**Appendix**

152 Induction Motors – Modelling and Control

voltage magnitude, as given in Fig. 2.

a. reactive loads decreasing

above100 kW,

above100 kW

10 kW;

kW,

kW.

the literature.

90%, respectively.

The results of the analysis presented in this chapter are summarised in the following.

The results of the research show dependencies of the input power and reactive load on

1. Decreasing voltage magnitude by 1%, by setting voltages in range of *Un*±5%, leads to:

b. power losses (and active input powers) decreasing/increasing (sign " - ")






2. On the basis of the investigation presented in this paper, it is confirmed that there are significant possibilities for energy savings by means of voltage magnitude setting, within values *Un*±5%, in networks with induction motors which are light loaded (<70%). 3. Setting voltage within band 0.9 *Un* - 0.95 *Un* is not recommended, even if it leads to reduced power losses and reactive loads, because starting and maximal torque are

4. It is explained that the rotor inverse resistance and rotor inverse reactance are higher for √2 times compared to the rotor resistance and rotor reactance in short-circuit mode, since current frequency of the negative sequence in the rotor winding is twice as high (*fr,NS* ≈ 2*f1*=2*fr,SC*), i.e. they are higher by 1.41 times than the corresponding values given in

5. Voltage unbalance causes increase of the motor heating, occurrence of inverse torques and a small increase in motor slip. Thus, for the voltage asymmetry of 2%, 3%, 4% and 5%, this causes an increase in power losses of 5.5%, 12%, 22% and 34% of motor nominal losses. Corresponding values of derating factors are 0.97, 0.94, 0.88 and 0.81, respectively, as noted in NEMA standards, so the acceptable voltage asymmetry is 2%. 6. Based on the actual calculation and analysis, it was found that the effects of an unbalance on power loss are smaller for motors of nominal power ≤ 10 kW. Thus,

7. Generally, motor operation is not allowed when voltage asymmetry is greater than 5%, because, in some cases, current and losses in one phase could be increased for 38% and

*A) Effect of voltage magnitude on motor power losses and motor reactive loads* 

decreased and it can also cause motor operation instability.

acceptable voltage asymmetry for these motors could be 3%.

*B) The most important conclusions regarding motor operation with the unbalanced voltage* 

**5. Summary** 

For deriving equations for electromagnetic torque and power, the equivalent Г-circuit, shown in Fig. 13, is used (Kostic, 2010):

**Figure 13.** Equivalent Г-circuit of induction machine

Equation (9) is completely derived in this Appendix.

1. Electromagnetic power (Pem,N) **at rated load**, i.e. at slip s=sN, can be expressed as following:

$$P\_{em,N} = T\_{em,N} \cdot \Omega\_1 = \frac{I\_L^2 \cdot \sigma\_s^2 R\_r}{s\_N} = \frac{\mathcal{U}\_1^2 \cdot \sigma\_s^2 R\_r / s\_N}{\left(\sigma\_s R\_s + \sigma\_s^2 R\_r\right)^2 + \left(\sigma\_s X\_s + \sigma\_s^2 R\_r / s\_N\right)^2} \tag{51}$$

For motors with power within the range of 1÷200kW, values for sN are 0.05÷0.01, respectively, and therefore: *s 2Rr/sN = (20÷100)·sRs* and *sXs + s 2Xr) ≈0.20·s 2Rr/sm.* 

$$P\_{em,N} = T\_{em,N} \cdot \Omega\_1 = \frac{I\_L^2 \cdot \sigma\_s^2 R\_r}{s\_N} \approx \frac{U\_1^2 \cdot \sigma\_s^2 R\_r / s\_N}{(1.15 \div 1.05)^2 \left(\sigma\_s^2 R\_r / s\_N\right)^2} = \frac{U\_1^2}{(1.15 \div 1.05)(\sigma\_s^2 R\_r / s\_N)}\tag{52}$$

2. **Regime with maximum input power,** i.e. at *s=sPm*, accrues when resistance (*sX*σ*s +s 2X*σ*r*) and reactance in load branch (*sRs+s 2Rr /sm*) are equal, i.e., and when the load branch impedance is *Z2,m=√*2*(sXσs+s 2Xσ*r). Corresponding electromagnetic power (*Pem,Pm*) on the resistance *s 2Rr /sm* is:

$$P\_{em,Pm} = T\_{em,Pm} \Omega\_1 = I\_L^2 \cdot \sigma\_s^2 \xrightarrow[S\_{Pm}]{} \frac{\mathcal{U}\_1^2 \cdot \sigma\_s^2 \mathcal{R}\_r / s\_{Pm}}{\mathcal{Z}(\sigma\_\sigma X\_{\sigma s} + \sigma\_\sigma^2 X\_{\sigma r})^2} \tag{53}$$

Effects of Voltage Quality on Induction Motors' Efficient Energy Usage 155

*Q P T T* (62)

then, based on equations (A-7), (A-9) and (A-10), it follows:

, ,

component of reactive power in load branch, *QLN=0.5·PN/(Tm/TN).* 

**Author details** 

Miloje Kostic

**6. References** 

, , , ,

*em Pm em Pm*

*LN N*

*em N em N*

*T T*

*T T*

(0.08 0.95) (1.15 1.05) (0.98 1.01) 0.5 / ( / ) <sup>2</sup> 0.95 0.99 2

, , 0.5

*em Pm em Pm*

*em N em N*

*T T*

(61)

*LN N N mN*

Since the maximum torque (*Tm≈Tem,m*), which is catalogue data, is greater up to 2% from mentioned torque (*Tem,Pm*) in the regime with maximum input power, i.e. *Tem,Pm*≤ 1.02 *Tm*, it might be concluded that the equation (9) sufficiently accurate for calculating the rating

Aníbal, T. de Almeida Fernando J. T. E. Ferreira; João Fong & Paula Fonseca (December 2007). *EUP Lot 11 Motors, Final Report*, ISR-University of Coimbra, Lot 11-8-280408. Boldea, I.S. & Nasar, A. (2002). *The Induction Machine Handbook*, 2002 by CRC Press LLC. Bonnett, A.H. (2000). An overview of how AC induction motors' performance has been affected by the October 24, 1997 Implementation of the Energy Policy Act of 1992, *IEEE* 

Fei, R.; Fuch, E.F.& Huang, H. (December 1989). Comparison of two optimization techniques as applied to three-phase induction motor design, *IEEE Transactions energy Conversion*,

Fink, D.G. (1983). *Standard Handbook for Electrical Engineers* (1983), 11th Edition McGraw-Hill

Hamer, P. S.; Love, D. M. & Wallace, S. E. (1997). Energy Efficient Induction Motors Performance Characteristic and Life Cycle Cost Comparison for Centrifugal Loads, *IEE* 

IEC 60034-31 (2010). *Guide for selection and application of energy-efficient motors including* 

Kostic, M (2010). *Energy Efficiency Improvement of Motors in Drives*, Electrical Engineering Institute Nicola Tesla, Belgrade, 2010, pp.325 (in Serbian), ISBN 978-86-83349-11-1. Kostic, M. & Kostic, B. (2011). Motor Voltage High Harmonics Influence to the Efficient Energy Usage, Invited paper for *15th WSEAS International Conference on Systems*, *Proc.* 

IEC 60034-30 (2008). *Efficiency classes of single speed three-phase cage induction motors*.

Ivanov-Smolensky, A. (1982). *Electrical Machines*, Vol. 2, Mir Publishers, 1982, pp. 464.

*Q P P TT*

2

*Electrical Engineering Institute "Nikola Tesla", Belgrade University, Belgrade, Serbia* 

*Transaction on Industry Applications*, Vol.36, No1, 2000, pp. 242-256.

Book Company, 1983, New York, pp. 2462, ISBN 0-07-020974-X

*Trans. Ind. Applications*, No. 5, 1997, pp. 1312-1320.

*pp.* 276-281, Corfu Island, Greece, July 2011.

Vol.4, pp. 651-660, December 1989.

*variable-speed applications.*

Since for motors with power within the range of 1÷200kW, values for corresponding slip are *sPm* = 0.25÷0.05, respectively, the skin effect in the bars of the squirrel-cage is minor (the depth of penetration *δr*(*smf1*) ≥ *Hb* -the bar (conductor rotor) height), so it is *s 2Rr /sm* = (5÷20) *sRs*. Consequently, it is:

$$\left(\sigma\_s^2 R\_r \text{s}\_{Pm} = (0.8 \div 0.95) \cdot \left(\sigma\_s R\_s + \sigma\_s^2 R\_r \text{ / s}\_{Pm}\right) = (0.8 \div 0.95) \cdot \left(\sigma\_s X\_{\sigma s} + \sigma\_s^2 X\_{\sigma r}\right) \tag{54}$$

and the electromagnetic power (*Pem,m*), in the regime with maximum input power, is:

$$P\_{cm,Pm} = T\_{cm,Pm} \cdot \Omega\_1 = \frac{I\_L^2 \cdot \sigma\_s^2 R\_r}{s\_{p\_{\rm pr}}} \approx \frac{\mathcal{U}\_1^2 \cdot (0.8 \div 0.95)(\sigma\_s X\_{\rm crs} + \sigma\_s^2 X\_{\rm cr})}{2 \cdot (\sigma\_s X\_{\rm crs} + \sigma\_s^2 X\_{\rm cr})^2} = \frac{\mathcal{U}\_1^2 \cdot (0.8 \div 0.95)}{2 \cdot (\sigma\_s X\_{\rm crs} + \sigma\_s^2 X\_{\rm cr})} \tag{55}$$

3. If *s 2Rr /sN* is expressed from (A-4), and (*sXσs+s 2Xσr*) is expressed from (A-5), then it is:

$$\left\{\sigma\_s^2 \mathcal{R}\_s + \sigma\_s^2 \mathcal{R}\_r / s\_N = \frac{\mathcal{U}\_1^2}{T\_{em,N} \cdot \Omega\_1 \{1.15 \div 1.05\}}\right\} \tag{56}$$

$$
\sigma\_s X\_{\sigma s} + \sigma\_s^2 X \sigma\_r = \frac{\mathcal{U}\_1^2 \cdot (0.8 + 0.95)}{2T\_{em, Pn} \cdot \Omega\_1} \tag{57}
$$

On the base of (A-6) and (A-7), it is obtained:

$$\frac{\sigma\_s X\_{\sigma s} + \sigma\_s^2 X\_{\sigma r}}{\sigma\_s^2 R\_s + \sigma\_s^2 R\_r / s\_N} = \frac{T\_{cm,N}}{2T\_{cm,Pm}} \cdot \text{(0.8 \div 0.95)} \cdot \text{(1.15 \div 1.05)}\tag{58}$$

Reactive power in the load branch of Г-circuit, under rated condition, *Q2N≈ QLN* (*QLN* – load component of reactive power), can be expressed in terms of the electromagnetic power, *Pem,N*

$$Q\_{2N} = P\_{em,N} \cdot \frac{\sigma\_s X\_{\sigma s} + \sigma\_s^2 X \sigma\_r}{\sigma\_s^2 R\_s + \sigma\_s^2 R\_r / s\_N} \approx Q\_{LN} \tag{59}$$

Since the relation between the electromagnetic power (*Pem,N*) and the rating power (*PN*) is:

$$P\_{em,N} = P\_N \cdot \frac{\sigma\_s^2 R\_s + \sigma\_s^2 R\_r / s\_N}{\{\sigma\_s^2 R\_r / s\_N\}} \cdot \frac{1}{1 - s\_N} \tag{60}$$

then, based on equations (A-7), (A-9) and (A-10), it follows:

$$Q\_{LN} = P\_N \cdot \frac{T\_{cm,N}}{2T\_{cm,Pm}} \cdot \frac{(0.08 \div 0.95) \cdot (1.15 \div 1.05)}{0.95 \div 0.99} = \frac{T\_{cm,N}}{2T\_{cm,Pm}} \cdot (0.98 \div 1.01) \approx 0.5 P\_N \; / \left(T\_m / T\_N\right) \tag{61}$$

$$Q\_{LN} \approx P\_N \cdot \frac{T\_{cm,N}}{2T\_{cm,Pm}} = 0.5 \cdot \frac{T\_{cm,N}}{T\_{cm,Pm}} \tag{62}$$

, ,

*em Pm em Pm*

Since the maximum torque (*Tm≈Tem,m*), which is catalogue data, is greater up to 2% from mentioned torque (*Tem,Pm*) in the regime with maximum input power, i.e. *Tem,Pm*≤ 1.02 *Tm*, it might be concluded that the equation (9) sufficiently accurate for calculating the rating component of reactive power in load branch, *QLN=0.5·PN/(Tm/TN).* 
