Control characteristics for quasi-optimal operation of switched
K. Tomczewski, P. Wach
Abstract ，This paper presents the control characteristics of switched reluctance (SR) motors defined for the maximum efficiency of the motor or the motor–converter
system and for the minimum ripple level of electromagnetic torque.Curves for control variables—switch-on and switch-off angles (or conduction angle) and average phase voltage—are obtained by computations from a simple mathematical model.This lumped -parameter model takes into account the magnetic saturation of the motor and the parameters of the power converter necessary to guarantee reliable results concerning power losses in the system. The investigations were carried out for two typical SRM with the number of teeth Ns/Nr=8/6 and 6/4 for a battery supply and for a 310-V rectifier supply. Time curves obtained from mathematical model and control characteristics resulting from numerous optimization computations were validated by thorough measurements performed on a special test rig.
Keywords， Reluctance motor, Modeling, Control
List of symbols
D viscous friction damping, Nms
e back EMF in the kth winding, V k
I current in the kth winding, A k
J moment of inertia, kg/m2
(L() phase winding’s inductance in unsaturated state H
(L(,i) phase winding’s inductance considering saturation H
m number of phases
N/N number of teeth: stator/rotor sr
n rotational speed, 1/s
R phase winding’s resistance, W
R current measurement resistor value, W i
R total resistance in the kth phase circuit, W k
R resistance of a power source, W s
R drain-source resistance of a transistor in the saturated state TDSat
r dynamic resistance of a diode, W D
T electromagnetic torque, Nm e
T load torque, Nm l
u voltage of the kth phase, V k
U phase voltage RMS value, V
U phase voltage average value, V av
switch-on angle, rad ：on
switch-off angle, rad ：off
=- conduction angle, rad_ ：：：zonoff
stroke angle of the motor, rad ，
? efficiency of a motor s
? efficiency of a motor–converter system u
( rotor position angle, rad
?((,i) saturation function of the winding’s inductance
m level of the torque ripples, % p
=2p/N rotor tooth pitch rad )rr
rotor position angle reduced to the kth tooth-pitch, rad ?k
(，(,i) flux linkage of a phase winding, Wb
; angular velocity, rad/s ?(，
;; angular acceleration, rad/s2 ?(，
Continuous improvements in construction and rapid development of power electronic elements and devices has intensified interest in the application of switched reluctance motors (SRM). Their characteristics, which are of a series d.c. motor type, make them applicable to vehicle drives. High efficiency in a wide range of angular velocity enables their application in high-power drives and battery-supplied drives. Simple and robust construction with a passive rotor makes them suitable for ultra-high-velocity drives. The other desirable feature of SRM, as in case of stepping motors, is the feasibility of direct control of the rotor position but, for SRM a torque control is still possible [2, 6, 7, 10]. SRM also have some weak points, such as relatively high levels of torque ripples and increased level of vibrations .
The most often used converter configuration for SRM control is the asymmetric half-bridge circuit presented in Fig. 4. Reluctance torque production does not require a change in direction of the current flow in stator windings and the motor can operate
;；?？Teasily in all four quadrants of a plane. The sense of rotation can be reversed e
by a different sequence of the switch-on pulses and the motoring/breaking mode changed by a proper value of the switch-on pulses that are advanced or delayed with respect to the pole axis. Angular velocity control and torque control are performed by
：：three variables: switch-on angle (), switch-off angle ()—or conduction angle onoff
：： =？—and a phase supply voltage U which is regulated in a pulse width ：onoffz
modulation (PWM) mode. These three control variables enable the same operation
;；?？Tpoint to be reached on the mechanical characteristics plane at their various
value combinations but resulting in different current, efficiency and torque ripple [4, 5, 9, 10]. So it is vital to find an optimum control characteristics in respect to selected required parameters of SRM drive.
In this paper, a quasi-optimum operation of the SRM drive is investigated to find the maximum efficiency and minimum torque ripple control characteristics. This is achieved by employing an original mathematical model, which is simple and efficient in numerous repeatable computations of dynamic courses, in a process of finding optimum control characteristics. This lumped-parameter model takes into account magnetic saturation and all necessary power-converter elements that cause power losses and therefore influence the efficiency of the drive.
1 Mathematical model
From the constructional point of view, the average torque and angular characteristics of the torque depend on angular curves of self and mutual inductance of the phase windings and their derivatives with respect to the rotor position angle. It is clear from various measurements and computations that the amplitude of mutual inductance of neighboring phase windings is quite low in comparison with phase self-inductance, and their influence on the motor’s performance is even more limited.
It could be demonstrated by measurements of SRM characteristics with a positive and negative coupling of neighbouring windings in the motor. There are mathematical models that take into account the mutual inductance of stator phase windings, e.g. , but they are much more difficult and expensive in computations of dynamics that are used in the optimization of control.
The lumped-parameter model [8, 9] used for this researchis based on the following assumptions:
– Complete magnetic and electrical symmetry of the motor.
– Mutual phase-to-phase inductance is neglected.
– Magnetic hysteresis and iron losses are neglected.
This model reflects the most important features of SRM as the phase inductance
(non-linearity with respect to the phase current i and angular position of the rotor , k
as well as power losses in the electronic converter and a voltage drop across the power source, are taken into account. The model was formulated by Lagrange’s equation method, and the Lagrange function of the motor is:
ikm1~~2; ;1； ,d;；L，，(ii？J(?kkk?2，1k0~(i;；，(,iiiin which = L(,) is the magnetic flux linkage. is the variable of kkkkkk
iintegration in contrast to the upper limit of the integral . k
The resulting equation for a phase winding current is
d;；；，L(,i，u？Ri！ k=1,….,m ;2； kkkkkdt
and the equation of the rotary motion:
;;;(，？？(JTTD ;3； cL
In the above, T is the electromagnetic torque ci~km(：Li;；,~~k ， ;4； Tiid?ekk?：(，1k0
The induction function L used in this model is assumed in the following k
(;；;；iL???,iL(,)= ;5； kkkkk
??1whereis the rotor position angle reduced to the kth ;；;；，？？?)frac(k1~kr??)r??
tooth-pitch, and frac(x) is the fractional part of the real number x.
This method of induction function decomposition is very effective in the estimation of parameters of induction function from measurements. It is also useful in investigation of influence that result from minor changes and corrections in magnetic circuit construction upon torque curves.
;；L?in the unsaturated state, Figures 1, 2 and 3 show the induction function
;；;；??？iL(,isaturation function and resultant phase windings inductance
respectively. After introducing this form (5) of induction function into current (2) and
;;，(，torque (4) equations, and taking into account that, one obtains: k
??dL,i：?;；kkk;??iuRiei,,/L,ii ;6； ，？？;；;；？；，?(?kkkkkkkkkk??dti：k??
ikm1~~; (7) ;；T，ci?(i,,d?ckkk?;(，1k0
Fig. 1. Phase winding’s inductancein unsaturated state for the MRV3 motor, ;；L?
measured and interpolated to a smooth curve
;；Fig. 2a, b. Saturation function??？iof a winding’s inductance as a function of: a
?current i; b angle .Curves measured for the MRV3 motor and interpolated by the first-order spline function (broken line) in respect to the phase current and cosine function in respect to the rotor position angle
In Eqs (6) and (7)
?((：,i;；：;；L??：Lkk;;((?((;；;；，，,？eioreiiL ;8； ??kkkkk(((：：：??(，?k
is back EMF generated in the winding. Expression (8) is the same as for the series-excited d.c. motors, which explains the similarities of mechanical characteristics of both types of motors.
Fig. 3. Resultant phase winding’s inductance as a function of a rotor position ;；L?,i
?angle and a phase current i, obtained according to (5) for MRV3 motor.
The induction functionand the saturation function for both tested ;；;；L???？i
motors were determined by measurements. The function was computed from ;；??？i
the recorded curves of the current during the step voltage test, for several positions of the rotor.
In the mathematical model, ;；is represented by a third-degree spline function L?
(Fig. 1), while ;；by the first-degree spline function with respect to the current I ??？i
?(Fig. 2a) and cosine function in respect of angle. (Fig. 2b).
Fig. 4. Converter phase circuit a and its equivalents for the three modes of operation:b
motoring; c free-wheeling; d breaking
The voltage supply and control variables were introduced to the model by a proper circuit connection of the power converter as it is presented in Fig. 4. The transistor commutation in the PWM switching was modelled in a simplified way by a cosine function approximation, which for the phase to phase switching is neglected due to the comparatively low frequency of that process. Taking into account these resistance parameters of the converter, responsible for major power losses in the system, it is necessary to get accountable results of the computed efficiency of the system.
2 Validation of the model
Experiments and measurements were carried out on a special test rig presented in Fig. 5. Figure 6 shows a cross-section of one of tested motors with the explanation of
(control variables, ,：,： and the rotor position angle . ：onoffz
Fig. 5. Test rig with MRV3 SRM and the control unit
Fig. 6. Cross-section of a four-phase, 8/6 teeth SR motor with the angles: switch-on
(：：(), switch-off (), conduction (： ) and rotor position angle . onoffz
Two different motors were tested. The first one, MRV3, has the following rated values: m=4, U=24 V; P=0.9 kW; n=3,500 rpm; phase resistance (20?C), R=0.043 NNN
(; insulation class H; maximum torque (locked rotor), 25 Nm; degree of protection, IP 21; nominal current, not specified; main dimensions, 200 mm?135 mm?135 mm;
；airgap width, =0.4 mm.
The second motor, EMS-71, is a three-phase motor designed for rectifier supply:
(m=3, U=310 V;P=0.75 kW; n=3,00 rpm; phase resistance (20?C), R=1.87 ; NNN
insulation class, H; nominal current, not specified; main dimensions, 200 mm?160
；mm?160 mm; airgap width, =0.35 mm.
The test rig consists of a separately excited d.c. generator connected to the tested SRM with an induction torque meter. The motor was supplied by a converter that enables the independent regulation of all three control variables: phase voltage Uav (by means of the PWM method), switchon angle ： and conduction angle . The ：onz
rotor position angle, and consequently the rotational speed, were measured by means of an incremental encoder with a resolution capacity equal to one degree. The control system was constructed on the basis of AT889C51 microprocessor. The measurements were carried out by the 12 channel power analyser Norma 6200. It enabled measurements of several dynamical courses necessary to verify the results obtained from mathematical modelling, including efficiency and the torque ripple level. The efficiency of the motor ? and the motor–converter system? was computed by the su
power analyser from the measured time curves of voltages currents, torque and speed.
The simulation program was developed in Object Pascal language in Borland Dephi 6.0 standard. For computation of dynamical courses, the Runge–Kutta RKF45
procedure was applied. In computations, all vital parameters of the converter and the whole system were included—according to the scheme presented in Fig. 4. The
computations based on mathematical modelling were carried out for about 1,000
：different combinations of control variables: Uav, and as well as different load ：onz
torque values to simulate the dynamic runs from the start of the motor until the steady state was reached. The efficiency and the torque ripple were computed for 1 s of steady operation at the final stage of each dynamic run.
The numerical performance of this model could be characterized by the ratio 1:50 of the real time of a dynamic course to the computation time used for the same course by a PC equipped with an Athlon XP 1600+ processor. The results of the computed steady states were interpolated with respect to the control variables by the third-degree spline functions to obtain smooth efficiency and torque ripple curves.
The efficiency of the motor and the motor–converter system were computed by
means of the particular losses method (9) taking into account the mechanical power of
！Pthe motor P, total losses of the motor and the losses in the converter and ms
supply circuit: ?！Pz
PPmm， ， ;9； ??suP？！PP？！P？?！Pmsmsz
In computations, the average resistance of all phase windings was taken into account at a temperature of 60?C. Approximately the same level of the winding temperature was maintained during measurements of the characteristics of the tested SRM.
During measurements and computations, the following curves were registered: phase currents, source current, rotational speed and torque; and in the steady state: SRM values of phase currents, phase and source voltage; average values of: speed, currents, phase voltage, torque, efficiency for the motor and for the motor–converter
system. Current spectra were also measured, computed and compared.
These investigations covered a wide range of change in the control variables and load for both tested motors. Exemplary results are shown in Fig. 7. The level of agreement of measured and computed results was examined for the steady-state values that were reached after stabilization of the dynamic courses. Because of the multitude of such results for various control parameter sets and load values (more than two hundred in total) a quasistatistical approach was adopted and the outcome is
presented in Table 1. The first row of the table shows the limit of discrepancy between measured and computed values, while the content of the table holds the percentage of single cases that keep within that limit—for velocity, efficiency and average as well as
the rms values of phase and source currents.
Fig. 7. Comparison of measured and computed time curves for MRV3 motor: a
：current for the steady state (=39.5?？=30?, Uav=22.8 V, n =21/s, T=5.7 Nm ); b ：lonz
：torque for a start up with:=19.5?, =15?, Uav=12.4 V and additional inertia ：onz2 0.072 kg/m
Table 1. Mathematical model validation. Level of agreement between measured and computed results for both tested SRM. The percentages in the table mean the cases
；that meet the discrepancy limit from more than 200 measured and computed cases
The differences between measured and computed steady-state values are higher
：in the two specific cases: for small values of theangle and for high values of off
：theangle in connection with a small value of conduction angle ：, which means onz
that the inductance–angle curves approximated by spline functions are not very precise at the border of the pole–pitch span.
3 Optimum control curves
The mathematical model was used for optimization computations to find the control parameters that yield the maximum level of efficiency and minimum level of torque ripples. The computations were carried out in a wide range of control
???：：：：：：parameters: 0U; <;. <;In these relations, U is NNonon maxz maxzz minz
the rated voltage of the motor, ：is the switch-on angle value above which the on max
velocity decreases; is the conduction angle in the case where only a ：z min
single-phase winding contains current;： is the conduction angle in the case z max
where two phase windings contain currents all the time. For each tested case, the dynamic states were computed until the steady state of the drive was reached. Example efficiency curves for a motor alone and the whole system including the source and the converter are presented in Fig. 8. The results of the computations for the optimum cases had been interpolated using the spline functions with respect to all three control parameters. The outcome for maximum efficiency is quite clear, and is presented in Fig. 9a for the torque load equal to 3 N m as an example for the MVR3 motor, and in Fig. 9b after generalization for
various loads and motor types.
Fig. 8. Computed efficiency curves of MRV3 motor alone a and with the drive system
b for constant load torque values
Fig. 9. Computed characteristics of maximum efficiency control variables: a for
MRV3 motor and torque load Tl=3 N m; b in the general
The maximum efficiency of SRM operating under a constant load is obtained by changing one of the three control variables, depending on speed range. For low speed,
：the average value of a phase voltage Uav is to be regulated with constant values of on
：：and angles. In the range of medium speed, is the variable that should be onoff
：regulated, while the voltage andangle remain constant. For the high-speed range, off
：the regulation is carried out by increasing the angle with constant phase voltage on
：and also constant conduction angle, which means an increase in theangle in ：offz
：accordance with. on
Fig. 10. n1 and n2 speed border values for optimum control characteristics as a
function of load torque. Results computed for MRV3 motor
The border values n and nof rotational speed that separate these ranges of 12
mode of optimum control depend on the load torque as presented in Fig. 10. The above results are also applicable for slow-changing dynamic courses (rotational 2acceleration below 8/s ) and could be applied in the design of the control system of a drive. For faster-changing operation of the motor, the highest efficiency is obtained
：for higher values of the switch-on angle than is in the case for the steady-state on