4. DYNAMIC THERMAL ANALYSIS WITH COUPLED ELECTROMAGNETIC−
4.4 Losses of the motor
Inputs in the thermal network model are the electrical losses of the motor, which can be divided into four groups:
• Copper losses PCu
• Iron losses PFe
• Additional losses PAdd
• Friction and windage losses PMech
Since the servomotors are always fed with a frequency converter, there are always voltage and current harmonics present (time harmonics), which increase the motor losses, and therefore the model must be able to take into account also the losses due to harmonics. Harmonics are also caused by the discrete distribution of the slots (winding harmonics), and by the slot openings, where there is a local permeance minimum under the slots (permeance harmonics).
4.4.1 Fundamental wave losses
The different loss components and their equations caused by the fundamental wave are briefly introduced below.
Copper losses
Copper losses form the largest proportion in machine total losses; they are generated in the stator and rotor windings according to Ohm’s law
Cu mRI2
P = , (4.30)
where m is the number of phases, R the resistance and I the current. Copper losses change with temperature, as the resistance increases directly proportional to temperature
(
T)
R
R= DC1+α∆ , (4.31)
where RDC is the DC resistance, α the resistance temperature coefficient, and T the temperature.
Stator copper losses must be separated into copper losses occurring in the copper in the slots and those in the end windings. A length of a single stator turn of a small low-voltage machine, required for resistance separation, is approximately (Jokinen 1979)
1
where L is the stack length and τp the pole-pitch. Similarly, the separation of the rotor copper losses between the bars and the end ring must be done. Separation can be done with the following equation (Kylander 1995)
Where Qr is the number of the rotor bars, lb and Ab are the length and the area of a rotor bar, respectively, and lr and Ar are the length and area of a ring segment between two bars.
Iron losses
Iron losses are generated in the conducting core laminations, and they can be divided into hysteresis, eddy-current and excess (known as anomalous) losses. An empirical equation for the iron losses per unit volume in the stator laminations is written as (Cedrat 2001)
( ) ( )
15 fwhere kh and ke are the hysteresis and the excess loss coefficients for the laminations, respectively, and kf is the lamination stacking factor. B is the flux density, f the frequency, σ the lamination conductivity, and d the lamination thickness. They are usually provided by the steel manufacturer, or they can be determined experimentally, for example with the Epstein-frame setup. The values of 153 Ws/T2/m3 for hysteresis loss coefficient kh and 2.32 W(T/s)-1.5/m3 for the excess loss coefficient were used in calculations. The value of 4×106 Ω-1m-1 was used for the lamination radial conductivity and 0.97 for the stacking factor.
Since in speed controlled induction motor drives the motor always operates on the linear part of the torque-speed curve, where the rotor frequency is low, typically few Hertz for the low-speed machines, it is justified to neglect the rotor iron losses for the fundamental frequency. During the field weakening, the rotor frequency can increase up to tens of Hertz, but as the flux density decreases correspondingly, the iron losses are further decreased, as the effect of the flux density is greater than that of the frequency for the iron losses. Rotor iron losses were thus neglected in this work. As all the iron losses are caused by the magnetizing flux, the flux densities in different parts of the machine, required in Eq. (4.34), can be calculated from the cross-sectional areas and the magnetizing flux of turns, ωs the angular frequency, ξ the winding factor, and Athe area.
Additional losses
Iron and eddy current losses caused by the leakage fluxes and by the high-frequency flux pulsations are called additional losses (or the stray-load losses), that are according to IEC 60034-2 standard defined as a fraction of the machine input power
in stray 0.005P
P = , (4.36)
There are basically six mechanisms that cause additional losses in the machine (Sen and Landa 1990)
• Eddy current loss in the stator copper due to slot leakage flux
• Loss in the motor end structures due to end leakage flux
• High-frequency rotor and stator surface losses due to tooth-tip leakage flux
• Tooth pulsation and rotor copper losses due to tooth-tip leakage fluxes • Rotor copper losses due to circulating currents induced by the leakage fluxes
• Iron losses with skewed motors due to skew-leakage flux
Even if the stator voltage is fully sinusoidal, there are always space harmonics present in the air gap flux due to permeance fluctuations caused by the slotting and the discrete winding distribution. These space harmonics induce eddy currents on the stator laminations and on the rotor bars (especially when fully open rotor bars are used to minimize the slot leakage), that are not included in either the fundamental iron or copper losses.
Mechanical losses
Mechanical losses may be divided into friction and windage losses. Friction losses are generated in the bearings, and they can be approximated analytically as (Gieras 1997)
π 3 100b r
b k m Ω
P = , (4.37)
where kb is a factor with value 1-3, mr is the mass of the rotor, and Ω is the mechanical speed.
Windage losses are generated at the air gap by the friction of the air, and also on the fan integrated to shaft, if there exists one. Windage losses for the machines of this frame size are very small compared to other loss mechanisms due to relatively small peripheral speed, and thus they are neglected here. Further, it would be very difficult to accurately divide the windage losses between the rotor and the stator.
4.4.2 Harmonic losses of the motor due to inverter supply
When the inverter supply is used, the voltage contains different harmonic frequencies, due to power stage switching operation, and the voltage can be expressed as
( ) ( ) ( )
phase angle, and n=0,1,2… Although Eq. (4.38) applies only for the synchronic pulse width modulation (PWM), where the switching-frequency is a fundamental frequency multiplied by an integer, it can be applied also for other modulation techniques as well (e.g. DTC), without making a significant error. Harmonics of the order [3n+1] rotate in the same direction as the main flux creating a positive torque, and those of the order [3n+2] rotate in the opposite direction creating a negative torque. Harmonics of the order [3n+3] produce no rotating mmf, and therefore they have no effect on torque. Besides that harmonic frequencies create harmful torques, they also have a disadvantageous impact on the machine losses, which causes that frequency converter suppliedmotors can withstand less continuous torque than the same motor with a sinusoidal supply. The
Where ns is the synchronous speed, nr the speed of the rotor, and s the fundamental slip. The + sign indicates that different harmonics rotate in different directions. Eq. (4.39) shows that the harmonic slip is approximately unity apart from the first two harmonics. The rotor thus appears almost stationary to harmonic frequencies, which means that the impedance of the magnetizing branch is much greater than the rotor impedance. Therefore the equivalent circuit similar to locked-rotor equivalent circuit, where the magnetizing branch is neglected can be used for the harmonics (Fig.
4.6).
Figure 4.6. Equivalent circuit used in the harmonic loss calculation The harmonic current of kth order is according to Fig. 4.6 therefore
(
sσ ')
2Harmonic voltages increase the saturation of the magnetic paths, thus increasing the iron losses.
Iron losses for the harmonics can be calculated with Eqs. (4.34) and (4.35) by substituting the harmonic voltages Uk and the corresponding frequency. It is assumed that the hysteresis and eddy current coefficients kh and ke remain the same for all frequencies. There is no significant effect on the stator resistance Rs of a small diameter, round-wire machine due to harmonics, since the skin effect is negligible, and the total stator copper losses can be calculated with Eq. (4.30) by substituting the value of the total RMS current
Since the depth of the rotor bars of induction machines in kW range is usually notably higher than the skin depth, the skin effect can have a significant effect on the rotor copper losses. With induction servomotors, the skin effect is further emphasized, since their nominal frequencies can be hundreds of Hertz. The influence of the skin effect on the resistance is taken into account with the skin effect factor kr
DC rR k
R= . (4.42)
When the geometry of the rotor bars is simple, for example rectangular as with the prototype machine, the skin effect factor kr can be calculated
Where hslot is the height of the slot and δ is the skin depth, calculated as
0f πµ
δ = ρ , (4.44)
where ρ is the resistivity, µ0 the vacuum permeability, and f the frequency. As the frequency of the rotor current during normal operating conditions is small, the skin effect is relatively small. For example at the nominal point of the prototype induction motor, the increase of the resistance of the rotor bars is only approx. 0.03 % due to fundamental frequency. Even when the motor occasionally operates at higher slip values during overloading, the skin effect is typically approx. 1
% at maximum, and thus it can be neglected for the fundamental wave. However, as the rotor appears almost stationary for all the harmonics, and the rotor frequency equals the harmonic frequency, all the stator harmonics create a strong skin effect on the rotor bars thus increasing the rotor bar resistance and rotor copper losses. For example, the rotor resistance of the prototype motor for the 7th harmonics is over four-times the DC resistance value according to Eq. (4.42) Iron losses caused by the harmonic frequencies can be approximated with the same equation as the fundamental wave losses, with the assumption that the hysteresis and eddy-current coefficients remain the same also at harmonic frequencies. Basically this is the case also with additional losses occurring at higher frequencies, but as their proportion is very small, they are neglected here. Also the inclusion of the additional losses at harmonic frequencies would require too profound investigation to be carried out in this study. Figure 4.7 shows the measured 30 first current and voltage harmonics of the prototype at the rated operation point (125 Hz). The measurement was carried out with the Yokogawa PZ4000 power analyzer in harmonic mode, and the motor was supplied with the ACS600 frequency converter.
0 2