**4.4. Harmonic losses when the motor is operating with rectangular shaped voltage**

When a motor is supplied by rectangular shaped voltage *Uh,i* (p.u.) = *1*/*hi*, *hi* = 1 to 37, it is required to calculate the correspondent approximate values of harmonic losses for motors with nominal powers from 5 kW to 400 kW (i.e. for the values of stator resistance from *Rs* = 0.045*ZN* to *R*s = 0.015*ZN* and correspondent approximate values in a short-circuit regime, *Rr,SC* ≈ 0.03, for each motor. For motors within the power range 3 kW-400 kW, for which parameters are given in chapter 4.3, power losses are determined.

The given results in Tab. 7 show that, in the specified harmonic content *h* = 1, 5, 7, 11, 13, 17, 19 ...35 and 37, the percentage of additional power losses, *PM,h* [%*PN*], is relatively high:


The literature (Radin et al., 1989) often states that the percentage of increase of power losses in the windings of stator and rotor is due to the harmonics in *PM,h* [%*PCu,N*]. Data from Tab. 6, column *PM,h* [%*PCu,N*], show that:


150 Induction Motors – Modelling and Control

Radin et al. (1989),

*PM,h* [%*PCu,N*]. Apparently:

RMh≈ Rr,h ≈Rr,SC·√h.

**voltage** 

power *PM,h* [%*PN*]), is relatively low:

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



Increments of harmonic losses are relatively small, as a percentage of nominal power losses

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

5 7

Equation (50) is derived by the following approximate assumptions: ZMh ≈ XMh≈ hXM,SC and

When a motor is supplied by rectangular shaped voltage *Uh,i* (p.u.) = *1*/*hi*, *hi* = 1 to 37, it is required to calculate the correspondent approximate values of harmonic losses for motors with nominal powers from 5 kW to 400 kW (i.e. for the values of stator resistance from *Rs* = 0.045*ZN* to *R*s = 0.015*ZN* and correspondent approximate values in a short-circuit regime, *Rr,SC* ≈ 0.03, for each motor. For motors within the power range 3 kW-400 kW, for which

The given results in Tab. 7 show that, in the specified harmonic content *h* = 1, 5, 7, 11, 13, 17, 19 ...35 and 37, the percentage of additional power losses, *PM,h* [%*PN*], is relatively high:



The literature (Radin et al., 1989) often states that the percentage of increase of power losses in the windings of stator and rotor is due to the harmonics in *PM,h* [%*PCu,N*]. Data from Tab. 6,

**4.4. Harmonic losses when the motor is operating with rectangular shaped** 

, , 40 *Cu h*

*h hi i*

*<sup>P</sup> h h for U U U Const forh P hh* (50)


corresponding decrease in efficiency is by about 0.12%.

following approximate equations is confirmed:

*Cuh*

decrease in efficiency is by 1%,

decrease in efficiency is by 1.7%.

column *PM,h* [%*PCu,N*], show that:

, 2 1 1

1 22

parameters are given in chapter 4.3, power losses are determined.


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

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


**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 motors, by extrapolation.
