Recent Developments of Electrical Drives - Part 11 pps

Characteristics of the asynchronous motor drive Asynchronous motor Frequency converter and the load torque on the shaft are given as input variables and the phase currents, electromagnet

:

: computationFEM (S-function)

phase voltage

load torque

phase current

flux linkage

electromagn torque angular speed angular position

Figure 4 Functional block of the FEM computation.

Table 1 Characteristics of the asynchronous motor drive

Asynchronous motor Frequency converter

and the load torque on the shaft are given as input variables and the phase currents, electromagnetic torque, rotor position, and the stator flux linkage are obtained as output variables

The mathematical coupling between the FEM model and SIMULINK is weak, which means that the internal variables of the subsystems are solved separately and updated to each other with one-step delay Accordingly, there is no need to use uniform step size in the whole model, which provides flexibility and computation-effective simulation due to the different timescales in the system model

Characteristics of the motor model

Table 1 presents the ratings and characteristics of the drive, including the asynchronous mo-tor and the frequency converter Because of symmetry, the finite element mesh of the momo-tor covers half of the cross section, comprising 13,143 nodes and 6,518 quadratic triangular elements The geometry of the modeled region is presented in Fig 5

Figure 5 Geometry of the asynchronous motor model.

Due to the stochastic nature of the DTC control strategy, direct comparison of the wave-forms doesn’t give much information Instead, the results are gathered from several cycles

of the fundamental frequency and Fourier analysis is performed to find out the harmonic content of the waveforms Fig 6 presents the spectrum of the line-to-line supply voltage ob-tained by FEM and analytical motor models in comparison with the measured spectrum, and Fig 7 presents the corresponding results for the phase current The fundamental components are scaled out from the figures in order to see the differences in higher harmonics In the volt-age spectrum, distinctive difference is seen between the FEM model and analytical model

in certain frequencies, but otherwise they follow each other closely and also correspond very well with the measured results In the current spectrum, the difference between FEM model and analytical model is significant in all harmonic components It is also seen that the current spectrum obtained by the FEM model agrees very well with the measured spectrum Good agreement between the simulated and measured results shows that the control model behaves correctly in the simulations and the weak coupling between frequency

Figure 6 Spectrum of the supply voltage obtained by the FEM and analytical models and compared

with the measured spectrum

Figure 7 Spectrum of the phase current obtained by the FEM and analytical models and compared

with the measured spectrum

converter and FEM models gives correct and reliable results Furthermore, the voltage spectrum reveals that the analytical model is adequate for modeling the control system in steady state, but the differences in the current spectrum clearly proves the better accuracy

of the FEM model over the analytical model in the harmonic analysis of the phase current This is also illustrated in Fig 8, which presents the impedance of the motor for the measured frequency range The impedance obtained by the FEM model follows closely the measure-ments until 4 kHz, whereas the analytical model shows two times higher impedance at the same frequency

The measured losses of the motor in steady-state operation were 58.8 kW, and the losses estimated by the FEM-based motor model were 60.6 kW, which shows excellent capability

Figure 8 Impedance of the motor obtained by the FEM model, analytical model, and measurements.

Figure 9 Electromagnetic torque and phase currents, when the torque is changed from zero to nominal

and from nominal to 0.5 pu

for loss prediction In FEM, the copper losses in the coils were determined from the resis-tance and current density, and the iron losses were determined from the harmonic compo-nents of the supply voltage and the loss factors provided by the iron sheet manufacturer

Transient operation using FEM model

After validating the drive simulator with steady-state measurements, the drive was simulated

in transient operation A torque step from 0 to 1.0 pu was applied, when the motor was run-ning at nominal speed After a while, the torque was changed to 0.5 pu The electromagnetic torque and the phase currents of the motor are presented in Fig 9

In another transient simulation, rotational speed was changed from nominal to 0.3 pu, while operating at no load conditions The inertia of the motor was reduced in order to have

a faster speed change The electromagnetic torque and rotational speed are presented in Fig 10

In both transient simulations, the control system responds well to reference changes Al-though not validated by measurements, the results show the capability to simulate transients

of the controlled drive system

Determination of the initial state

Traditionally, finding out the correct initial state for the FEM computation has been problem-atic Especially with static frequency converter models, simulation of the startup transient may take hours of computation time Even if the simulation is started from an initial field ob-tained by sinusoidal supply, several periods of fundamental frequency must be time stepped, until the transient has stabilized in the motor

In the presented simulation environment, the initial transient converged remarkably faster than in the previous studies This is due to the calculated initial states for all state variables

Figure 10 Electromagnetic torque and rotational speed, when the speed reference was changed from

nominal to 0.3 pu

and accurate closed-loop model of the control system, which estimates the magnetic state

of the motor and controls the supply voltage to set the motor in the required operating point

as quickly as possible In other words, the simulation model operates exactly as the real drive system

Conclusion

This paper presents a drive simulator system comprising a three-level inverter, speed/torque control by DTC algorithm and an analytical or FEM-based motor model The FEM model

of the motor is coupled with SIMULINK using indirect approach This means that different parts of the drive system can be simulated simultaneously, but using different time steps

An asynchronous machine drive was simulated using analytical and FEM-based motor models and the results were compared with measurements In the supply voltage spectrum, agreement with measured results was excellent for both motor models In the current spec-trum, agreement with the measurements was clearly better with the FEM-based model

In transient simulation, the control system responds very well to the changes in reference values

Using the proposed methodology, the FEM model of the motor and the frequency con-verter model can be designed separately and easily combined for coupled simulation With the developed simulation environment, the initial states for the analytical motor model and the FEM computation are achieved very rapidly Based on the results, analytical motor model is suitable for control design, but FEM model is needed for detailed analysis

of the saturation and frequency dependence of the motor parameters As well, the motor losses obtained by the FEM computation agree very well with the measurements In general, the simulation results with the FEM model are very accurate and reliable, which leads to benefits in the design and development of advanced control algorithms

I-9 AN INTUITIVE APPROACH TO THE ANALYSIS OF TORQUE RIPPLE

IN INVERTER DRIVEN INDUCTION MOTORS

¨

O G¨ol1, G.-A Capolino2and M Poloujadoff3

1Electrical Machines and Drives Research Group, University of South Australia, Australia GPO Box

2471, Adelaide SA-5001, Australia

ozdemir.gol@unisa.edu.au

2Energy Conversion and Intelligent Systems Laboratory, Universit´e de Picardie Jules Verne 33, rue

Saint Leu, 80039 Amiens Cedex 1, France

gerard.capolino@u-picardie.fr

3Universit´e de Pierre et Marie Currie—Case 252, 4 place jussieu, 75252 Paris, France

mpo@ccr.jussieu.fr

Abstract An intuitive approach of parasitic effects with particular emphasis on torque ripple has

been proposed successfully It is shown that a good approximation can be achieved in predicting the nature and the magnitude of torque ripple by the use of a relatively simple time-domain model

Introduction

It is well known that, when an induction motor is driven from a non-sinusoidal supply, problems may arise due to the presence of supply harmonics For instance it is well known that the use of a six-step inverter may lead to the creation of parasitic effects such as torque pulsations accompanied by noise and vibration It is less well known that torque ripple along with associated disturbances can also be present in the case of drive systems which emulate a sine wave, such as field orientation control schemes if and when they are driven into overmodulation

Various methods of analysis have been proposed to assess the extent of the effect of supplying a motor from a non-sinusoidal source [1–3] Of these, methods which are based

on frequency domain analysis yield results which provide no interpretation of time-domain results, thus not allowing the significance of supply harmonics in terms of parasitic behavior

to be appreciated when a harmonic-riddled source is used

This paper proposes an intuitive approach to the analysis of parasitic effects with par-ticular emphasis on torque ripple The approach is based on the notion of space phasor modeling [4] It is shown that a good approximation can be achieved in predicting the nature and the magnitude of torque ripple by the use of a relatively simple time-domain model

S Wiak, M Dems, K Kom˛eza (eds.), Recent Developments of Electrical Drives, 93–100.

2006 Springer.

The space phasor concept

The transition from a direct phase model to a space phasor model can be effected by be-stowing “vector” attributes upon the time-variant electromagnetic quantities of the machine Thus the sum of stator and rotor phase currents for a three-phase machine in space phasor notation become

˜S =2 3



I SA + ˜aI SB + ˜a2I SC



(1)

˜S= 2 3



I SA + ˜aI SB + ˜a2I SC



(2) Stator phase voltages can also be expressed as a single space phasor quantity as

˜

U S = 2 3



U SA + ˜aU SB + ˜a2

U SC



(3) Similar considerations apply to flux linkages, namely

˜

λ S =2 3



λ SA + ˜aλ SC + ˜a2λ SC



(4) where

˜a = e j2π

˜a2 = e j4π

It must be emphasized that the complex j -operator used in the definition of the unit space phasors ˜a and ˜a2 has a completely different connotation from the one used in electrical circuit analysis: it designates a spatial shift of the quantity with which it is associated

Equations (1) to (4) imply that a single space phasor can be constructed on the basis of individual phase windings of the polyphase motor Alternatively, especially if the transition

is from a transformed model as in the case of orthogonal models, the aggregate stator and

rotor currents in space phasor notation can also be expressed as

Similar considerations apply to both the stator and rotor phase voltages and flux linkages, namely

˜

˜

˜

˜

The machine model

With the foregoing considerations, a space phasor model describing the electromagnetic behavior of the entire machine can be devised; remarkably, consisting of a single model equation for stator and rotor phase windings respectively, that is

˜

˜

where ˜U R= 0 for the singly excited induction motor

In terms of electrical circuit model parameters the equations can also be written as

˜

U S = R S˜S + L S p ˜ I S+3m

2 p R S

0= R R˜R + L !R



p ˜ I R + jpϑ ˜I R

 +3m 2



p ˜ I S + jpϑ ˜I S



(16) Together with the equation of motion given below, this deceptively simple model can be deployed to analyze the behavior of a polyphase induction motor in the time domain

In the above equations, p denotes the time derivative of the variable it precedes.

The electromagnetically developed torque can be obtained as:

Telec= 3m

2 ˜R˜∗

S



(18)

The supply model

If the induction motor is to be operated in a variable speed drive, then the non-sinusoidal nature of the supply voltage must be taken into account in modeling the drive to reflect the effect on machine performance of the harmonic content of the supply voltage In the case

of a voltage source inverter configured in six-step mode, illustrated in Fig 1, the terminal

voltages V A , V B , and V C are as depicted in Fig 2 For the purposes of this discussion, the inverter model shown here assumes ideal switches Fig 2 depicts the resultant voltages at

VC

E/2

E/2

Figure 2 Terminal inverter voltages.

the inverter terminals, leading to “six-step” voltages across the stator phase windings of the motor

The space phasor form of the resultant “six-step” voltage applied to the motor terminal can be conveniently obtained in terms of orthogonal components as

˜

U α (t )=

 2 3



V A−V B + V C

2



(20)

U β (t )= √1

The Fourier expansions of U α and U βgive

U α =

 2 3

2E π

∞



k=1

sin(2k− 1)π

6 + sin(2k − 1) π

2

U β = √1

2

4E π

∞



k=1

cos(2k− 1)π

6

Obviously, not all harmonics in (22) and (23) are significant in terms of causing parasitic behavior Only those harmonics which are significant and can profoundly affect performance need be considered in the supply model