5.4 Comparison between the Classical and the Proposed Method
5.4.2 Stator Current Performance
The performance of the stator current have been assessed by means of the Total Harmonic Distortion (THD). The THD is defined as follows:
T HD = Ih If Ih = q I2 2 + I32+ . . . + In2 (5.7)
where In is the RMS value of the nth harmonic and If is the RMS value
of the fundamental current.
0 1000 2000 3000 0 0.5 1 0 2 4 Speed (rpm) Classical method T e (Nm) THD 0 1000 2000 3000 0 0.5 1 0 2 4 Speed (rpm) Proposed method T e (Nm) THD
Figure 5.36: Comparison between the classical and the proposed method. Stator current performance.
The stator current performance over all the machine operating points is analyzed in Fig. 5.36. It is noted that the proposed method shows better performance at any operating point. As in the case of the stator flux ripple, it should be emphasize the difference between both systems at low torque values as the rated speed is approached.
Table 5.4 summarizes the improvements achieved by the proposed method in terms of stator flux and electromagnetic torque ripples, relative error of the
torque mean value and the stator current THD. The maximum, minimum, and the average improvement in % is calculated for every improvement vari- able.
Table 5.4: Improvements of the proposed method.
Improvement σTe (%) Teerror (%) σ|ψs|(%) THD (%)
max 50.27 37.13 21.22 62.92
min 16.62 0.14 4.12 7.33
average 28.28 7.21 9.64 21.09
It can be seen that higher improvements are achieved in terms of electro- magnetic torque ripple. However, it should be noted from Figs. 5.31, 5.33, 5.35 and 5.36 that higher improvements can be achieved by other variables depending on the machine operating point.
Table 5.5 shows the improvements achieved by every variable at different values of speed and torque.
Table 5.5: Improvements dependency of speed and torque.
Speed/Torque σTe Teerror σ|ψs| THD
Low/Low Medium Small High Medium Low/High Medium Small Medium Small High/Low High High Medium High High/High High Medium Small High
5.5
Conclusions
The use of MC implementing a DTC to drive a PMSM has been investigated in this chapter. A new DTC scheme has been developed and compared with the classical DTC using MC pointing out the benefits of the former. The proposed method employs not only the large voltage vectors of the MC but also the small ones. This fact makes possible the inclusion of a four-level
hysteresis comparator allowing the DTC to distinguish between small and large electromagnetic torque errors and, hence, reducing the electromagnetic torque and stator flux ripples.
The proposed scheme also improves the voltage transfer ratio of the MC by 36% (50% for the classical method and 86% for the proposed method) in the worst case. This fact allows the proposed method to reach the rated operating point of the machine with much less steady state error than the classical DTC using MC. Moreover, the improvement in the voltage transfer ratio will make the system to response faster to a torque transient.
Reducing the Common Mode
Voltage using Matrix
Converters
It is well known that one of the main sources of early motor winding failure and bearing deterioration is the Common Mode Voltage (CMV) produced by modern power converters. Moreover, high frequency components and large amplitudes of the CMV at the motor neutral point have been shown to gen- erate high frequency currents to the ground path and induced shaft voltage [69]. Several methods to reduce the CMV have been proposed in the liter- ature [70], [71], however, these methods are designed for three-phase Pulse Width Modulation (PWM) VSI systems. In [50], a new modulation strategy of the converter which reduces the CMV at the output of the MC is pre- sented. Based on the Indirect Space Vector Modulation (ISVM) technique, the method selects a medium value phase voltage as a zero vector and places it in the center or on the both sides of the sampling period without changing the active voltage vectors. The harmonic content of the input current is also reduced. However, these advantages are achieved at the expense of deterio-
rating the output harmonic content.
An approach to reduce the CMV in a DTC-PMSM drive using Matrix Converters is presented in this chapter. Firstly, the CMV in MC is mathe- matically analyzed. Based on this analysis, a very simple algorithm to reduce the CMV is proposed and investigated.
6.1
Common Mode Voltage in Matrix Con-
verters
Fig. 6.1 shows a MC connected to a PMSM where the impedance Zcm repre-
sents the leakage current (icm) path between the machine’s neutral point (n)
and ground (N). The CMV vcm is defined as the voltage between these two
points:
vcm= vnN (6.1)
Taking into consideration the CMV vcm, the voltage expressions of the
PMSM can be written as:
vaN − vcm= Rsia+ Ls dia dt + d dtψP Mcosθr (6.2) vbN − vcm = Rsib+ Ls dib dt + d dtψP Mcos(θr− 2π 3 ) (6.3) vcN − vcm = Rsic+ Ls dic dt + d dtψP Mcos(θr− 4π 3 ) (6.4)
where vaN, vbN, and vcN are the MC output voltages with respect to
ground. Rs and Ls are the resistance and inductance at each phase of the
PMSM and ψP M is the flux produced by the permanent magnets. Since
Figure 6.1: 3 × 3 Matrix Converter.
induced voltage system ψP Mcosθr+ ψP Mcos(θr−2π3 ) + ψP Mcos(θr−4π3 ) = 0,
the CMV can be found adding (6.2), (6.3) and (6.4): vcm=
vaN + vbN + vcN
3 (6.5)
As it can be seen in table 3.2 from chapter 3, when active vectors are selected in a MC, there are two output phases connected to the same input phase. Thus, the CMV when active vectors are delivered by a MC va
cm can
be written as:
vcma = vi+ 2vj
3 (6.6)
where i and j represents the two input phases involved when an active vector is present at the output of the MC.
In MCs, the input filter is designed in such a way that the phase voltages vA, vB, and vC can be considered equal to the phase voltages of the mains
vSA, vSB, and vSC respectively [41]. Assuming a balanced voltage system at
the mains and, hence, at the input side of the MC,
vSA = vA= Vpsin (ωt) (6.7) vSB = vB = Vpsin (ωt − 2π 3 ) (6.8) vSC = vC = Vpsin (ωt + 2π 3 ) (6.9)
where Vp is the voltage peak value per phase, (6.6) can be written as:
va cm= 1 3Vpsin (ωt) + 2 3Vpsin (ωt ± 2π 3 ) (6.10)
Rearranging (6.10), a simpler form of va
cm can be obtained: va cm= 1 √ 3Vpsin (ωt ± π 2) (6.11)
Equation (6.11) states that the instantaneous value of the CMV when any active vector is delivered by the MC va
cm, depends on the instantaneous value
of the input voltages selected to deliver such vector. Hence, the maximum instantaneous value of the CMV when active vectors are delivered by the MC va cm will be: va cmmax = ± 1 √ 3Vp (6.12)
The contribution to the CMV of each active vector is shown in Fig. 6.2. The frequency of voltages vSA, vSB, and vSC is 50Hz. In every input vol-
tage period (T = 0.02s), a different output vector is delivered by the MC (+1 . . . + 9 ; −1 . . . − 9). It can be noted that, for every active vector, the CMV is at its maximum value, ±√1
3Vp, every kπ radians (k = 0, 1, 2 . . . n).
When a zero vector is selected in a MC, the three output phases are connected to the same input phase. Thus, when a zero vector is delivered by a MC, the CMV vz
Figure 6.2: Common mode voltage when active vectors are delivered by a MC.
vz
cm= vi (6.13)
where i represents the selected input phase.
Since vi = Vpsin (wt ± φ), the maximum CMV vcmz when a zero is selected
will be:
vz
cmmax = ±Vp (6.14)
Fig. 6.3 shows the contribution to the CMV of each zero vector (a different zero vector is selected in every input voltage period T = 0.02s). As it can be seen, the CMV follows the selected phase to deliver the zero vector, being the maximum CMV equal to the peak value of that phase.
Figure 6.3: Contribution of zero vectors to the common mode voltage.