Ratio Correction Factor and Phase Displacement at Arbitrary Burdens in Instrument Transformers

1. Introduction

Voltage transformers (VTs) and current transformers (CTs) are normally used in the power industry for high voltage, heavy current, and power loss measurements [1-3]. The method most widely used to calibrate the transformers is to compare the transformer under test with a reference transformer that has the same nominal ratio and higher accuracy [4, 5]. According to the International Electrotechnical Commission (IEC) [6] and American National Standard Requirements for Instrument Transformers (ANSI) [7], the ratio correction factor and phase displacement of an instrument transformer under test should be measured by connecting an external burden to the secondary of the transformer under test. The measurements of ratio correction factor and phase displacement of the instrument transformers at two different burdens are required. However, the values of burden required in IEC are different from that in ANSI.

The users in industry are usually using their instrument transformers at the different burden conditions compared with the burdens calibrated according to requirement of IEC or ANSI. Therefore, it is very useful that the ratio correction factor and phase displacement of instrument transformer are obtained at the real burden complying with the environment of industry without any additional measurements.

In this study, we have developed the methods for obtaining the ratio correction factor and phase displacement at arbitrary any burdens without any additional accuracy measurement, from the ratio correction factor and phase displacement of instrument transformer measured at two different burdens according to IEC or ANSI. This burden effects are based on a theoretical analysis of the equivalent circuits of the VT and CT. The validity of this method was verified by comparing the theoretical value for several instrument transformers with measured value.

 

2. Burden Effects on VT

2.1 Calculation of ratio correction factor and phase displacement at arbitrary burdens

An equivalent circuit for a VT with a zero burden is shown in Fig. 1 [8]. Z0 in Fig. 1 is the leakage output impedance of the secondary of the VT:

Fig. 1.Equivalent circuit for a VT with a zero burden

The complex ratio of the primary voltage vector (Vp) to the secondary voltage vector (Vs) of the VT with high accuracy at a zero burden is given by [8, 9].

In Eq. (2), NVT is the rated transformation ratio of VT. α0 is the ratio error at a zero burden, and β0 is the phase displacement between the secondary voltage vector and the primary voltage vector at a zero burden, defined as β0 ≡ βs − βp . The RCF0 is the ratio correction factor at a zero burden, defined as the actual transformation ratio (Na) divided by the rated transformation ratio (NVT). The relationship between RCF0 and α0 is as follows;

An equivalent circuit for the VT with an external burden, Zb, is shown in Fig. 2 [8]. The VT burden consists of a serial connection of the resistance and the inductor, expressed as:

Fig. 2.Equivalent circuit for a VT with burden, Zb

In a similar manner to Eq. (2), the complex ratio of the primary voltage vector (Vp) to the secondary voltage vector (Vb) of a VT with high accuracy in the presence of an external burden is:

where αb is the ratio error with the external burden, and βb is the phase displacement between the secondary voltage vector and the primary voltage vector with the external burden. RCFb is the ratio correction factor with the external burden, written as RCFb ≡ 1 − αb . The ratio error and the phase displacement of the VT arise from the secondary burden, the leakage inductance of the primary and secondary, the winding resistance of the primary and secondary, the magnetizing current, and the core loss [10, 11].

Equating the currents passing through Z0 and Zb, as shown in Fig. 2, we obtain the following equation.

This can be rewritten as:

Eq. (7) can be changed into Eq. (8) using Eqs. (1), (2), (4) and (5) as:

By taking the real part of Eq. (8), we can obtain the ratio correction factor with the external burden as:

where Gb and Bb are defined as follows.

: conductance of the VT burden : susceptance of the VT burden

In a similar manner to Eq. (9), we can obtain the phase displacement with the external burden by taking the imaginary part of Eq. (8):

The leakage output impedance, Z0, of the secondary of the VT is written by substituting subscript t for b in Eq. (7) as follows;

βt and β0 is less than 10-3 order in Eq. (11), thus the quadratic and higher order terms in the exponential expansion of e−j(βt−β0) are less than 1×10–6, which is neglected. Therefore, the resistance ( R0 ) and reactance ( X0 ) of Z0 are given by as follows;

By measuring the ratio correction factor/phase displacement at the zero, t burden and impedance of t burden, we can obtain both R0 and X0 according to eqs. (12) and (13), respectively. Using values of R0 and X0 , the ratio correction factor and phase displacement at arbitrary b burden could be theoretically obtained without any additional accuracy experiment according to eqs. (9) and (10), respectively.

2.2 Comparison between calculated and measured values for ratio correction factor and phase displacement at an arbitrary burden

For study of burden effects, the ratio correction factor and phase displacement at four different burdens (zero, t, b and b’ burden) including zero burden are measured for three different types of VTs (VT 1, VT 2, and VT 3) with different accuracy. The VT 1, VT 2 and VT 3 have the accuracy of 0.005 %, 0.02 % and 0.2 %, respectively. The values of resistance/reactance of t, b and b’ burden are summarized in table 1. Here t, b and b’ burden correspond to 4.87 VA with unity power factor, 5.05 VA with 0.80 lagging power factor and 15.02 VA with 0.80 lagging power factor, respectively.

Table 1.The values of resistance/reactance of t, b and b’ burdens

The measurement results of ratio correction factor/phase displacement at zero, t and b burdens for three different types of VTs are represented in the second, third and fourth row of table 2, respectively. From the errors measured at zero/t burdens and the impedance values of t burden, the values of R0 and X0 for three different types of VTs using eqs. (12) and (13), respectively, are calculated as follows ;

Table 2.* : measured value + : calculated value

R0 = 0.29 Ω, X0 = 0.10 Ω for VT 1 R0 = 0.21 Ω, X0 = 0.22 Ω for VT 2 R0 = 1.56 Ω, X0 = 0.51 Ω for VT 3

The ratio correction factor and phase displacement at arbitrary b burden are calculated using eqs. (9) and (10), respectively, as shown in the fifth row of table 2. The differences between calculated and measured values at arbitrary b burden are represented in the sixth row of table 2, which are less than 2×10-6 and shows good consistency. Meanwhile, the ratio correction factor and phase displacement are measured at arbitrary b’ burden for three VTs, as shown in the seventh row of table 2. The ratio correction factor and phase displacement at arbitrary b’ burden according to eqs. (9) and (10) are calculated using R0 and X0 obtained already at zero and t burdens, as shown in the eighth row of table 2. The differences between calculated and measured values at arbitrary b’ burden are represented in the last row of table 2, which are less than 11 × 10-6. The expanded uncertainty (k = 2) for the errors of the VTs was estimated to be within 30 × 10-6 in ratio and 30 microrad in phase [12]. This implies that the measured values at arbitrary any burdens for VTs with different accuracy are good consistent with calculated value within the expanded uncertainty.

 

3. Burden Effects on CT

3.1 Calculation of ratio correction factor and phase displacement at arbitrary burdens

An equivalent circuit for a CT with an external burden, Zb, is shown in Fig. 3 [13-14]. The parameters shown in figure 3 are defined as follows:

Fig. 3.Equivalent circuit for a CT with external burden, Ze.

Zm = Rm + jXm : magnetizing impedance. Z1 = R1 + jX1 : primary leakage impedance. Z2 = R2 + jX2 : secondary leakage impedance. Ze = Re + jXe : impedance of external burden. N1 : number of primary windings. N2 : number of secondary windings. Ip : actual primary current. Ie : actual secondary current with an external burden. I : ideal current with infinite magnetization impedance. Im : magnetization current.

The complex ratio of the primary current vector (Ip) to the secondary current vector (Ie) in a CT with an external burden is given by [13].

where NCT is the rated transformation ratio of CT. εe is the ratio error with the external burden and δe is the phase displacement between the secondary current vector and the primary current vector with the external burden. RCFe is the ratio correction factor with the external burden, defined as RCFe ≡ 1 −εe.

By taking the real and imaginary parts of (Z2 + Ze)/Zm in Eq. (14), we obtain the ratio correction factor ( RCFe ) and the phase displacement ( δe ), respectively, with the external burden as follows:

where is the conductive component of the excitation admittance, and is the susceptive component of the excitation admittance.

When a non-reactive resistor with an AC–DC difference less than 10–5 [15], i. e., (Xe/Re) < 10–5, is used as the external burden of the CT under test, then Eqs. (15) and (16) become:

Both the ratio correction factor ( RCFe ) and the phase displacement ( δe ) of the CT under test in Eqs. (17) and (18) are proportional to the resistance of external burden, Re, because of the constant values of Gm, Bm, R2, and X2 for a fixed secondary current. By measuring RCFe and δe at two resistive burdens, the slopes in Eqs. (17) and (18) give Gm and Bm, respectively. From values of Gm and Bm obtained from measurements of two resistive burdens, the ratio correction factor and phase displacement at arbitrary any e burden can be calculated according to Eqs. (15) and (16), respectively, without any additional accuracy measurement.

3.2 Comparison between calculated and measured values for ratio correction factor and phase displacement at an arbitrary burden

The ratio correction factor and phase displacement at four different burdens are measured for three different types of CTs (CT 1, CT 2, and CT 3) with different accuracy. The CT 1, CT 2 and CT 3 have the accuracy of 0.001 %, 0.01 % and 0.2 %, respectively. The values of resistance/reactance of Ra, Rb, e and e’ burden and secondary winding resistance (R2) for three different types of CT are summarized in table 3. Ra and Rb indicate resistive burden with unity power factor. The e and e’ indicate burden composing as a serial connection of a resistance and inductance. In general, the primary leakage reactance (X1) in a CT is much larger than secondary leakage reactance (X2), and the secondary windings are next to the core in a single layer. Therefore, it is reasonable to neglect X2 entirely [10, 16].

Table 3.CT 1 : 5 kA Tettex company CT, accuracy : 0.001 %, rated burden : 5 VA CT 2 : 2 kA Cte-tech company CT, accuracy : 0.01 %, rated burden : 5 VA CT 3 : 1.5 kA Yokogawa company CT, accuracy : 0.2 %, rated burden : 15 VA

The measurement results of ratio correction factor/phase displacement at Ra and Rb resistive burdens for three types of CTs are represented in the second and third row of Table 4, respectively. From the ratio correction factor/phase displacement obtained at Ra and Rb resistive burdens, the values of Gm and Bm are calculated using eqs. (17) and (18), respectively. The calculated values of Gm and Bm for three different types of CTs are presented as follows ;

Table 4.* : measured value + : calculated value

Gm = 0.000 001 S, Bm = 0.000 001 S for CT 1 Gm = 0.000 036 S, Bm = -0.000 114 S for CT 2 Gm = 0.000 659 S, Bm = -0.000 657 S for CT 3

The ratio correction factor and phase displacement are measured at arbitrary e burden for three CTs, as shown in the fourth row of table 4. The ratio correction factor and phase displacement at arbitrary e burdens are calculated using eqs. (15) and (16), respectively. The calculated ratio correction factor and phase displacement at arbitrary e burden are presented in the fifth row of table 4. The differences between calculated and measured values at arbitrary e burden for three CTs are shown in the sixth row of table 4, which are less than 14 × 10-6 for CT 1, CT 2 and 84 × 10-6 for CT 3. Meanwhile, the ratio correction factor and phase displacement are measured at arbitrary e’ burden for three CTs, as shown in the seventh row of table 4. The ratio correction factor and phase displacement calculated at arbitrary e’ burden using eqs. (15) and (16), respectively, are presented in the eighth row of table 4. The differences between calculated and measured values at arbitrary e’ burden for three CTs are shown in the last row of table 4, which are less than 14 × 10-6 for CT 1, CT 2 and 75 × 10-6 for CT 3.

The expanded uncertainty (k = 2) for the errors of the CTs was estimated to be within 30 × 10-6 in ratio and 30 microrad in phase for the case of CT with higher accuracy [17]. This implies that the measured values at arbitrary any burdens for CT 1 and CT 2 are good consistent with calculated value within the expanded uncertainty. However, the relative large difference between calculated and measured values for CT 3 implies CT 3 with lower accuracy of 0.2 % has lager measurement uncertainty than CT 1 and CT 2.

 

4. Conclusions

From the ratio correction factor and phase displacement measured at two different burdens, we have developed the methods for obtaining the ratio correction factor and phase displacement of instrument transformers at arbitrary any burdens without any additional accuracy measurement. The methods were applied the three VTs with the accuracy of 0.005 % to 0.2 % and three CTs with the accuracy of 0.001 % to 0.2 %. The validity of these methods was verified by comparing the theoretical with experimental values. The calculated values for the several instrument transformers are consistent with measured values within the expanded uncertainty. For additional accuracy measurement of transformer under test in industry, they need a reference transformer with higher accuracy, which is only located in National Measurement Institute. Thus, it is more difficult to get the accuracy data precisely without reference transformer in industry itself. The proposed method could be possible it obtain the accuracy data at any arbitrary burden without the reference transformer in industry. This is also the merit of the proposed method. Consequently, this paper is a small contribution to understanding such burden effects on instrument transformers.