Estimation of On-Fly Phase Resistance of on 8 / 6 Switched Reluctance Motor for Sensorless Control

Given that most sensorless methods use the “flux linkage” algorithm for their control, this paper deals with possibilities to improve this sensorless method. When using the sensorless “flux linkage” method it is necessary to know not only the values of phase voltages and currents, but also the correct value of the phase resistance. During the operation of the switched reluctance motor, the value of the phase resistance is significantly changing due to the temperature dependence. Such resistance error causes an inaccurate estimation of the flux linkage. For this reason, it was desirable to implement a sensorless control algorithm for estimating phase resistance. In this paper, the experimental results of the implemented algorithm for estimating the phase resistance under the sensorless operation of a switched reluctance motor are presented. The practical implementation of control of the switched reluctance motor is carried out using a control system with a 16-bit digital signal processor. DOI: http://dx.doi.org/10.5755/j01.eee.20.5.7093


I. INTRODUCTION
The switched reluctance motor (SRM) is topologically and electromagnetically similar to stepper electric motors with variable reluctance.The main differences of these two motors lie in the design proposal, control methods, as well as performance and application properties.The switched reluctance motor is typically controlled by feedback from the position of the shaft so as to synchronize the commutation of phase currents with the position of the rotor, while the stepper electric motor usually operates in an open-loop, i.e. without any feedback from the position of the shaft.While SRMs are designed for high power conversion efficiency at high speeds, stepper electric motors are designed for greater torque motors with limited speed range.Basic differences are in the geometry of the motor, power electronics and control [1].

II. OPERATION PRINCIPLES OF SRM
The switched reluctance motor is an electric machine.
Movement of the SRM may be either rotating or sliding and design of the rotor part internal or external.The stator winding consists of coils wound on each stator pole that are electrically isolated from each other, and can be supplied separately or together.The rotor is made of steel plates.SRM has a number of suitable combinations of stator and rotor poles, while the number of poles on the stator and rotor is always different and in general, there is valid relationship between Nr = Ns − (Ns / m) where m is the number of phases.There is a number of Ns / Nr combinations..
Figure 1 shows one of the most widely used four-phase motors (m = 4) in the configuration of 8/6, which has 24 cycles per revolution and a torque pulse angle of 15°.Winding of each of the two opposite stator poles are connected in series so that their fluxes add and generate a S-J pole pair -the phase.
A pair of coils marked A-A´, B-B´, C-C´and D-Dŕ epresent one phase.Figure 1 shows that each SRM phase requires one power switching circuit.In the case of switching the power transistors T1 and T2 DC voltage is connected to the winding of the relevant SRM phase, and the current starts flowing through the circuit.If the power switches open, the current starts to flow through the zero diodes D1 and D2, the voltage on the winding changes its polarity and the current drops down to zero [1], [2].
where uj is phase voltage, ij is phase current are measured with a digital oscilloscope.Ψj is phase flux linkage.An asymmetric bridge of one phase of an inverter, which enables switching (+Udc /−Udc) and thus a steeper decrease in current, is used for a power circuit.Measurements were carried out for only one connected phase at the time so as to minimize the effect of other phases, of which winding has been disconnected.From the measured characteristics of flux linkage, the matrix ψ (i, θ) was generated, where the rows of the matrix represent individual positions and the columns of the matrix represent selected values of the currents.A current value of the flux linkage for a given position  and the current i are obtained by quadratic interpolation in the direction i, and by linear interpolation in the direction . Figure 2 shows a threedimensional graph of the measured magnetization SRM characteristics ψ (i, θ), approximated by cubic spline [3], [4].

III. SENSORLESS CONTROL OF SRM
In the past few years, a number of interesting sensorless methods were designed for the switched reluctance motor [5]- [8].Despite great advances made in the sensorless control strategy of SRM, none of the current proposals was able to fully replace the mechanical position sensor without this resulting in different limitations in the control of SRM.
The fundamental principle of most sensorless methods is based on the assumption that information about the rotor position can be obtained from measurements of the stator circuit variables or their derived parameters.In other words, information about the position is in fact obtained from the magnetization characteristics of the machine.
When applying the sensorless method based on the evaluation of flux linkage, an incorrectly determined resistance of stator winding affects the accuracy of the estimation of rotor position.The motor operation results in the heating of the stator winding, and thus in the change of its resistance value.The change in stator winding resistance can reach up to 30 % of its nominal value [9]- [12].

A. Estimation of stator winding resistance in SRM
Estimation of stator winding resistance is based on calculation of the flux linkage, which is also used to determine the rotor position.If we neglect the mutual effects of the individual phases, and the current ij of a given phase is equal to zero, then the flux linkage ψj must also be equal to zero.If this condition has not been met, the (1) evidently shows that the main parameter affecting the flux linkage calculation is the value of winding resistance.Calculation of the flux linkage at the time of current conduction through the given phase is described by the (2) as follows where uj and ij are the phase currents and voltages, RjAssume is the assumed value of the phase resistance of the stator winding, and ton is the moment of the voltage conduction into the measured phase of SRM.Estimated value of the phase winding resistance is defined as the sum of the current value of the phase resistance Rj and its variation RjError, as described in (3).Variation of the phase resistance can be caused, for example by a measurement error, an incorrectly defined initial value of the phase resistance or its change in the SRM operation caused by a temperature influence Figure 4 shows the courses of flux linkage for different values of variation of phase resistance during one work cycle of SRM.At the beginning of the work cycle in time ton, DC-Bus voltage is conducted to the given phase, and phase current increases from zero.At the moment of commutation tc the phase stops being supplied with power and current drops down to zero.Similarly, this can be applied to flux linkage, which increases from zero in the <ton, tc> interval, and from the moment tc it begins to decline until time toff, in which the phase current ceases to exist.If all parameters of (2) are correctly specified and variation of phase resistance RError is equal to zero, then the composite magnetic flux in time toff must be equal to zero.The described situation is indicated by the course of the flux linkage ψ2Estim in Fig. 3.
The algorithm for estimating the phase resistance is based on the fact that when the phase current is equal to zero, then the flux linkage must be equal to zero at time toff.Under this assumption, it is possible to express the relationship for flux variation in time toff from the (3) follows off on jEstim off jError off jError j To simplify the process, we further assume that the rate of change of phase resistance during one working cycle SRM is very small.Then the variation of the phase resistance can be written in the form of (5) off on jError off jError off j In this implementation of sensorless control, flux linkage calculation is performed in each switching period through numerical integration using the trapezoidal method.The same method is used to calculate variation of flux linkage and integral of the phase current, by which the variation of phase stator winding resistance is determined [13], [14].

IV. ESTIMATION OF ROTOR POSITION OF SRM
With the use of spline functions in the Matlab software, magnetization characteristics were modified so that they can be applied in the algorithm of SRM sensorless control.The m-file that generates the table of magnetization curves was created in Matlab.Each table defines the relationship between rotor position and flux linkage for a given current value.Adjusted magnetization curves for the accurately specified value are shown in Fig. 4. Their number agrees with the number of the tables, which were stored in an external programme memory located on the board MSK2407, which also includes a digital signal processor TMS320LF2407.Blue colour is used for indicating an appropriate area of magnetization characteristics to evaluate the rotor position, and is further divided into three welldefined sectors according to the current strength.This method maximized the accuracy of the evaluation of the rotor position [15]- [18].The output values of the block represent the values of the current rotor position related to different phases, and are determined from the magnetization curves using linear interpolation.Interpolation of the position is performed only in the active SRM phases.In the block entitled "Selection of a suitable phase", the phase from which the rotor position will be evaluated is selected according to the given criteria.Preference is always an active phase, of which the value of an estimated rotor position is located in the appropriate area of evaluation.If there are simultaneously two phases in the appropriate area, or none of them, the phase through which a higher current is conducted is always selected to estimate the rotor position.In order to eliminate the greater inaccuracies in estimating the rotor position, the block entitled "Correction of the estimated position" has been established.The input variables of this block include: a current estimated rotor position of the selected phase θjEstim(k), a value of converted estimated rotor position from the previous switching period θEstim(k−1), and change of the estimated position θEstim(k−1), which was calculated in the previous switching period from the most suitable phase.

V. FLUX LINKAGE CALCULATION IN THE DISCRETE DOMAIN
During motor operation, the phase current and the phase voltage are measured regularly in rates of the sampling frequency.The flux linkage estimation is based on these measurements.In order to increase the precision of the integration a trapezoidal integration is used.Equation ( 2) can be converted to j( ) j( -1) j( ) PWM where TPWM is the sampling period, gj(k) is calculated according to (7) and ψ j(N) is the calculated magnetic flux linkage for the measurement cycle N.The flux linkage is calculated regularly in each sampling cycle from the beginning of the commutation stroke (ton).The sampling period TPWM is constant.Because the sampling period is kept constant, the division can be considered as a scaling factor j( ) where d1(k) is calculated regularly in each sampling cycle from the equation d1(k) = t1(k)/TPWM (t1 is time when both transistors T1 and T2 are turned-on, as shown in Fig. 1), d2(k) from the equation d2(k) = t2(k)/TPWM (t2 is time when transistor T1 is turned-off and T2 is turned-on) and d3(k) from the equation d3(k) = t3(k)/TPWM (t3 is time when both transistors T1 and T2 are turned-off and both diodes conduct the current).Udc(k) and ij(k) is the sampled DC-Bus voltage and the phase current.The phase voltage is not measured directly.The DC-Bus voltage is measured instead and the phase voltage is calculated according to the applied PWM duty cycle.The integral of the phase current according to ( 5) can be also expressed as a sum of individual samples over the current pulse j( ) j( 1) j( ) PWM Then ( 5) can be expressed as jError j( -1) j( ) PWM 1 j( 1) j( ) PWM 1 j( -1) j( ) j( 1) j( ) where N is the last sample taken when the phase current disappears to zero.The actual phase resistance can be expressed as where RAssumed is the assumed phase winding resistance.Equation ( 10) can be efficiently used for the correct resistance calculation.

VI. EXPERIMENTAL RESULTS OF STATOR WINDING RESISTANCE ESTIMATION MEASURED ON REAL SRM DRIVE
This chapter presents the estimate results of stator winding phase resistance during the SRM operation.An algorithm providing the current value of the phase resistance is, in order to improve the estimate of the rotor position used in a sensorless method, based on the evaluation of the flux linkage.
Control algorithms were tested with the use of MSK2407 development kit equipped with a TMS320LF2407 signal processor.The MSK2407 kit contains an external memory, which was also used to save up to 8 selected variables of the application software (i.e.motor speed, actual and estimated rotor position, flux linkage, etc.).These variables were investigated with the use of DMC Developer Pro software, and then saved as text files and drawn in Microsoft Excel spreadsheet.All graphs presented in this chapter were obtained in this way.
Experimental measurement was carried out on a real drive equipped with a switched reluctance motor SR90.This is a 4-phase 8/6 SRM.Its rated power is Pn = 400 W, rated speed nn = 3000 rpm and Mn = 0.3 Nm.This SRM was powered by a frequency converter, where DC link voltage Udc = 100 V and switching frequency fPWM = 10 kHz.
Figure 6 (above) shows the phase current courses and calculated flux linkage with the set reference value of the phase resistance to be 20 % higher than its actual value.The measurements are performed in a stable state nRef = 300 rpm and feedback from the algorithm estimating phase resistance is intentionally not implemented.The course of the flux linkage also shows an influence of the value of the phase resistance on its exact determination.The resistance estimation algorithm is based on the fact that when phase current equals zero, magnetic flux has to equal zero as well.Resistance error leads to the flux estimation error.Since resistance estimation error is positive (+20 %), flux linkage error is negative for time ton, as described in Chapter 3. Figure 6 (below) compares the courses of actual and estimated rotor positions, which is determined from the magnetization characteristics of SRM using measured phase current and calculated flux linkage.Estimated position θEstim graph makes it possible to observe the effect of incorrectly selected phase resistance value on the estimation of SRM rotor position, using the flux linkage based sensorless method.The picture shows that when SRM was loaded with nominal torque, phase resistance increased by 0.9 Ω in 14 minutes, which means an increase by 22 %.This demonstrates very clearly the effect of coil heating during SRM operation on its coil resistance.
Figure 9 to Fig. 12 show the actual position and the estimated position, the estimated rotor position error, the estimation of rotor position of the individual phases and the number of the phase from which the rotor position is evaluated during sensorless control of the SRM at nRef = 420 rpm.The graph in Fig. 10 shows that the variation θError fluctuates in a range of ±2°.The principle of determining a suitable phase that is selected to estimate rotor position is described in Chapter 4.    Figure 14 shows that phase currents i2 and i4 are limited, so they do not exceed a maximum 4 A value.The functionality of this applied algorithm determining the phase resistance of SRM stator winding, has been proved by practical results.

VII. CONCLUSIONS
The real 8/6 switched reluctance motor with a power of 400 W was used for experimental measurements.For such a low power motor, the price of a position sensor is comparable to the price of the motor.Therefore, efforts were made to design, with the use of microprocessor technology, a control algorithm to control SRM without the position sensor.
The paper deals with the sensorless control of the switched reluctance motor, particularly with the influence and estimation of phase resistance during motor operation.This control only uses the DC-Bus voltage and phase currents to estimate the rotor position, which can be easily measured from the motor connection terminals.The phase resistance estimation algorithm was developed based on the phase flux linkage calculation and its relation to the phase current.Experimental results confirm that without the proper phase resistance estimation, it is very difficult to control the sensorless SR drive according to the optimal motor characteristics linked to the rotor position due to the fact that resistance significantly changes with the temperature of the motor.Using the phase resistance estimation algorithm, the rotor position estimation error at middle and lower speeds is below 2 %.An applied "flux linkage" sensorless method is suitable for the area of middle and higher revolutions per minute, where there are no inaccuracies in the calculation of the flux linkage caused by a large number of integration steps.

Fig. 1 .
Fig. 1.Four-phase 8/6 SRM with a switched circuit for one phase.Magnetization characteristics of the SRM define the relationship between the flux linkage and the current for any rotor position.To measure the magnetization characteristics,

Fig. 3 .
Fig. 3. Effect of the value of phase winding resistance on the course of flux linkage.

Figure 5
Figure 5 shows a block diagram of the applied sensorless control algorithm designed to estimate the rotor position.The measured values of the phase currents and calculated

5 .
Simplified block diagram of the algorithm to estimate the rotor position.

6 .
Effect of winding resistance on calculation of the flux linkage and on the rotor position estimate accuracy when nRef = 300 rpm and RRef = +20 % without resistance correction.

Figure 7 7 .
Figure7shows the phase current courses and calculated flux with the set reference value of the phase resistance to be 20 % lower than its actual value and with nRef = 300 rpm.Because the resistance estimation error is negative (-20 %), the flux linkage error is positive for time ton, as described in

Fig. 8 .
Fig. 8. Estimate of winding resistance during SRM operation when nRef = 600 rpm and a preset load ML = MN.

Figure 8
Figure8shows an estimated winding resistance during the SRM operation over a longer time period and a preset load of working mechanism ML = MN.The curve courses show an initial value of the phase resistance RRef, variation of phase resistance R1Error, calculated each business cycle in the phase 1, corrected phase variation R1Error[av], and an estimated value of the phase resistance R1Assume[av].The picture shows that when SRM was loaded with nominal torque, phase resistance increased by 0.9 Ω in 14 minutes, which means an increase by 22 %.This demonstrates very clearly the effect of coil heating during SRM operation on its coil resistance.Figure9to Fig.12show the actual position and the estimated position, the estimated rotor position error, the estimation of rotor position of the individual phases and the number of the phase from which the rotor position is evaluated during sensorless control of the SRM at nRef = 420 rpm.The graph in Fig.10shows that the variation θError fluctuates in a range of ±2°.The principle of determining a suitable phase that is selected to estimate rotor position is described in Chapter 4.

Fig. 9 .
Fig. 9. Actual position and estimated position during sensorless control of the SRM at nRef = 420 rpm.

Fig. 12 .
Fig. 12. Number of the phase from which the rotor position is evaluated during sensorless control of the SRM at nRef = 420 rpm.

Figure 13 to
Figure13 to Fig.14show the actual position and the estimated position, actual phase currents i2 and i4 during sensorless control of the SRM motor at nRef = 250 rpm.Figure14shows that phase currents i2 and i4 are limited, so they do not exceed a maximum 4 A value.

Fig. 13 .
Fig. 13.Actual position and estimated position during sensorless control of the SRM at nRef = 250 rpm.

Figure 15 to
Figure15 to Fig.18show the actual position and the estimated position, actual speed and estimated speed, phase current i2 and flux linkage ψ2 during sensorless control of the SRM at nRef = 0 to 1750 rpm.Figure16shows that Figure15 to Fig.18show the actual position and the estimated position, actual speed and estimated speed, phase current i2 and flux linkage ψ2 during sensorless control of the SRM at nRef = 0 to 1750 rpm.Figure16shows that

Fig. 15 .
Fig. 15.Actual position and estimated position during sensorless control of the SRM at nRef = 0 to 1750 rpm.

Fig. 16 .
Fig. 16.Actual speed and estimated speed during sensorless control of the SRM at nRef = 0 to 1750 rpm.