A New Definition for Total Harmonic Distortion in Inverter-fed Induction Motors Including Loss Considerations

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A New Definition for Total Harmonic Distortion in Inverter-fed Induction Motors Including Loss Considerations H. Ghafoori Fard Department of Electrical Engineering Amir Kabir University of Technology Tehran-Iran S. Kaboli H. Oraee*

School of Electrical Engineering Sharif University of Technology Tehran-Iran

A New Definition for Total Harmonic Distortion in Inverter-fed Induction Motors Including Loss Considerations

Summary: Harmonic information is necessary tools in the estimation of motors life and performance. The conventional expression of harmonic amount is carried out by total Harmonic distortion. This distortion factor is only related to amplitude of harmonics while the motor loss is affected by frequency domain of them. In this paper a new definition of this factor has been presented. This definition includes the frequency of harmonics as well as their amplitudes. Experimental results verify the validity of this new definition. Keywords: Harmonics, Core loss, Derating, Induction motor

I. Introduction Several industrial and domestic loads comprising static power converter, such as arc melting furnaces, induction heating devices, switch mode power supplies inject current harmonics in the power system. Such phenomena as well as iron saturation in distribution transformers when operated at voltages exceeding the nominal value lead to voltage waveform distortion. Induction motors constitute the most popular energy converters and are sensitive to harmonic voltages. Both their efficiency and performance can be considerably affected by the power quality of the supply. On the other hand, a considerable proportion of induction motors are fed by voltage source inverters. The output voltage waveform of an inverter has a non sinosuidal shape and includes some harmonics. Estimation of motor life and its performance requires calculation of voltage harmonics and their effect of motor temperature rise. One of the most popular tools for explaining the amount of harmonics is the total harmonic distortion (THD) [1], [2], [3]. The common definition for this parameter does not include the frequency content of
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harmonics and only deals with the amplitude. On the other hand, motor loss is a function of both frequency and amplitude of the supply voltage. Therefore, the definition of THD can not predict the temperature rise of motor carefully [4]. In this paper a new definition of THD has been presented which is included not only the amplitude of harmonics but also the frequency of them. This new expression scales the amplitude of harmonics versus their frequency. This study has been performed on the voltage waveform because access to voltage information of motor is simpler than its other parameters. Some experimental results will be presented that verify the validity of this new expression of THD in estimation of motor temperature rise. II. Calculation of Flux Density caused by Voltage Harmonic When a sinosuidal voltage V1(t) is applied to stator of an induction motor shown in figure (1), the relation between the motor and voltage parameters are presented by E1 = 4.44 N s fB1 A (1)

where E1 and f are the amplitude and frequency of applied voltage, Ns is the number of turns in stator winding, A is the stator cross section and B1 is the amplitude of magnetic flux density [4]. If another voltage, Vh(t) which has frequency equals to h*f and amplitude as Eh is applied to stator then relation (2) is changed to E h = 4.44 N s (h * f ) Bh A where Bh is the amplitude of flux density created by harmonic voltage Eh [5], [6]. (2)

Power Supply

Induction Motor

Fig. (1). Induction motor and its power supply Therefore the relation between the amplitude of flux density in this case versus amplitude of flux density for fundamental component (B1) is equal to
Bh = B1 Eh h * E1

(3)

III. Core loss for Voltage Harmonic The hysteresis loss and the eddy current loss are lumped together as the core loss: Pc = Ph + Pe The power loss in the core due to the hysteresis effect is [7]: (4)

Ph = K h B n f

(5)

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where B is the maximum of flux density, Kh is a constant and f is the frequency. The eddy current loss in a magnetic core subjected to a time varying flux is [7]:

Pe = K e B 2 f 2 (6)
Thus, The hystersis loss of fundamental component, PH1, is given by

PH 1 = K H B1 f
The hystersis loss for h-th harmonic (Vh(t)), PHh, is given by
PHh = K H Bh (h * f )
n

n

(7)

(8)

Therefore, hystersis loss for hth harmonic versus hystersis loss of fundamental component is calculated from (3), (7) and (8):

PHh

 Eh  = K H  B1  h * E  (h * f )  1  
E  1 = K H B f  h  n −1 E  h  1

n

(

n 1

)

n

E  1 = PH 1  h  n −1 (9) E  h  1 Similarly, The eddy current loss of fundamental component, PE1, is given by PE1 = K E B1 f 2 The eddy current loss for hth harmonic (Vh(t)), PEh, is given by
PEh = K E Bh (h * f ) 2
2

n

2

(10)

(11)

Therefore, eddy current loss for hth harmonic versus eddy current loss of fundamental component is calculated from (3), (10) and (11):  Eh  2 PEh = K E  B1   h * E  (h * f ) 1   E = KE B f  h E  1
2

(

2 1

2

)

   

2

E = PE1  h E  1

   

n

(12)

Therefore, harmonic core loss caused by h-th voltage component, Vh(t), is calculated from (9) and (12): PCh = PHh + PEh

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E = PH 1  h E  1

 1 E   n −1 + PE1  h   h E    1

n

n

(13)

Each of fundamental hysteresis loss, PH1, and fundamental eddy current loss, PE1, consist certain proportion of fundamental core loss PCh as the following: PH 1 = a1 * PC1 PE1 = a 2 * PC1 where a1 and a2 are two constant so that a1+a2=1. Substituting of (14) into (13) leads to: E PCh = PC1 [ a1  h E  1  1 E  n −1 + a 2  h  h E   1
n

(14)

  ] (15)  

n

IV. New THD Definition Total harmonic distortion is defined for a waveform which has some harmonics on the fundamental. For a voltage waveform THD is given by

(16) V1 where V1 is the amplitude of fundamental component and V2 , V3 , … are the amplitude of the other harmonics. Comparison of (15) and (16) shows that harmonic order h which is an important parameter in calculation of core loss does not enter in the conventional definition of THD. Therefore, THD presented in relation (16) is not a suitable scale for evaluation of harmonic core loss. Total harmonic core loss is the summation of harmonic core loss as follows: PC − Harmonics = ∑ PCh
h

THD =

V 22 + V 32 + ...

E  1 E  = PC1 * {[ a1  h  n −1 + a 2  h  ]} (17) E  h E  h  1  1 The fundamental core loss,Pc1, is constant for a motor and then modified definition of THD (MTHD) is given by:



n

n

MTHD =


h

E {[ a1  h E  1

 1 E   n −1 + a 2  h  ]}  h E    1

n

n

(18)

In this definition, the most important parameters are coefficients of hysteresis and eddy current loss. It can be seen that these coefficients are suppose to be constant. However, these coefficients are a function of motor operating point. To show this dependence some experiments have been performed on some ferromagnetic core. The experimental setup, shown in Fig. 2 consists of a transformer, harmonic generator equipment, and measuring equipment for power loss. Two different transformer cores are used in the tests. The specifications of transformers used in the tests are shown in table I. Figs. 3 and 4 show the variation of hysteresis loss coefficient versus frequency. Since the distinct determination of Kh and n is difficult, the merged form of these coefficients, KhBn, is studied for a constant flux density (B=1T). It can be seen that these coefficients increase with frequency. Fig. 5 and 6

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show the eddy current loss coefficient decreases with frequency. Therefore, these coefficients can not be assumed constant in modified THD definition. TABLE I:
CHARACTERISTICS OF TRANSFORMER A

Parameter Rated Power Rated voltage Shape TABLE I:

Value 1 KW 220/24 Toroid

CHARACTERISTICS OF TRANSFORMER B

Parameter Rated Power Rated voltage Shape

Value 100 W 220/12 E-type

Fig.2. Experimental setup

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4.2 4 3.8 3.6 Kh*Bn 3.4 3.2 3 2.8 2.6 0 100 200 300 Frequency(Hz) 400 500

Fig.3. Variation of hysteresis loss coefficients with frequency in core A
1 0.9 0.8 Kh*Bn 0.7 0.6 0.5 0.4 0 200 400 600 Frequency(Hz) 800 1000

Fig.4. Variation of hysteresis loss coefficients with frequency in core B

0.016 0.014 Ke*B2 0.012 0.01 0.008 0.006 0.004 0 200 400 600 Frequency(Hz) 800

Fig.5. Variation of eddy loss coefficients with frequency in core A

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1.4 1.2 1 Ke*B2 0.8 0.6 0.4 0.2

x 10

-3

0 0

200

400 600 Frequency(Hz)

800

1000

Fig.6. Variation of eddy loss coefficients with frequency in core B V. Experimental Results To justify the theoretical results some tests are carried out an induction motor. The motor specifications are given in the following. Table II. Motor Specifications PARAMETER Nomonal Power Nominal Voltage Nominal Frequency Number of Slots Depth of slots Length of Slots VALU E 2.2 KW 220 V 50 Hz 30 20 mm 240 mm

Figure (7) shows two voltage waveforms applied to the motor. These waveforms have the same fundamental component and different harmonic content. Thus, fundamental core loss for the both waveform is equal but harmonic core loss is different. The conventional THD for waveform shown in figure (7)(a) is equals to 31% and for waveform shown in figure (7)(b) is equals to 42%. Therefore, it is predicted that motor temperature rise for the second waveform is more than the first waveform. But MTHD factor for the first waveform is greater than the second and it expresses that motor temperature rise for the first waveform is greater than the other voltage. Figure (8) shows the motor temperature rise for these voltage waveforms. It is obvious that the first voltage waveform has more temperature rise. So its harmonic loss is more than the second voltage waveform.

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Fig. (7): Applied waveform of motor

Fig. (8): Temperature rise of motor for two voltage waveform V. Conclusion In this paper a new definition of THD has been presented that includes motor loss. This modified THD is a convenient tool for estimation of temperature rise in the motor. This new expression related to not only amplitude of harmonics but also frequency of them.

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Experimental results verify the validity of this new definition in the evaluation of motor harmonic loss. REFERENCES [1]. Y. Bertin, E. Videcoq, “ Thermal Behavior of an Electrical Motor Through a Reduced Model”, IEEE Transaction on Energy Conversion, VOL. 15, NO. 2, JUNE 2000. [2]. A. L. Shenkman, M. Chertkov, “ Experimental Method for Synthesis of Generalized Thermal Circuit of Polyphase Induction Motors”, IEEE Transaction of Energy Conversion, VOL. 15, NO. 3, JUNE 2000. [3]. K. Yamazaki, “Harmonic Copper and Iron Losses Calculation of Induction motor Using Nonlinear Time-Stepping Finite Element Method”, IEEE International Electric Machines and Drives Conference, 2001, pp. 551-553. [4]. J. P. G. Abereu, A. E. Emanuel, “ Induction Motor Thermal Aging Caused by Voltage Distortion and Imbalance: Loss of Useful Life and Its Stimated Cost”, IEEE International Electric Machines and Drives Conference, 2001, pp. 105-114. [5]. N. Stranges, R. D. Findlay, “Methods for Predicting Rotational Iron Losses in Three Phase Induction Motor Stators”, IEEE Transaction on Magnetics, VOL. 36, NO. 5, 2000, pp. 3112-3114. [6]. J. Saitz, “Computation of the Core Loss In an Induction Motor Using the Vector Preisach Hystresis Model Incorporated in Finite Element Analysis”, IEEE Transaction on Magnetics, VOL. 36, NO. 4, 2000, pp. 769-773. [7]. S. Cuming, " Harmonic Field Effects in Induction Machines", Czechoslovak Academy of Siences,1977.

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