Incorporating capacitance into partial-core transformer models to determine first natural resonant frequencies

M.C. Liew and P.S. Bodger

Abstract: Capacitive components are added to partial core transformer models to allow resonant analysis at harmonic frequencies (50-5 kHz). The harmonic frequency responses of partial core transformers with relatively low turns ratio arc analysed with the model. Capacitive loads of various magnitudes are then connected to the secondary terminals in order to calculate loaded resonant frequencies. The calculated values are validated with test results. The harmonic frequency responses of two high-voltage partial-core transformers with large turns ratio are presented. Calculations of the transformers’ first natural resonant frequency were validated with test results. Additionally, a capacitive load was placed across each of the secondary terminals and the loaded resonant frequencies, calculated by the model, are compared to the measured values.

1 Introduction

A reverse-design-transformer equivalent-circuit model has been introduced [l], which derives circuit components from the characteristics and dimensions actually used to build transformers. From the equivalent circuit, the transformer performance can be determined. This is essentially the opposite of the conventional design ap- proach, hence the name reverse-design approach. This approach has shown improved accuracy in predicting performance as compared to conventional transformer modelling [2].

This reverse-design approach has since been applied to partial-core transformers [3], where the return yokes and limbs of a full-core transformer have been removed. Partial-core transformers are being studied because the size of their cores can be dramatically reduced albeit by an increase in winding turns. The combination gives better magnetisation than a coreless transformer and maintains the leakage flux at an acceptably low level. The combination also means that the overall weight of the partial-core units is significantly reduced, and they are easier to manufacture than the conventional units. The calculated and measured operational performances of these transformers at normal operating temperatures have shown good agreement [3 ] . In addition, the operational performances of these transformers at liquid nitrogen temperatures have also shown good agreement [4].

In this paper, capacitive components are added to the model to determine the first natural resonant frequencies of partial-core transformers.

0 IEE, 2002 IEE Proceedings online no. 20020725 doi: IO. 1049/ip-gtd:20020725 Paper first received 18th July 2001 and in revised form 3rd May 2002 The authors are with the Department of Electrical and Electronic Engineering, University of Canterbury, Private Bag 4800. Christchurch, New Zealand

2 individual capacitive components

Consideration is given to the axial view of the partial-core transformer shown in Fig. 1. The transformer windings are

center line

I- I

I

1

-IF c2 T

ik

Fig. 1 tunces

Axiul view of’ the trunsjormer showing individual cupnci-

wound in the helical configurations as shown. Five capacitive components can be identified:

1. C,. = capacitance between the core and the first primary layer, 2. Cl,,l = self-capacitance of the primary winding, 3 . C12 = intenvinding capacitance, 4. = self-capacitance of the secondary winding, 5. C2T= capacitance between the last secondary layer and the transformer tank.

746 IEE Proc.-Gmer. Tronsm. Dbtrih.. Vol. 149, Nu. 6, Noi;einber 2002

With respect to the capacitance between the outmost layer of the secondary winding and an earthed plane, the plane's orientation may be horizontal (usually just the groundplane) or vertical (in this case the transformer tank). In Fig. 1, a tank is incorporated to act merely as an earthed plane at some uniform distance away from the transformer's outennost winding.

2.1 Calculation of C,, The calculation of any capacitance can be done using the parallel-plate theory [5]. This is strictly true, only if the electric field can be considered linear between the plates. This is a reasonable approximation if the electrode surface separation dimension is small compared to the electrode dimensions. Consider two adjacent surfaces shown in Fig. 2. The voltage of surface A is assumed to be varying

core

"AP -

"A0

Fig. 2 potential distribution

Two u4acent conductive szirfuws. eucli liming CI lincar

linearly from VAu at the lower end to VAp at the top end. Similarly, the voltage of surface B varies linearly from VBo to VBp. The voltage difference between the two surfaces should vary linearly along the length 1, from V,= V&- Vao at the bottom to I f p = = VBp- VAp at the top. The voltage difference at the element dx is

dv = vo + ( V p - vo)f (1)

The associated electrical energy within the element dx is 1

d W = - dCd V 2 2

I & x 2 ( 2 )

-Cl - (vu+(vP- 2 1 vu,J

C, is the capacitance between the two surfaces within the elemental length dx.

IEE Proc.-Gencr. Trmsi?i. Distrib., Vol. 149, No. 6, NoL;eiiibrr 2002

The total energy is calculated by integrating over the whole length 1:

+( v, - v&} I' dx

If one of the surfaces represents a layer of a winding having a total voltage of I, across it, then the capacitance C appearing across the terminals of the winding, due to the distributed capacitance of that layer, can be calculated by equaling energies:

therefore,

A primary winding with Lyl layers is shown in Fig. 3a. If Cnz is the parallel-plate capacitance between the first layer

Drimary

and the core, and C, is the capacitance between layers, then the primary winding equivalent capacitance is

layer 1 layer 2 layer Lyl

(6) Hence, the capacitance between the core and the first layer of the primary winding is

(7)

To find C,,,, consider enlarged portion of the transformer showing the core, the first two layers of the primary winding, and the space between the core and the winding, as depicted in Fig. 3b. C,, is a series of capacitances caused by

141

different insulation layers:

where

Crcl = capacitance of the insulation between the core and

CwIl = capacitance of primary winding wire insulation

Each of these insulation capacitances are calculated using

the primary winding ZC1,

W j l .

the parallel-plate capacitance formula:

I a y e r s l k

core P 0

a

€,.€,A e=- d (9)

e ) V l ( where

E,. = the corresponding relative permittivity of the insulating

8, = permittivity of free space = 8.854 x A = area of the insulation surface, d= effective thickness of the insulation.

material Fm-',

\- core P 0

b

secnndarv

2.2 Calculation of C,, The self-capacitance of the primary winding is calculated using the second term of (6): primary

C

Fig. 4 u Lyl odd h Lyl even c Space between the primary and secondary windings

where

C,, = capacitance of the primary-secondary interwinding

CwR = capacitance of secondary winding wire insulation

Determining the interwinding cnpncitunce

insulation ZI2

)"/2.

2.4 Calculation of Cw, In a similar way to C,,Jl, the secondary winding self- capacitance is calculated as

From Fig. 313, Cl is calculated as 1

Cl = '+2 Cll (C:,,I> -

2.3 Calculation of CI2 The calculation of the intenvinding capacitance depends on whether the number of layers in the primary winding is odd or even. Using (5) and the schematic diagrams of Fig. 4u and h, CI2 can be determined:

Cl for the secondary winding, referring to Fig. 4c, is given by

+(z - 6)'] forLy, odd 2.5 Calculation of C,, To calculate the capacitance between the last layer of the secondary winding and the tank, consider the diagram of a secondary winding with L.v2 layers. Using (5) and the schematic of Fig. 5a, C2, is calculated as [6]

Ct 3 5

c2z =+v; + vovp + v;) Ct LY2 - 1

=- 3 v; (52 + (r) 5=

+( r*) li)?) Examining the enlarged portion of the interwinding space between the primary and secondary windings shown in Fig. 4c, the parallel-plate capacitance C,, between the two windings is

748 IEE ProcGmer. Trunsin. Distrih., Vol. 149, No. 6, Norenihrr 2002

W'2

secondary b

Fig. 5 Calcuhtiny C,,

+ v, CL=

-

Once again, an enlarged portion of the transformer showing the last two layers of the secondary winding, the tank, and the space between the last layer and the tank is shown in Fig. Sb.

The parallel-plate capacitance of C, is, thus,

(17) 1 c, = ~ l + L

cll'l? c/Z7

II

+ c2 == "2

-

where C,2T= capacitance of the insulations between secondary winding and the tank 12T.

3 Equivalent transformer capacitive components

Consideration is given to the transformer equivalent circuit of Fig. 6. Current in the interwinding capacitance is obtained from

=wC,2 (v, - 5)

where

= secondary voltage referred to the primary

Alternatively, the interwinding capacitance current enter- ing the secondary side may be expressed as

When referred to the primary side, (1 9) becomes

From Fig. 6, referring the secondary capacitance to the primary, and using (1 8) and (20) to refer the interwinding

capacitance to the primary side, yields the equivalent circuit shown in Fig. 7.

Fig. 7 to the primary

Transformer equivalent circuit, with 01'1 coinponetits referred

The total primary winding capacitance is

a - 1 cl, = CI + c 1 2 (T) The primary winding capacitive reactance is, thus,

1 XC, =-

w c; The total secondary winding capacitance is, thus,

and the secondary winding capacitive reactance is 1

xc2 = __ WC;

The referred interwinding capacitance is

CI 2 Ci2 = 7 and the invenvinding capacitive reactance is

4 Verification of the components

With the incorporation of the capacitive components into the reverse-design partial-core model, it has become possible to carry out harmonic-frequency analysis of the partial-core transformers. Three partial-core transformers were used to compare the harmonic-frequency-response results between model calculations and experiments. The equivalent-circuit parameters of the transformers calculated by the model are summarised in Table 1.

To confisni the validity of the model developed for frequencies up to S kHz, capacitive loads were connected across the secondary terminals in order to force the resonant frequencies of the transformers down to within power-system harmonic-frequency ranges. The experimen- tal circuit is shown in Fig. 8. The tests were conducted at three different capacitive loads for the transformers, to force resonant frequencies at 1 kHz, 2 kHz and S kHz, respec- tively. Table 2 shows the capacitive load C, values used for the transformers to give the three frequencies.

Together with the component values in Table 1, each of the C, values determined from the tests was then entered into a circuit simulation package to determine the corresponding calculated resonant frequency. Table 3 shows the resonant frequencies calculated.

149 IEE Psoc-Gener. Trurisn. Di.vril?., Vol. 149. Nu. 6. Nowmber 2002

Table 1: Equivalent circuit parameters of the three transfor- mers

Circuit parameters #I #2 #3

Resistance, R

Rcore

Ri

4

Inductance, mH

Lrn

L1

L;

Capacitance, nF

c; c; CI?

223.08 1.03 2.26

76.66 11.37 11.37

5.81 0.97 1.02

680.47 1.18 2.84

116.91 7.0% 7.08

5.79 1.04 1.15

613.68 0.70 1.32

69.47 7.95 7.95

5.86 1.41 1.81

0-1 MHz _ - - - A

Fig. 8 Frequency response test circuit

Table 2 C, values to give specified frequencies

36 48 19 10 13 5 1.9 2.3 0.9

Table 3: Calculated frequencies

Measured Calculated frequency (kHz) frequency (kHz)

#I #2 #3

1.05 1.04 1.05 2.02 2.00 2.05 5.00 4.70 4.77

From Table 3 it is evident that the calculated frequencies are very accurate. However, verification of the harmonic model by putting a capacitive load on the secondary winding does not necessarily imply that the developed inherent capacitive components of the transformer are correct. This is because the magnitude of the transformer’s inherent capacitances are much smaller than the capacitive loads C,. The accuracy of the results obtained here mostly confirms the validity of the inductive reactance components,

750

especially the leakage reactance. This is because the leakage reactance usually determines the transformer’s first natural resonant frequency [7]. It is therefore necessary to check the accuracy of the capacitive components developed against experiment results on transformers which have significant internal capacitance.

From (23), it can be seen that, if a transformer with a very high secondary to primary turns ratio is designed, the transformer’s turns ratio (u) becomes a very small value. As a result, the secondary winding capacitance, when referred to the primary, becomes significantly large. This affects and therefore lowers the transformer’s first natural resonant frequency [7]. To investigate the accuracy of predicting the resonant performance of high turns ratio transformers, two high-voltage partial-core transformers have been built. They have the specifications shown in Table 4.

Table 4 Specifications and design data for the transformers

Transformer HVPCl HVPQ

Rating:

Primary voltage, V 14 Secondary voltage, kV 4.56 VA rating, VA 620 Number of primary turns 45 Number of secondary turns 14,605 Core:

Length, m m 110 Diameter, m m 76 Core/LV winding insulation thickness, m m 6 LV winding:

Number of layers 2 Wire diameter, m m 3.5 lnterlayer insulation thickness, m m 0

LV/HV winding insulation thickness, m m

HV winding:

0.5

Number of layers 20 Wire diameter, m m 0.1 1 lnterlayer insulation thickness, m m 0.15

12 6

1000 50

24,853

132 64 6

2 4.25 0 0.8

29 0.11 0.1

Both transformers were designed as scale models of a high-voltage transformer for testing power-system compo- nents at harmonic frequencies. The equivalent-circuit parameters, as determined by the model, are shown in Table 5.

The referred secondary winding capacitance Ci are significantly larger in magnitude than Cl, and Ci2. The open-circuit frequency responses of each transformer are simulated and plotted in Figs. 9a and loa.

From Fig. 9a, the natural frequency of HI/ PCl is found to be 4.4kHz. The measured frequency response is also plotted in Fig. 9u. The measured resonant frequency was 4.5 kHz, which matched that of the calculated value. The measured amplitude of the resonant peak is much lower than that calculated from the model’s expected outcome. This is because in all tuned filters using magnetic core inductances, the losses increase considerably near the resonant frequency. The quality factor of the circuit, hence the magnitude of the peak, is decreased [SI. This has not been modelled.

IEE Pr.oc.-Gener. Trmsn?. Disrrih.. V d 149. No. 6. N17ceniher 2002

Table 5: Equivalent circuit parameters

4 -

3 - .- m 0 -0

E 2 -

0

1 -

Circuit parameters HVPCI HVPC?

Resistance, R R,,,

R1

R;

Inductance, mH

Lrn

L1

4

44.4 0.024 0.088

0.474 0.014 0.01 4

1.675 x

0.272 x IO-^ 47.2

12.8 0.016

0.058

0.487 0.018 0.018

0.819 x

156

0.216 x

0 1000 2000 3000 4000 5000 frequency, Hz

a

4 ,

0 4 0 1000 2000 3000 4000

frequency, Hz b

Fig. 9 a Open-circuit response b Loaded response

Frequency response o j HVPCI

7

5000

The open-circuit frequency response of HVPC;! was simulated and is depicted in Fig. 1Ou. The simulated resonant frequency is 2.1 kHz. The frequency response of HVPC2 was measured, and is also plotted in Fig. 10n. The simulated and measured results again match, confirming the validity of the capacitive components developed under open-circuit conditions.

A capacitive load of IOOOpF was then connected to the secondary winding of each transformer. 1000 p F was chosen

0 500 1000 1500 2000 2500 3000 frequency, Hz

a

R

- -5 - simulated r ,

0 4 0 1000 2000 3000

frequency, Hz b

Fig. 10 u Open-circuit response b Loaded response

Frequency response of HVPC2

to represent the scaled loading effect of a capacitive voltage transformer [9]. When referred to the primary, the load becomes

e; 1

= (F) x 1000 pF

=105.3 ,uF

for H VPCl , and

~ 2 4 7 . 1 pF

for HVPC2. The frequency response of HVPCl with the capacitive

load was simulated, and it is shown in Fig. 9n. Again, it can be seen that the magnitude of the measured resonant peak is much lower than that of the model. The resonant frequency was estimated to be 2.4 kHz. The loaded frequency response was also measured and is plotted in Fig. 96. The measured resonant frequency was 1.9 kHz.

Finally, the frequency response of HV PC2 under the loaded condition was simulated. The results are shown in Fig. 106. The simulated frequency is approximately 1.3 kHz. The loaded frequency response was measured, and the resonant frequency was determined to be 1.1 kHz. The result is also plotted in Fig. 10h for comparison. The simulated and measured resonant frequencies are close enough to be useful in determining the transformer bandwidths. Again, the magnitude of each of the measured resonant peaks is much lower than that of the model, due to

751 IEE Proc -Gener. Tvmsm. Distrib., VoL 149. No. 6. Nouendxr 2002

reduced quality factors of the circuits. The closeness of the calculated and measured results for the transformers validates the usefulness of the models derived in this paper.

5 Conclusions

Capacitive components have been added into the reverse- design partial-core transformer model. The harmonic frequency response of three partial-core transformers with relatively low turns ratio have been analysed. Capacitive loads were connected to the secondary terminals in order to calculate loaded resonant frequencies. Acceptable matches between the calculated and the test values were achieved. Two high-voltage partial-core transformers with large turns ratios were designed, built and analysed. Calculations of the transformer’s resonant frequencies, under both open-circuit and loaded conditions, were verified by experimental results. This has strengthened the use of the reverse-design partial- core model developed as an entry-level design tool, from which more accurate designs can be made.

6 References

1

2

BODGER, P.S., and LIEW, M.C.: ‘Reverse as-built transformer design method’, in Znt. J. Elect. Eny. Educ., 2002, 39, ( I ) , pp. 42-53 BODGER, P.S., LIEW, M.C., and JOHNSTONE, P.T.: ‘A compar- ison of conventional and reverse transformer design’. Australasian universities power engineering Conference (AUPEC), Brisbane, Aus- tralia, 2000, pp. 80-85 LIEW, M.C., and BODGER, P.S.: ‘Partial core transformer design using reverse modelling techniques’, in IEE Proc., Electr. Power App/. ,

LIEW, M.C., and BODGER, P.S.: ‘Operating partial core transfor- mers under liquid nitrogen conditions’, in IEE Proc., Electr. Power

5 SNELLING, E.C.: ‘Soft Ferrite: Properties and Applications’, 2nd Edn. (Butterworth & Co. Ltd., UK, 1988)

6 MACFAYDEN, K.A.: ‘Small Transformers and Inductors’, (Cliap- man and Hall Ltd., London, England, 1953)

7 JIANG, Q., and BODGER, P.S.: ‘Harmonic response and terminal resonances of high voltage transformers’, it? Int. J. Electr. Eng. Edic., 1991, 28, (2), pp. 144156 OLIVIER, G., BOUCHARD, R.P., GERVAIS, Y., and MUKHED- KAR, D.: ‘Frequency response of HV test transformers and the associated measurcment problems’. in IEEE Truns Poiver Appur. Syst., 1980, 99, (l), pp. 141-145 BRADLEY, D.A., BODGER, P.S., and HYLAND, P.R.: ‘Hamionic response tests on voltage transducers for the New Zealand power system‘, in IEEE Truizs. Power Appur. Sysr., 1985, 104, (7), pp. 175&1756

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2001, 148, (6), pp. 513-520 4

A/>pl., 2001, 148, (4), pp. 293-298

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