. The paper considers the issue of a current stabilizer in case of a short circuit in one of the phases. The equations approximating the nonlinear magnetization of the ferromagnetic core of the inductor and describing the processes in abnormal operating modes of the current stabilizer are derived.
Keywords: current stabilizer, short circuit, approximation, magnetization, core, ferromagnetic, inductor.
Introduction. A ferroresonance current stabilizer is a static device in which the phenomenon of ferroresonance currents is used to convert unstable mains voltage and load current into a current whose effective value is almost constant. It can be used in automatic installations, to power consumer electronics, to convert a single-phase voltage system into a symmetrical three-phase one [1].
One of the most important properties of ferroresonant current stabilizers is their almost inertia-free action. Changes in the load current within the operating range only lead to changes in the shape of the output voltage curve: the effective (or half-cycle average) value of the latter remains practically unchanged. They can be used for devices sensitive to sudden short-term (over several half-cycles) changes in the supply voltage. The disadvantages are: dependence of the stabilized current on the frequency of the power source, non-sinusoidal shape of the output current curve, sensitivity to the type of load, large weight per unit of output power [2]. The physical processes in such stabilizers can be compared to a swing. A swing that has been pumped up to a certain strength is difficult to stop or suddenly force to swing faster. When riding a swing, you don't have to push off every time - the energy of vibration makes the process inertial. It is also difficult to increase or decrease the oscillation frequency - the swing has its own resonance. In ferroresonant stabilizers, electromagnetic oscillations occur in the oscillatory circuit of capacitance and inductance [3].
Ferroresonant current stabilizer with three-phase input in single-phase short circuit mode, i.e. at ZA= 0, shown in Fig. 1. has certain advantages compared to analogues.
When analyzing the physical processes of a ferroresonant current stabilizer, we will make the following assumptions:
1. The magnetization curve of a nonlinear element is approximated by an incomplete third-
order power function:
i = ay + b^3
2. Losses in linear reactive elements are not taken into account;
3. The nonlinear element is represented by an equivalent circuit consisting of parallel-connected nonlinear inductance and constant active conductance, which takes into account losses in the core [4].
Fig.1. Schematic diagram of a ferroresonant current stabilizer
Ih[a]
C2=9 mkf const Ci=VAR
75 mkf
490 mkf
/
60 mkf
40 mkf
Ubx[V]
20 40 60 80 100 120 140 160 180 200 220 240 260 280
Fig. 1 Adjustment characteristics /h[4] = f(UBX)
IH[A]
C2=9 mkf const C1=VAR
75 mkf
490 mkf
/
60 mkf
40 mkf
Ubx[V]
20 40 60 80 100 120 140 160 180 200 220 240 260 280
Fig 2 Adjustment characteristics /h[4] = f(UBX)
C03tf>rr| CDSCp
RH[Q]
Fig. 3 Dependence of efficiency " q" and asymmetry coefficient " Ku" from load " Rh".
We use the following notation: Uab'Ubc'Uca — external influence; y- flux linkage in the core of a nonlinear element; i-A^B^c - currents flowing in each phase of a three-phase system;
i1'i2i3- currents flowing respectively through the windings of a nonlinear element through conductivity g and capacitor C2.
For the scheme under consideration, the following relations are valid based on Kirchhoffs
laws;
uab = -1jiBdt
Ubc = IfiBdt-^-L^
UcA = + W
dt
dt
when
iA + h + ic = 0 ic + iq + iy = 0
q
^ dt
(1.1)
(1.2) (13)
(14) (1.5)
C2
d2\p It2
= aty + bty3
Taking (1.6) into account, we can write that
d^
d2y
ic = a xp + bxp6 + q—+ C2 —
dt2
(16)
(1.7)
From (1.2), (1.1) and (1.4) we obtain, respectively;
dU
BC
dt
dUAB _ ib dt c1
lb Cl
d2ty d2ty d2i
dt2
L
dt2
L
dt2 ib =
Cl-
dUAB dt
dUBC
dt dt dt2 r dt v r J dt2 ^ dt3 2 dt4 Let us accept the following assumptions:
UAB = sin(r) UBC = Umsin(r - 120°) ty = tymsin(r +
(1.8)
(1.9)
c
R
c
when r = tot
Thus, we will consider the operating mode of the nonlinear inductance at the fundamental
harmonic, taking into account (1.9) from (1.8) we obtain
t/m cos(r — 1200) = — t/m cos(r) + ^mwsin(r +
— sin(r + • cos(r +
mJ 2t
[a + 36^ sin2(A + ^n)]
toL^m sin(r + - qL^mto2 cos(r + + LC2 ^mto3 sin(A + Whence it follows that
ym(cos(r) • cos(1200) + sin(r) sin(1200) = —ymcos(r) + +^mto sin(r + — sin 2(t + — aLto^m sin(r +
—3&Lto^ sin(r + — $L^mto2 cos(r + + LC2to3^m sin(r + From the last expression we get;
i/m V3
-1- • COs(t) + T sin« = c°s« + ^(sinM cos(^ +
+ cos(r) sin(^n) — sin 2(t + — aLto^m(sin(r) cos +
+ sin — sin 2(r + — aLto^m(sin r cos + sin cos r)
3 _ 3 .....1
1
(1.10)
(1.11)
3ötoL^[(-sin(r) cos^n — -sin^n cos(r)) — (-sin(3T) cos(3^n)
— — cos 3t sin(3^n)) ] — qL^mto2(cos(r) cos(^n) — sin(r) sin(^n)) +
4
+LC2to3^m(sin(r) cos(^n + sin(^n cos(r)) Without taking into account higher harmonics from (1.12), we obtain
(112)
aLto^m + -ötoL^ + LC2to3^m) sin(^n) + qL^mto2 cos(^n)
4
T =
I V3 9 — o o
— = qL^ mto sin(^n) + (to^m — aLto^n + -öto^m + LC2to ) cos(^J
(113)
By introducing the notation:
9
to^m — aLto^m + -öLto^ + LC2to3^m = ^ 4
qL^mto2 = B
^ = K : 2
Taking into account the notation we obtain: = ^ sin( + B cos(^n) (N = B sin( + ^ cos(^n) From system (1.14) it follows that
42-S2
f =
(1.14) (115)
cos(^n) =
(1.16)
From (3.15), (3.16) we obtain:
V42-s2/ + (42-s2 / = 1
-sw
^B2
Considering that Then from (1.17) we get:
(1.17)
K = — : N =
22
^f(A — 43B)2+^f(43A — B)2 = (A2—B2)2 (1.18)
U2
-f(A2 — 243AB + 3B2 + 3A2 — 243AB + B2) = (A2 — B2)2 U2
-m4(A2 — 43AB + B2) = (A2 — B2)2
Let us assume that (for a first approximation)
(A2 — 43AB + B2) ~(A — B)2 43 = 1.74 « 2
Then
Um = (A + B) T.e (1.19)
9
Um = —bteLipm + — aLw + LC2m3 + gLo>2) 4
KA — BN A— 43B
= arc tan ———-— = arc tan^-
rn AN — KB 43A — B
As a result of the analysis, equations were derived that approximate the nonlinear
magnetization of the ferromagnetic core of the inductive element. In the event of a short circuit in
phase A and the corresponding parameters of phase C, ferroresanance of currents occurs. Thus,
the current is stabilized in phase C.
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