**2. Circuit modelling of voltage transformers**

Classical voltage transformer (VT) is a two or three winding transformer with a relatively high transformation ratio and low rated power, intended to supply only measuring inputs

Voltage Harmonics Measuring Issues in Medium Voltage Systems 91

pass *LC* filter is mainly correlated to VT primary side parameters (*RHPF, LHPF*) and the pass band characteristic of low pass *LC* filter is mainly correlated to parameters of secondary side

Fig. 2. Theoretical transfer ratio wideband characteristic of VT modelled by classical circuit model

Based on this assumption the low corner frequency *flow* of VT transfer characteristic can be

2 *HPF*

2 *LPF*

*HPF*

*LPF*

*<sup>L</sup>* (1)

*<sup>L</sup>* (2)

Fig. 3. High pass and low pas filter representation of VT circuit model

easily defined by formula (1) and high corner frequency *fhigh* by formula (2).

*low HPF*

*high LPF*

*<sup>R</sup> f f*

*<sup>R</sup> f f*

(*RLPF, LLPF*).

of metering apparatus or protection relays extensively used in power system. VT are mostly used in medium voltage (MV) and high voltage (HV) systems for separation of the measuring and protecting circuit from high voltage hazard.

Rated primary voltages of VTs, typically used in power system, have to correspond to rated voltages of MV and HV transmission lines in particular power system. Secondary rated voltage levels usually used in a typical measuring and protection systems are: *100 V, 100/3 V, 100/3 V* what results with transformation ratios of the order from few tenth up to few hundredths for MV VT and more than thousand for HV VT. Such a high transformation ratio and low rated power of VT influence significantly its specific parameters, especially related to performance in wide frequency range.

The classical equivalent circuit model of two windings transformer widely used for modelling VT for power frequency range is presented in Fig.1. This model consists of leakage inductances of primary winding *Lp* and secondary winding *Ls* and magnetizing inductance *Lm*. Corresponding resistances represent VT losses in magnetic core *Rm* and windings *Rp*, *Rs*.

Fig. 1. Classical equivalent circuit model of a voltage transformer

For VT operated under power frequency and rated load presented circuit model can be simplified radically because magnetizing inductance *Lm*. is usually many times higher than leakage inductances *Lm.>>Lp, Ls* and VT nominal load impedance *Zload= Rld+jLload* is usually much higher than secondary leakage impedance *Zs= Rs+jLs* (*Zld>>Zs*). This assumption cannot be adopted for frequencies varying far from power frequency range because VT reactance change noticeably with frequency what results with VT transformation ratio change.

Based on this model, which characterizes two not ideally coupled inductances, frequency dependant transfer characteristic for frequencies higher than the nominal (50 or 60 Hz) can be estimated as well. Theoretical wideband transfer characteristic of VT modelled by using classic circuit model is presented in Fig.2 where low corner frequency of pass band *flow* and high corner frequency of pass band *fhigh* can be defined based on *3 dB* transfer ratio decrease margin assumption.

Low and high frequency response of VT can be determined analytically based on VT classic circuit model parameters. For wideband analysis simplification classical circuit model of VT can be represented as a serial connection of high pass filter (HPF), ideal transformer and low pass filter (LPF) (Fig.3). According to this simplification, the pass band characteristic of high

of metering apparatus or protection relays extensively used in power system. VT are mostly used in medium voltage (MV) and high voltage (HV) systems for separation of the

Rated primary voltages of VTs, typically used in power system, have to correspond to rated voltages of MV and HV transmission lines in particular power system. Secondary rated voltage levels usually used in a typical measuring and protection systems are: *100 V,* 

few hundredths for MV VT and more than thousand for HV VT. Such a high transformation ratio and low rated power of VT influence significantly its specific parameters, especially

The classical equivalent circuit model of two windings transformer widely used for modelling VT for power frequency range is presented in Fig.1. This model consists of leakage inductances of primary winding *Lp* and secondary winding *Ls* and magnetizing inductance *Lm*. Corresponding resistances represent VT losses in magnetic core *Rm* and

For VT operated under power frequency and rated load presented circuit model can be simplified radically because magnetizing inductance *Lm*. is usually many times higher than

cannot be adopted for frequencies varying far from power frequency range because VT reactance change noticeably with frequency what results with VT transformation ratio

Based on this model, which characterizes two not ideally coupled inductances, frequency dependant transfer characteristic for frequencies higher than the nominal (50 or 60 Hz) can be estimated as well. Theoretical wideband transfer characteristic of VT modelled by using classic circuit model is presented in Fig.2 where low corner frequency of pass band *flow* and high corner frequency of pass band *fhigh* can be defined based on *3 dB* transfer ratio decrease

Low and high frequency response of VT can be determined analytically based on VT classic circuit model parameters. For wideband analysis simplification classical circuit model of VT can be represented as a serial connection of high pass filter (HPF), ideal transformer and low pass filter (LPF) (Fig.3). According to this simplification, the pass band characteristic of high

*Ls* (*Zld>>Zs*). This assumption

*Lload* is usually

*3 V* what results with transformation ratios of the order from few tenth up to

measuring and protecting circuit from high voltage hazard.

Fig. 1. Classical equivalent circuit model of a voltage transformer

much higher than secondary leakage impedance *Zs= Rs+j*

leakage inductances *Lm.>>Lp, Ls* and VT nominal load impedance *Zload= Rld+j*

related to performance in wide frequency range.

*100/3 V, 100/*

windings *Rp*, *Rs*.

change.

margin assumption.

pass *LC* filter is mainly correlated to VT primary side parameters (*RHPF, LHPF*) and the pass band characteristic of low pass *LC* filter is mainly correlated to parameters of secondary side (*RLPF, LLPF*).

Fig. 2. Theoretical transfer ratio wideband characteristic of VT modelled by classical circuit model

Fig. 3. High pass and low pas filter representation of VT circuit model

Based on this assumption the low corner frequency *flow* of VT transfer characteristic can be easily defined by formula (1) and high corner frequency *fhigh* by formula (2).

$$f\_{low} = f\_{HPF} = \frac{R\_{HPF}}{2\pi L\_{HPF}}\tag{1}$$

$$f\_{\text{high}} = f\_{\text{LPF}} = \frac{R\_{\text{LPF}}}{\text{2\pi L}\_{\text{LPF}}} \tag{2}$$

Voltage Harmonics Measuring Issues in Medium Voltage Systems 93

2 2 2

Summarizing, based on classical circuit model analysis, low frequency response of VT is mostly dependant on its leakage to magnetizing impedance ratio which limits transfer characteristic in low frequency range. In applications where voltage source impedance is relatively high and cannot be neglected an extra pass band limitation is observed due to its influence (Fig.4). High frequency response of VT is dependant mainly on its leakage impedance to load impedance ratio which limits transfer characteristic in high frequency range. Increase of load impedance extends bandwidth of VT towards higher frequencies. Theoretically, for very low VT load its pass band can be very wide from the magnetic coupling point of view, nevertheless parasitic capacitances usually limit noticeably VT pass

Modelling of VT in a wide frequency range using classical circuit model is usually not adequate enough. The foremost reasons for the inadequacy of the classical circuit model are the parasitic capacitances of windings and frequency dependant voltage source and VT load impedances. Parasitic capacitances existing in windings change noticeably the

Parasitic capacitance is an effect of proximity of windings and its sections to each other and to other conductive usually grounded elements, like for example magnetic core, electric shields and other conductive elements of VT. Parasitic capacitances of VT windings are usually unwanted and unluckily unavoidable; there are only various techniques used to reduce its values or change distribution. Parasitic capacitances of VT change radically its behaviour in high frequency range usually reduce evidently the pass band bandwidth with flat transfer characteristic. Parasitic capacitances of VT windings have distributed nature strictly correlated with particular winding arrangement, therefore their identification and modelling is problematic (Vermeulen at al., 1995, Islam at al., 1997, Luszcz, 2004a, 2004b,

*BW f f LLL <sup>L</sup>* 

*high low*

Fig. 4. Influence of source and load impedance on the VT pass band

transformation ratio characteristic, particularly in high frequency range.

**3. Broadband modelling of voltage transformers** 

*BW*

band in high frequency range.

*Load p source*

(11)

(12)

*R R R*

*p s Load m*

 

 

 

2 2 2 2 2 <sup>2</sup> %2 2 *high low m load <sup>p</sup> s Load <sup>p</sup> source*

*f f LR L L L R R*

*f f LR L L L R R*

*high low m load p s Load p source*

Wideband analysis of VT transfer characteristic requires taking into account also external impedances of measured voltage source and VT load. Therefore, equivalent resistance of high pass filter *RHPF* can be defined as a sum of VT primary winding resistance and primary voltage source resistance *Rsource* (3). Respectively high pass filter equivalent inductance *LHPF* is a sum of VT magnetizing inductance *Lm*, primary winding leakage inductance *Lp* and primary voltage source inductance *Lsource* (4).

$$R\_{HPF} = R\_p + R\_{source} \tag{3}$$

$$L\_{HPF} = L\_p + L\_m + L\_{source} \tag{4}$$

Analogous equivalent parameters for LPF are as follows:

$$R\_{LPF} = R\_p + \eta^2 R\_s + \eta^2 R\_{load} \tag{5}$$

$$\mathcal{L}\_{\rm LPF} = \mathcal{L}\_p + \eta^2 \mathcal{L}\_s + \eta^2 \mathcal{L}\_{\rm load} \tag{6}$$

For *RL* type low and high pass filters *3 dB* pass band margin is obtained for the frequency at which magnitudes of filter resistance *R* is equal to magnitude of filter reactance *XL=2fL*. According to this formula, the corner frequency of low pass equivalent filter *fLPF* determines the lowest signal frequency *flow* transformed by VT (7) and the corner frequency of high pass equivalent filter *fHPF* determines the highest signal frequency *fhigh* transformed by VT (8), where *=NP/NS* is a VT winding ratio.

$$f\_{low} = \frac{R\_p + R\_{source}}{2\pi \left(L\_p + L\_m + L\_{source}\right)}\tag{7}$$

$$f\_{l\_s \text{light}} = \frac{R\_p + \eta^2 R\_s + \eta^2 R\_{\text{load}}}{2\pi \left(L\_p + \eta^2 L\_s + \eta^2 L\_{\text{load}}\right)}\tag{8}$$

Assuming that magnetizing inductance *Lm* of a typical VT is much higher than leakage inductance *Lm* of primary winding (*Lm>> Lp*) and also much higher than primary voltage source inductance (*Lm>> Lsource*) the equation (7) can be simplified to (9). Similarly, because resistance of secondary winding *Rs* is usually much lower than load resistance *Rload* (*Rs << Rload*) the equation (8) can be simplified to (10).

$$f\_{low} \approx \frac{R\_p + R\_{source}}{2\pi \cdot L\_m} \tag{9}$$

$$f\_{\rm high} \approx \frac{\eta^2 R\_{\rm load}}{2\pi \left(L\_p + \eta^2 L\_s + \eta^2 L\_{\rm load}\right)}\tag{10}$$

Finally, VT bandwidth *BW* can be estimated by using formula (11) and relative bandwidth *BW [%]* using formula (12).

Wideband analysis of VT transfer characteristic requires taking into account also external impedances of measured voltage source and VT load. Therefore, equivalent resistance of high pass filter *RHPF* can be defined as a sum of VT primary winding resistance and primary voltage source resistance *Rsource* (3). Respectively high pass filter equivalent inductance *LHPF* is a sum of VT magnetizing inductance *Lm*, primary winding leakage inductance *Lp* and

> 2 2 *RRRR LPF <sup>p</sup>*

2 2 *LLLL LPF <sup>p</sup>* 

For *RL* type low and high pass filters *3 dB* pass band margin is obtained for the frequency at which magnitudes of filter resistance *R* is equal to magnitude of filter reactance *XL=2*

According to this formula, the corner frequency of low pass equivalent filter *fLPF* determines the lowest signal frequency *flow* transformed by VT (7) and the corner frequency of high pass equivalent filter *fHPF* determines the highest signal frequency *fhigh* transformed by VT (8),

> 2 *p source*

*R R*

 *LL L* 

2 2 2 *p s load*

 

Assuming that magnetizing inductance *Lm* of a typical VT is much higher than leakage inductance *Lm* of primary winding (*Lm>> Lp*) and also much higher than primary voltage source inductance (*Lm>> Lsource*) the equation (7) can be simplified to (9). Similarly, because resistance of secondary winding *Rs* is usually much lower than load resistance *Rload* (*Rs <<* 

> 2 *p source*

*<sup>R</sup> <sup>f</sup> LLL* 

*L*

2 2 2 2

 

Finally, VT bandwidth *BW* can be estimated by using formula (11) and relative bandwidth

*R R*

*m*

*p s load*

*load*

*RRR <sup>f</sup> LLL* 

*p m source*

*p s load*

 

2 2

*low*

*low*

*f*

*high*

*f*

*high*

 

 

*R RR HPF <sup>p</sup> source* (3)

*L LL L HPF <sup>p</sup> m source* (4)

*s load* (5)

*s load* (6)

(7)

(8)

(10)

(9)

*fL*.

primary voltage source inductance *Lsource* (4).

*=NP/NS* is a VT winding ratio.

*Rload*) the equation (8) can be simplified to (10).

*BW [%]* using formula (12).

where 

Analogous equivalent parameters for LPF are as follows:

$$BVV = f\_{high} - f\_{low} = \frac{\eta^2 R\_{Load}}{L\_p + \eta^2 L\_s + \eta^2 L\_{Load}} - \frac{R\_p + R\_{source}}{L\_m} \tag{11}$$

$$\text{BVV}\left[\%\right] = 2 \cdot \frac{f\_{high} - f\_{low}}{f\_{high} + f\_{low}} = 2 \cdot \frac{\eta^2 \text{L}\_m \text{R}\_{load} - \left(\text{L}\_p + \eta^2 \text{L}\_s + \eta^2 \text{L}\_{Load}\right)\left(\text{R}\_p + \text{R}\_{source}\right)}{\eta^2 \text{L}\_m \text{R}\_{load} + \left(\text{L}\_p + \eta^2 \text{L}\_s + \eta^2 \text{L}\_{Load}\right)\left(\text{R}\_p + \text{R}\_{source}\right)}\tag{12}$$

Summarizing, based on classical circuit model analysis, low frequency response of VT is mostly dependant on its leakage to magnetizing impedance ratio which limits transfer characteristic in low frequency range. In applications where voltage source impedance is relatively high and cannot be neglected an extra pass band limitation is observed due to its influence (Fig.4). High frequency response of VT is dependant mainly on its leakage impedance to load impedance ratio which limits transfer characteristic in high frequency range. Increase of load impedance extends bandwidth of VT towards higher frequencies. Theoretically, for very low VT load its pass band can be very wide from the magnetic coupling point of view, nevertheless parasitic capacitances usually limit noticeably VT pass band in high frequency range.

Fig. 4. Influence of source and load impedance on the VT pass band

#### **3. Broadband modelling of voltage transformers**

Modelling of VT in a wide frequency range using classical circuit model is usually not adequate enough. The foremost reasons for the inadequacy of the classical circuit model are the parasitic capacitances of windings and frequency dependant voltage source and VT load impedances. Parasitic capacitances existing in windings change noticeably the transformation ratio characteristic, particularly in high frequency range.

Parasitic capacitance is an effect of proximity of windings and its sections to each other and to other conductive usually grounded elements, like for example magnetic core, electric shields and other conductive elements of VT. Parasitic capacitances of VT windings are usually unwanted and unluckily unavoidable; there are only various techniques used to reduce its values or change distribution. Parasitic capacitances of VT change radically its behaviour in high frequency range usually reduce evidently the pass band bandwidth with flat transfer characteristic. Parasitic capacitances of VT windings have distributed nature strictly correlated with particular winding arrangement, therefore their identification and modelling is problematic (Vermeulen at al., 1995, Islam at al., 1997, Luszcz, 2004a, 2004b,

Voltage Harmonics Measuring Issues in Medium Voltage Systems 95

 partially distributed – lumped parasitic capacitances are specified for most representative internal parts of winding, like for example windings layers, winding

 fully distributed – windings are modelled as a series and parallel combination of inductances and capacitances which form ladder circuit with irregular parameter

Generally more detailed parasitic capacitances representation allows obtaining higher level of model adequacy in wider frequency range. Nevertheless because of identification problems the model complexity should be kept within a reasonable level to allow achieving

Fig. 6. Parasitic capacitances arrangement in shielded VT windings

Fig. 7. Typical circuit representations of winding parasitic capacitances

shields,

high usefulness.

distribution.

Mohamed at al., 2008). Consequences of parasitic capacitances are especially significant for multilayer windings with high number of turns which is characteristic for high voltage and low power transformers like VT. The most essential categories of partial parasitic capacitances occurring in typical VT windings are presented in Fig. 5.

Identification of not equally distributed partial parasitic capacitances for particular VT require detailed specification of winding arrangement, is extremely elaborative and usually do not provide adequate enough results. Difficulties of parasitic capacitances identification can be reduced by defining lumped equivalent capacitances which represent groups of many partial capacitances related to entire winding or part of windings; for example single layer of winding. Noticeable simplification of parasitic capacitances distribution in VT winding can also be achieved by changing winding arrangement and introducing windings' shields. Example of influence of windings' shields on parasitic capacitances distribution is presented in Fig. 6 where the inter-winding capacitance, usually most noticeable, is radically reduced by introducing additional winding-to-shield capacitances.

Fig. 5. Major categories of parasitic capacitances occurring in VT windings: *Cwg* – winding to ground, *Cil* – interlayer, *Cit* – inter-turn, *Ciw* – inter-winding

Use of lumped representation of parasitic capacitances allows reducing winding model complexity and consequently simplifies noticeably its parameters identification process. A possible to apply winding model simplification level should be closely correlated with the expected adequacy in a given frequency range and depends evidently on particular winding arrangement complexity. Commonly, three methods of winding parasitic capacitances circuit representations are used to model transformer windings (Fig. 7):

 winding terminals related – where all defined lumped equivalent capacitances are connected to windings' terminals only,

Mohamed at al., 2008). Consequences of parasitic capacitances are especially significant for multilayer windings with high number of turns which is characteristic for high voltage and low power transformers like VT. The most essential categories of partial parasitic

Identification of not equally distributed partial parasitic capacitances for particular VT require detailed specification of winding arrangement, is extremely elaborative and usually do not provide adequate enough results. Difficulties of parasitic capacitances identification can be reduced by defining lumped equivalent capacitances which represent groups of many partial capacitances related to entire winding or part of windings; for example single layer of winding. Noticeable simplification of parasitic capacitances distribution in VT winding can also be achieved by changing winding arrangement and introducing windings' shields. Example of influence of windings' shields on parasitic capacitances distribution is presented in Fig. 6 where the inter-winding capacitance, usually most noticeable, is radically

Fig. 5. Major categories of parasitic capacitances occurring in VT windings: *Cwg* – winding to

Use of lumped representation of parasitic capacitances allows reducing winding model complexity and consequently simplifies noticeably its parameters identification process. A possible to apply winding model simplification level should be closely correlated with the expected adequacy in a given frequency range and depends evidently on particular winding arrangement complexity. Commonly, three methods of winding parasitic capacitances

winding terminals related – where all defined lumped equivalent capacitances are

ground, *Cil* – interlayer, *Cit* – inter-turn, *Ciw* – inter-winding

connected to windings' terminals only,

circuit representations are used to model transformer windings (Fig. 7):

capacitances occurring in typical VT windings are presented in Fig. 5.

reduced by introducing additional winding-to-shield capacitances.


Generally more detailed parasitic capacitances representation allows obtaining higher level of model adequacy in wider frequency range. Nevertheless because of identification problems the model complexity should be kept within a reasonable level to allow achieving high usefulness.

Fig. 6. Parasitic capacitances arrangement in shielded VT windings

Fig. 7. Typical circuit representations of winding parasitic capacitances

Voltage Harmonics Measuring Issues in Medium Voltage Systems 97

Concise analysis of the exemplary magnetizing and leakage impedance characteristics allow estimating characteristic frequency ranges correlated with the VT different performance. Firstly, measured impedances exhibit characteristic serial resonances which appear in frequency range around *fr1≈150 Hz* and around *fr2≈200 kHz* for magnetizing and leakage impedances respectively. On the magnetizing impedance characteristic, between these two frequencies *fr1* and *fr2*, serial resonance is clearly visible for the frequency range around *fr3≈2 kHz*. These three particular resonance frequencies *fr1*, *fr2*, *fr3* divide the frequency spectrum into three sub-ranges closely related to VT behaviour. Secondly, in the frequency range between around *3 kHz* and *20 kHz* several less meaningful resonances can be observed which are correlated with local resonances appearing in winding internal subsections.

<sup>10</sup> <sup>100</sup> 1k 10k 100k 1M 10M 30M -90

Frequency (Hz)

Fig. 10. Measured magnetizing impedance-frequency characteristics of investigated VT

<sup>10</sup> <sup>100</sup> 1k 10k 100k 1M 10M -90

Fig. 11. Measured leakage impedance-frequency characteristics of investigated VT

Frequency (Hz)


> -45 0 45

Impedance (Degree)

Impedance (absolut) (

)

100m 1 10 100 1k 10k

Impedance (Degree)

Impedance (Absolut) (

)

The influence of winding parasitic capacitances on the VT transfer ratio also depends on winding grounding method used in measuring application. In Fig. 8 two mostly used VT winding configurations are presented, where primary winding is connected to measured voltage in a different way. VTs configured as one-side grounded primary and secondary windings are commonly used for phase voltage measurement in power system, while VT floating primary winding allows for direct measurement of inter-phase voltages (Fig.9). The analysis of VT transfer characteristic for VT with both windings grounded is noticeably simpler, therefore presented further analysis based on the proposed circuit model and experimental tests have been limited to this case.

Fig. 8. Different configurations of VT primary winding grounding

Fig. 9. Typical VT connection for phase and inter-phase voltage measurement in tree phase power system

#### **4. Wideband parameters identification of voltage transformer**

Particular voltage transformation ratio of VT in high frequency range is closely related to impedance–frequency characteristics of primary and secondary windings. Therefore measurement results of VT magnetizing and leakage impedances within the investigated frequency range are the fundamental data resources for analysis its broadband behaviour. Measurement of VT impedances can be done similarly as in a typical no load and short circuit tests recommended for power frequency with the use of sweep frequency excitation. Examples of magnitude and phase characteristics of the magnetizing impedance of investigated VT are presented in Fig. 10 and leakage impedance in Fig. 11.

The influence of winding parasitic capacitances on the VT transfer ratio also depends on winding grounding method used in measuring application. In Fig. 8 two mostly used VT winding configurations are presented, where primary winding is connected to measured voltage in a different way. VTs configured as one-side grounded primary and secondary windings are commonly used for phase voltage measurement in power system, while VT floating primary winding allows for direct measurement of inter-phase voltages (Fig.9). The analysis of VT transfer characteristic for VT with both windings grounded is noticeably simpler, therefore presented further analysis based on the proposed circuit model and

experimental tests have been limited to this case.

power system

Fig. 8. Different configurations of VT primary winding grounding

Fig. 9. Typical VT connection for phase and inter-phase voltage measurement in tree phase

Particular voltage transformation ratio of VT in high frequency range is closely related to impedance–frequency characteristics of primary and secondary windings. Therefore measurement results of VT magnetizing and leakage impedances within the investigated frequency range are the fundamental data resources for analysis its broadband behaviour. Measurement of VT impedances can be done similarly as in a typical no load and short circuit tests recommended for power frequency with the use of sweep frequency excitation. Examples of magnitude and phase characteristics of the magnetizing impedance of

**4. Wideband parameters identification of voltage transformer** 

investigated VT are presented in Fig. 10 and leakage impedance in Fig. 11.

Concise analysis of the exemplary magnetizing and leakage impedance characteristics allow estimating characteristic frequency ranges correlated with the VT different performance. Firstly, measured impedances exhibit characteristic serial resonances which appear in frequency range around *fr1≈150 Hz* and around *fr2≈200 kHz* for magnetizing and leakage impedances respectively. On the magnetizing impedance characteristic, between these two frequencies *fr1* and *fr2*, serial resonance is clearly visible for the frequency range around *fr3≈2 kHz*. These three particular resonance frequencies *fr1*, *fr2*, *fr3* divide the frequency spectrum into three sub-ranges closely related to VT behaviour. Secondly, in the frequency range between around *3 kHz* and *20 kHz* several less meaningful resonances can be observed which are correlated with local resonances appearing in winding internal subsections.

Fig. 10. Measured magnetizing impedance-frequency characteristics of investigated VT

Fig. 11. Measured leakage impedance-frequency characteristics of investigated VT

Voltage Harmonics Measuring Issues in Medium Voltage Systems 99

Fig. 13. Broadband circuit model of the VT with lumped parasitic capacitances referenced to windings terminals: *Cp* – primary winding, *Cs* – secondary winding, *Cps* – inter-winding

characteristics (Fig. 10 and 11). Impedance characteristic below resonance frequencies allows to determine adequate inductances and corresponding parasitic capacitances which can be calculated based on the identified resonance frequencies (13), (14). Winding inductances and parasitic capacitances vary slightly with frequency increase therefore matching

> <sup>1</sup> 1

 <sup>2</sup> 1

For analytical evaluation of proposed circuit model two-port network representation can be efficiently used (15) and the final formula (16) for the VT transfer function characteristic vs. frequency can be determined. Transmission matrix representation of the defined two-port network model of VT allows furthermore for consideration of the influence of source impedance (17) and load impedances (18) which have to be considered in broadband

*L LL C C*

 

*source load source load source*

*p s p s V j Aj Bj V j I j Cj Dj I j*

*s load*

*Z R jL source source source*

*U j Z H j U j AZ B CZ Z DZ*

 

 

 

(16)

*LC C*

*m p s*

*p sm s ps*

(13)

(14)

(15)

(17)

<sup>2</sup> *<sup>r</sup>*

*f*

<sup>2</sup> *<sup>r</sup>*

 

( ) ( ) ( )

*f*

analysis.

approximations should be taken into account for simplification.

Simplified representation of VT magnetizing and leakage impedances is presented in Fig. 13, where characteristic resonance frequencies *fr1*, *fr2*, *fr3* and related to them frequency bands *B1, B2,* and *B3* are emphasized. The frequency band *B1* below *fr3* can be characterized as a VT pass band where the magnetizing impedance is much higher than the leakage impedance; therefore in this frequency range between primary and secondary windings magnetic coupling effect is dominating. In the frequency band *B2*, between *fr2* and *fr3*, the magnetizing impedance values are comparable to the leakage impedance, what weakening noticeably the magnetic coupling effect and the influence of VT load impedance on the transfer ratio characteristic became significant. In the frequency range above *fr2* (band *B3*) the capacitive character of magnetizing and leakage impedances of VT is dominating, what means that capacitive type of coupling between VT windings is predominant.

Fig. 12. Simplified representation of VT magnetizing and leakage impedance-frequency characteristics

The developed wideband circuit model of VT is based on the classical circuit model of the magnetic coupling presented in previous subsection. Distributed parasitic capacitances of VT windings are modelled by the lumped capacitances connected only to windings terminals (Fig.13). This assumption reduces noticeable model complexity and allows determining parasitic capacitances based on the measured windings impedances. In the analysed case primary and secondary windings of investigated VT are one side grounded which limits furthermore the number of lumped capacitances necessary to be determined.

Detailed analysis of VT magnetizing and leakage impedance-frequency characteristics and identification of specific resonance frequencies allows estimating parameters of the VT circuit model presented in Fig.13 in the analysed frequency range. The method of determination parasitic lumped capacitances is based on identification of resonance frequencies which are usually possible to determine using the measured impedance

Simplified representation of VT magnetizing and leakage impedances is presented in Fig. 13, where characteristic resonance frequencies *fr1*, *fr2*, *fr3* and related to them frequency bands *B1, B2,* and *B3* are emphasized. The frequency band *B1* below *fr3* can be characterized as a VT pass band where the magnetizing impedance is much higher than the leakage impedance; therefore in this frequency range between primary and secondary windings magnetic coupling effect is dominating. In the frequency band *B2*, between *fr2* and *fr3*, the magnetizing impedance values are comparable to the leakage impedance, what weakening noticeably the magnetic coupling effect and the influence of VT load impedance on the transfer ratio characteristic became significant. In the frequency range above *fr2* (band *B3*) the capacitive character of magnetizing and leakage impedances of VT is dominating, what

Band 1 Band 2 Band 3

fr1 fr3 log f

Fig. 12. Simplified representation of VT magnetizing and leakage impedance-frequency

The developed wideband circuit model of VT is based on the classical circuit model of the magnetic coupling presented in previous subsection. Distributed parasitic capacitances of VT windings are modelled by the lumped capacitances connected only to windings terminals (Fig.13). This assumption reduces noticeable model complexity and allows determining parasitic capacitances based on the measured windings impedances. In the analysed case primary and secondary windings of investigated VT are one side grounded which limits furthermore the number of lumped capacitances necessary to be determined. Detailed analysis of VT magnetizing and leakage impedance-frequency characteristics and identification of specific resonance frequencies allows estimating parameters of the VT circuit model presented in Fig.13 in the analysed frequency range. The method of determination parasitic lumped capacitances is based on identification of resonance frequencies which are usually possible to determine using the measured impedance

fr2

means that capacitive type of coupling between VT windings is predominant.

Zleak

characteristics

Zmag

Fig. 13. Broadband circuit model of the VT with lumped parasitic capacitances referenced to windings terminals: *Cp* – primary winding, *Cs* – secondary winding, *Cps* – inter-winding

characteristics (Fig. 10 and 11). Impedance characteristic below resonance frequencies allows to determine adequate inductances and corresponding parasitic capacitances which can be calculated based on the identified resonance frequencies (13), (14). Winding inductances and parasitic capacitances vary slightly with frequency increase therefore matching approximations should be taken into account for simplification.

$$f\_{r1} \approx \frac{1}{2\pi\sqrt{L\_m\left(\mathcal{C}\_p + \mathcal{C}\_s\right)}}\tag{13}$$

$$f\_{r2} \approx \frac{1}{2\pi\sqrt{\left(L\_p + L\_s\right)L\_m\left(C\_s + C\_{ps}\right)}}\tag{14}$$

For analytical evaluation of proposed circuit model two-port network representation can be efficiently used (15) and the final formula (16) for the VT transfer function characteristic vs. frequency can be determined. Transmission matrix representation of the defined two-port network model of VT allows furthermore for consideration of the influence of source impedance (17) and load impedances (18) which have to be considered in broadband analysis.

$$
\begin{bmatrix}
\underline{V}\_p(jo) \\
\underline{I}\_p(jo)
\end{bmatrix} = \begin{bmatrix}
\underline{A}(jo) & \underline{B}(jo) \\
\underline{C}(jo) & \underline{D}(jo)
\end{bmatrix} \begin{bmatrix}
\underline{V}\_s(jo) \\
\underline{I}\_s(jo)
\end{bmatrix} \tag{15}
$$

$$\underline{\mathbf{H}}(jo) = \frac{\underline{\mathbf{U}}\_s(jo)}{\underline{\mathbf{U}}\_{source}(jo)} = \frac{\underline{\mathbf{Z}}\_{load}}{\underline{\mathbf{A}}\underline{\mathbf{Z}}\_{load} + \underline{\mathbf{B}} + \underline{\mathbf{C}}\underline{\mathbf{Z}}\_{source}\underline{\mathbf{Z}}\_{load} + \underline{\mathbf{D}}\underline{\mathbf{Z}}\_{source}}\tag{16}$$

$$\underline{\mathbf{Z}\_{source}} = \mathbf{R}\_{source} + \mathbf{j} \circ \mathbf{L}\_{source} \tag{17}$$

Voltage Harmonics Measuring Issues in Medium Voltage Systems 101

 Z mag Z leak

10 100 1k 10k 100k 1M 10M 30M

Frequency (Hz)

Fig. 14. Magnetizing and leakage impedances of the evaluated VT calculated using

characteristics calculated for different resistive loads are presented in Fig. 15.

load can intensify adverse effects of any resonances which might arise in VT.

inductive (*PF=0.7 @ 50 Hz*) increases the *3 dB* cut-off frequency few times.

and magnetic coupling between windings becames less meaningful.

Despite the relatively low accuracy, the developed VT circuit model can be used for simulation analysis of the influence of the VT parameters and its load on the voltage transfer ratio frequency characteristic. The exemplary simulation results of VT voltage transfer ratio

Based on the presented simulation results it can be noticed that the VT voltage transfer characteristic change essentially for frequencies higher than the main resonance frequency observed on the leakage impedance, which is about *100 kHz* for the evaluated case. Above this frequency VT voltage transfer ratio depends mainly on winding parasitic capacitances

Simulation results demonstrate that in frequency range close to leakage impedance resonance VT load has the major influence on the VT transfer characteristic. Increase of resistive VT load reduces significantly VT voltage transfer ratio around this frequency. Obtained simulation results confirm that according to the analytical investigation Eq. (8), VT load rate has significant influence on VT performance in high frequency range and limits usually its pass band. According to the presented simulated transfer characteristics it is possible to expand the pass band width by lowering the VT load. Unfortunately, low VT

VT load character, capacitive or inductive, has also essential influence on the voltage transfer ratio frequency characteristics. Simulation results for VT loaded with the same impedance determined for power frequency and different power factor (1 resistive, 0.7 inductive and 0.7 capacitive) is presented in Fig. 16. The obtained results demonstrate positive impact of inductive character of VT load on the pass band width. Inductive VT load broadens the pass band towards the resonance frequency and causes effects of resonance to be more sharp. In the evaluated case, the change of the VT load from purely resistive to

100m

1

10

Modulus of impedance [

developed circuit model

]

100

1k

10k

$$\underline{\mathbf{Z}\_{load}} = \mathbf{R}\_{load} + jo\mathbf{L}\_{load} \tag{18}$$

The *A,B, C* and *D* coefficients of transmission matrix *T* (19) can be used for fully charactering electrical performance of the VT in any external condition and its frequency characteristics can be defined by measurement for a specific load condition. Based on the relationship resulting from (15); *A(j)* is a complex voltage transfer function at no load (20), *B(j)* is a complex transfer impedance with the secondary winding shorted (21), *C(j)* is a complex transfer admittance at no load (22), *D(j)* is a complex current transfer function with the secondary winding shorted (23).

$$T = \begin{bmatrix} \underline{A}(joo) & \underline{B}(joo) \\ \underline{C}(joo) & \underline{D}(joo) \end{bmatrix} \tag{19}$$

$$A(joo) = \frac{V\_p(joo)}{\underline{V\_s}(joo)}\bigg|\_{I\_s=0} \tag{20}$$

$$B(jco) = \frac{V\_p(jco)}{\underline{I\_s}(jco)}\bigg|\_{V\_s=0} \tag{21}$$

$$\mathbf{C}(jco) = \frac{I\_p(jco)}{\underline{V}\_s(jco)}\bigg|\_{I\_s=0} \tag{22}$$

$$D(joo) = \frac{I\_p(joo)}{\underline{I\_s}(joo)}\bigg|\_{V\_s=0} \tag{23}$$

#### **5. Simulation investigation of voltage transformer circuit model**

The investigated circuit model of VT can be examined by simulation in any PSpice compatible environment in the conducted disturbance propagation frequency range up to *30 MHz*. The essential verification of VT model adequacy has been done by determining magnetizing and leakage impedance characteristics which allows verifying model representation adequacy of magnetic coupling between windings. Obtained impedance characteristics are presented in Fig.14 and should be compared to the corresponding measurement results presented in Fig.10 and Fig.11.

Obtained simulation results confirm generally broad-spectrum impedance variations within few *dB* accuracy margin but many greater than few *dB* discrepancies can be observed especially close to the resonance frequencies and in frequency range above a few kHz. It should be underlined that the expected compliance level cannot be too high because of many simplifications implemented in the model. The most significant limitation of the evaluated circuit model is associated with lumped representation of winding parasitic capacitances with relation to windings terminals only.

*Z R load load load j*

The *A,B, C* and *D* coefficients of transmission matrix *T* (19) can be used for fully charactering electrical performance of the VT in any external condition and its frequency characteristics can be defined by measurement for a specific load condition. Based on the relationship

> 

> > ( )

*<sup>s</sup> <sup>I</sup>*

( )

*<sup>s</sup> <sup>V</sup>*

( )

*<sup>s</sup> <sup>I</sup>*

( )

*<sup>s</sup> <sup>V</sup>*

 

 0

0

0

0

*s*

*s*

*s*

*s*

complex transfer impedance with the secondary winding shorted (21), *C(j*

*Aj Bj <sup>T</sup> Cj Dj* 

( ) ( )

( ) ( )

( ) ( )

( ) ( )

*D j I j*

**5. Simulation investigation of voltage transformer circuit model** 

measurement results presented in Fig.10 and Fig.11.

capacitances with relation to windings terminals only.

*p*

The investigated circuit model of VT can be examined by simulation in any PSpice compatible environment in the conducted disturbance propagation frequency range up to *30 MHz*. The essential verification of VT model adequacy has been done by determining magnetizing and leakage impedance characteristics which allows verifying model representation adequacy of magnetic coupling between windings. Obtained impedance characteristics are presented in Fig.14 and should be compared to the corresponding

Obtained simulation results confirm generally broad-spectrum impedance variations within few *dB* accuracy margin but many greater than few *dB* discrepancies can be observed especially close to the resonance frequencies and in frequency range above a few kHz. It should be underlined that the expected compliance level cannot be too high because of many simplifications implemented in the model. The most significant limitation of the evaluated circuit model is associated with lumped representation of winding parasitic

*I j*

*C j V j*

*p*

*I j*

*B j I j*

*p*

*V j*

*A j V j*

*p*

*V j*

resulting from (15); *A(j*

transfer admittance at no load (22), *D(j*

secondary winding shorted (23).

*)* is a complex voltage transfer function at no load (20), *B(j*

*L* (18)

*)* is a complex current transfer function with the

(20)

(21)

(22)

(23)

(19)

*)* is a

*)* is a complex

Fig. 14. Magnetizing and leakage impedances of the evaluated VT calculated using developed circuit model

Despite the relatively low accuracy, the developed VT circuit model can be used for simulation analysis of the influence of the VT parameters and its load on the voltage transfer ratio frequency characteristic. The exemplary simulation results of VT voltage transfer ratio characteristics calculated for different resistive loads are presented in Fig. 15.

Based on the presented simulation results it can be noticed that the VT voltage transfer characteristic change essentially for frequencies higher than the main resonance frequency observed on the leakage impedance, which is about *100 kHz* for the evaluated case. Above this frequency VT voltage transfer ratio depends mainly on winding parasitic capacitances and magnetic coupling between windings becames less meaningful.

Simulation results demonstrate that in frequency range close to leakage impedance resonance VT load has the major influence on the VT transfer characteristic. Increase of resistive VT load reduces significantly VT voltage transfer ratio around this frequency. Obtained simulation results confirm that according to the analytical investigation Eq. (8), VT load rate has significant influence on VT performance in high frequency range and limits usually its pass band. According to the presented simulated transfer characteristics it is possible to expand the pass band width by lowering the VT load. Unfortunately, low VT load can intensify adverse effects of any resonances which might arise in VT.

VT load character, capacitive or inductive, has also essential influence on the voltage transfer ratio frequency characteristics. Simulation results for VT loaded with the same impedance determined for power frequency and different power factor (1 resistive, 0.7 inductive and 0.7 capacitive) is presented in Fig. 16. The obtained results demonstrate positive impact of inductive character of VT load on the pass band width. Inductive VT load broadens the pass band towards the resonance frequency and causes effects of resonance to be more sharp. In the evaluated case, the change of the VT load from purely resistive to inductive (*PF=0.7 @ 50 Hz*) increases the *3 dB* cut-off frequency few times.

Voltage Harmonics Measuring Issues in Medium Voltage Systems 103

Experimental investigations have been done for voltage transformers typically used in MV power system with primary and secondary windings grounded. Exemplary measurement results presented in this chapter have been obtained for VT of *50 VA* rated power and *20 kV/0.1 kV* nominal transformation ratio. Parameters of the proposed VT circuit model for simulation have been identified by analysis of secondary windings impedance-frequency characteristics measured for no load condition (magnetizing inductance – Fig.10) and short circuit condition (leakage inductance – Fig.11). Measurements have been done in frequency range from *10 Hz* up to *30 MHz*, which is a range typically used for the analysis of conducted disturbances in power system. Particular attention has been paid to the frequency range below *10 kHz* which is obligatory for power quality analysis, especially for analysis of

The measured voltage transfer characteristics of evaluated VT for nominal load and no load conditions are presented in Fig. 17. Based on these results the *3 dB* high frequency pass band of the evaluated VT can be estimated to be about *2 kHz*. For frequencies higher than 2 kHz a number of less meaningful resonances are clearly visible on the VT voltage transfer characteristic which are not adequately characterized by the evaluated circuit model. This inadequacy is associated with extra internal resonances appearing in windings which cannot be properly represented by lumped parasitic capacitances referenced only to windings terminals. In order to model these phenomena more complex circuit model are required which take into account more detailed distributed representation of partial parasitic

10Hz 100Hz 1kHz 10kHz

 No load Nominal load

Frequency [Hz]

Fig. 17. Normalized voltage transfer ratio of the VT for no load and nominal resistive load

**6. Experimental tests of voltage transformer transfer characteristic** 

power system voltage harmonics related phenomena.

capacitance of VT primary winding.

0.0

0.5

1.0

1.5

Normalized voltage transfer ratio [-]

2.0

2.5

3.0

Capacitive character of VT load induces opposite effect and narrows VT pass band width significantly. Similarly, the change of VT load from purely resistive to capacitive (*PF=-0.7 @ 50 Hz*) decrease of VT pass band width is about tens times.

Fig. 15. Simulation results of the influence of resistive load of VT on voltage transfer ratio frequency characteristic

Fig. 16. Influence of the character of VT load on voltage transfer ratio frequency characteristic – simulation results

Capacitive character of VT load induces opposite effect and narrows VT pass band width significantly. Similarly, the change of VT load from purely resistive to capacitive (*PF=-0.7 @* 

> Under Load (0.1) Nominal Load (1) Over Load (10)

10 100 1k 10k 100k 1M 10M 30M

Frequency (Hz)

R

 RL, PF=0.7 RC, PF=0.7

10 100 1k 10k 100k 1M 10M 30M

Frequency (Hz)

Fig. 16. Influence of the character of VT load on voltage transfer ratio frequency

Fig. 15. Simulation results of the influence of resistive load of VT on voltage transfer ratio

*50 Hz*) decrease of VT pass band width is about tens times.

10m

10m

characteristic – simulation results

100m

1

10

100

100m

Normalized voltage transfer ratio []

frequency characteristic

Modulus of impedance [

]

1

10
