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Proc. of Int. Conf. on Advances in Electrical & Electronics 2010

Influence of Skin and Proximity effects on Surge Voltage Distribution in Transformers B.V.Sumangala

G.R.Nagabhushana

Proessor and Head Dr.Ambedkar Institute of Technology, Bangalore Sumangala_bv@yahoo.com

Former Professor, HV Engineering Indian Institute of Science, Bangalore

Abstract:-The paper presents an investigation on transient voltage distribution in transformers which is of significance to all power system engineers. A better knowledge of this is possible with today’s digital computers, for which a good modeling of the transformer is necessary. Using this information, the design engineers can develop a more reliable and possibly economic insulation structure which is the main issue affecting the cost of the transformer. Here an attempt has been made to study the impulse response of the transformer windings, with the best resolution that is turn resolution rather than using lumped coils as is usually adopted. Also, the effect of Skin and Proximity effects due to high frequency pulses are also studied. In this work, a network of distributed inductances, capacitances of individual turns, models the transformer winding. The studies are carried out on a model transformer with steep fronted pulses of different rise times and various possible types of surges including fast transients. The transformer winding is modeled with turn resolution to compare the results with experimental values, thus validating the model. The digital simulation is carried out using OrCAD/PSpice software.

However, to be assured of reasonable correctness, a small model transformer (27 turns, wound as 3 coils, each coil with 3 layers of 3 turns each) was built and the voltage distributions at the turn-to-turn level were experimentally measured. These measurements were under sharp pulses with small but different rise times. Pulse width was 0.65ms while rise times of 40ns, 100ns and 200ns were considered in view of the small (only 27 turns) winding. Also, the response under damped sinusoidal pulse of 10MHz frequency (representing VFTO) was used. Subsequently, a circuit model was developed with turn resolution. The simulation and experimental results were in excellent agreement validating the method of modeling. The model involved 1. Self and Mutual Inductances of each turn. 2. Resistance of each turn considering both skin and proximity effects. 3. Turn to turn capacitances. 4. Turn to ground capacitances where appropriate.

Keywords:-Transformer Model, Surge distribution, Skin effect, Proximity effect

II. MODEL TRANSFORMER DETAILS

I. INTRODUCTION The power transformer is perhaps the most vital and expensive equipment in a power system. The operational security of the transformer is therefore of decisive importance for reliable power supply. Apart from the continuous rated power frequency voltage, the power transformer faces overvoltages in the field due to lightning as well as switching operations. Lightning overvoltages having very sharp rise-times as low as 1 ms, pose considerable hazard to the transformer insulation because of the non-uniform voltage distribution with possibly a dangerous concentrations in certain regions. Because of this, considerable work has been carried out to study the voltage distribution along the winding for unit step and standard impulse excitations. However, perhaps because of the computational complexities, such work has been carried out at section-wise resolution of the winding, which is not really adequate. Only limited information is available on the voltages at each of the turns with respect to ground, voltages across various discs (disc winding transformer winding) and voltage between physically adjacent turns (not consecutive due to interleaving arrangement of the turns) of transformer winding.

The model transformer used in the experiments is a core type transformer with a three - stepped core and having two windings (one on each limb) each having 27 turns wound as three coils of 9 turns each. The conductor has a rectangular cross section (5mm´10mm) and rounded at the corners. The conductors are paper insulated. Leads are taken out at each turn for purposes of measurements.

Proc. of Int. Conf. on Advances in Electrical & Electronics 2010

stress is maximum at the line end. 60% of the applied voltage appear across the first 35% of the winding from the line end. But towards the ground end, the voltage distributions follow almost uniform distribution.

Rise time of Pulses 40ns 100ns 200ns

100

Turn Voltage, Volts

80 60 40 20

Fig 1 Details of Model Transformer

15 20 Turn Number

III. TYPES OF HIGH FREQUENCY PULSES AND DOMINATING 1.0

Following are the different kinds of high frequency pulses used other than the lightning impulse to study the transient behavior of the Model Transformer.

0.8 Node Voltage, pu

FREQUENCY COMPONENTS

0.0

15 20 Turn Number

Fig 2 Node Voltage Distribution Fig1 (a) Rectangular Pulse Fig 2(b) Damped Sinusoidal Pulse

3.2 SOURCE II—Damped Sinusoidal Waveform (DSW) It is reported [1] that the fast transients in Gas Insulated Substations (GIS) have frequency components in 10MHz range which may lead to significant internal resonance voltages if the excitation closely matches with one of the natural frequencies of the winding. It is also reported [2] that the rise time of such pulses is less than 100ns and its waveform comprises of damped oscillations of frequency of a few MHz lasting for tens of microseconds.

V. TRANSFORMER MODELING

IV. EXPERIMENTAL PROCEDURE AND RESULTS The above mentioned high frequency pulses are applied at the line end of the transformer winding and the other end of the winding is grounded. The measurements are made (with a Digital Storage Oscilloscope) at each of the taps provided. The node voltage distribution for Rectangular Pulses with 40ns, 100ns and 200ns rise times are shown in Fig 2(a) for a constant input voltage magnitude of 100V at the line end. It is to be noted here that, all the voltages are peak values with respect to ground occurring at different instants of time. As may be expected, the rise time of the pulses play an important role in the voltage distribution. The voltage distribution is more non-uniform for the pulse with 40ns rise time compared to the pulses with 100ns and 200ns rise times. The node Voltage distribution for Damped Sinusoidal Pulse with respect to ground for a magnitude of 1000V is shown in Fig 2(b). Here, it is seen that, the distribution is below the linear distribution line unlike the voltage distribution with rectangular pulse. The voltage

The transformer is a frequency sensitive impedance. For system studies it is sufficient to model the transformer rather coarsely such that the terminal impedance characteristic is duplicated within the frequency range of concern. On the other hand, when internal transient response of the transformer is required, it is necessary to use a much more detailed model in which all regions of critical electric stress can be identified. Internal transient response is a result of the distributed electrostatic and electromagnetic characteristics of the transformer windings. For all practical winding structures, this interaction is quite complex and can only be realistically investigated by constructing a very detailed distributed parameter model and carrying out a numerical solution for the transient response. 5.1 Computation of Various R, L and C Parameters 5.1.1 Resistances To start with, only the dc resistance is considered. But it was found that the results are not in agreement with the experimental values. As the circuit is excited by an alternating voltage, the power loss is higher due to skin and proximity effects. Also, internal transformer damping is extremely important [3] in limiting internal response voltages. For excitation near a transformer natural frequency, this is the only protection available. As the transients involve high frequency components, increase in

119

0.4 0.2

3.1 SOURCE I - Rectangular Pulse Generator This pulse generator gives, on actuation, a single pulse of 500V magnitude (variable magnitude) with an adjustable pulse width in the range 0.65msec to 5msec and variable rise times in the range of 40ns to 200ns.

0.6

Proc. of Int. Conf. on Advances in Electrical & Electronics 2010

resistance due to skin and proximity effects need to be considered. Further, the skin effect is more prominent in the case of square or rectangular conductors compared to circular conductors. The estimation of resistance is made in two stages. In the first step, only skin effect is considered and the following assumptions are made: • Conductor is assumed to be isolated (this excludes the proximity effect) • Conductor is assumed to be tubular and circular. The formula for the resistance ratio of a tubular conductor is given by [4],

The resistance values thus computed for different frequencies are shown in the Table 1 for the Model Transformer winding. It may be noted here that the resistance values are calculated only for the principal frequencies as seen from the frequency spectrum of the pulses. The frequency spectrum for the Rectangular Pulse and the Damped Sinusoidal Wave are shown in Figs 3(a) and 3(b). The principal frequency component is 1MHz and 10MHz for Rectangular and Damped Sinusoid Wave respectively.

⎡ ⎫ ⎤ ⎧ 3 1 t2 ⎛ t 3 1 t2 r r2 ⎞ ⎜⎜ 7 − 6 + 3 2 ⎟⎟ + ....⎬ ⎥ + + ......+ 2ε −2r cos2τ ⎨1 − ⎢1 + 2 2 2 τ r 16 4 τ qr q τ r q ⎠ Rac ⎝ t ⎞⎢ ⎭ ⎥ ⎩ ⎛ = τ ⎜1 − ⎟⎢ ⎥ Rdc ⎝ 2r ⎠⎢ ⎫ ⎧ 3 1 t2 ⎛ r r2 ⎞ −2 r −4 r ⎜ 7 − 6 + 3 2 ⎟⎟ + .......⎬ + terms− of − ε + .....⎥ + 2ε sin 2τ ⎨1 + 2 2 ⎜ ⎢ ⎥ 64 q τ r q ⎝ ⎠ ⎭ ⎩ ⎣ ⎦

Where, Rac = Resistance under ac of frequency f Rdc = Resistance for dc q = inner radius of the tube, in cm. r = outer radius of the tube, in cm. t = thickness of the tube = r – q. mt and m = 2π 2 μf , for copper m = 0.2142 τ= 2

Fig. 3(a) Rectangular Pulse

Table 1 Resistances of the winding at different frequencies with skin and proximity effects

Sl.No. Frequency

For strong skin effect, that is, for values of τ greater than 3, the high frequency resistance per unit length of the conductor is given by, fσ ⎛ 1 1 t ⎞ Ω per cm. R= ⎜1 + ⎟ r ⎝ 2τ r⎠ For copper (σ = 1.6 * 10-7 Ω cm), this becomes 4.15 −8 ⎛ 1 1 t ⎞ 10 ⎜1 + R= ⎟ f Ω per cm. r ⎝ 2τ r⎠

It may be noted that in the above formula, value of σ is considered for pure copper at 200C. For commercially available copper and for different temperatures, appropriate corrections are required. In the second stage, the following corrections are applied to take care of the assumptions made (above in the first step). • A multiplier of 7.87/4.3 = 1.83 is used to correct for the actual resistivity of the material used in the conductor of the physical model transformer. The actual resistance was 7.87mΩ where as for pure copper, the resistance value would be 4.3 mΩ for the same length and cross section of the conductor. • A multiplier of 1.3 is used as a correction since the actual conductor is of rectangular cross section [4]. • A multiplier of 2.5 is used to take care of the proximity effect for the resistance at 10MHz [4,5].

1. 2.

DC Value 250 kHz.

Resistance of the total winding, mΩ 7.87 163

1MHz

323.5

10MHz

2034

Relevance 1.2/50μsec standard impulse Rectangular Pulse of 0.65μsec pulse width Damped Sinusoidal Wave

5.1.2 Self-Inductances It is seen from the literature [1,6,7] that the inductance does not depend on frequency after 100kHz and the transformer behaves like an air core transformer above this frequency. Therefore, air core inductances have been used for calculation of transients due to lightning and other high frequency pulses in transformer. The formulae compiled by Grover [8] are employed for the calculations. The inductance of a circular turn (i.e., single turn coil) depends on (a) The radius (b) Area of cross-section In the present model, the coils (turns) are treated as thin coils (turns) since the coil’s radial thickness is less than the axial thickness c < b as shown in Fig 4. Self-inductance is calculated using the well known Nagaoka’s formula for rectangular cross section [8]. ⎛ 2a ⎞ L = 0.019739⎜ ⎟ N 2 a cm K ' ⎝ b ⎠ Where, K ' = (K – k) Here,

Fig. 3(b) Damped Sinusoidal Wave

Proc. of Int. Conf. on Advances in Electrical & Electronics 2010

a = Radius of the winding, cm b = Thickness of the conductor in the axial direction, cm c = Thickness of the conductor in the radial direction, cm K is the constant in the Nagaoka’s formula for a solenoid and is tabulated as a function of 2a/b or b/2a, depending on the cross section of the coil [8].

Fig. 4 Turn arrangements for calculation of L

[( A − a ) k'= [( A + a )

•

k is the constant which takes into account the decrease of inductance due to separation of the turns in the radial direction. This is taken from table 22 [8], where it is tabulated as a function of and c or b the two parameters c b

80 60 40 20 0 5

Turn Voltage, Volts

14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

f aA

μH

1 ⎛ ⎞ f = 0.014468⎜ log − 0.53307 ⎟ k' ⎝ ⎠

15 20 Turn Numbers

Fig 6 Comparison of Turn Voltages – 27turn Model transformer – RP – 40ns front time

15 20 Turn Numbers

With Skin effect Without Skin effect

Fig 5 Comparison of Node Voltages for Rectangular Pulse of 40ns front time on the Model transformer

The quantities 'K' and 'k' depend upon a, b and c, so that, to use the tables, a double interpolation is necessary. 5.1.3 Mutual Inductances The mutual inductance for circular coils of rectangular cross section to a first approximation is given by Grover [8]. M = N1 N2 M0 M0 is the mutual inductance of the central filaments of the coils. N1 and N2 are the number of turns in the coils. ‘a’ and ‘A’ are the mean radii of the coils and ‘d’ is the distance between the median planes of the coils (Fig 3.4). Further,

With skin effect(Simul) Without skin effect(simul) Experimental Data

100

b 2a

It is interesting to know the extent of the influence of these skin and proximity effects. Towards this, voltage distribution was also computed considering only the DC resistance of the winding i.e., neglecting the skin and proximity effects. The results are shown in Figs. 5 and 6. It is seen that except at the line and ground ends, at other zones the voltages are about 5% to 20% higher. This is for the case of the rectangular pulse with 40ns rise time. For higher rise times the influence would be less. Obviously for 1.2/50μs wave, the effect may be even less. However, it is good to know that when skin and proximity effects are considered, the agreement with experimental results is very good. It is clearly seen that the voltage stress is slightly reduced at very high frequencies due to the increase in resistance of the winding.

Node voltage, Volts

β =

] + (d ) ] 2

VI. RESULTS AND CONCLUSIONS

Since, c « 2a, interpolation in Table 36 [8], becomes some what uncertain. Therefore, its calculation is recommended directly by,

Where,

+ (d )

Proc. of Int. Conf. on Advances in Electrical & Electronics 2010

On the whole, the computed results are very close to the experimental results with turn resolution of transformer model. Particularly, it is seen that the influence of skin and proximity effects are significant at high frequencies greater than 1 MHz. This may not be very significant for analysis with lightning impulse but surges due to VFTO, skin and proximity effects play a major role. REFERENCES [1]

G.B.gharehpetian, H.Mohseni, K.Moller, “Hybrid Modelling of Inhomogeneous Transformer Windings for Very Fast Transient Overvoltage Studies”, IEEE Transactions on Power Delivery, Vol. 13, No. 1, pp. 157-163, January 1998. [2] S.Fujita, N.Hosokawa, Y.Shibuya, “Experimental Investigation of High Frequency Voltage Oscillation in Transformer Windings”, IEEE Transactions on Power Delivery, Vol. 13, No. 4, pp. 1201-1207, October 1998. [3] R.C.Degeneff, “The Transient Voltage Interaction of Transformers and Transmission Lines”, pp16-31, 1983. [4] Fredrick Emmons Terman, “Radio Engineers Handbook”.

[5] Ashraf W.Lotfi, Pawal M. Gradzki, fred C. Lee, “Proximity Effects in Coils for High Frequency Power Applications”, IEEE Transactions on Magnetics, Vol. 28, No. 5, pp. 2169-2171, September 1992. [6] A. Miki, T. Hosoya, K.Okuyama, “A Calculation Method for Impulse Voltage Distribution and Transferred Voltage in Transformer Windings”, IEEE Transactions on Power Apparatus and Systems, Vol. PAS-97, No. 3, pp. 930-939, May/June 1978. [7] Enrique E.Mombello, “Impedances for the Calculation of Electromagnetic Transients Within Transformers”, IEEE Transactions on Power Delivery, Vol. 17, No. 2, pp. 479-488, April 2002. [8] Fredrick W. Grover, “Inductance Calculations”, Dover Publications, Inc., New York. [9] Enrique E. Mombello, “Impedances for the Calculation of Electromagnetic Transients within Transformers”, IEEE Transactions on Power Delivery, Vol. 17, No. 2, pp. 479-488, April 2002. [10] Delfino, F. Procopio, R. Rossi, M.High “Frequency EHV/HV Autotransformer Model Identification from LEMP Test Data”, IEEE Transactions on Power Delivery, Vol. 25, No 4, pp 99, February 2010.