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Combined analytical and FEM methods for parameters calculation of detailed model for dry-type transformer
Simulation Modelling Practice and Theory 18 (2010) 390–403
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Combined analytical and FEM methods for parameters calculation of detailed model for dry-type transformer M. Eslamian, B. Vahidi *, S.H. Hosseinian Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 3 June 2009 Received in revised form 7 December 2009 Accepted 9 December 2009 Available online 16 December 2009 Keywords: Cast-resin dry-type transformer Detailed model FRA FEM Parameter calculation
a b s t r a c t Non-flammable characteristic of cast-resin dry-type transformers make them suitable for different kind of usages. This paper presents a method of how to obtain high frequency model of these transformers. For this purpose a detailed model is used and parameters of the model including inductances, capacitances and resistances are calculated using FEM (finite elements method). The effect of the frequency is considered in the inductance calculation using FEM. In order to validate the model, a setup was constructed for testing on highvoltage winding of dry-type transformer. The simulation results were compared with the experimental data measured from FRA (frequency response analysis) and impulse tests. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Transformers in operation are subject to various kinds of overvoltages caused by lightning strikes, switching operations or system disturbances. Generally, under normal conditions in a steady-state, the voltage is distributed linearly inside the winding. For impulse voltage, the voltage distribution in the winding is not linear and for the calculation of impulse voltage distribution, the windings should be simulated in terms of an equivalent circuit consisting of lumped R, L, C elements. Cast-resin dry-type transformers are the most suitable transformers for distribution of electricity in high degree of safety. Dry-type transformers compared with oil-immersed are lighter and non-flammable. They also do not have contaminating substances such as oil. Non-flammable characteristic of these transformers make them suitable for residential and hospital usages. Fig. 1 shows the structure of one phase of a cast-resin dry-type transformer consisting of low- and high-voltage windings. Low-voltage winding of this transformer is made of a full-height aluminum sheet wounded simultaneously with insulation layer. High-voltage winding is constructed from several series disks that every disk is made of aluminum foils interleaved with insulating layers. After the high-voltage winding is wounded, it is placed in a mold and cast in a resin under vacuum pressure. Lower sound levels are realized as the winding is encased in solid insulation. Filling the winding with resin under vacuum pressure eliminates voids that can cause corona. With a solid insulation system, the winding has superior mechanical and short-circuit strength and is impervious to moisture and contaminants. 2. Transformer transient modeling Transformer transient modeling has been a subject of investigation and research for a century. In the 1954, Lewis [1] proposed that the transient behavior of a transformer winding can be studied with an equivalent ladder-type network * Corresponding author. Tel.: +98 21 64543330; fax: +98 21 66406469. E-mail address: [email protected] (B. Vahidi). 1569-190X/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.simpat.2009.12.005
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Fig. 1. Schematic view of cast-resin dry-type transformer.
composed of a finite number of uniform sections. Each section is composed of lumped capacitive and inductive components which represent the distributed parameters of actual winding. Lewis’s model is applicable only to a uniform winding. Furthermore, the representation of inductive coupling effect was included by modifying the self inductance value. McWhirter et al. [2] studied the same problem based on an equivalent circuit approach. Their model still suffers the restrictions arising from the size and symmetry of the equivalent circuit used to represent the winding. Dent et al. [3] used an equivalent circuit of the same general form as the proposed by Lewis, but with certain differences. Dent’s model can represent a non-uniform winding and the effect of inductive coupling between sections is taken into account. After Dent’s paper was published, most of the researchers in this area, concentrated their attention on the calculation of parameters of the equivalent circuits. Okuyama [4] calculated the self and mutual inductances of transformer winding through introduction of some correction factors obtained from experiment. Stein [5] and Kawaguchi [6] proposed a method to calculate equivalent series capacitance by computing the electrostatic energy stored in the coils. Fergestad and Henriksen [7,8] calculated self and mutual inductance of sections of windings by taking certain effects of iron core into account. Wilcox et al. [9,10] derived a set of formulae from Maxwell’s equations to calculate self and mutual inductive parameters. De Leon and Semlyen [11] used the image method to calculate turn to turn leakage inductance and the charge simulation method to calculate the capacitance between turns and from turn to ground. They also proposed a detailed model of losses [12] for accurate calculations. Also Transformer transient has been investigated by analytical methods in the last years [13–15]. Using the analytical methods is one of the designing tools that may yield acceptable results depending on the geometry of problem. But this method is not flexible. Nowadays, with respect to difficulties of analytical methods, the FEM as an accurate and reliable method becomes more popular. The most important advantage of FEM is that any complicated geometry is solvable by this method because the formulation of FEM is independent of the geometry. FEM has many applications in simulation of transformer behaviors in different situations. These applications include the calculation of energy, leakage fluxes and electromagnetic forces. By using FEM and considering the structural details of transformer, it is possible to calculate the parameters of winding detailed model. The advantage of this method over the analytical method is that there may be unequal amounts of inductances and capacitances in the equivalent circuit. Till now, calculation of lumped parameters of power transformers by means of FEM was accomplished [16–20]. All works done on transformer transients are for oil filled transformers and this was not done for dry-type transformers yet. As it is clear from its name, dry-type transformer do not have any oil and instead a solid dielectric material such as resin is used in its structure. The shape of conductor, types of dielectric materials and the structure of windings of dry-type transformers are different from oil filled transformers. Also the shape of electrical field is rather different. On the other hand, the surrounding air as an insulator and cooler is running in the space between windings and tips of bars. Because of large difference of air dielectric coefficient with other solid insulators of dry-type transformers (about 4 times), the large amount of voltage in this kind of transformers is dropped on air spaces and with respect to low resistance of air in comparison with other solid materials, it is important to have information about the size and the way of electrical field distribution in transients to make sure about tolerance of air spaces. Therefore, study and research on dry-type transformers seems to be important and useful. In this paper detailed parameters and transient behavior of the cast-resin dry-type transformer as a new generation of distribution transformer is calculated.
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Fig. 2. Equivalent lumped parameter circuit.
3. The equivalent circuit of dry-type transformer Equivalent circuit used for modeling of dry-type transformers is shown in Fig. 2. In this case, each layer of LV winding which is placed between two air canals and every disk of HV winding is considered as a branch in the equivalent circuit. Every branch is composed of parallel connection of self inductance of Li and series capacitance of Ki. There is a mutual inductance between every pair of branches (Lij, Lik, . . .). Regarding that the last layer of LV is creating an equipotential surface in front of other layers of LV, only the capacitances of the last layer of LV to HV disks are considered which are shown with Cp. The Ce is the capacitance of the first layer of LV to the core which is grounded. Ri is ohmic losses of each branch and Gi, Gp and Ge are conductances which are representing the dielectric losses. Dielectric losses are not included in this work because of their lower effect in comparison with ohmic losses. As an approximation, a constant value based on low-frequency measurements can be used. 4. Parameter calculation 4.1. Inductance calculation Traditionally, the magnetic flux is composed of two components of leakage and common fluxes. The leakage flux is not very dependent on frequency however the common flux which is passing through the core is highly frequency dependent and in high frequencies, it is completely displaced from the middle of the core. So it is the leakage inductance that is the determining factor for internal transients in transformer windings and the effect of iron core could be neglected in transformer high frequency modeling. As it was mentioned, in dry transformers, HV winding is composed of several disks with foil conductors and LV winding is composed of full-height foil layers. By increasing of the frequency, the skin effect in the foils is increased and it causes a considerable change in the flux distribution. Therefore in order to calculate the inductances correctly in high frequencies, effects of all HV and LV winding foils must be considered simultaneously. Because of long height of the LV foils, effect of LV layers in change of flux distribution is more than HV disks and in high frequencies LV layers prevent the penetration of the flux into the LV winding. Generally, if there are n conductors in the space, with respect to linearity of the system, the following equation could be written in each frequency.
V f ¼ jxLf If
ð1Þ
j kT j kT In which If ¼ I1f ; I2f ; . . . ; Inf is the vector of net current phasors in frequency of f, V f ¼ V 1f ; V 2f ; . . . ; V nf is the vector of voltage drop phasors in frequency of f, Lf is the system’s inductance matrix in frequency of f and x ¼ 2pf . Assume each element of Lf is lijf . By applying the current phasor of I to conductor i and forcing a zero net current to other conductors, after calculation of produced voltage drop in all conductors, the ith column of Lf is obtained as follows:
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Fig. 3. Upper part of the FEM-model established in the FEMM4.2.
2
l1if 6 6 l2if 6 6 . 6 .. 4 lnif
3 7 ( ) 7 Vf 1 7 Imag 7¼ 7 jx I 5
ð2Þ
In order to obtain voltage drops, suppose that a zero current is applied to all conductors except conductor i. After FEM analysis, current density, J, and vector potential, A is given in each point. By means of Maxwell equations, the electric field could be calculated as follows:
E¼
@A rV @t
ð3Þ
By taking integral of both side of Eq. (3) with respect to length in the cylindrical coordinates and with regard to relation of J ¼ rE, the produced voltage drop on the j conductor due to the applying current to conductor i could be calculated as follows:
V j ¼ 2pr
J rj
r
! þ j x Ar j
ð4Þ
In which Jrj and Arj are the values of current density and vector potential in a desired point on the conductor j and with the radius of r. r is conductivity of the j conductor. As it was explained in Section 3, in the equivalent circuit of the dry transformer, each disk of HV winding and every part of LV winding which is placed between two air canals are considered as a branch. Each HV disk is composed of many foils which are in series together. So the total voltage drop on each disk is equal to sum of the voltage drops on all foils of the disk. This is also true for every part of the LV winding. Generally, in order to calculate the ith column of the inductance matrix by using FEM, a current source equal to I is applied to each foil of part i and a current source equal to zero is applied to all foils of the other parts. Then the total voltage drop in each part is calculated. By having the values of voltage drops, the elements of ith column are calculated by using Eq. (2). This method is also repeated for calculation of the other columns and so the inductance matrix, Lf in each frequency could be calculated. Basically discretisation level of a winding depends on the investigated frequency range with regard to the conductor dimension. The skin depth of aluminum at 50 Hz and 75 °C is equal to 13.2 mm. By increasing of the frequency, the skin depth will decrease so that at 100 kHz and 1 MHz, it will be equal to about 0.3 and 0.1 mm, respectively. Thickness of each HV foil is 0.2 mm. Since the investigated frequency range is from 1 kHz to 1 MHz, all foils must be modeled completely. Since in FEM analysis there should be at least one element in one skin depth so the number of elements will increase and the solution especially in high frequencies will be time consuming. Since inductances have a predictable frequency-dependency, it is possible to interpolate between as low as 5–10 different frequencies across four decades, depending on the type of interpolation (linear interpolation requires larger number of samples). The upper part of the FEM-model established in the FEMM4.2 [21] is shown in Fig. 3. Inductance results according to frequency for disk 7 of HV winding are shown in Fig. 4. It can be seen that inductance is reduced by increasing the frequency. As seen from Fig. 4 the inductance decreased from 0.012 H in 0 Hz to 0.007 H in 1 MHz.
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Fig. 4. Inductances of middle disk (disk 7) of HV winding according to frequency.
4.2. Capacitance calculation The most accurate method for calculating capacitance is using FEM. In order to calculate the capacitances between transformer windings using FEM, first all foils and all different dielectric materials are modeled in a FEM tool and for each set of different voltages applied to the conductors the total resultant energy is computed. Then the energy equations are solved simultaneously and the capacitances are obtained. By this method all the foils capacitances are calculated. But since all the capacitances are not actually used in the detailed model (because of negligible value) and also because of long solution time of this method it is better to apply substructuration principles to reduce the model order. For example with respect to construction of LV winding in dry transformers, the equation of plane capacitor could be used for calculation of lumped capacitance between two adjacent foils while the capacitance between two non-adjacent foils could be ignored because it is insignificant. The arrangement of conductors and dielectric materials such as air and resin and also the capacitance network is shown in Fig. 5. The method of calculation of capacitances in the detailed model is described in the following. 4.2.1. Calculation of capacitance between core and first layer of LV winding With respect to large height of LV winding and ignoring the fringing effect of field in edges, this capacitance could be calculated by means of the equation of cylinder capacitor. Since the core is grounded, this capacitance is actually the capaci-
Fig. 5. The arrangement of conductors and dielectric materials between HV and LV windings.
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Fig. 6. The model used for calculation of capacitances between HV and LV windings.
tance of first layer of LV winding with the earth. Regarding that the 1st layer will create an equipotential surface in front of the core leg, other layers will have ignorable capacitance with earth and so could be ignored in the detailed model. 4.2.2. Calculation of capacitance between LV winding layers With respect to the large height and internal diameter of LV winding and also the close distance between the LV winding layers, the equation of plane capacitor could be used for calculation of the capacitance between two adjacent LV layers. 4.2.3. Calculation of capacitance between HV and LV windings Since the last layer of LV will create an equipotential surface in front of other layers of LV, only the capacitance of last layer of LV to HV disks will be considered and the capacitance of other layers of LV to HV disks will be ignored, because they are very small. Also there is a similar condition in disks of HV, and just the capacitance between the last layer of LV and the first layer of HV disks is important and the capacitance between the last layer of LV and the other layers of HV disks is ignorable. Therefore, only capacitances between the last layer of LV and the first layer of HV disks are calculated. The arrangement of conductors and dielectric materials such as air and resin is shown in Fig. 6. With respect to the arrangement of conductors, the precise calculation of capacitances between conductors by analytical method is impossible. To calculate the capacitances correctly it is necessary to consider the influence of all conductors simultaneously. By using FEM, it is possible to model all conductors with several dielectric materials and calculate the mutual capacitances with high degree of accuracy. 4.2.4. Calculation of series capacitance All of the disks in HV winding are wounded in one direction that is from inside to outside. By applying a voltage difference across a pair of disks terminals, the equipotential lines in the space between disks will be as shown in Fig. 7(a). As shown in Fig. 7(a), the equipotential lines are in the form of oblique lines. For each disk except the first and the last one, the potential lines could be considered as shown in Fig. 7(b). If the energy of a disk (WT) to be known, then the disk’s series capacitance, K, could be calculated as:
K¼
2W T
ð5Þ
ðU 1 U 2 Þ2
In which U1 and U2 are disk’s terminal voltages. The energy of each disk is composed of the energies stored in the three regions which are shown in Fig. 7(b). If the energy stored in the regions 1, 2 and 3 is named by W1, W2 and W3, respectively, then the total energy of each disk will be:
WT ¼ W1 þ W2 þ W3
ð6aÞ
It is noticeable that in the case of the first and last disks only one of the W2 or W3 energies is considered. If the number of layers in the disk is N, then W1 will be:
W1 ¼
N X t¼1
Wt ¼
! 2 N 1 X U1 U2 Ct 2 t¼1 N
ð6bÞ
In which Ct is the capacitance between layers t and t + 1 of the disk. With respect to large height of aluminum foil (50 mm) and small thickness of insulator layer (0.046 mm) and also long inside diameter of HV winding (216 mm), Ct could be calculated by means of the plane capacitor equation. In order to calculate W2 and W3, the FEM method is used. To do so, the
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Fig. 7. Equipotential lines (a) between a pair of disks and (b) around each disk.
Fig. 8. Voltage boundary conditions for calculation of energy in the above (a) and below (b) of each disk.
stored energy in the above and below of each disk is calculated by defining the boundary value conditions as it is shown in Fig. 8. The initial voltage distribution from test and simulation are shown in Fig. 9. The presented results in Fig. 9 are for a situation that the LV winding from both sides and the HV winding from its end are grounded. The test results are derived from the results of a steep-front impulse test in a time about 0.2 ls. Since in times less than 1 ls, the winding inductances have no effect on the initial voltage distribution (current in an inductance cannot be established instantaneously), the voltage distribution is predominantly decided by the capacitances.
Table 1 Comparison of some capacitances calculated by the proposed method and the FEM results.
Capacitance between layer 1 and core Layer 1 series capacitance Layer 2 series capacitance Disk 1 series capacitance Disk 2 series capacitance Disk 3 series capacitance Capacitance between disk 1 and layer 2 Capacitance between disk 2 and layer 2 Capacitance between disk 3 and layer 2
Proposed method
FEM
Error (%)
2.91E10 1.18E08 1.41E08 3.45E10 3.88E10 3.88E10 1.59E11 1.62E11 1.48E11
2.96E10 1.19E08 1.42E08 3.42E10 3.9E10 3.89E10 1.57E11 1.59E11 1.47E11
1.69 0.84 0.70 0.88 0.51 0.26 1.27 1.89 0.68
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Fig. 9. Initial voltage distribution.
4.2.5. Comparing the calculated capacitances with the FEM results For calculation of capacitances using FEM, the conductors which the capacitances related to them are not needed are defined as float conductors and then the capacitances between desired conductors are calculated by the method explained before (energy method). Hence, in addition to considering the effects of all conductors, there is no need to calculate all capacitances and so the solution time is decreased considerably. Since electric field in a conductor is zero and energy is stored only in dielectrics, conductors are not meshed in the FEM method. Although this will reduce the number of elements but thickness of insulators in HV disks is too small and it is difficult to have very small elements inside each insulator. To overcome this problem, each two or more insulators (depending on required precision) are modeled as one insulator. In Table 1 some capacitances obtained from proposed and FEM methods are compared. Table 1 results show that the proposed method has enough accuracy for calculation of capacitances of the detailed model. 4.3. Losses calculation Increase in frequency will lead to increase in eddy current losses in the foils and so the resistances should be calculated according to frequency. The magnetic field in direction of r and z are shown in Figs. 10 and 11, respectively. As shown in Figs. 10 and 11, the magnetic field in both directions of r and z is a function of both r and z variables and therefore using the analytical formula for calculation of losses will be some inaccurate especially in the LV winding layers and the end disks of the HV winding. On the other hand, using analytical formula especially in frequencies above 200 kHZ lead to erroneous results. This is the frequency where the skin-depth is equal to the dimensions of conductors. So for high frequencies the use of FEM is more accurate even if FEM computations are time consuming.
Fig. 10. Magnetic field in the z direction (axial field).
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Fig. 11. Magnetic field in the r direction (radial field).
For calculation of losses in each frequency, after calculation of the inductances which explained in the Section 4, rated winding currents are applied to the foils so that the resultant ampere-turns to be equal to zero. After solution, the losses of each foil are obtained. The disk’s (or layer’s) total losses are equal to sum of the losses of its foils. The calculated resistances for LV layers and two first disks of HV winding are shown in Figs. 12 and 13, respectively. Simulation results shows that the field distribution in all disks except the end disks is approximately analogous but in the first and the last disks, radial component of the field is greater which leads to increase in losses. The current density distribution in a LV foil in frequency of 100 Hz is shown in Fig. 14. According to Fig. 14, the current density at the foil ends is much more than the uniform current density and so the eddy current losses in ending parts of the foil will be more. By increasing of the frequency, the current density in ending parts increase and almost all the losses occur in the foil ends. There is a rather similar condition in the foils of HV winding especially in foils of the end disks due to the large radial component of the magnetic field in the winding ends. 5. System description The system description can be used to investigate internal resonances of the transformer winding, or to determine transfer-functions between terminals. The nodal admittance matrix (without considering the inductive branches) and the branch impedance matrix are as follows:
Y N ðjxÞ ¼ GðxÞ þ jxCðxÞ
ð7Þ
Z B ðjxÞ ¼ RðxÞ þ jxLðxÞ
ð8Þ
Fig. 12. Calculated resistances for LV layers.
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Fig. 13. Calculated resistances for two first HV disks.
Fig. 14. Current density distribution in each foil of LV winding in 100 Hz.
The branch-matrix Z B ðjxÞ is then transformed to a nodal form by means of a transformation matrix:
Y B ðjxÞ ¼ AZ B ðjxÞ1 AT
ð9Þ
The transformation-matrix A, describes the relation between nodal currents and branch currents. The system admittance matrix is then given as:
Y SYS ðjxÞ ¼ Y N ðjxÞ þ Y B ðjxÞ
ð10Þ
6. Calculation of node voltages At the first step, HðjxÞ (the voltage transfer function of each node to input) is calculated using detailed model. At the second step, the input impulse voltage in frequency domain XðjxÞ can be obtained by applying the FFT (Fast Fourier Transformation) to the known input voltage. By multiplying XðjxÞ by HðjxÞ the desired node voltage in frequency domain UðjxÞ is obtained:
UðjxÞ ¼ XðjxÞ HðjxÞ
ð11Þ
As a last step, node voltages in time domain can be calculated by applying IFFT (Inverse Fast Fourier Transformation) as below:
uðtÞ ¼ IFFT½UðjxÞ
ð12Þ
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Fig. 15. Test object.
Fig. 16. Measuring circuit (a) impulse test and (b) FRA test.
Fig. 17. Measured disk voltages to ground at full lightning impulse.
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Fig. 18. Measured disk voltages to ground at chopped lightning impulse.
Fig. 19. FRA results for disk 3.
Fig. 20. FRA results for disk 4.
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402
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Fig. 21. Simulated disk voltages to ground at full lightning impulse.
7. Experiments and results In order to find the impulse voltage distribution in the dry transformers, some tests are performed on a high-voltage winding. This winding was manufactured in Iran-Transfo Company in such a way that there are outputs from its every disk. Accessibility of outlets from each disk facilitated voltage measuring from every disk and so the impulse voltage distribution could be found inside the winding. In order to perform the tests, the high-voltage winding was installed with a low-voltage winding in the form of a complete phase in the middle leg of a three-leg core. The voltage ratio in windings was 20/0.4 kV. The test object is shown in Fig. 15. The impulse and FRA tests were done on the test object. The impulse tests include full and chopped-wave lightning tests. The measuring circuit of impulse tests is shown in Fig. 16. In the full lightening test, the impulse voltage of 1.2/50 ls was applied to the HV winding input and then the output voltage from each disk was measured and stored by means of a digital oscilloscope. These tests were also repeated for chopped lightning impulse. Measurement results for full and chopped impulse tests are shown in Figs. 17 and 18, respectively. In FRA test, a sinusoidal voltage with various frequencies is applied to transformer input and in each frequency, the amplitude and phase of the desired output sinusoidal voltage are measured and then by using the amplitude and phases of input and output voltages, the amplitude and phase of transfer function in that frequency is calculated. In order to perform FRA tests, an FRA analyzer device fabricated in the OMICRON Company was used. This device has the capability of sweeping the input sinusoidal voltage from 1 kHz to 5 MHz and also it can measure the desired output voltage. This device has a connection with the computer and a specific software can instantly calculate and draw output to input transfer function. The
Fig. 22. Simulated disk voltages to ground at chopped lightning impulse.
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input signal was applied to the HV winding input and then the frequency response of each disk was measured. In the FRA tests, as shown in Fig. 16, output signals are measured from the top of a 50 X shunt resistance which is grounded. 8. Simulation results To verify the proposed method, parameters of detailed model were calculated for the test object. After establishing the model it was employed in MATLAB and FRA and impulse voltage analysis were implemented. Terminal conditions in all simulations are similar to the test arrangement. FRA results from test and simulation for two disks (disks 3 and 4) are shown in Figs. 19 and 20. The impulse voltage results from simulation for full and chopped lightning impulses are shown in Figs. 21 and 22, respectively. Comparison between simulation results and experimental results shows that the proposed method is suitable for high frequency modeling of dry-type transformers. 9. Conclusion In this paper high frequency behavior of cast-resin dry-type transformers was simulated. First a detailed model proposed for these transformers and then parameters of the model were calculated using FEM. Using the FEM-approach, parameters of detailed model could be calculated considering the constructional details of transformer. The effect of frequency on inductance value was considered and capacitances were calculated considering several dielectric materials. To verification the model’s accuracy a lab transformer was constructed and FRA and impulse tests were implemented. Comparison between simulation and experimental results shows that the proposed method is well suited for modeling the transient behavior of cast-resin dry-type transformers. References [1] T.J. Lewis, The transient behavior of ladder networks of the type representing transformer and machine windings, IEE Proc. 101 (Pt. II) (1954) 541–553. [2] J.H. McWhirter, C.D. Fahrnkopf, J.H. 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Okuyama, A calculation method for impulse voltage distribution and transferred voltage in transformer windings, IEEE Trans. Power Apparat. Syst. PAS-97 (1978) 930–939. [19] H. Yamashita, V. Cingoski, E. Nakamae, A. Namera, H. Kitamura, Design improvements on graded insulation of power transformers using transient electric field analysis and visualization technique, IEEE Trans. Energy Convers. 14 (4) (1999) 1379–1384. [20] A.M. Miri, N.A. Riegel, A. Kuhner, Finite element models for the computation of the transient potential and field distribution in the winding system of high voltage power transformers, IEE Conf. High Voltage Eng. 2 (467) (1999) 39–42. [21] D.C. Meeker, Finite Element Method Magnetics, Version 4.0.2. (03.12.2006 Build).
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Simulation Modelling Practice and Theory journal homepage: www.elsevier.com/locate/simpat
Combined analytical and FEM methods for parameters calculation of detailed model for dry-type transformer M. Eslamian, B. Vahidi *, S.H. Hosseinian Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 3 June 2009 Received in revised form 7 December 2009 Accepted 9 December 2009 Available online 16 December 2009 Keywords: Cast-resin dry-type transformer Detailed model FRA FEM Parameter calculation
a b s t r a c t Non-flammable characteristic of cast-resin dry-type transformers make them suitable for different kind of usages. This paper presents a method of how to obtain high frequency model of these transformers. For this purpose a detailed model is used and parameters of the model including inductances, capacitances and resistances are calculated using FEM (finite elements method). The effect of the frequency is considered in the inductance calculation using FEM. In order to validate the model, a setup was constructed for testing on highvoltage winding of dry-type transformer. The simulation results were compared with the experimental data measured from FRA (frequency response analysis) and impulse tests. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Transformers in operation are subject to various kinds of overvoltages caused by lightning strikes, switching operations or system disturbances. Generally, under normal conditions in a steady-state, the voltage is distributed linearly inside the winding. For impulse voltage, the voltage distribution in the winding is not linear and for the calculation of impulse voltage distribution, the windings should be simulated in terms of an equivalent circuit consisting of lumped R, L, C elements. Cast-resin dry-type transformers are the most suitable transformers for distribution of electricity in high degree of safety. Dry-type transformers compared with oil-immersed are lighter and non-flammable. They also do not have contaminating substances such as oil. Non-flammable characteristic of these transformers make them suitable for residential and hospital usages. Fig. 1 shows the structure of one phase of a cast-resin dry-type transformer consisting of low- and high-voltage windings. Low-voltage winding of this transformer is made of a full-height aluminum sheet wounded simultaneously with insulation layer. High-voltage winding is constructed from several series disks that every disk is made of aluminum foils interleaved with insulating layers. After the high-voltage winding is wounded, it is placed in a mold and cast in a resin under vacuum pressure. Lower sound levels are realized as the winding is encased in solid insulation. Filling the winding with resin under vacuum pressure eliminates voids that can cause corona. With a solid insulation system, the winding has superior mechanical and short-circuit strength and is impervious to moisture and contaminants. 2. Transformer transient modeling Transformer transient modeling has been a subject of investigation and research for a century. In the 1954, Lewis [1] proposed that the transient behavior of a transformer winding can be studied with an equivalent ladder-type network * Corresponding author. Tel.: +98 21 64543330; fax: +98 21 66406469. E-mail address: [email protected] (B. Vahidi). 1569-190X/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.simpat.2009.12.005
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Fig. 1. Schematic view of cast-resin dry-type transformer.
composed of a finite number of uniform sections. Each section is composed of lumped capacitive and inductive components which represent the distributed parameters of actual winding. Lewis’s model is applicable only to a uniform winding. Furthermore, the representation of inductive coupling effect was included by modifying the self inductance value. McWhirter et al. [2] studied the same problem based on an equivalent circuit approach. Their model still suffers the restrictions arising from the size and symmetry of the equivalent circuit used to represent the winding. Dent et al. [3] used an equivalent circuit of the same general form as the proposed by Lewis, but with certain differences. Dent’s model can represent a non-uniform winding and the effect of inductive coupling between sections is taken into account. After Dent’s paper was published, most of the researchers in this area, concentrated their attention on the calculation of parameters of the equivalent circuits. Okuyama [4] calculated the self and mutual inductances of transformer winding through introduction of some correction factors obtained from experiment. Stein [5] and Kawaguchi [6] proposed a method to calculate equivalent series capacitance by computing the electrostatic energy stored in the coils. Fergestad and Henriksen [7,8] calculated self and mutual inductance of sections of windings by taking certain effects of iron core into account. Wilcox et al. [9,10] derived a set of formulae from Maxwell’s equations to calculate self and mutual inductive parameters. De Leon and Semlyen [11] used the image method to calculate turn to turn leakage inductance and the charge simulation method to calculate the capacitance between turns and from turn to ground. They also proposed a detailed model of losses [12] for accurate calculations. Also Transformer transient has been investigated by analytical methods in the last years [13–15]. Using the analytical methods is one of the designing tools that may yield acceptable results depending on the geometry of problem. But this method is not flexible. Nowadays, with respect to difficulties of analytical methods, the FEM as an accurate and reliable method becomes more popular. The most important advantage of FEM is that any complicated geometry is solvable by this method because the formulation of FEM is independent of the geometry. FEM has many applications in simulation of transformer behaviors in different situations. These applications include the calculation of energy, leakage fluxes and electromagnetic forces. By using FEM and considering the structural details of transformer, it is possible to calculate the parameters of winding detailed model. The advantage of this method over the analytical method is that there may be unequal amounts of inductances and capacitances in the equivalent circuit. Till now, calculation of lumped parameters of power transformers by means of FEM was accomplished [16–20]. All works done on transformer transients are for oil filled transformers and this was not done for dry-type transformers yet. As it is clear from its name, dry-type transformer do not have any oil and instead a solid dielectric material such as resin is used in its structure. The shape of conductor, types of dielectric materials and the structure of windings of dry-type transformers are different from oil filled transformers. Also the shape of electrical field is rather different. On the other hand, the surrounding air as an insulator and cooler is running in the space between windings and tips of bars. Because of large difference of air dielectric coefficient with other solid insulators of dry-type transformers (about 4 times), the large amount of voltage in this kind of transformers is dropped on air spaces and with respect to low resistance of air in comparison with other solid materials, it is important to have information about the size and the way of electrical field distribution in transients to make sure about tolerance of air spaces. Therefore, study and research on dry-type transformers seems to be important and useful. In this paper detailed parameters and transient behavior of the cast-resin dry-type transformer as a new generation of distribution transformer is calculated.
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Fig. 2. Equivalent lumped parameter circuit.
3. The equivalent circuit of dry-type transformer Equivalent circuit used for modeling of dry-type transformers is shown in Fig. 2. In this case, each layer of LV winding which is placed between two air canals and every disk of HV winding is considered as a branch in the equivalent circuit. Every branch is composed of parallel connection of self inductance of Li and series capacitance of Ki. There is a mutual inductance between every pair of branches (Lij, Lik, . . .). Regarding that the last layer of LV is creating an equipotential surface in front of other layers of LV, only the capacitances of the last layer of LV to HV disks are considered which are shown with Cp. The Ce is the capacitance of the first layer of LV to the core which is grounded. Ri is ohmic losses of each branch and Gi, Gp and Ge are conductances which are representing the dielectric losses. Dielectric losses are not included in this work because of their lower effect in comparison with ohmic losses. As an approximation, a constant value based on low-frequency measurements can be used. 4. Parameter calculation 4.1. Inductance calculation Traditionally, the magnetic flux is composed of two components of leakage and common fluxes. The leakage flux is not very dependent on frequency however the common flux which is passing through the core is highly frequency dependent and in high frequencies, it is completely displaced from the middle of the core. So it is the leakage inductance that is the determining factor for internal transients in transformer windings and the effect of iron core could be neglected in transformer high frequency modeling. As it was mentioned, in dry transformers, HV winding is composed of several disks with foil conductors and LV winding is composed of full-height foil layers. By increasing of the frequency, the skin effect in the foils is increased and it causes a considerable change in the flux distribution. Therefore in order to calculate the inductances correctly in high frequencies, effects of all HV and LV winding foils must be considered simultaneously. Because of long height of the LV foils, effect of LV layers in change of flux distribution is more than HV disks and in high frequencies LV layers prevent the penetration of the flux into the LV winding. Generally, if there are n conductors in the space, with respect to linearity of the system, the following equation could be written in each frequency.
V f ¼ jxLf If
ð1Þ
j kT j kT In which If ¼ I1f ; I2f ; . . . ; Inf is the vector of net current phasors in frequency of f, V f ¼ V 1f ; V 2f ; . . . ; V nf is the vector of voltage drop phasors in frequency of f, Lf is the system’s inductance matrix in frequency of f and x ¼ 2pf . Assume each element of Lf is lijf . By applying the current phasor of I to conductor i and forcing a zero net current to other conductors, after calculation of produced voltage drop in all conductors, the ith column of Lf is obtained as follows:
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Fig. 3. Upper part of the FEM-model established in the FEMM4.2.
2
l1if 6 6 l2if 6 6 . 6 .. 4 lnif
3 7 ( ) 7 Vf 1 7 Imag 7¼ 7 jx I 5
ð2Þ
In order to obtain voltage drops, suppose that a zero current is applied to all conductors except conductor i. After FEM analysis, current density, J, and vector potential, A is given in each point. By means of Maxwell equations, the electric field could be calculated as follows:
E¼
@A rV @t
ð3Þ
By taking integral of both side of Eq. (3) with respect to length in the cylindrical coordinates and with regard to relation of J ¼ rE, the produced voltage drop on the j conductor due to the applying current to conductor i could be calculated as follows:
V j ¼ 2pr
J rj
r
! þ j x Ar j
ð4Þ
In which Jrj and Arj are the values of current density and vector potential in a desired point on the conductor j and with the radius of r. r is conductivity of the j conductor. As it was explained in Section 3, in the equivalent circuit of the dry transformer, each disk of HV winding and every part of LV winding which is placed between two air canals are considered as a branch. Each HV disk is composed of many foils which are in series together. So the total voltage drop on each disk is equal to sum of the voltage drops on all foils of the disk. This is also true for every part of the LV winding. Generally, in order to calculate the ith column of the inductance matrix by using FEM, a current source equal to I is applied to each foil of part i and a current source equal to zero is applied to all foils of the other parts. Then the total voltage drop in each part is calculated. By having the values of voltage drops, the elements of ith column are calculated by using Eq. (2). This method is also repeated for calculation of the other columns and so the inductance matrix, Lf in each frequency could be calculated. Basically discretisation level of a winding depends on the investigated frequency range with regard to the conductor dimension. The skin depth of aluminum at 50 Hz and 75 °C is equal to 13.2 mm. By increasing of the frequency, the skin depth will decrease so that at 100 kHz and 1 MHz, it will be equal to about 0.3 and 0.1 mm, respectively. Thickness of each HV foil is 0.2 mm. Since the investigated frequency range is from 1 kHz to 1 MHz, all foils must be modeled completely. Since in FEM analysis there should be at least one element in one skin depth so the number of elements will increase and the solution especially in high frequencies will be time consuming. Since inductances have a predictable frequency-dependency, it is possible to interpolate between as low as 5–10 different frequencies across four decades, depending on the type of interpolation (linear interpolation requires larger number of samples). The upper part of the FEM-model established in the FEMM4.2 [21] is shown in Fig. 3. Inductance results according to frequency for disk 7 of HV winding are shown in Fig. 4. It can be seen that inductance is reduced by increasing the frequency. As seen from Fig. 4 the inductance decreased from 0.012 H in 0 Hz to 0.007 H in 1 MHz.
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Fig. 4. Inductances of middle disk (disk 7) of HV winding according to frequency.
4.2. Capacitance calculation The most accurate method for calculating capacitance is using FEM. In order to calculate the capacitances between transformer windings using FEM, first all foils and all different dielectric materials are modeled in a FEM tool and for each set of different voltages applied to the conductors the total resultant energy is computed. Then the energy equations are solved simultaneously and the capacitances are obtained. By this method all the foils capacitances are calculated. But since all the capacitances are not actually used in the detailed model (because of negligible value) and also because of long solution time of this method it is better to apply substructuration principles to reduce the model order. For example with respect to construction of LV winding in dry transformers, the equation of plane capacitor could be used for calculation of lumped capacitance between two adjacent foils while the capacitance between two non-adjacent foils could be ignored because it is insignificant. The arrangement of conductors and dielectric materials such as air and resin and also the capacitance network is shown in Fig. 5. The method of calculation of capacitances in the detailed model is described in the following. 4.2.1. Calculation of capacitance between core and first layer of LV winding With respect to large height of LV winding and ignoring the fringing effect of field in edges, this capacitance could be calculated by means of the equation of cylinder capacitor. Since the core is grounded, this capacitance is actually the capaci-
Fig. 5. The arrangement of conductors and dielectric materials between HV and LV windings.
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Fig. 6. The model used for calculation of capacitances between HV and LV windings.
tance of first layer of LV winding with the earth. Regarding that the 1st layer will create an equipotential surface in front of the core leg, other layers will have ignorable capacitance with earth and so could be ignored in the detailed model. 4.2.2. Calculation of capacitance between LV winding layers With respect to the large height and internal diameter of LV winding and also the close distance between the LV winding layers, the equation of plane capacitor could be used for calculation of the capacitance between two adjacent LV layers. 4.2.3. Calculation of capacitance between HV and LV windings Since the last layer of LV will create an equipotential surface in front of other layers of LV, only the capacitance of last layer of LV to HV disks will be considered and the capacitance of other layers of LV to HV disks will be ignored, because they are very small. Also there is a similar condition in disks of HV, and just the capacitance between the last layer of LV and the first layer of HV disks is important and the capacitance between the last layer of LV and the other layers of HV disks is ignorable. Therefore, only capacitances between the last layer of LV and the first layer of HV disks are calculated. The arrangement of conductors and dielectric materials such as air and resin is shown in Fig. 6. With respect to the arrangement of conductors, the precise calculation of capacitances between conductors by analytical method is impossible. To calculate the capacitances correctly it is necessary to consider the influence of all conductors simultaneously. By using FEM, it is possible to model all conductors with several dielectric materials and calculate the mutual capacitances with high degree of accuracy. 4.2.4. Calculation of series capacitance All of the disks in HV winding are wounded in one direction that is from inside to outside. By applying a voltage difference across a pair of disks terminals, the equipotential lines in the space between disks will be as shown in Fig. 7(a). As shown in Fig. 7(a), the equipotential lines are in the form of oblique lines. For each disk except the first and the last one, the potential lines could be considered as shown in Fig. 7(b). If the energy of a disk (WT) to be known, then the disk’s series capacitance, K, could be calculated as:
K¼
2W T
ð5Þ
ðU 1 U 2 Þ2
In which U1 and U2 are disk’s terminal voltages. The energy of each disk is composed of the energies stored in the three regions which are shown in Fig. 7(b). If the energy stored in the regions 1, 2 and 3 is named by W1, W2 and W3, respectively, then the total energy of each disk will be:
WT ¼ W1 þ W2 þ W3
ð6aÞ
It is noticeable that in the case of the first and last disks only one of the W2 or W3 energies is considered. If the number of layers in the disk is N, then W1 will be:
W1 ¼
N X t¼1
Wt ¼
! 2 N 1 X U1 U2 Ct 2 t¼1 N
ð6bÞ
In which Ct is the capacitance between layers t and t + 1 of the disk. With respect to large height of aluminum foil (50 mm) and small thickness of insulator layer (0.046 mm) and also long inside diameter of HV winding (216 mm), Ct could be calculated by means of the plane capacitor equation. In order to calculate W2 and W3, the FEM method is used. To do so, the
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Fig. 7. Equipotential lines (a) between a pair of disks and (b) around each disk.
Fig. 8. Voltage boundary conditions for calculation of energy in the above (a) and below (b) of each disk.
stored energy in the above and below of each disk is calculated by defining the boundary value conditions as it is shown in Fig. 8. The initial voltage distribution from test and simulation are shown in Fig. 9. The presented results in Fig. 9 are for a situation that the LV winding from both sides and the HV winding from its end are grounded. The test results are derived from the results of a steep-front impulse test in a time about 0.2 ls. Since in times less than 1 ls, the winding inductances have no effect on the initial voltage distribution (current in an inductance cannot be established instantaneously), the voltage distribution is predominantly decided by the capacitances.
Table 1 Comparison of some capacitances calculated by the proposed method and the FEM results.
Capacitance between layer 1 and core Layer 1 series capacitance Layer 2 series capacitance Disk 1 series capacitance Disk 2 series capacitance Disk 3 series capacitance Capacitance between disk 1 and layer 2 Capacitance between disk 2 and layer 2 Capacitance between disk 3 and layer 2
Proposed method
FEM
Error (%)
2.91E10 1.18E08 1.41E08 3.45E10 3.88E10 3.88E10 1.59E11 1.62E11 1.48E11
2.96E10 1.19E08 1.42E08 3.42E10 3.9E10 3.89E10 1.57E11 1.59E11 1.47E11
1.69 0.84 0.70 0.88 0.51 0.26 1.27 1.89 0.68
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Fig. 9. Initial voltage distribution.
4.2.5. Comparing the calculated capacitances with the FEM results For calculation of capacitances using FEM, the conductors which the capacitances related to them are not needed are defined as float conductors and then the capacitances between desired conductors are calculated by the method explained before (energy method). Hence, in addition to considering the effects of all conductors, there is no need to calculate all capacitances and so the solution time is decreased considerably. Since electric field in a conductor is zero and energy is stored only in dielectrics, conductors are not meshed in the FEM method. Although this will reduce the number of elements but thickness of insulators in HV disks is too small and it is difficult to have very small elements inside each insulator. To overcome this problem, each two or more insulators (depending on required precision) are modeled as one insulator. In Table 1 some capacitances obtained from proposed and FEM methods are compared. Table 1 results show that the proposed method has enough accuracy for calculation of capacitances of the detailed model. 4.3. Losses calculation Increase in frequency will lead to increase in eddy current losses in the foils and so the resistances should be calculated according to frequency. The magnetic field in direction of r and z are shown in Figs. 10 and 11, respectively. As shown in Figs. 10 and 11, the magnetic field in both directions of r and z is a function of both r and z variables and therefore using the analytical formula for calculation of losses will be some inaccurate especially in the LV winding layers and the end disks of the HV winding. On the other hand, using analytical formula especially in frequencies above 200 kHZ lead to erroneous results. This is the frequency where the skin-depth is equal to the dimensions of conductors. So for high frequencies the use of FEM is more accurate even if FEM computations are time consuming.
Fig. 10. Magnetic field in the z direction (axial field).
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Fig. 11. Magnetic field in the r direction (radial field).
For calculation of losses in each frequency, after calculation of the inductances which explained in the Section 4, rated winding currents are applied to the foils so that the resultant ampere-turns to be equal to zero. After solution, the losses of each foil are obtained. The disk’s (or layer’s) total losses are equal to sum of the losses of its foils. The calculated resistances for LV layers and two first disks of HV winding are shown in Figs. 12 and 13, respectively. Simulation results shows that the field distribution in all disks except the end disks is approximately analogous but in the first and the last disks, radial component of the field is greater which leads to increase in losses. The current density distribution in a LV foil in frequency of 100 Hz is shown in Fig. 14. According to Fig. 14, the current density at the foil ends is much more than the uniform current density and so the eddy current losses in ending parts of the foil will be more. By increasing of the frequency, the current density in ending parts increase and almost all the losses occur in the foil ends. There is a rather similar condition in the foils of HV winding especially in foils of the end disks due to the large radial component of the magnetic field in the winding ends. 5. System description The system description can be used to investigate internal resonances of the transformer winding, or to determine transfer-functions between terminals. The nodal admittance matrix (without considering the inductive branches) and the branch impedance matrix are as follows:
Y N ðjxÞ ¼ GðxÞ þ jxCðxÞ
ð7Þ
Z B ðjxÞ ¼ RðxÞ þ jxLðxÞ
ð8Þ
Fig. 12. Calculated resistances for LV layers.
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Fig. 13. Calculated resistances for two first HV disks.
Fig. 14. Current density distribution in each foil of LV winding in 100 Hz.
The branch-matrix Z B ðjxÞ is then transformed to a nodal form by means of a transformation matrix:
Y B ðjxÞ ¼ AZ B ðjxÞ1 AT
ð9Þ
The transformation-matrix A, describes the relation between nodal currents and branch currents. The system admittance matrix is then given as:
Y SYS ðjxÞ ¼ Y N ðjxÞ þ Y B ðjxÞ
ð10Þ
6. Calculation of node voltages At the first step, HðjxÞ (the voltage transfer function of each node to input) is calculated using detailed model. At the second step, the input impulse voltage in frequency domain XðjxÞ can be obtained by applying the FFT (Fast Fourier Transformation) to the known input voltage. By multiplying XðjxÞ by HðjxÞ the desired node voltage in frequency domain UðjxÞ is obtained:
UðjxÞ ¼ XðjxÞ HðjxÞ
ð11Þ
As a last step, node voltages in time domain can be calculated by applying IFFT (Inverse Fast Fourier Transformation) as below:
uðtÞ ¼ IFFT½UðjxÞ
ð12Þ
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Fig. 15. Test object.
Fig. 16. Measuring circuit (a) impulse test and (b) FRA test.
Fig. 17. Measured disk voltages to ground at full lightning impulse.
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Fig. 18. Measured disk voltages to ground at chopped lightning impulse.
Fig. 19. FRA results for disk 3.
Fig. 20. FRA results for disk 4.
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402
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Fig. 21. Simulated disk voltages to ground at full lightning impulse.
7. Experiments and results In order to find the impulse voltage distribution in the dry transformers, some tests are performed on a high-voltage winding. This winding was manufactured in Iran-Transfo Company in such a way that there are outputs from its every disk. Accessibility of outlets from each disk facilitated voltage measuring from every disk and so the impulse voltage distribution could be found inside the winding. In order to perform the tests, the high-voltage winding was installed with a low-voltage winding in the form of a complete phase in the middle leg of a three-leg core. The voltage ratio in windings was 20/0.4 kV. The test object is shown in Fig. 15. The impulse and FRA tests were done on the test object. The impulse tests include full and chopped-wave lightning tests. The measuring circuit of impulse tests is shown in Fig. 16. In the full lightening test, the impulse voltage of 1.2/50 ls was applied to the HV winding input and then the output voltage from each disk was measured and stored by means of a digital oscilloscope. These tests were also repeated for chopped lightning impulse. Measurement results for full and chopped impulse tests are shown in Figs. 17 and 18, respectively. In FRA test, a sinusoidal voltage with various frequencies is applied to transformer input and in each frequency, the amplitude and phase of the desired output sinusoidal voltage are measured and then by using the amplitude and phases of input and output voltages, the amplitude and phase of transfer function in that frequency is calculated. In order to perform FRA tests, an FRA analyzer device fabricated in the OMICRON Company was used. This device has the capability of sweeping the input sinusoidal voltage from 1 kHz to 5 MHz and also it can measure the desired output voltage. This device has a connection with the computer and a specific software can instantly calculate and draw output to input transfer function. The
Fig. 22. Simulated disk voltages to ground at chopped lightning impulse.
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input signal was applied to the HV winding input and then the frequency response of each disk was measured. In the FRA tests, as shown in Fig. 16, output signals are measured from the top of a 50 X shunt resistance which is grounded. 8. Simulation results To verify the proposed method, parameters of detailed model were calculated for the test object. After establishing the model it was employed in MATLAB and FRA and impulse voltage analysis were implemented. Terminal conditions in all simulations are similar to the test arrangement. FRA results from test and simulation for two disks (disks 3 and 4) are shown in Figs. 19 and 20. The impulse voltage results from simulation for full and chopped lightning impulses are shown in Figs. 21 and 22, respectively. Comparison between simulation results and experimental results shows that the proposed method is suitable for high frequency modeling of dry-type transformers. 9. Conclusion In this paper high frequency behavior of cast-resin dry-type transformers was simulated. First a detailed model proposed for these transformers and then parameters of the model were calculated using FEM. Using the FEM-approach, parameters of detailed model could be calculated considering the constructional details of transformer. The effect of frequency on inductance value was considered and capacitances were calculated considering several dielectric materials. To verification the model’s accuracy a lab transformer was constructed and FRA and impulse tests were implemented. Comparison between simulation and experimental results shows that the proposed method is well suited for modeling the transient behavior of cast-resin dry-type transformers. References [1] T.J. Lewis, The transient behavior of ladder networks of the type representing transformer and machine windings, IEE Proc. 101 (Pt. II) (1954) 541–553. [2] J.H. McWhirter, C.D. Fahrnkopf, J.H. Steele, Determination of impulse stresses within transformer windings by computers, AIEE Trans. 75 (Pt. III) (1957) 1267–1279. [3] B.M. Dent, E.R. Hartill, J.G. Miles, A method of analysis of transformer impulse voltage distribution using a digital computer, IEE Proc. 105 (Pt. A) (1958) 445–459. [4] K.A. Okuyama, A numerical analysis of impulse voltage distribution in transformer windings, Electr. Eng. Jpn. 87 (1) (1967) 80–88. [5] M. Stein, A study of the initial surge distribution in concentric transformer windings, AIEE Trans. (1964) 877–892. [6] Y. Kawaguchi, Calculation of the circuit constants for computing internal oscillating voltage in transformer windings, Electr. Eng. Jpn. 89 (3) (1969) 44– 53. [7] P.I. Fergestad, T. Henriksen, Transient oscillations in multi-winding transformers, IEEE Trans. Power Apparat. Syst. PAS-93 (1974) 500–509. [8] P.I. Fergestad, T. Henriksen, Inductances for the calculation of transient oscillations in transformers, IEEE Trans. Power Apparat. 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