Power Transformer Modeling based on Wide Band Impedance and Admittance Measurements

Júnio César S. Silva; Alberto De Conti; Danilo G. Silveira; José Luiz Silvino LRC – Lightning Research Center

UFMG – Federal University of Minas Gerais Belo Horizonte, Brazil

Abstract— This paper evaluates the transient behavior of a single-phase, two-winding, 127/12 V, 12 VA signal transformer according to two different high-frequency models. The first model is based on the direct measurement of the transformer’s impedance matrix, while the second model consists in a combination of the admittance matrix and voltage transfer measurements. Both are alternatives to obtain a model with more information on the transformer’s behavior at frequencies ranging from 10 Hz to 2 MHz. The admittance matrix resulting from the inversion of the measured impedance matrix (in the first model) and the modified admittance matrix (in the second model) are approximated by rational functions via the vector fitting technique, which leads to two equivalent RLC models that are stable and ready to be used in EMTP-like programs. The obtained models are first compared in the frequency domain with measurements carried out at the transformer terminals. Then, in order to validate the developed models in the time domain, comparisons between the theoretical and measured results of the transferred voltages from the primary to the secondary windings of the transformer are made. These comparisons consider distinct load conditions and the application of impulse waves with different time characteristics.

Keywords - High-frequency transformer modeling; wide-band measurements; electromagnetic transients; transferred voltages.

I. INTRODUCTION

The surge transfer from the primary to the secondary windings of distribution transformers is one of the most frequent phenomena resulting from the lightning interaction with distribution networks. Transferred surges propagate toward the low voltage network, often leading to permanent damage to consumer loads and causing stresses to transformer windings. Human beings and animals may also be injured by transferred surges. Since lightning surges contain frequencies up to a few MHz, the determination of the surge transfer from the primary to the secondary windings of distribution transformers requires a previous understanding of the transformer high-frequency behavior [1, 2]. Such behavior is determined by the geometry of the coils, by the magnetic core and by the metal fittings that surround its active part (e.g., clamp bars, tank, leads, etc.). All these elements together can be modeled as a complex network of resistances, inductances and capacitances that are responsible for a nonuniform voltage distribution along the windings and between the windings during a transient event.

In the last decades, many high-frequency transformer models have been proposed. Some models, known as white box models, are obtained from constructive aspects of the transformer [3-6]. However, they require detailed information that is the manufacturer’s property. Other models, known as black box models, are obtained from wide-band measurements carried out at the transformer terminals in order to characterize its behavior in a specific frequency range [7-9].

The objective of this study is to derive a pair of black-box models for a single-phase signal transformer. One of these models is obtained by approximating the admittance matrix indirectly obtained from measurements of the transformer impedances in a wide frequency range. The other model is obtained by resorting to directly-measured transformer admittances plus the transformer voltage ratio. Equivalent RLC transformer models are derived by approximating both matrices as a sum of rational functions [8, 10, 11]. Such models are validated through comparisons with measured data for surge transfer tests involving the application of impulse voltage waves under distinct load conditions.

II. METHODOLOGY

A transformer can be modeled as a multiport element. If core saturation effects are neglected, its behavior is best characterized in the frequency domain, where the relations between the port variables take the form of linear algebraic equations. A two-port circuit, as is the case of a single-phase transformer, is entirely defined by four variables, namely two terminal voltages (V1 and V2) and two terminal currents (I1 and I2), as shown in Fig. 1. In traditional approaches, the admittances of the circuit are determined according to the ratio of a current by a voltage measured in frequency domain, while either port 1 or port 2 of the circuit in Fig. 1 is short-circuited.

Nevertheless, by short-circuiting one of the terminals of a single-phase, two-winding transformer, the effect of the core is essentially removed. As a consequence, the frequency response of the transformer is dominated by the windings and metal fittings [12]. This can be best understood by verifying in Fig. 1 that the magnetizing circuit of the transformer is bypassed if terminals 1 or 2 are short-circuited. In addition, models elaborated in this manner tend to be inaccurate in low-frequency studies and open-circuit applications, such as the evaluation of transferred voltages [8]. For this reason, this study uses the direct measurement of the impedance matrix and

2013 International Symposium on Lightning Protection (XII SIPDA), Belo Horizonte, Brazil, October 7-11, 2013

978-1-4799-1344-2/13/$31.00 ©2013 IEEE 291

a combination of the admittance matrix and voltage ratios, for deriving two wide-band models suitable for the calculation of transferred lightning overvoltages.

Figure 1. Transformer two-port circuit [12]

A. Model based on Impedance Measurements

The first model is described by a 2x2 impedance matrix, whose elements are determined from measurements performed at the transformer terminals as described in Fig. 2.

1

111 I

VZ = (1)

(a)

2

222 I

VZ = (2)

(b)

1

221 I

VZ = (3)

(c)

2

112 I

VZ = (4)

(d)

Figure 2. (a) Measurement of Z11 in open-circuit test. (b) Measurement of Z22 in open-circuit test. (c) Measurement of Z21 in transferred voltage test. (d) Measurement of Z12 in transferred voltage test [12].

According to the figures shown in Fig. 2, it is possible to represent the measured data in the form [V]=[Z][I] where

[ ] ��

���

�=

2221

1211

ZZ

ZZZ (5)

is a tri-dimensional 2x2xn impedance matrix and n represents the number of sampled frequencies in the frequency range defined by fmin and fmax. The measured impedance matrix can then be inverted for obtaining the corresponding admittance matrix:

1

2221

1211

2221

1211)(−

��

���

�=�

�

���

�=

ZZ

ZZ

YY

YYsY (6)

The next step is to approximate the transformer admittance matrix by a series of rational fractions, which can be done using the vector fitting technique described in [11]. The approximation is obtained by replacing a common set of initial poles by an improved set of poles using the relocation method based on the least square problem [11]. The order of the approximation is equal to the number of starting poles. The simulations have shown that at least 23 stable pairs of complex poles are enough for a reasonable agreement with the experimental results.

Fig. 3 shows both measured (obtained through inversion of the measured impedance matrix) and fitted elements through the vector fitting technique [11]. In the measurement and fitting procedures it was assumed that fmin=10 Hz, fmax=2 MHz and n=213. It can be seen that the agreement between measured and fitted curves is satisfactory.

According to [13], the passivity enforcement technique available in the vector fitting routine has the effect of mitigating instabilities from the time domain simulations with the burden of introducing a minimum fitting error. However, in a previous study [14], a considerable divergence with regard to experimental results was observed. Theoretically, an actual passive system, whose frequency response is approximated by the vector fitting technique, would not require any passive enforcement. In this study, time domain simulations were seen to be stable even in the absence of the passive enforcement procedure, which is the reason this technique was avoided in the fitting procedure.

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Frequency (Hz)

Mag

nitu

de (

p.u.

)

measured

fitteddeviation

Y21

Y22 Y11

(a)

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0

50

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Frequency (Hz)

Pha

se a

ngle

(de

g)

measured

fittedY22

Y11

Y21

(b)

Figure 3. (a) Magnitude and (b) phase angle of the 23th order frequency response for the indirect admittance matrix approximation (Yindirect).

B. Model based on Admittance Measurements

The second transformer model is obtained from the direct measurement of the transformer short-circuit admittances in a wide frequency range as described in Fig. 4.

1

111 V

IY = (7)

(a)

2

222 V

IY = (8)

(b)

Figure 4. (a) Measurement of Y11 in short-circuit test. (b) Measurement of Y22

in short-circuit test [12].

As previously mentioned, short-circuit measurements exclude information that is relative to the transformer core and, as a consequence, overvoltage studies tend to be unreliable in low-frequency and open-circuit applications [8]. In order to improve the model’s accuracy in low-frequency and open-circuit conditions, reference [8] proposes that the transfer admittances Y12 and Y21 be corrected by the voltage ratio of the transformer. It is important to mention that only the correction of Y21 is enough to increase the accuracy of the data set, once the vector fitting routine admits that Y(s) is a symmetric matrix and only the upper triangle of this matrix is subjected to the fitting.

The measured voltage ratio is introduced into the model by making the following substitution in the element Y21 of the directly-measured admittance matrix [8]:

1

22221 V

VYY ×−=′ (9)

Fig. 5 shows both measured and fitted elements of the directly-measured admittance matrix through the vector fitting technique [11]. One can observe that the agreement between the measured and fitted curves is very good. The passive enforcement procedure was avoided in the fitting procedure of the matrix corrected by (9) for the same reasons mentioned in section II-A.

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Frequency (Hz)

Mag

nitu

de (

p.u.

)

measured

fitteddeviation

Y22Y21 Y11

(a)

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-50

0

50

100

150

200

Frequency (Hz)

Pha

se a

ngle

(de

g)

measured

fitted

Y11

Y22

Y21

(b)

Figure 5. (a) Magnitude and (b) phase angle of the 23th order frequency response for the directly-measured matrix approximation considering the correction in (9) (Ycorrected).

III. EXPERIMENTAL VALIDATION AND ANALYSIS

In order to validate the implemented black-box models, a lightning transient study is presented in this section for two practical conditions, namely the transformer operating under no load conditions and the transformer operating with loads of different sizes. Initially, a comparison between the voltage transfer characteristic obtained with the actual transformer and those obtained with the models is presented in the frequency domain. Then, the measured and calculated time domain results are evaluated when impulsive voltage waveforms of different time characteristics are transferred from the high-voltage to the low-voltage transformer windings. For both studies, the fitted admittance elements associated with Figs. 3 and 5 were used as

293

input parameters for deriving RLC networks that were implemented in ATPDraw [10].

A. Frequency Domain Results

In the first case study, both black-box models derived in Section II were compared with experimental data in terms of their frequency responses. The equivalent RLC networks elaborated for each model were used to simulate transferred voltages between the high-voltage to the low-voltage terminals in the frequency range from 10 Hz to 2 MHz. The ratio between the measured voltage at the low-voltage terminal (with the low-voltage terminal left open) and the sinusoidal signal with a peak value of 10 V applied to the high-voltage terminal determines the frequency response of the models and the transformer. The obtained results are shown in Fig. 6, which indicates a satisfactory agreement between the model’s predictions and measured data, except for the model obtained by the direct measurement of the admittance matrix without the correction of Y21 using (9). This can be best observed by considering details 1 and 2 shown in Fig. 6. Based on such information it is possible to note how much the corrected model response comes closer to the transformer’s response.

Frequency (Hz)

Mag

nitu

de (

p.u.

)

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0.5

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Frequency (Hz)

Mag

nitu

de (

p.u.

)

Detail 1

105

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Mag

nitu

de (

p.u.

)

Detail 2

106

0

0.2

0.4

0.6

0.8

1

Yindirect

Ydirect

Ycorrected

Transformer

Figure 6. Comparison of frequency response analysis, in which Yindirect represents the response of the model obtained through the direct measurement of the impedance matrix, Ydirect represents the response of the model obtained through the direct measurement of the admittance matrix, Ycorrected represents the response of the model obtained through the direct measurement of the admittance matrix corrected by the transformer voltage ratio and Transformer represents the measured response of the transformer under test.

In relation to the model response obtained by the direct measurement of the impedance matrix one can observe through detail 1 shown in Fig. 6 a small divergence of about 9% in the frequency response at approximately 18 Hz, possibly attributed to errors caused by the inversion of the impedance matrix, inaccuracies associated with the use of shunt resistors as current transducers and to the effect of the nonlinear behavior of the transformer core at low frequencies.

It is important to mention that in addition to the lack of information about the magnetic core, another limitation of using the model obtained by the direct measurement of the admittance matrix Ydirect is the loading effect observed during the measurement of the Y21 element. Due to its extremely low ohmic resistance, the LV winding looks like a short-circuit for the signal generator. This effect is more pronounced until 10

kHz. As the frequency increases, the winding resistance also increases and the Ydirect model response tends to be similar to the other responses.

B. Time Domain Results in No-load Condition

This section evaluates the time-domain responses of the high-frequency black-box transformer models described in Section II assuming the transformer LV terminal to be open-circuited. Three cases corresponding to the application of 1.2/50 μs, 0.5/50 μs, and 0.5/10 μs voltage waveforms with amplitude of 10 V at the HV transformer terminal were considered. The voltage waves transferred to the low-voltage terminal were then compared with computed results. Fig. 7 presents the results, which shows that the model’s predictions present a good agreement with measured data, except again for the model obtained by the direct measurement of the admittance matrix without the correction of Y21 using (9). One can note, however, that the poor agreement between this particular model and the measured results is mostly confined to the tail of the curves. Most likely, these discrepancies are associated with the lack of information about the magnetic core in the low-frequency range, as well as with difficulties in measuring the Y21 element, as mentioned in section II and Fig. 6. One can therefore conclude that the introduction of the transformer’s voltage ratio into the directly-measured admittance matrix (curves labeled as Ycorrected in Fig. 7) leads to results that are more accurate than those obtained with the direct measurement of the admittance matrix (curves labeled as Ydirect in the Fig. 7), at least for the transformer considered in this paper.

Time (μs)

Vol

tage

(V

)

3 4 5 6 7 8 9 10

0

0.5

1

1.5

Yindirect

Ydirect

Ycorrected

Transformer

(a)

Time (μs)

Vol

tage

(V

)

6 7 8 9 10 11 12 13 14

0

0.2

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1.2

1.4

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1.8Yindirect

Ydirect

Ycorrected

Transformer

(b)

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Time (μs)

Vol

tage

(V

)

3 4 5 6 7 8 9 10

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6Yindirect

Ydirect

Ycorrected

Transformer

(c)

Figure 7. Transferred voltages to the LV transformer terminal in no-load conditions for (a) 1.2/50 μs, (b) 0.5/50 μs, and (c) 0.5/10 μs voltage waveforms applied at HV terminal.

C. Time Domain Results in Load Condition

In the third case study, the voltage waves considered in Section III-B were now applied at the transformer high-voltage terminals while the secondary was connected to resistive loads of 10 �, 100 � or 1000 �. Fig. 8 presents three sample results of transferred voltage to the low-voltage terminals, all referring to the application of 1.2/50 μs voltage waves.

Time (μs)

Vol

tage

(V

)

0 5 10 15 20 25 30 35 40 45 50-0.1

0

0.1

0.2

0.3

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0.8Ycorrected

Ydirect

Yindirect

Transformer

(a)

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Vol

tage

(V

)

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0

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1

Ydirect

Yindirect

Transformer

Ycorrected

(b)

Time (μs)

Vol

tage

(V

)

0 2 4 6 8 10 12 14 16 18 20

0

0.2

0.4

0.6

0.8

1

Transformer Ycorrected

YdirectYindirect

(c)

Figure 8. Transferred voltages to the low-voltage terminal assuming (a) 10 �, (b) 100 �, and (c) 1000 � loads for the application of a 10 V-1.2/50 μs voltage waveforms at the high-voltage terminal.

Fig. 8 shows that the model corresponding to the indirect determination of the admittance matrix from the measurement of the matrix impedance (curves labeled as Yindirect in Fig. 8) is more sensitive to the presence of loads than the model determined by the direct measurement of the admittance matrix considering the correction in (9) (curves labeled as Ycorrected in Fig. 8). One possible reason for this sensitivity related to loads connected to the secondary terminal is the fact that while the Ycorrected model loses information about the magnetizing branch, the Yindirect model loses information related to the windings. This can be confirmed by superimposing Fig. 3 (a) and Fig. 5 (a) and observing that until 100 kHz the Ycorrected admittances are about 26 times greater than the Yindirect admittances, as shown in Fig. 9. This frequency limit is in fact the limit from which the frequency response is completely decoupled from the magnetic core and dominated only by the windings and by the transformer’s metal fittings [12].

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Frequency (Hz)

Mag

nitu

de (

p.u.

)

Yindirect

Ycorrected

Y11

Y11 Y21 Y22

Y21 Y22

Figure 9. Comparison of Yindirect and Ycorrected magnitudes along the frequency range of 10 Hz and 2 MHz.

A similar behavior can be noted for the 1.2/10 μs, 0.5/50 μs, 0.5/10 μs and 5/50 μs transferred voltages, although not shown here. As the ohmic value of the load increases, the system response approaches the no-load condition discussed in

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the previous section, and consequently the models Yindirect and Ycorrected behave more similarly.

IV. CONCLUSIONS

This paper discusses two different approaches for developing high-frequency black-box models for a two-winding, 127/12 V, 12 VA single-phase transformer with the objective of calculating transferred voltages to the secondary terminal. The validation of the models was obtained through comparisons between experimental and calculated results.

The model corresponding to the indirect determination of the admittance matrix from the measurement of the impedance matrix, Yindirect, and the model corresponding to the direct measurement of the admittance matrix corrected by the transformer voltage ratio, Ycorrected, proved to adequately represent the transferred voltage calculation taking the no-load condition into account. The results considering different loads connected to the low-voltage terminal, although less satisfactory, are considered acceptable. Among the evaluated models, the model obtained from the direct measurement of the transformer admittance matrix with the correction of Y21 using (9), Ycorrected, was seen to lead to results that are in closest agreement with experimental data for both load and no-load conditions. This agreement between Ycorrected and experimental data responses is explained by the fact that in Fig. 6 the correction of the admittance element Y21, according to (9), had caused a minimum divergence between the calculated and measured frequency-domain responses.

The errors introduced by the inversion of the impedance matrix associated with the use of shunt resistors as current transducers, as well as by the core nonlinear effects and the loss of winding short-circuit characteristics, might have affected the low-frequency response of the model elaborated from the direct measurement of the transformer impedance matrix, Yindirect.

The obtained results illustrate the validity of the adopted modeling approaches, which are based on fundamentals mainly discussed in [8], and in principle can be readily extended to power transformers of any size and rating.

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