**4.3. Current harmonics and harmonic losses in motors supplied from the network with voltage harmonics**

The harmonic fields induce a current in the rotor and, as a result of interaction, are given corresponding asynchronous torques. The direction of these harmonic torques coincides with the fundamental torque direction, when *h*=6*n*+1, and harmonic torque is the opposite to the fundamental torque direction when *h*=6*n*-1. In a motor regime with slip equal to *s* = 0.01– 0.06, in relation to the rotating fields of harmonics, the slip is approximately equal to 1, *sh* = 1 ± 1/*h* ≈ 1.

In Radin et al. (1989), the following assumptions (partly wrong ones) are often listed:


while the reactance values are somewhat reduced, *Xsh* ≤ *hX*s, *Xrh* ≤ *hXr*, since the values of corresponding inductances are decreased, *Lr(hf)* < *Lr* and *Ls(hf)* ≤ *Ls*.

The truth however is slightly different (Kostic, 2010):


By this and (43), an expression for determining the value of the motor resistance *RM,h* for harmonic order *h* ≥ 5 is obtained:

$$R\_{Mh} = \frac{4}{3}R\_s + 0.03 \cdot \sqrt{h} \tag{49}$$

## **Example 1**

148 Induction Motors – Modelling and Control

**network with voltage harmonics** 

formula *Rrh* ≈ *hRr*, and

shown in Kostic (2010),

harmonic order *h* ≥ 5 is obtained:

*e -* ≈ *Rr* /3 ≈ *Rs* /3.

according to formula *Rsh* ≈ *hRs* and *Rrh* ≈ *hR*r,

The truth however is slightly different (Kostic, 2010):

voltage motor powers of 100–300 kW,

corresponding inductances are decreased, *Lr(hf)* < *Lr* and *Ls(hf)* ≤ *Ls*.

*sh* = 1 ± 1/*h* ≈ 1.

Commonly, these power losses are calculated as a percentage of nominal power losses in motor windings (%*PCuN*). Thus, assuming that losses *PCuN* make up one half of the total

<sup>2</sup> <sup>2</sup> <sup>100</sup>

**4.3. Current harmonics and harmonic losses in motors supplied from the** 

In Radin et al. (1989), the following assumptions (partly wrong ones) are often listed:

the values of the stator and rotor leakage reactance (*Xsh* = *hXs, Xrh* = *hXr*),




while the reactance values are somewhat reduced, *Xsh* ≤ *hX*s, *Xrh* ≤ *hXr*, since the values of




By this and (43), an expression for determining the value of the motor resistance *RM,h* for

<sup>4</sup> 0.03

*RR h Mh s* (49)

3

cos 1 *M,h M,h Cu,h CuN R I <sup>η</sup> P %P*

The harmonic fields induce a current in the rotor and, as a result of interaction, are given corresponding asynchronous torques. The direction of these harmonic torques coincides with the fundamental torque direction, when *h*=6*n*+1, and harmonic torque is the opposite to the fundamental torque direction when *h*=6*n*-1. In a motor regime with slip equal to *s* = 0.01– 0.06, in relation to the rotating fields of harmonics, the slip is approximately equal to 1,

*η η* 

(48)

power losses in the motor, their value can be determined from the formula:

For motors with power 5 kW - 400 kW, in the same order, the ranges of parameter values are given (Kravcik, 1982) for:


and the


Application of the suggested method is illustrated by Tab. 6. The results show the amounts of increase in power losses due to the presence of harmonics in a given amount (*Ui* = 5%, *i* = 5, 7, .., 37) in the supply voltage.


**Table 6.** Values of harmonic resistances (*RM,h*), reactances (*XM,h*) and impedances (*ZM,h*) and corresponding currents and harmonic losses for motors > 100 kW (left) and < 5 kW (right), for the given values of voltage harmonics *Uh,i* (p.u.) = 5%

The results in Tab. 6 show that, at the maximum permitted content of harmonics in supply voltage (*Uh,i* = 5%, *i* = 1–37), the percentage of harmonic losses, (in units of the nominal motor power *PM,h* [%*PN*]), is relatively low:

Effects of Voltage Quality on Induction Motors' Efficient Energy Usage 151

[%*I <sup>n</sup>* ] [%*P <sup>n</sup>* ] [%*P Cun* ]

*ΣP M,h ΣP M,h* 0.94 -1.68 34.92-19.05



as compared to those values for the fundamental harmonic in short-circuit mode.

increases √h times, for the harmonics of order *h*.

1 1.00 0.015-0.05 0.030 0.045-0.080 0.161 0.167-0.180

Total *THD u =* 30.3% *THD <sup>i</sup>* =31.5%

motors, by extrapolation.

This last figure corresponds to the values that are found in the literature, while the value of 19.05%*PCu,N*, for motors of lower power (3–10 kW), is much higher than the figure which is referred to in the literature (by about 5–10%). The reason for this lies in the fact that it is (wrongly) believed that the resistance of the rotor does not change for harmonic frequencies, i.e. that is identical for all harmonics (*Rr,h* = *Rr,*<sup>1</sup> = *Rr* = *Const*), which brings the difference mentioned above - and error. However, things are different because the rotor resistance is variable: *Rr,h* > *Rr,SC* > *Rr,*1, (Kostic, 2010; Kostic M. & Kostic B., 2011)*.* To be precise, the values of rotor slot resistance are higher and the values of rotor slot inductance are √h times lower

Some examples from the literature can be used as proof of the view that the rotor resistance changes for low power motors. Specifically in Vukic (1985), the influence of harmonics on the motor of low power (1.6 kW) was tested. The calculation results, which were carried out assuming that *Rr* = *Const*, gave an increase in power losses of 12.6%, while the experimental measurements showed that the actual increase in losses was 18.5%. Our calculations give rise to losses of 19%, which slightly differs from the measured values. The accuracy of our calculations has been increased with respect to the fact that slot reactance of the rotor

*h=f/f <sup>1</sup> U h,i R <sup>s</sup> R r,h R M,h X M,h Z M,h I M,h P M,h P M,h*

5 0.20 0.015-0.05 0.067 0.072-0.117 0.735 0.739-0.744 26.990 0.618-1.184 23.447-13.418 7 0.14 0.015-0.05 0.079 0.094-0.129 1.018 1.022-1.053 13.790 0.213-0.341 7.668-3.864 11 0.11 0.015-0.05 0.099 0.114-0.144 1.579 1.583-1.586 7.010 0.069-0.099 2.504-1.122 13 0.08 0.015-0.05 0.108 0.123-0.158 2.416 2.419-2.421 3.180 0.015-0.022 0.556-0.252 17 0.06 0.015-0.05 0.124 0.129-0.174 2.694 2.643-2.646 2.230 0.007-0.012 0.244-0.136 19 0.05 0.015-0.05 0.131 0.146-0.181 3.249 3.252-3.254 1.600 0.005-0.007 0.180-0.079 23 0.04 0.015-0.05 0.144 0.159-0.194 3.526 3.534-3.536 1.220 0.003-0.004 0.106-0.046 25 0.04 0.015-0.05 0.150 0.165-0.200 3.833 3.836-3.838 1.040 0.002-0.003 0.075-0.036 29 0.03 0.015-0.05 0.162 0.177-0.212 4.080 4.084-4.086 0.830 0.001-0.002 0.036-0.025 31 0.03 0.015-0.05 0.167 0.182-0.217 4.357 4.361-4.362 0.730 0.001-0.002 0.036-0.025 35 0.03 0.015-0.05 0.177 0.192-0.227 4.910 4.919-4.920 0.590 0.001-0.002 0.036-0.025 37 0.03 0.015-0.05 0.182 0.197-0.232 5.187 5.191-5.192 0.520 0.001-0.002 0.036-0.025

**Table 7.** Values of harmonic resistances (*RM,h*), reactances (*XM,h*) and impedances (*ZM,h*); as harmonic currents (*IM,h*) and harmonic losses (*PM,h*) for motors with power > 100 kW (left) and lower power, 3–10 kW (right), when the motor is supplied by the rectangular voltage, i.e. by voltage with harmonics *Uhi = 1/hi*.

As Rs=0.050÷0.015, respectively, for motors of power 3÷200 kW, the given results are useful for the evaluation of harmonic currents (*IM,h*) and harmonic losses (*PM,h*) for all mentioned


Increments of harmonic losses are relatively small, as a percentage of nominal power losses *PM,h* [%*PCu,N*]. Apparently:


By the results in Tab. 6, for *Uh,i* = Const (example *Uh,i* = 5%, *hi* = 1–37, as in Tab. 6), the following approximate equations is confirmed:

$$\frac{P\_{Cu,h2}}{P\_{Cuh1}} \approx \frac{h\_1}{h\_2} \sqrt{\frac{h\_1}{h\_2}}, \quad \text{for } \mathcal{U}\_{h5} = \mathcal{U}\_7 = \mathcal{U}\_{hi} = \text{Const}, \text{ for} \\ h\_i \le 40 \tag{50}$$

Equation (50) is derived by the following approximate assumptions: ZMh ≈ XMh≈ hXM,SC and RMh≈ Rr,h ≈Rr,SC·√h.
