Current distribution in the low-voltage winding of the furnace transformer

Electrical Power and Energy Systems 43 (2012) 1251–1258

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Current distribution in the low-voltage winding of the furnace transformer Bojan Stojcˇic´ a, Damijan Miljavec b,⇑ a b

Kolektor ETRA, Šlandrova ulica 10, Ljubljana, Slovenia University of Ljubljana, Faculty of Electrical Engineering, Trzaska 25, Ljubljana, Slovenia

a r t i c l e

i n f o

Article history: Received 30 January 2012 Received in revised form 16 May 2012 Accepted 17 May 2012 Available online 20 July 2012 Keywords: A. Furnace transformer B. Current distribution C. Magnetic-field energy D. Foil-wound transformer E. Reactive power

a b s t r a c t The paper describes a mathematical model to be used in calculating the current distribution among the coils in the low-voltage winding of the furnace transformer. The method is derived from the leakage magnetic-field distribution in the transformer based on the minimum magnetic-field energy. The energy is calculated by using the principle of the minimum reactive power in the transformer window. The results are validated by using the finite-element model. The knowledge of the proper current distribution is very useful in designing the furnace-transformer cooling system to avoid local overheating. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction In the metal industry, melting furnaces are used in production and recycling of metals. These furnaces are supplied with high electric currents [1] from the furnace transformer (FT) connected to the electric power grid [2,3]. The low-voltage winding (LW) in a FT of approximately 4 MVA (and above) is designed with several coils connected in parallel, because of the very high nominal currents in the FT LW. The current distribution in LW can be calculated by solving the transformer equivalent circuit [4–6]. The main disadvantage of the method [4] is in the need of having the coil voltage assumed. On the other hand, by using the principle of the minimum reactive power in the transformer window, this deficiency can be avoided. In every transformer there is a leakage magnetic field [7] distributing the currents among parallel connected coils thus minimizing the leakage-field energy. The paper describes a method to calculate the current distribution in the FT LW based on the minimum stray-field energy in the transformer [8]. Our new method to be used in calculating the current distribution among the coils in the low-voltage winding is demonstrated on a furnace transformer. The main advantages of the method are its simplicity, as it enables the analysis to start from the magneto-static field calculation (inductance calculation), and the possibility to use it also in studying the rectifier transformers.

⇑ Corresponding author. E-mail address: [email protected] (D. Miljavec). 0142-0615/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2012.05.030

The authors are not aware of any such theoretical approach and they have not found any publication using such a method or even a similar analytical approach in the recent literature. 2. The mathematical model The mathematical model of the transformer is derived from a simple sketch shown in Fig. 2. Based on Fig. 2, the set of the basic equations can be written as: inductance of winding no. 1:

L1 ¼

N1 U1 N1 Ur1 N1 U12 ¼ þ I1 I1 I1

ð1Þ

inductance of winding no. 2:

L2 ¼

N2 U2 N2 Ur2 N2 U12 ¼ þ I2 I2 I2

ð2Þ

and the mutual inductance between winding nos. 1 and 2 is defined as:

M¼

N 2 U12 N1 U12 ¼ I1 I2

ð3Þ

where N1 is the number of turns in winding no. 1, N2 is the number of turns in winding no. 2, I1 is the current in winding no. 1, I2 is the current in winding no. 2, Ur1 is the leakage flux of winding no. 1, Ur2 is the leakage flux of winding no. 2, U12 is the common magnetic flux in the transformer core, U1 is the magnetic flux of winding no. 1 and U2 is the magnetic flux of winding no. 2.

B. Stojcˇic´, D. Miljavec / Electrical Power and Energy Systems 43 (2012) 1251–1258

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winding 1 Φ12 Φ1=Φσ1+Φ12

Φ1 Φ σ1

N1

U1 Φ σ2

Φ2=Φσ2+Φ12

I1

N2

I2 U2

Φ2 magnetic core (Fe) winding 2

Fig. 1. Active part of the furnace transformer.

Fig. 2. Basic model of the transformer.

Using (1) and (3), leakage inductance Lr1 is determined as:

I1 L1 I1 L1 I1 M  U12 ¼  N2 N1 N1

ð4Þ

  N1 Ur1 N1 I1 L1 I1 M N1 ¼ L1  M ¼  N2 I1 I1 N 1 N2

ð5Þ

Ur1 ¼ Lr1 ¼

R1

X1

R2

R0

X0

X2

=

Leakage inductance Lr2 is derived in the same manner as (5):

Lr2 ¼ L2  M

N2 N1

The transformer can be treated as a T equivalent circuit with the elements of the secondary side (winding no. 2) reduced to the primary side (winding no. 1) [9]:

L0r2 ¼ Lr2

(a)

ð6Þ

2.1. Transformer equivalent circuit

 2 N1 N1 ¼ L02  M N2 N2

N1 N2

ð8Þ

2.2. Equation parameters

k ¼ 1; 2 . . . n;

Z lk ¼ Z kl

ð9Þ

For example, the matrix form of the circuit presented in Fig. 3b is:



U1 U2



 ¼

Z 11 Z 21

   I1 Z 12  Z 22 I2

ð10Þ

where U1 and U2 are primary and secondary voltages, respectively. The same is valid for the currents I1 and I2. In FT, the magnetizing current can be neglected [4] and the following rule applies: n X Ik ¼ 0 k¼1

(b)

When measuring short circuit currents at the manufacturer’s laboratory, the primary side is connected to a generator (U1 = Uks) and the secondary side is short-circuited (U2 = 0). On the basis of 11 and 10 can be presented as follows:



U ks



 ¼

Z 11 Z 12

   In  ; Z 22 In Z 12

I1 ¼ I2 ¼ In

ð12Þ

And the solution is:

In transformers designed with several windings each winding has the main impedance (winding ‘‘l’’ has main impedance Zll). Other windings affect this winding by their mutual impedance (the ‘‘k’’ winding affects ‘‘l’’ through the mutual inductance expressed as Zkl). A general form of the matrix voltage equation of a transformer with ‘‘n’’ windings can be written as:

l ¼ 1; 2 . . . n;

Z22

Fig. 3. Transformer equivalent circuits. The elements in figure are R1 is the resistance of the primary winding, R02 is the reduced resistance of the secondary winding, X1 is the leakage reactance of the primary winding (jxLr1), X 02 is the 0 reduced leakage reactance of secondary winding  ðjxLr2 Þ, R0 is the core-losses resistance, X0 is the excitation reactance jM NN12 , Z11 is the open-circuit primary impedance (R1 + jxL1), Z22 is the open-circuit secondary impedance ðR02 þ jxL02 Þ and Z12 is the impedance relaying to the mutual inductivity.

0

½U l  ¼ ½Z lk ½Ik ;

Z12

ð7Þ

With (5) and (7), the overall leakage inductance Lr can be defined as:

Lr ¼ Lr1 þ L0r2 ¼ L1 þ L02  2M

Z11

ð11Þ

U ks ¼ ðZ 11 þ Z 22  2Z 12 ÞIn ¼ Z 12 In ; Z 12 ¼ Z 11 þ Z 22  2Z 12

ð13Þ

Based on (8) and (13), it can be concluded that Z 12 represents the stray magnetic-field impedance or the leakage impedance. In FTs the resistance is very small compared to the leakage reactance. The Joule component of the leakage impedance in (13) can therefore be ignored and ½X  ½Z. So:

X 12 ¼ X 11 þ X 22  2X 12

ð14Þ

2.3. Derivation of the basic equations When measuring the transformer load loss, the primary side is powered and the secondary side is short-circuited with all the power consumed in the transformer. The equation for this complex power is:

P þ jQ ¼ I U ¼ ½Il ½Z lk ½Ik 

ð15Þ

where P stands for real power and Q represents reactive power. The complex power consumed by the transformer with ‘‘n’’ windings can be expressed as:

B. Stojcˇic´, D. Miljavec / Electrical Power and Energy Systems 43 (2012) 1251–1258

P þ jQ ¼

n X n X Z lk Ik Il

I0

ð16Þ

l¼1 k¼1

n X n 1X  2Z lk Ik Il 2 l¼1 k¼1

U0

ð17Þ

According to the rule laid down in (11), the following equations will be used in later derivations:



n X n n n X 1X 1X Z ll Ik Il ¼  Z ll Il Ik ¼ 0; 2 l¼1 k¼1 2 l¼1 k¼1

n X Ik ¼ 0

ð18Þ

k¼1

and



N1 I1

I1

N2 I 2

I2

N3 I3

I3

N0 I 0

and further as

P þ jQ ¼ 

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n X n n n X 1X 1X Z kk Ik Il ¼  Z kk Ik Il ¼ 0; 2 l¼1 k¼1 2 k¼1 l¼1

n X Il ¼ 0

ð19Þ

Fig. 4. Presentation of a simple FT model with three parallel coils in LW and the primary winding. The symbols in figure denote N0 is the number of turns in the primary winding, N1 is the number of turns in the LW coil no. 1, N2 is the number of turns in the LW coil no. 2, N3 is the number of turns in the LW coil no. 3, I0 is the RMS value of the current in primary winding, I1 is the RMS value of the current in the LW coil no. 1, I2 is the RMS value of the current in the LW coil no. 2 and I3 is the RMS value of the current in the LW coil no. 3.

l¼1

Based on 13, 18, 19 and 17 can be transformed into: n X n n X n 1X 1X P þ jQ ¼  ðZ ll þ Z kk  2Z lk ÞIk Il ¼  Z lk Ik Il 2 l¼1 k¼1 2 l¼1 k¼1

The short-circuit impedance can now be defined as:

ð20Þ

u2ks ¼ u2r þ u2x

ð28Þ

It can now be concluded that the real power P and reactive power Q in (20) can be expressed with the leakage impedance of all the transformer windings and winding currents. Since Zlk = Zkl, the real power and reactive power of a pair of windings are:

where ur is the Joule component of the short-circuit impedance and ux is the reactive component of the short-circuit impedance. As the characteristics of FT is ux P ur , uks ffi ux and (26) can be presented the below per-unit form:

1 Plk þ jQ lk ¼  Z lk ðIk Il þ Ik Il Þ 2

q¼

ð21Þ

The currents in (20) can be denoted as:

Ik ¼

I0k

þ

00 jIk

ð22Þ

00

ð23Þ

Il ¼ I0l þ jIl Ik Il

þ

Ik Il

¼2

ðI0k I0l

þ

I00k I00l Þ

ð24Þ

thus making the power equation to be:

P þ jQ ¼ 

n X n 1X Z lk ðI0k I0l þ I00k I00l Þ 2 l¼1 k¼1

ð25Þ

Having ignored the Joule component of the leakage impedance, our attention will be paid to the reactive power in (25):

Q ¼

n X n 1X X lk I00k I00l 2 l¼1 k¼1

U ks 100 UP

ð29Þ

where uxlk is the reactive component of the short-circuit impedance between the ‘‘l’’ and ‘‘k’’ windings, ik is the per-unit current in winding ‘‘k’’ and il is the per-unit current in winding ‘‘l’’. 2.4. Demonstration on a simple model Derivation of a matrix equation will be presented with its solution resulting in per-unit coil-current values. The primary winding will be indexed with ‘‘0’’ and the parallelly connected LW coils with the numbers from ‘‘1’’ to ‘‘n’’ (n – number of parallel coils, in our case n = 3). As the model shown in Fig. 4 is a per-unit model the value of the per-unit current of the primary winding is set to 1 and so is also the sum of the per-unit currents of the LW coils (30). The first to be laid down is the rule for the transformer shown in Fig. 4:

ð26Þ N1 ¼ N2 ¼ N3 ¼ N

From here on, the system of equations shall be converted into a per-unit system. The most common per-unit component in the transformer design is the short-circuit impedance [8–13]. It is calculated from the transformer leakage magnetic field. The finiteelement model and the magneto-static finite element analysis [14] were used to calculate the geometry-dependant short-circuit impedance for each coil. In the industrial practice, the Lyle method of the equivalent filaments [11] is also used for its fast computational performance. The short-circuit impedance presents the voltage measured on the transformer primary side, while its secondary side is short-circuited and the transformer primary windings are loaded with the rated current. So:

uks ¼

n X n 1X uxlk ik il 2 l¼1 k¼1

ð27Þ

where uks is the transformer short-circuit impedance, Uks is the voltage measured on the primary side (high-voltage winding) while on the secondary side (LW) it is short-circuited, and UP is the nominal primary voltage.

i0 ¼ 1

ð30Þ

i3 ¼ 1  i1  i2 where i0 represents the per-unit value of the primary-winding current, i1, i2 and i3 are the per-unit value of the current in the LW coil nos. 1, 2 and 3, respectively. As the reactive power is a time derivate of the transformer stray magnetic-field energy, the current distribution among the LW coils can be calculated by partial derivatives of the per-unit reactive power with respect to the per-unit current in the LW coils. The (29) for the case shown in Fig. 4 is presented in form:

q ¼ ux01 i0 i1 þ ux02 i0 i2 þ ux03 i0 i3  ux12 i1 i2  ux13 i1 i3  ux23 i2 i3 ð31Þ To write (31) for the partial derivatives, one of the unknown coil currents should be expressed with the other unknown coil currents, as in (30). Using (30) and (31), the reactive power in the per-unit form is:

B. Stojcˇic´, D. Miljavec / Electrical Power and Energy Systems 43 (2012) 1251–1258

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q ¼ &ux01 i1 þ ux02 i2 þ ux03  ux03 i1  ux03 i2  ux12 i1 i2  ux13 i1 þ

2 ux13 i1

þ ux13 i1 i2  ux23 i2 þ ux23 i1 i2 þ

to hundreds of thousand amperes which imposes an enormous challenge on LW designers. It is for this reason that LW is made of small coils with their turns connected in parallel so as to sum up the high load currents. Basically, FT operates in the way very similar to that of the foil-wound transformer. The elements of the general matrix form based on (36) for the FT winding assembly shown in Fig. 5 are given with the below Eqs. (40)–(42).

2 ux23 i2

ð32Þ By partial derivation of (32) regarding to i1 and i2 the matrix of the short-circuit impedance reactive components can be set:

@q ¼ ux01  ux03  ux12 i2  ux13 þ 2  ux13 i1 þ ux13 i2 þ ux23 i2 ¼ 0 @i1 ð33Þ

2 6 6 6 6 6 6 ½U XX  ¼ 6 6 6 6 6 4

2ux1n

ux2n þ uxn1  ux12



uxin þ uxn1  ux1i

   uxðn1Þn þ uxn1  ux1ðn1Þ

3

ux1n þ uxn2  ux21 2ux2n ux3n þ uxn2  ux23    uxðn1Þn þ uxn2  ux2ðn1Þ 7 7 7 7 .. .. .. .. 7 . . . . 7 7 ux2n þ uxni  uxi2  2uxin    uxðn1Þn þ uxni  uxiðn1Þ 7 ux1n þ uxni  uxi1 7 7 .. .. .. .. 7 . . . . 5 2uxðn1Þn ux1n þ uxnðn1Þ  uxðn1Þ1    uxin þ uxnðn1Þ  uxðn1Þi    uxðn2Þn þ uxnðn1Þ  uxðn1Þðn2Þ ð40Þ

@q ¼ ux02  ux03  ux12 i1 þ ux13 i1  ux23 þ ux23 i1 þ 2  ux23 i2 ¼ 0 @i2 ð34Þ In (33) and (34), the minimum of the system reactive power is derived and presented in the following matrix form:



2ux13

ux23 þ ux31  ux12

ux13 þ ux32  ux21 2ux23   ux13 þ ux03  ux01 ¼ ux23 þ ux03  ux02



i1



i2

3

6i 6 2 6. 6. 6. ½i ¼ 6 6i 6 i 6. 6. 4.

7 7 7 7 7 7 7 7 7 7 5

ð41Þ

in1 ð35Þ 2

When written in a general matrix form we get:

½U XX ½i ¼ ½U XY 

i1

2

ð36Þ

where [UXX] is the matrix of the short-circuit impedances on the left side of (35), [UXY] is the matrix of the short circuit impedances on the right side of (35) and [i] is the per-unit value vector of the ampere-turns in a single LW coil. The current distribution in the LW coils is determined by solving the equation system (36) giving the following results:

I1 ¼ i1

N 0 I0 N

ð37Þ

I2 ¼ i2

N 0 I0 N

ð38Þ

ux1n þ ux0n  ux01

6 6 6 6 6 ½U XY  ¼ 6 6 6 6 6 4

ux2n þ ux0n  ux02 .. . uxin þ ux0n  ux0i .. .

3 7 7 7 7 7 7 7 7 7 7 5

ð42Þ

uxðn1Þn þ ux0n  ux0ðn1Þ By solving the general Eq. (36) using (40)–(42), the current distribution in the FT LW can be calculated for numerous parallel coils. The current distribution in LW of a 5MVA FT, calculated by

I0 NI1

I1

NI2

I2

3. Generalized equations

NIi

Ii

Fig. 5 is a mathematical presentation of FT shown in Fig. 1. The ‘‘0’’ indexed winding presents the FT primary winding, while the ‘‘1, 2 . . . , n’’ indexed windings present the LW coils of the FT connected in parallel. The load connected to the FT secondary side (LW side) operates at the voltage range of down to 70 V and at the load current of up

NIn

In

I3 ¼ ð1  i1  i2 Þ

N0 I0 N

N0 I 0

ð39Þ

U0

Fig. 5. Presentation of a general assembly of the FT winding.

B. Stojcˇic´, D. Miljavec / Electrical Power and Energy Systems 43 (2012) 1251–1258

Fig. 6. Current distribution among the LW coils (distribution denoted with triangles obtained by using new generalized equations; distribution denoted with crosses obtained by dividing the load current by the number of parallelly connected coils).

using the above generalized equations, is in Fig. 6 depicted with triangles, while the curve with crosses denotes the LW nominal current divided by the number of parallelly-connected FT coils. Current distribution in the coils is calculated based on the stray magnetic field energy in the transformer window. What about the high harmonic components in the current when the current shape in an electric arc furnace is in the form (for example) of a triangle wave (in which there are high harmonic components)? The currents of quite the same shape also flow in the furnace transformer parallel LW coils. These currents would produce a stray magnetic field in the transformer window in the form of a triangle (timedependent form). It can be concluded that the additional power losses due to high harmonics are homogeneously distributed across the transformer windings, and since the dimensions of a single conductor are relatively small, the skin effect is due to its small depth not so prominently expressed in the end coils of the LW winding. Therefore, the proportions between additional (high harmonic) losses in individual LW coils remain roughly the same. The presented method can be used for large furnace transformers in which the magnetic energy of the stray magnetic fields is much higher than the energy of the Joule losses. So, a further increase in the additional Joule losses (due to high harmonics) does not significantly affect current distribution in parallel connected LW coils. Increased losses in the windings, due to high harmonics in the current, increase the winding temperature. The problem is solved by having the cooling system properly dimensioned, thus keeping the hot-spot temperature on the desired level. In the furnace transformers, the current high harmonic components appear in the melt starting phase, and at the time when the melt originate the current stabilization occurs (high harmonic components are reduced).

1255

Fig. 7. Cross section of the FT window axis-symmetric geometry (dimensions are in mm). The elements in figure are: high-voltage winding with 228 turns, coarse winding [8] with 150 turns, tapped winding [8] with 150 turns, LW composed of 32 parallel coils with eight turns each.

former oil being 1, the Neumann boundary conditions [9] were set to the inner edge of FT window. The left window edge (Fig. 7) presents the core limb, the upper and lower window border describe the core yoke and the right window side depicts the tank wall. Since the FT windings take a concentric form, the axis-symmetric definition of the problem was used in the finite-element analysis. As the FT windings are made of conductors of a small cross section carrying currents of the frequency of 50–60 Hz, the skin-effect due to the leakage magnetic-flux density was neglected. Therefore, the windings were modeled as solid rectangle coils of the electrical conductivity equal to zero. The nominal voltage of the high-voltage winding (primary side) was 10 kV and that of the low-voltage side (LW) varied from 151.5 V up to 350.9 V. The current distribution shown in Figs. 6 and 9 provides the condition at which FT operates with the lowest number of turns of the high-voltage winding and 350.9 V on the low-voltage side. The value of the load resistor was chosen so as to achieve the same load current (the sum of coil currents) as the one used in the proposed analytical method. Fig. 8 illustrates the magnetic-flux density distribution in the transformer window at the time the currents reach the amplitude value. The radial (r-coordinate) component of the magnetic flux density in the region of the upper low-voltage coil increases, whereas, in the region of all other coils its axial (z-coordinate) component prevails.

4. Validation of proposed method A thorough finite-element analysis of FT was made. To evaluate and validate the proposed method, the geometry of the analyzed FT is shown in Fig. 7. The finite-element model connected to an external circuit supplied by a voltage source and the time-domain magneto-transient finite-element analysis [14] were used to calculate the currents in each coil. The following was taken into account in the FT finite-element analysis. The transformer core being made of a high-permeability grain-oriented silicon steel and the relative permeability of trans-

Fig. 8. Magnetic flux density distribution in the transformer window at currents reaching their amplitude value.

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Fig. 9. Current distribution among the LW coils (distribution marked with triangles obtained by using new generalized equations; distribution marked with stars obtained by using time domain finite element analysis).

Fig. A1. Cross section of the used axis-symmetric geometry of the transformer window [17].

Appendix A By using the time-domain magneto-transient finite-element analysis, the current distribution in the low voltage coils was obtained (Fig. 9). In the central region, the distribution is the same as the one calculated by the new generalized equations (Fig. 6). The difference is observed in the area of the upper and lower coils and it is due to better description of the leakage inductance in the FT window when using the finite element method. The current distribution among all the coils takes the same shape irrespective of the used method. The finite-element analysis was made for both current distributions (Fig. 6) in the FT parallel LW coils. Since the leakage magnetic flux density in the FT window is not uniform (as seen from Fig. 8), the currents among the coils are not distributed evenly but in the way providing the lowest leakage-field energy. The magnetic field energy in the transformer window calculated for evenly distributed LW currents is 161.34 J. In the same window but with the current distribution calculated by using the proposed method it is 160.69 J. Comparing the values of the leakage magnetic-field energy shows a small difference between them, but there is a great difference in the RMS current distributions among the LW coils (Fig. 6). Thus performed validation of the obtained results demonstrates correctness of the proposed method. The main advantage of using the new method is in its determination of the current distribution in the FT LW, thus enabling the manufacturers to optimally design the FT cooling system to avoid the critical local-overheating issue. 5. Conclusion The main advantage of the method described and proposed in this paper is, that it can be used without exactly knowing the coils voltages. And as such can be used for calculating the current distribution in regulating winding of the double-stacked transformer and other various type of power rectifier transformer [12]. Also the method can be efficiently used in designing the winding cooling system to prevent local overheating [15,16]. To achieve this target, it is important that the current is correctly distributed in the power transformer. Undetected and unresolved winding hotspots can increase the winding-insulation deterioration and can consequently dramatically speed-up the transformer ageing process. The method which was also verified by measurements (Appendix), gives good and useful results also when used in smoothing the current distribution by coil positioning.

Having no opportunity to build up and measure the current distribution in the parallel coils of a real FT. Our work was based on a reference paper [17]. This helped us to test our analytical approach with confirmation using the time-domain finite-element method and finally to demonstrate correctness of the method. In the reference paper, the authors built-up a system of a high-voltage coil and low-voltage parallel coils wound around the central limb of a three-limb transformer core (Fig. A1), fully specifying the geometric dimensions and electric parameters of the coil and core system. Based on the above paper we built a finite-element model linked to an external circuit supplied by a voltage source. To calculate the current density in each of the sixteen parallelly connected coils we used the time-domain magneto-transient analysis (TDFEA). The results are compared in Table A1. Fig. A2 illustrates the magnetic flux density distribution in the transformer window at the time the currents reach their amplitude value. The radial (rcoordinate) component of the magnetic flux density in the region of the first four upper low-voltage coil increases, whereas in the region of all other coils its axial (z-coordinate) component is uniform. As seen from the above table, the differences between the current density distributions in the central coils between the measured [17], calculated [17] and those determined by using our finite element model are very small. Still, there are differences both

Table A1 Comparison between the values obtained by calculating [17], measuring [17] and those determined by using our time-domain finite-element analysis. No. of coil

Calculated [12]

Measured [12]

Determined by using TD-FEA

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

2.208 1.203 1.828 1.643 1.686 1.681 1.686 1.675 1.686 1.684 1.679 1.687 1.644 1.827 1.203 2.21

2.318 1.143 1.778 1.713 1.616 1.611 1.596 1.775 1.611 1.724 1.739 1.737 1.594 1.747 1.183 2.26

2.0047 1.5936 1.5921 1.6249 1.6352 1.6405 1.6433 1.6444 1.6444 1.6434 1.6409 1.6362 1.6271 1.5964 1.6078 2.0554

B. Stojcˇic´, D. Miljavec / Electrical Power and Energy Systems 43 (2012) 1251–1258

1257

Fig. A4. Current density distribution in unevenly positioned low-voltage coils.

Fig. A2. Magnetic flux density distribution in the transformer window [17] after the currents have reached their amplitude value.

coils evenly positioned and the upper ones (at the top of transformer window) pushed aside. The space between the coil nos. 2 and 3 and between the coil nos. 3 and 4 was increased to measure 9 mm. The results are shown in Fig. A4. As shown in Fig. A4, the current distribution is strongly affected by the coil mutual positioning. When the parallelly connected lowvoltage coils are distributed evenly (the distances between them are equal), the current distribution in coils varies from its maximum to its minimum value and a quite smoothly to its constant value (Figs. 6 and 9 and the right side of Fig. A4). Meanwhile, an unevenly mutual coil changes the current distribution. The current values vary from their maximum to minimum and again to their lower maximum and to lower minimum. The current distribution is somehow wavy (left side of Fig. A4). The difference in the current distribution shown in Fig. A3 is now clear. The measured results [17] and the above explanations confirm correctness and applicability of the developed analytical method described and proposed in this paper. References

Fig. A3. Calculated current density distributions among low-voltage coils, squares – results from [17], circles – time domain finite element analysis.

in the value and shape of the current density distributions in the upper and lower low-voltage coils. The results are shown in Fig. A3. To allow for a rough approximation, FT may be compared with the foil-wound transformer. The evenly distributed parallelly connected low-voltage coils can be treated as a conductive foil. In [18– 20], the current distribution in the low-voltage coil presented by a foil is much more similar to current distribution presented here than in [17]. The question arises, why is the current distribution in [17] so wavy? The increase in the current of the extreme upper and lower coil (Figs. 6 and 7) is due to the increase of the radial component of the magnetic flux density (Fig. 8) in the area of these two coils in the transformer window. This component decreases when going in to the central coils region and the current distribution too, becomes more uniform. To investigate the impact of the mutual coil position, we built a finite element model with lower

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