Characteristics of specialized single-phase high voltage doubler rectifier Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

UDC 621.317.3

doi: 10.20998/2074-272X.2018.5.09

V.O. Brzhezytskyi, R.V. Vendychanskyi, Ye.O. Trotsenko, Ya.O. Haran, O.M. Desyatov, V.I. Khominich

CHARACTERISTICS OF SPECIALIZED SINGLE-PHASE HIGH VOLTAGE DOUBLER RECTIFIER

Introduction. To obtain a high voltage direct current, voltage multipliers with a number of cascades of three or more are widely used. At the same time, for voltage levels of 100...200 kV there are several advantages of using a specialized single-phase high voltage doubler rectifier. Problem. The main difficulty is that at the moment mathematical modeling has not been worked out for describing modes that use the built-in R, C-filter, as well as a nonlinear load in the form of Zener diodes. Goal. Generalization of the results of the authors' previous publications on the development of an analytical method for calculating the modes of a typical high-voltage direct current installation based on a specialized single-phase voltage doubler rectifier. Methodology. Compilation of a system of algebraic linear and nonlinear equations that describe the current and voltage modes in the elements of a typical high-voltage direct current installation with a nonlinear load. Results. It is shown that with the use of linearization of the current-voltage characteristics of Zener diodes used in the load circuits of a typical high-voltage direct current installation, an analytical solution for the voltages and currents in its elements can be obtained. Originality. The theoretical basis of the complex solution of the system of equations for the currents, voltages and power of the elements of a typical high-voltage direct current installation with the account of nonlinear pulsations is formulated for the first time. Practical value. The obtained theoretical results can be used for calculations, design, optimization of the modes for a wide range of high-voltage direct current installations of technical, technological, and measuring purposes in the range up to 100...200 kV. References 16, tables 2, figures 3. Key words: voltage doubler rectifier, high-voltage Zener diode, current-voltage characteristic.

Цель. Обобщение результатов предыдущих публикаций авторского коллектива по разработке аналитического метода расчёта режимов типовой установки высокого напряжения постоянного тока на основе специализированного однофазного выпрямителя с удвоением напряжения. Методика. Составление системы алгебраических линейных и нелинейных уравнений, описывающих режимы тока и напряжения в элементах типовой схемы установки высокого напряжения постоянного тока с нелинейной нагрузкой. Результаты. Показано, что с применением линеаризации вольт-амперных характеристик стабилитронов, используемых в цепях нагрузки типовой установки высокого напряжения постоянного тока, может быть получено аналитическое решение для напряжений и токов в её элементах. Научная новизна. Впервые сформулирован теоретический базис комплексного решения системы уравнений для токов, напряжений и мощности элементов типовой установки высокого напряжения постоянного тока с учётом нелинейных пульсаций. Практическая значимость. Полученные теоретические результаты могут быть использованы для расчётов, проектирования, оптимизации режимов широкого спектра установок высокого напряжения постоянного тока технического, технологического, а также измерительного предназначения в диапазоне до 100.200 кВ. Библ. 16, табл. 2, рис. 3.

Ключевые слова: выпрямитель с удвоением напряжения, высоковольтный стабилитрон, вольт-амперная характеристика.

Introduction. The variety of high voltage applications for steady-state modes of technological equipment (electrostatic precipitators of coal-fired power plants, electro-coloring and coating sputtering devices, electric separators) necessitates the improvement of their power supplies to a level of 100...200 kV. In connection with this, recently interest in various variants of the improvement of the classical Cockcroft-Walton direct current voltage generator [3] with a number of stages of three or more [1, 2] has appeared. At the same time, since high-voltage diodes [4] are of high quality, in order to obtain the above voltage level, it is more efficient to use the Cockroft-Walton generator with only doubling the rectified voltage, and to reduce output voltage ripple - to supplement it with the «built-in» R, C - filter [5].

It should be noted that to date no rigorous mathematical model of the Cockroft-Walton generator has been created. The available publications on this topic give different results on the magnitude of voltage ripples, and there are also no analytical expressions for the shape of pulsing voltage, etc.

In this connection, a new development of the authors' team [6-8] on the creation of elements of the theory of voltages and currents of the Cockroft-Walton generator with voltage doubling, together with the integrated R, C - filter,

and, in addition, the possibility of attaching to its load nonlinear elements such as Zener diodes.

The goal of the paper is to generalize the results of previous publications of the authors' group, to formulate and analyze the final analytical expressions for both voltages and currents in the rectifier circuit with voltage doubling in Fig. 1, and for the powers of the elements of the Cockroft-Walton generator with voltage doubling and its integrated R, C - filter, taking into account the nonlinear pulsation modes. To reduce the terminology, we will call such a generator a specialized single-phase high voltage doubler rectifier.

The subject of the research is a specialized highvoltage single-phase rectifier with voltage doubling, its generalized circuit is shown in Fig. 1.

In Fig. 1: VT - high-voltage step-up transformer; VD1, VD2 - high-voltage diodes; C1, C2, C3 - capacitors; Rf, Rlv, r - resistors. The branch of n Zener diodes ZDi,...ZD„, resistor r and voltmeter V forms the «built-in» high-precision measuring group of the load voltage ULV. In this case, the voltage source in the circuit diagram in Fig. 1 is of interest for both technological applications and measuring equipment [9, 10]. The peculiarity of this voltage source is that by changing the parameters of the elements it is possible to adjust the amplitude and shape

© V.O. Brzhezytskyi, R.V. Vendychanskyi, Ye.O. Trotsenko, Ya.O. Haran, O.M. Desyatov, V.I. Khominich

Fig. 1. Functional diagram of rectifier with voltage doubling [5]

of the output voltage ripple ULV in a wide range. The problems of the synthesis of circuits with capacitive energy storage devices, including those using nonlinear electrical loads, are considered in modern publications [11-13]. However, the features of the high-voltage source in Fig. 1 in the well-known publications of other authors have not been studied. The calculated relationships for the voltages and currents of the circuit in Fig. 1 were first obtained in [6-8].

The initial prerequisites for the research are based on a number of conditions:

• a typical current-voltage characteristic of a Zener diode has the form shown in Fig. 2, where u0, I0 denote the selected point of its operating mode;

• a differential resistance of a Zener diode rd = duZD IdizD in its operating domain is far less than the impedance u0/I0 (number 1 denotes the linearized current-voltage characteristic of a Zener diode);

duzd

a capacitance current C

zd

dt

of a Zener diode is

far less than its through-current I0 (here CZD is the inter-electrode capacitance of a Zener diode);

• placement of a Zener diode in a metal casing (Fig. 3) completely shields its internal active element from the influence of external electric fields [5].

Fig. 2. Current-voltage characteristic of the Zener diode D818D

When these conditions are fulfilled, a series connection of the same type of Zener diodes in the steady state is characterized by the flow of the same current I0 through their circuit with the voltage operating point

Uo = U01 + U02 + ... + Uoi + ... + Uon,

as well as the total differential resistance

Rd = rd1 + rd2 + ... + rdi + ... + rdn, that corresponds to the conclusion of [4] on the admissibility of a serial connection of any number of Zener diodes. The typical «high-voltage» design of insulation of a series of similar Zener diodes eliminates the need to take into account the corona and other phenomena of distributed currents [9].

Derivation of the initial expressions. Then for the instantaneous load voltage uLV(t), one can write:

ULV (t) = U0 + v + (i (t) -10 + r), (1) where: i(t) - instantaneous current through a Zener diode and resistor r branch. From here one can get:

• (tX uLV (t) - U0 - J0r , J

• (t ) =---+ J 0

Rd + r

(2)

The expressions (1) and (2) are valid within the stabilized domain of the current-voltage characteristic of Zener diodes (Fig. 2) and, therefore, are applicable up to a current ripple level of ~50 % (when I0 is selected in the middle part of this domain). The peculiarity of the expressions (1) and (2) is also that they can provide (with an appropriate selection of the parameters U0, Rd) any variants of a series connection of Zener diodes and a resistor r, up to the limit: «only the Zener diode» load or «only resistive» load.

Let us write

uLV(t) = U0 +10r + Au(t), where Au(t) - load voltage ripple as a function of time.

Wherein

1

Jàu(t)dt = 0 ,

where T =1/f - voltage period of sinusoidal voltage U (t) = Um • sin(ct), c = 2f - angular frequency, f -voltage frequency.

Let us write z'3 = C3 • duLv (t)/ dt (taking into account the losses in the capacitors of the circuit in Fig. 1 slightly refines the results obtained [6]) and

iLV = ULV (t)/ RLV .

Then the current flowing through the resistor Rf is:

. , Au(t) + U0 +10r + Au(t) + C duLv (t)

If = 10 +---1---+ C3- ,

f 0 Rd + r Rlv dt

and voltage is:

UC2(t) = ULV (t) + ifRf .

In its turn, i2 = C2 • duC2 (t) / dt and the total current at the input of the right-hand side of the circuit is

iIN = if + i2 =

1 1

= j0 + eollol + àu (t )

dÀu (t )

RLV

/

v rlv

Rj + r

dt

C3 + C

1 +

R

f

R

f

R

LV

R; + r

(3)

+

Fig. 3. Photograph of the Zener diode D818D

+ C2C3 Rf

d 2Àu (t ) dt2

4-

4-

4

Since the process occurs cyclically, let us assume that at time instant t1, the diode VD1 «opens», and the current i\=iiN flows (the resistance of the diode VDi in the open state is neglected). At time instant t2>tb the diode VD1 «closes» (the resistance of the diode in the closed state is assumed to be infinitely large). In the time interval t2 < t < T+ti, the current iIN = 0. Proceeding from this for a given period of time, from (3) one can obtain an equation for a function of Att1 (t) in the form:

2

d Am (t) dAu (t)

- + flj-i--+ O^Au^t) =

dt

2

dt

I0 +(( 0 +10 r )

(4)

R

LV

C2C3 Rf

where

C3 + C2

1+-

R

R

f + Rf ^ Rd + r

LV

C2C3Rf 1 1

R

LV

Rd + r

a2 = "

C2C3 Rf

A study of the roots p1, p2 of the characteristic equation p2 + a1p + a2 = 0 shows that its discriminant D > 0. Thus, one can find the solution for Au^t) in the form:

Au1 (t) = A1ep1t + A2ep2t +Au1s, (5)

where Au1s - steady-state voltage.

For the industrial power supply frequencies of the circuit in Fig. 1, one can ignore the inductance of its elements. Therefore, in the open state of VD1 (during the time interval t1 < t < t2) one can obtain [6]: iiN(t) = i1(t),

1 '

UC 2 = Um ) + Um - - J«t ».

'1

After differentiating this expression, one can get:

duC 2(t) / s. 1

C, =®Um cos(ot)-—i1(t) dt C1

and, consequently:

i1 (t) = C1aUm cos(at)- C1

due 2 (t ) dt

(6)

d Au2(t) , dAu2(t) , , -^ + bi-+ b2 Au2 (t ) =

dt2

dt

Ci^Um cos(at) Rlv

10 (( 0 +10 r )

(7)

Rf C3 (Ci + C2 ) RfC3 (1 + C2 )

where

b1 =

C3 +(C1 + C2 ) + -f + 3 w 2 \ Rlv Rd + r

RfC3 (Ci + C2 ) 1 1

b2

Rd + r

2 RfC3 (C1 + C2 )

The discriminant of the characteristic equation in this case is also greater than zero, and the solution for AU2 (t) is found in the form:

Au2 (t) = A3 sin(®t + i/z) + A4ep3t + A5ep4t + Au2s, (8) where Au2s - steady-state voltage, p3, p4 - roots of the characteristic equation p2 + 61p + b2 = 0. Values of A3 and fare given by following expressions:

C1^Um

A3 =

R fC3 ((1 + C2 )b2«2 +(b2-a2 )

y = arctan

2 ^

b2

b1®

Comparing (5) and (8), one can find:

10 +-

Au]_s = Au2s = Aus = —

R

— (( 0 +10 r )

l2s

LV

rlv Rd + r

Using the expression uc2 (t) = uLV (t) + if (t) • Rf

and performing its differentiation, and also substituting (6) in (3), one obtain the equation for the function Au2 (t) during the time period t1 < t < t2:

where Aus - continuous component of the ripple voltage.

Using the invariance of uC2 and uC3 at the time instants t1, t2, as well as the determination of t1 from the condition Um(l + sin(®t1 )) = uc2((),and the time instant

t2 from the condition i1(t2) = 0, and also the expression t+t1 t2

|Au1 (()t + |Au2 (()dt = 0, one can get a system of

t2 ti

seven algebraic equations (9) - (15) with seven unknowns: A1, A2, A3, A4, A5, t1, t2

coA3 cos(®At + ^)+ p3A4eP3At + p4A5eP4At =

= Pi Ai + P2 A2;

®A3 cos(^) + P3A4 + P4A5 = (10)

= P1 A1ePi(T "At ) + P2 A2eP2 (t "At ); (10)

A3sin(^)+A4 + A5 = A1ePi(T ~At )+ A2eP (t ); (11) A3 sin(®At + ^) + A4eftAi + A5eP4At = A1 + A2 ; (12)

(9)

A3

a

(cos(y)- cos(aAt + y)) = -AusT -

A (pi (-A0-i)-il (p2(-A0-1)- (13)

Pi P2

A (P3A - i)- il (P4Aî - i);

P3 P4

1

ti =— arcsin

a

UL -1

V Um J

(14)

a =

1

1

t2 =— arccos

c

F2

\cU m J

(15)

In (14) F1 is given by the expression:

F = 10 Rf +((0 +10 r X) +

R

■LV

+ (Aus + A3 sin(/)+A4 + A5) 1 + —— +-f

Rf

R

LV

Rd + r

+ C3R f (A3 cos/)+ P3 A4 + P4A5).

In its turn, in (15) F2 is given by the expression: F2 = C3Rf (-c2A3 sin(cAt + /) + p32A4eP3At +

\ f Rf Rf | + P42 A5eP4A )+l 1 l(cA3 cos(cAt + /) +

\ RLV Rd +r J

+ P3 A4eP3At + P4 A5eP4At).

It should be noted that At = t2 -11, while the relation between A3 and Um is defined above.

Equations (9) - (13) are linear with respect to A1, ... Aj, ... A5, and therefore the system of equations (9) - (15) can be reduced to three equations with three unknowns t1, t2, Aj. Our experience in calculating (9) - (15) confirms the possibility of obtaining in each particular case a unique solution of the system in a set of real numbers.

Development of theory. The initial expressions (1) - (15) obtained above are derived from the publications of the authors [6-8] and are necessary for the further presentation of the materials in this article.

The advantage of the obtained analytical solution of this problem implies its logical conclusion in the derivation of expressions for the power of the elements of the circuit in Fig. 1 taking into account the voltage and current ripple in these elements (without the assumption of a limitation of their smallness).

The power losses in the RLV load can be found as follows:

=/ f +f]

ulv ((;

•LV

R

-dt,

(16)

LV

R

'■fz- J(1)=((1)) (+11 -12)+A- >

e2PY (T+tj-t2)_ 1

2V0(1) A1

A2

2 P2

P1 (T+t1 -t2)-1

P1

;P2 (T +t1 -t2 )- 1

,2 P2 (+tj-t2 )-

2V0(1) A2

2 AA

RLVPLV

P1 + P2

P2

,(P1 + P2 +t1 -t2 )- 1

f

t2 -11 1 —-1 +-

2 4c

V(1) )* ((2 -11) + A2:

[sin(2/) - sin(2( - h) + /))]

A4

,2 P3 (t2-t1 )-

2 P3 2V0(1) A3

A2

2 P 4

,2P4 (t2-tj )-

[cos(/) - cos[(t2 -11)+/]] +

2Vp(1) A4 P3

,P3 •(t2-t1 )-

2A3A4

2V0(1) A5

P4

,P4 •((2-t1 )- 1

c

P3 | +1

c

P3 ((2-t1/ilsin[c((2 -11)+/]- (19)

\ c

- cos[ ((2 -11)+/])- — sin/)+cos/)

2A3A5

P± l +1

c

,P4 ((2-t1 )

—4sin[c((2 -11) + /]-cos[c((2 -11 ) + /]|-

c

—4 sin/)+cos/)

c

2A4A5

,(P3 + P4 )2-t1 )-

where uLV (t) = U 0 + I0r + Au1(t )for time interval t2 < t < T+t1, and uLV(t) = U0 +10r + Au2(t) for time interval t1 < t < t2.

Then let us transform (16) to the following form:

>(1) f R P >(2)

P3 + P4

The expression for the power losses in the group of

elements ZD1.ZDn, r, taking into account the previous consideration, will look like:

P =

T+tj f ]

[0 + AU1(< ))•

Au1(t)

RLVPLV

RLVPLV f

RLVPLV f

f

T+t, ' t2 ' ' (I7)

= ] [[ + Au1 (()]2 dt + ] [[ + Au2 (()]2 dt, t2 t1 where V0 = U0 +10r.

Let us rewrite formula (17), integrating each component and assuming V0(1) = V0 +Aus, then one can

obtain the following expressions, that allow computing the power losses of PLV taking into account the voltage and current ripple on the load RLV:

f ]

[¿0 +AU2 (t )]•

Rd + r

AU2 (( )

Rd + r

-+10

+10

dt.

dt +

(20)

Next, let us transform expression (20) to the form:

N f'tPf-

t+t1

V0I0 +Au1(()

(

I0 +-

V0

Rd + r

(AU1 (())2

Rd + r

dt + (21)

-2 i

V010 +AU2 (()

I0 +

V0

Rd + r

(AU2 (())2

Rd + r

dt.

x

+

+

x

+

X

+

e

+

+

2

CO

CO

X

+

2

2

+

2

After integrating each component, the following expressions can be obtained, that allow calculating power losses in the group of elements ZD1 ...ZDn, r:

r ,(1) I (1)A1

4 I = F0I0( +11 -12)+ 10 A1

P

f

I (1) An I0 A2

P1

P (T +'1 -t2 )-

The power losses in the active resistance of the filter Rf can be determined by the formula:

T+i h\ 1

Pf = f ibf 1)2 Rf dt + f j[(if2)2 Rf d , (24)

t2

t1

P2

p (T+h-t2 )-

+ l0^AUs (( +11 -12 ) +

where the values of ifu ip have the following form:

U0 + I0r , Au (t)f 1 _ + 1 A

if 1 = I0 +

R

+ Au1(t)

V(2) A?

2 P1

,2P2 (T +Î1 -t2 )-

,2 P1(T+Î1 -f2 )- 1

V(2) A22

LV

2 P2

+ K(2) (Aus )2 (T +11 -12 ) +

(22)

+ C

dAu (( )

V RLV Rd + r y

dt

2V0(2) AA

,(P1 + P2 +t1 -t2 )- 1

P1 + P2

,P1(T+Î1 -f2 )- 1

2V0(2)Aus A1

2Vq2)AUsA2

P2

P1

P2 (T+t1 -t2 )- 1

r \ = V010((2 -11 ) + [cos(f)-

f y —

T (1) A r / X ■

-cos[—((2 -11 )+^U + eP3((2-t1)-1

n(2)

I ® A3

P3

I ® A

0 5

P4

P4((2-t1)-1]+101)Aus((2 -11 ) + V0(2)A32:

1 ((2 - t1 ) + 11-[sin(2^)-sin(2 (®((2 - t1)+^)

2 4—

V0(2) A42

2 P3

,2 P3 ((2-t1 )-

V0(2) A52

2 P4

(2),

,2 P4 ((2-t1 )-

+ V0(2)(Aus)2((2 -11 )+ V A3A% [eP3((2^)

P3 | +1

P3 •

sin[—((2 -11 )+f ]-cos[—((2 -11 ) + f ] |-

-—sin(f ) + cos(f )

2V0(2) A3A5

P4 | +1

—

,P4 ((2-t1)

—sin[—((2 -t1)+f]-cos[—((2 -11 )+f]|-

—

■—sin(f ) + cos(f )

2Vq2)AUsA3

-cos[—((2 -11

2V0(2) A4 A5

[cos(^)-

,(P3+P4 X^2-t1 )- 1

2V0(2) Aus A4

P3

2V0(2)Aus A5

P3 + P4

P3 ((2-t1) - 1

P4

,P4 ((2-t1 )- 1

i f 2 = I0 +

U 0 + V

R

+ Au2(t )

LV

+ C

3"

dAu2 (( )

' 1 1 I -+-

V RLV Rd + r y

dt

P

Then let us transform expression (24) to the form:

( n A(1) ( n I(2)

f

P

f

+

P

f

Rff V Rff y I Rff

T+t1 i

\ j / \ j / ( ^

J I02) +Au1(t )

t2 V

V RLV Rd + r y

J |i02) + u2(t )

' 1 1 I -+-

RLV Rd + r y

+c

+ c3 ¿MO |2 dt +

dt y 2

dAu2 (( )

dt

dt,

where I02) = I0 +(u0 + VVRlv .

After integrating each component, one can obtain the following expressions that allow calculating the power losses in the active filter resistance of the installation:

( „ I(1)

P3

Rff

X 2

:(t 03) )2 ( +11 -12 )+ Ip. [e 2 P1(T+t1 ^ 2 P1

-1

2 P2

,2 P2 (T+t1 -12 )-

2103) X 4

2103) X 5

P2

P1

,P2 (T +t1-t2 )- 1

,P1(T +t1 -t2 )-

Pf

|(2)

:[e(P1+P2 )(T +t1 -t2 )- 1

2 X 4 X5 +-4-5 X

P1 + P2

(25)

=(3) ) ((2 -11 )+X12 [ 1 ((2 -11 )+

Rff

+—[sin(2f )- sin2[—((2 -t1) + f]

4—

+ ((3—C3 )2

■2((2 - t1) + ~7— [sin(2[— ((2 - t1 ) + sin(2^)]

2 4—

X22

2 P3

2

2P3 ((2-t1 )- 1 + e 2P4 ((2-t1 )- 1

2 P4

+ 2I0 X1 [cos(f)-cos[—(t2 -11 ) + f]] +

—

(23)

+ 2l03)A3C3 [sin[—((2 -11 ) + f ] - sin(f )] +

where I01} = I0 + Vj((d + r), V(2) = + r).

2I03) X 2 P3

+

x

+

X

+

+

X

cy

+

X

+

IO

—

X

+

cy

X

+

,P3

(f2-h )_1

2I03) X 3

P4

,P4 ((2)_

XA3C3

+ —1 3 3 X

: [cos(2/) _ cos([a (t2 _ t1 ) + /])] +

2 XiX 2

a

+1

P3

((2 -1 K

x 3 sin[a((2 _ t1) +/] _ cos[a((2 _ t1) +/] | _ _ —3sin/) + cos/)

2 X1X 3

P4

a

+1

,P4

((2 _t1 )

• sin[((2 _t1 ) + /]_cos[((2 _t1 )+/]|_

v 3 )

P4 •

—— sin/ + cos/

a

2 X 2 A3C3

,P3-(t2 _t1 K

P-1 +1

a

—• cos[a •(( _t1 )+/] + sin[a•(( _t1 )+/] |_

a)

_—cos/)_ sin/)

a

2 X 3 A3C3

P± a

,P4 (t2 _t1 K

+1

a

-^4cos[a(t2 _t1)+/] + sin[a(t2 _t1) + /]|_ _—cos/)_ sin/)

2 X2 X3

+-x

P3 + P4

where

x e

103) = 102) +A«s

,(P3 + P4 )(2 _t1 )_

(26)

' 1 1 I -+-

vrlv Rd + r )

(

X1 = A3

(

R

11

■ + -

LV nd

Rd + r

X 2 = A4

11

P3C3 +-+

A

X3 = a5

R

P4C3 +--+

Rd + r 1

LV nd • ■ )

1 1 I

X 4 = A1

X 5 = A2

rlv Rd + r

11

P1C3 +-+

A

V

P2C3 +

rlv Rd + r )

rlv Rd + r

shown in Fig. 1 were assigned the following parameters that correspond to the installation of DETU 08-04-99 in the modes of rated voltages V0 from 1 to 180 kV: C1 - charging capacitor (0.1 ^F); C2, C3 - filter capacitors (0.072 ^F); Rf - filter resistance (1.78 MŒ); ZDU _ ZD,, _ ZDn are Zener diodes of the D818D type; RLV is the resistance of the resistive voltage divider.

For the D818D Zener diodes, the value of stabilized current I0 = 5 mA was selected for 27 different values of the rated voltages V0 on the load, according to Table 1. The voltage divider has four values of input rated voltages V0: 180 kV; 90 kV; 60 kV; 30 kV for which the current of the divider voltage is ILV = 2.5 mA. For the other 23 input voltages of the voltage divider V0, its current decreases in proportion to the input voltage.

Table 1

Calculation results for 27 power modes of the DETU 08-04-99 installation

V0, kV Um, kV Ai, V A2, V An, % I0+ILV, mA

1 5.97 3.54 -4.08 0.381 5.083

2 6.61 3.81 -4.88 0.217 5.167

3 7.25 4.45 -5.69 0.169 5.250

4 7.89 5.04 -6.49 0.144 5.333

5 8.52 5.57 -7.27 0.128 5.417

6 9.16 6.04 -8.04 0.117 5.500

7 9.80 6.44 -8.81 0.109 5.583

8 10.44 6.78 -9.54 0.102 5.667

9 11.08 7.07 -10.26 0.096 5.750

10 11.72 7.31 -10.97 0.091 5.833

20 18.27 11.41 -20.65 0.080 6.667

30 24.66 13.09 -24.15 0.062 7.500

40 29.27 11.71 -22.07 0.042 6.667

50 35.22 12.46 -23.82 0.036 7.083

60 41.17 13.18 -25.51 0.032 7.500

70 46.09 12.18 -23.86 0.025 6.944

80 51.90 12.63 -24.97 0.023 7.222

90 57.71 13.09 -26.05 0.021 7.500

100 62.04 11.11 -22.32 0.017 6.389

110 67.70 11.39 -22.88 0.016 6.528

120 73.36 11.67 -23.43 0.015 6.667

130 79.02 11.95 -23.97 0.014 6.806

140 84.68 12.22 -24.51 0.013 6.944

150 90.34 12.49 -25.03 0.012 7.083

160 96.00 12.76 -25.55 0.012 7.222

170 101.66 13.02 -26.07 0.012 7.361

180 107.32 13.29 -26.59 0.011 7.500

Approbation of the obtained theoretical results

was performed using the calculations of the parameters of the standard installation DETU 08-04-99, that is used in the State verification scheme for means of measuring the direct current electric voltage in the range 1.. .180 kV [14].

To the high voltage direct current power supply circuit

Calculations were performed to solve the system of equations (9) - (15) for the parameters r = 10 kQ for the modes V0 = 1.10 kV and r = 60 k^ for the V0 = 20.180 kV modes. The value of rd was assumed according to [4] equal to 22 Q for each Zener diode, and Rd = nrd, where n is the number of Zener diodes corresponding to each mode V0. This quantity is determined based on the average value of the D818D stabilization voltage u0 = 9 V.

Based on the calculation results, the maximum positive pulsation values Au(t) = A1 and minimum negative pulsation values Au(t ) = A2 were determined, as well as the pulsation amplitude coefficient:

A P = A1 _A2 100,

2V0

(27)

The obtained results of calculations are shown in Table 1. Table 1 also shows the value of the total average

X

+

2

CO

X

2

X

+

2

X

1

load current I0 + ILV (mA) for each DETU 08-04-99 installation operating mode.

From the data in Table 1 it follows that with increase in load voltage V0, the pulsation amplitude coefficient Ap decreases accordingly. In the V0 = 1 kV mode, the pulsation amplitude coefficient Ap = 0.381 %, and in the V0 = 180 kV mode the pulsation amplitude coefficient is Ap = 0.011 %. The above values of ripple in different operating modes of the installation differ by a factor of 35.

Using the values of PLV, P,, Pf obtained above, one can determine the energy efficiency coefficient of the DETU 08-04-99 installation with nonlinear load: Plv + p, 1

EFF--

-100 =

Plv + p, + pf 1 + pf/ {Plv + p,)

100, %. (28)

Table 2 shows the calculated values of EFF (28) for the DETU 08-04-99 installation for V0 modes from 1 to 30 kV. For the V0 = 180 kV mode, the EFF value is 93.1 %.

The calculated results for the DETU 08-04-99 installation, given in Table 1 and Table 2 are confirmed by the data of installation experimental study.

Table 2

The values of the energy efficiency coefficient of the DETU 08-04-99 installation for the V0 values from 1 kV to 30 kV

V0, kV Plv, W Pi, W P, W EFF, %

1.0 0.09 5.25 46.15 10.38

2.0 0.35 10.25 47.68 18.19

3.0 0.78 15.25 49.20 24.57

4.0 1.37 20.25 50.77 29.86

5.0 2.13 25.25 52.37 34.33

6.0 3.05 30.25 53.99 38.15

7.0 4.14 35.25 55.63 41.45

8.0 5.40 40.25 57.30 44.34

9.0 6.83 45.25 59.00 46.88

10.0 8.42 50.25 60.72 49.14

20.0 34.34 101.50 79.85 62.98

30.0 76.51 151.50 100.97 69.31

The discussion of the results. When determining the energy efficiency coefficient of the high-voltage installation, let us consider the power P(Mt = PLV+Pi as a net power, while the power Pf represents the additional power losses in the filter resistance. At the same time, an increase in the resistance of the filter Rf is a means of decreasing the amplitude of the voltage ripple in the installation load when increasing voltage Um at the input of the circuit [5].

The peculiarity of developing a mathematical model for the typical high-voltage installation modes (Fig. 1) is that it determines the necessary parameters of the installation in the opposite direction - given average voltage drop across the group of Zener diodes U0 for a given average current I0 through it. The proposed solution sets the numerical value of the time instant ti - the start of the charging of the installation capacitor C2 and the time instant t2 - the «disconnection» of the right part of the installation from the capacitance C1, and also determines the parameters A1, A2, A3, A4, A5, p1, p2, p3, p4, y, Aus, Um depending on the values of U0, I0, C1, C2, C3, RLV, T, r, Rd, Rf by the analytical method, and is new.

The use of Zener diodes in the measuring group of a high-voltage direct current installation allows significantly reducing the pulsation amplitude (up to 3 and more times) and improve the voltage quality on the load [8].

It should also be noted that nowadays professional and demonstration versions of various circuit simulation programs are widely used to simulate voltage multiplication schemes, as well as processes in electrical equipment insulation. To simulate, for example, the phenomenon of partial discharges in the insulation of highvoltage equipment, demonstration versions of the programs are sufficient enough [15, 16]. However, the number of Zener diodes in the operating DETU 08-04-99 installation is tens of thousands of pieces. In this regard, on the one hand, a complete simulation of such a scheme requires expensive professional circuit simulation programs. On the other hand, as shown in this article, there is no need for a detailed circuit simulation of such a complex scheme. Taking these factors into account, the group of authors made a choice in favor of a generalized analytical solution of the problem posed in this article.

Conclusion.

1. An analytical method for solving equations for a complex system of a typical high voltage direct current installation based on a voltage doubler rectifier with an integrated R, C - filter, and a measuring group has been developed.

2. The solution obtained is generalized for the case of insertion of Zener diodes into the measuring group, while it is valid within the linearized domain of their current-voltage characteristic.

3. The use of Zener diodes in the load circuits of the installation makes it possible to significantly reduce the amplitude of voltage ripple, and also to improve the quality of the output voltage of direct current voltage multiplier.

4. A theoretical basis is developed not only for voltages and currents, but also for the electrical power of the elements of a typical high-voltage direct current installation with allowance for nonlinear voltage pulsations.

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Received 30.08.2018

V.O. Brzhezytskyi1, Doctor of Technical Science, Professor,

R.V. Vendychanskyi2, Deputy Chief of the Research Department

of Measurements of Electrical Values,

Ye.O. Trotsenko1, Candidate of Technical Science, Associate

Professor,

Ya.O. Haran1, Assistant, O.M. Desyatov1, Engineer,

V.I. Khominich1, Candidate of Technical Science, Senior Researcher,

1 National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute»,

37, Prosp. Peremohy, Kyiv, 03056, Ukraine, phone +380 44 2367989, e-mail: [email protected]

2 State Enterprise «All-Ukrainian State Research and Production Center For Standardization, Metrology, Certification and Consumers Rights Protection» (SE «Ukrmetrteststandard»),

4, Metrolohichna Str., Kyiv, 03168,Ukraine, phone +380 44 5263485 e-mail: [email protected]

How to cite this article:

Brzhezytskyi V.O., Vendychanskyi R.V., Trotsenko Ye.O., Haran Ya.O., Desyatov O.M., Khominich V.I. Characteristics of specialized single-phase high voltage doubler rectifier. Electrical engineering & electromechanics, 2018, no.5, pp. 54-61. doi: 10.20998/2074-272X.2018.5.09.