Open access peer-reviewed chapter

Zero and Low-Speed Sensorless Control of Induction Machines Using Only Fundamental Pulse Width Modulation Waveform Excitation

Written By

Qiang Gao, Greg Asher and Mark Sumner

Submitted: November 3rd, 2017 Reviewed: February 23rd, 2018 Published: November 5th, 2018

DOI: 10.5772/intechopen.75892

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Abstract

This chapter presents a position sensorless method for induction machines that only relies on the fundamental pulse width modulation (PWM) waveforms to excite saliency. Position signals can be synthesized through the measurement of the derivatives of the line currents induced by the PWM voltage vectors. This method is essentially saliency detection based, and therefore derivation of the rotor position is possible at low and zero speeds. In addition, it works also at higher speeds without the need of the knowledge of the machine’s fundamental model. Experimental results showing fully sensorless induction motor control at low and higher speeds validate the principle of this method.

Keywords

  • sensorless control
  • induction machine
  • saliency
  • PWM excitation
  • current derivative

1. Introduction

Due to the incapacity of the fundamental model-based sensorless rotor position estimation at zero and low frequencies, alternative methods have been intensively studied. These methods exploit the anisotropy or saliency of the machine resulting from either saturation or geometric variation on the rotor. They can be classified into two categories according to the detection method for the anisotropy (or saliency) position. One category relies on the continuous injection of high-frequency voltage signals and then measuring the response of the high-frequency (hf)current [1, 2, 3, 4, 5, 6, 7]. Demodulation of the hfcurrent signal enables the extraction of the rotor angle. The second category makes use of the line-current transient response to a PWM switching state. This can be realized by injecting special voltage test vectors [8, 9, 10, 11, 12, 13] or by modifying the normal pulse width modulation waveforms [14], which may increase the hazardous common mode current in the machine [15]. The induced current transient response during a test vector reflects the spatial variation of the stator leakage inductances due to the anisotropy. Therefore, it is possible to acquire the rotor position, or rotor flux angle, through the measurement of the transient current derivative in response to the test vector.

In this chapter, a method belonging to the second category is described. Instead of using extra test vectors or modifying the standard modulation scheme, this method integrates the test vectors with the standard PWM waveforms [16]. In the following paragraphs, the theory of the method will be presented first, then, its application on a 4-pole 30 kW Δ-connected cage machine having 56 open slots will be demonstrated. Other implementation issues related to the speed sensorless operation, such as the noise filter of the position signals, will also be introduced.

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2. Position estimation with the fundamental wave PWM

When a three-phase, delta-connected induction machine has its stator leakage inductances modulated by the anisotropies introduced by either the main flux saturation or the rotor slotting, they can be assumed to vary according to:

lσa=l0+ΔlcosnanθanE1
lσb=l0+Δlcosnanθan2π/3E2
lσc=l0+Δlcosnanθan4π/3E3

where l0 is the average inductance and Δl is the amplitude of inductance variation caused by the anisotropy (nan = 2 for saturation-induced anisotropy or nan = nrs = Nr/p for rotor slotting, where Nr is rotor slot number and p the pole pairs).

The standard space vectors of Figure 1 are applied. Figure 2 shows the equivalent circuit when the machine is applied with vector u1 from which the following equations can be derived:

Ud=iabu1rs+lσadiabu1dt+eau1E4
0=ibcu1rs+lσbdibcu1dt+ebu1E5
Ud=icau1rs+lσcdicau1dt+ecu1E6

Figure 1.

Definition of space vectors.

Figure 2.

Equivalent circuit withu1being applied.

By the application of the null vector u0 or u7, one has:

0=iabu0rs+lσadiabu0dt+eau0E7
0=ibcu0rs+lσbdibcu0dt+ebu0E8
0=icau0rs+lσcdicau0dt+ecu0E9

If the instants of applying u1 and u0 are close enough, it is viable to assume that:

eau0eau1,ebu0ebu1,ecu0ecu1.

Additionally, the voltage drops across the stator resistance and can be ignored due to their small values compared with Ud.

Hence, the subtraction of the Eqs. (4)(7), (5)(8) and (6)(9) yields:

diabu1dtdiabu0dt=UdlσaE10
dibcu1dtdibcu0dt=0E11
dicau1dtdicau0dt=UdlσaE12

From the relationship between phase currents and line currents, for example, ia = iab- ica, one has:

diau1dtdiau0dt=lσa+lσclσalσcUdE13
dibu1dtdibu0dt=1lσaUdE14
dicu1dtdicu0dt=1lσcUdE15

Considering Eqs. (1) to (3), one has:

diau1dtdiau0dt=Udl02+Δll0cosnanθan2π3E16
dibu1dtdibu0dt=Udl01Δll0cosnanθanE17
dicu1dtdicu0dt=Udl01Δll0cosnanθan4π3E18

from which three balanced position scalars pa, pband pccan be defined as follows:

pa=1+c1dibu1dtdibu0dtE19
pb=2+c1diau1dtdiau0dtE20
pc=1+c1dicu1dtdicu0dtE21

where c1=l0/Ud. If both c1and didtare known, it is possible to construct the position vector pdirectly via:

p¯=pa+apb+a2pcE22

where a=ej2π/3. However, c1consists of an unknown coefficient l0, which may vary with the saturation level of the main flux. This uncertainty can be avoided by looking at the current response to another voltage vector u2. Following the same as earlier, another three position scalars can be defined as:

pa=2c1dicu2dtdicu0dtE23
pb=1c1dibu2dtdibu0dtE24
pc=1c1diau2dtdiau0dtE25

It should be noted that because the application of u1 and u2 results in the same position scalars, they are defined with the same terms, pa, pband pc.

By referring to (19), (20), (21) and (23), (24), (25), it is possible to define pthrough the combination of (19), (24) and (25):

p¯=pα+jpβ=pa+apb+a2pc=c1dibu1dtdibu0dtadibu2dtdibu0dta2diau2dtdiau0dtE26

Therefore,

pα=c1dibu1dtdibu0dt+12dibu2dtdibu0dt+12diau2dtdiau0dtE27
pβ=32c1diau2dtdiau0dtdibu2dtdibu0dtE28

whereby the position can be derived through the arctan operation as shown in (29).

θan=arctanpβ/pαE29

Such a combination between two adjacent voltage vectors, and a null vector, also exists in other five sectors. Therefore, the position estimation is achievable using only the fundamental PWM sequence. Table 1 gives the position scalars corresponding to all the vectors.

Table 1.

Definition of position scalars of all voltage vectors in a delta‐connected IM.

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3. Experimental implementation

3.1. Practical considerations

It can be seen that the correct position estimation relies on the precise measurement of the di/dt signals. For this aim, air-cored Rogowski [9], ferrite-cored Rogowski [18], or air-cored coaxial cable-typed [14] transducers can be used. Or direct digital calculation, di/dt = i(t2)−i(t1)/(t2−t1), can be employed instead. Figure 3 shows a typical di/dt signal along with ADC trigger signal when an air-cored coaxial cable-typed transducer is used.

Figure 3.

ADC trigger signal (1,V) and di/dt signal (2,V) measured by a air-cored coaxial cable type transducers.

Another important issue comes from the fact that due to the common/differential mode voltages produced by the inverter, high-frequency oscillation exists in the phase currents, which poses a challenge for accurate di/dt measurement. This is true when dwell times of the voltage vectors are too short or when the reference voltage vectors pass the boundaries of sectors. Therefore, a minimum dwell time, tmin, is imposed on voltage vectors for di/dt measurement. When the original dwell time of a voltage vector, t, is shorter than tmin, an opposite vector with a dwell time, tmin- t, is added to maintain the volt-second. This procedure can be realized simply by the edge-shifting technique, which is illustrated in Figure 4 when the reference voltage lies in Sector I.

Figure 4.

Edge shifting of PWM waveforms when the dwell times of u1 and u2 are smaller thantmin.

3.2. System schematic

Figure 5 shows the system schematic of the sensorless speed control. Three di/dt sensors are connected in series with the lines of the induction motor, whose parameters are given in Table 2. The position vector formation block synthesizes the position vector pαβaccording to the measured di/dts and the sector index (Si) of the reference voltage vector. For example, when Si = 1, Eqs. (27) and (28) can be used. Each of the other five sectors has its own set of equations like (27) and (28) such that the continuous position signal pαβ can be constructed. Ideally, pαβcan be used for position/speed estimation, but due to the presence of disturbance signals, which are mainly due to the saturation of the machine, such as the 2feand 4fecomponents [3], the estimated position will become distorted if pαβis used directly. This disturbance cannot be eliminated by filtering since the disturbance frequencies converge to the wanted signal frequencies as the excitation frequency approaches zero. As a result, a memory-type filter is required. In this work, an adaptive disturbance identifier (ADI) [17] is employed to separate these disturbance signals, i.e., pdαβ_m, which are filtered out from pαβ. This identifier does not require a pre-commissioning phase. Rather it initiates a learning type sequence in which the resolver disturbance signals are estimated and continually refined as the machine passes through the appropriate torque-speed space [17]. The pdαβ_mdisturbance signals are stored in memory and are subtracted from the uncompensated signals to give the compensated signal prsαβ. The resulting position vector prsαβis fed to the Speed/Position Calculation block for the speed and position estimation.

Figure 5.

System schematic of the sensorless speed control (ADI).

Table 2.

Parameters of the IM.

The Speed/Position Calculation block further refines the position signals. Because the ADI-compensated rotor slot position still consists of a speed dependent disturbance signal rotating at Nrωr/p-2ωi, which is attributed to the inter-modulation effect between the slotting and saturation in the machine [11], and can be conveniently removed by a side band filter [11]. The filtered position signal is shown in Figure 6.

Figure 6.

Position estimation at 15 rpm with 75% rated load (fe=1.0 Hz). Top: filtered position signal; and bottom: estimated rotor position (rad).

The position signals obtained so far can be used for position acquisition in the way shown in (29). However, a mechanical observer similar to [19] is utilized to reduce the noise. The schematic is given in Figure 7.

Figure 7.

Schematic of the modified mechanical observer (Jis the moment of inertial andkis a design constant, s = d/dt).

Figure 8 shows an improved estimated rotor position signal before and after the observer. An offset angle can be observed between these two rotor positions. This is due to the fact that the mechanical observer’s estimation was aligned to the encoder’s angle initially, whereas the estimate from the SBF only yields the incremental position.

Figure 8.

Rotor position estimation (rad) at −210 rpm, no load top: before the observer, bottom: after the observer.

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4. Experimental results

The following experiments show the operation of the drive under sensorless speed control at low and higher speeds. The induction motor drive is under speed sensorless control. The load machine is under torque control. The rating of the dynamometer converter is such that only a loading of 80% rate can be applied to the induction motor. It is emphasized that all the experimental waveforms were recorded on an experiment rig.

Figure 9 shows that under no load condition the IM is reversed at 6 rpm, corresponding to the excitation frequency fe = 0.2 Hz. Good rotor estimation can be seen. In Figure 10, the drive performs a speed reversal at ±12 rpm under 70% rated load condition. At –12 rpm, this load condition corresponds to the drive under braking at zero excitation frequency. Since a constant torque is applied to the DC load machine, under speed reversal, the amplitude of isqchanges due to the losses involved.

Figure 9.

Speed reversal at 6 rpm, no load top: measured speed (rpm), bottom: estimated rotor position (rad).

Figure 10.

Speed reversal at ±12 rpm under 70% rated load top: measured speed (rpm), bottom: rotor flux angle (1, rad) and filtered Isq (2, A).

In Figure 11, the drive was taken between ±210 rpm under no load. This test confirms the capability of the drive to perform larger speed transients.

Figure 11.

Speed reversal at ±240 rpm, no load top: measured speed (1, rpm) and filtered Isq (2, A); bottom: estimated rotor position (rad).

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5. Conclusion

A sensorless position scheme for AC machines is presented relying on the line-current derivative measurements in response to a fundamental PWM switching sequence. The rotor position angle is derived due to the tracking of rotor slotting and the signal is used for sensorless control. If the anisotropy caused by the main flux saturation is tracked, for example, in a permanent magnet machine, this method is applicable throughout a very wide speed range. If rotor slotting is tracked, then the maximum speed is limited by the Nyquist frequency associated with the rotor slot passing frequency. In principle, however, the method can work over the entire torque-speed envelope.

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Written By

Qiang Gao, Greg Asher and Mark Sumner

Submitted: November 3rd, 2017 Reviewed: February 23rd, 2018 Published: November 5th, 2018