Table 3.2: Rated data and parameters of the induction machine
Pn Nominal power 3 kW
nn Nominal rated speed 18/24 krpm f1n Rated frequency 300 Hz
VLL,rms Nominal voltage 380 V
Iph,rms Nominal current 7.7 A
τn Rated torque 1.5 Nm
Rs Stator resist. 1.125 Ω
Rr Rotor resist. 0.85 Ω
Lls Stator leak. induct. 25 mH Llr Rotor leak. induc. 14 mH Lm Magnetizin ind. 84.82 Ω
J Inertia 5 kgcm2
Table 3.3: Controller parameters KpΩ Speed cont. prop. gain 1.5
TiΩ Speed cont. time constant 0.085 s Kpiq q-axis cont. prop. gain 2
Tiiq q-axis cont. time constant 8 ms Kpid d-axis cont. prop. gain 2
Tiid d-axis cont. time constant 9 ms
speed is maintained close to the constant reference speed with a good accuracy (the relativ error is less than 0.5%).
Figure 3.11(b) and 3.12(b) show the trace of the simulated and measured stator current vectoris, Figure 3.11(c) and 3.12(c) show the trace of the simulated and measured stator flux vectorΨsin steady state whenTload = 0.6 Nm. The simulation and experimental result are in good agreement, but it can be seen the ripples in the current signal is higher in the measured signal. By comparing the two sampling techniques it can be concluded that the difference between RS and DS is negligible.
By increasing the reference speed Ω∗ and keepingfcconstant both the ratio of sampling to fundamental frequency mf and carrier to fundamentalF decreases. Whenmf orF is low the difference between RS and DS becomes crucial. Figure 3.13 shows the simulated time function of mechanical speed and simulated and measured time functions of the electric torque during transient when Ω∗ = 1005 rad/s (mf =F ≈8.75) and RS technique is applied. As it can be seen in the figure that the operation of the drive applying RS becomes unstable when the load torque was increased. During the laboratory analysis the over-current protection was activated and it shut down the system to protect the machines from damages.
By applying DS instead of RS the stability range of the drive could be extended: by recalculating the reference signals vref,i (i = a, b, c) of the SVM at the negative peak of the carrier signal (Fig.3.7(b)) where no current sampling took place, the performance of the drive was improved. Figure 3.14(a) shows the simulated mechanical speed Ω and the simulated and measured electric torque Te during transient when Ω∗ = 1005 rad/s and DS sampling technique was applied. It can be seen, contrary to RS, the response was stable. Figure 3.14(b) and 3.14(c) show the simulated and measured stator current is and stator flux Ψs. As it was expected at lower frequency ratio the ripples both in the currents and fluxes increased resulting in high total harmonic distortion (THD). As the frequency ratio is a low non-integer value, subharmonic components are generated both in the current and flux values, however, their amplitudes are negligible compared to the fundamental ones (see Fig.1.21(a) in Chapter I).
Again the simulated and measured results were in good agreement.
Figure 3.15(a) and 3.16(a) show the simulated time function of mechanical speed Ω and simulated and measured time function of the electric torque Te during transient when Ω∗ = 1100 rad/s (mf = F ≈ 8) and Ω∗ = 1162 rad/s (mf = F ≈ 7.6), respectively. Based on
(a) Time function of Ω andTe (b)isin steady state (c)Ψsin steady state
Figure 3.11: Simulated and measured responses for ramp like torque change applying RS SVM, Ω∗ = 440 rad/s,F =mf ≈20, ∆Tload = 0.6 Nm
(a) Time function of Ω andTe (b)isin steady-state (c)Ψsin steady state
Figure 3.12: Simulated and measured responses for ramp like torque change applying DS SVM, Ω∗ = 440 rad/s,F =mf ≈20, ∆Tload = 0.6 Nm
the figures the conclusion is that the control loop was stable even for such a low mf and F values. Figure 3.15(b) and 3.16(b) show the trace of the stator current vector is and 3.15(c) and 3.16(c) presents the trace of the stator flux vectorΨs. Due to the lowmf value the current and flux signals have high harmonic content.
By further increasing the reference speed the response of the drive applying DS sampling technique becomes also unstable. Figure 3.17 shows the time function of the simulated me-chanical speed Ω and the measured and simulated electric torque Te when Ω∗ = 1193 rad/s (F = mf ≈ 7.3). In the simulation model the torque starts to oscillating after the
load-Figure 3.13: Simulated and measured responses for ramp like torque change applying RS SVM, Ω∗ = 1005 rad/s, F =mf ≈8.75, ∆Tload = 0.6 Nm
(a) Time function of Ω andTe (b)isin steady state (c)Ψsin steady state
Figure 3.14: Simulated and measured responses for ramp like torque change applying DS SVM, Ω∗ = 1005 rad/s, F =mf ≈8.75, ∆Tload = 0.6 Nm
ing torque increases. However, the oscillations settles down, such a large torque pulsation is unacceptable in practical drives. The laboratory result is again in good agreement with the simulated one: the response becomes unstable and the over-current protection was activated which shut-down the machines.
Based on the simulation and experimental results it can be concluded it is worth to resample the SVM at the negative peak of the carrier signal as more robust control performance can be obtained. The stability range is extended. The stability border of the control loop for the given set of controller parameters given in Table 3.3 is expanded from Ω∗ = 160 rad/s to Ω∗ = 190 rad/s. It should be emphasized that the sampling frequency is the same both for RS and DS.
(a) Time function of Ω andTe (b)isin steady state (c)Ψsin steady state
Figure 3.15: Simulated and measured responses for ramp like torque change applying DS SVM, Ω∗ = 1100 rad/s, F =mf ≈8, ∆Tload = 0.6 Nm
(a) Time function of Ω andTe (b)isin steady state (c)Ψsin steady state
Figure 3.16: Simulated and measured responses for ramp like torque change applying DS SVM, Ω∗ = 1162 rad/s, F =mf ≈7.6, ∆Tload= 0.6 Nm
Speed sensor-less FOC control methods must be robust to plant parameter variations. Fig-ure 3.18 and 3.19(a) show the simulated mechanical speed Ω and the simulated and measFig-ured electric torque Te during transient for RS and DS, respectively. Now 20 % smaller mutual inductance value Lm,est was used during the estimation than the actual one and the stator resistance Rs,est was set to be only 40 % of the real one (the parameters of the motor can be found in Table 3.2). The reference speed was Ω∗ = 942 rad/s ( F = mf ≈ 9.3)in 3.18 and 3.19. By using RS sampling technique for the SVM the response was oscillatory even at no-load. By increasing the load torque from 0 Nm to 0.6 Nm in a ramp shape, the amplitude
Figure 3.17: Simulated and measured responses for ramp like torque change applying DS SVM, Ω∗ = 1193 rad/s, F =mf ≈7.3, ∆Tload= 0.6 Nm
of the oscillations was increased and the drive practically was becoming unstable. During the laboratory test similar phenomenon was obtained: oscillations in the electric torque were de-veloped and the over-current protection shut-down the drive. By applying DS instead of RS the response was stable (Fig.3.19(a)). By comparing the time functions of Ω in Fig.3.19(a) and in Fig.3.14(a) (the mf and F value were practically the same in both cases) the conclusion is that the magnitude of the ripples in the mechanical speed and the relative error of the speed controller were increased when inaccurate machine parameters were applied. However, the response remained stable and was still acceptable.
Figure 3.19(b) and 3.19(c) show the trace of the stator current vectorisand the stator flux vector Ψs, respectively. The current and flux signals has a high harmonic distortion due to the low mf. Again the measured and simulated trajectories are in good agreement.
The simulation and experimental results clearly demonstrated that the DS is more robust at low frequency ratios even for variations in the motor parameters.
Figure 3.18: Simulated and measured response for torque change applying RS SVM, Lm,est= 0.8Lm,Rs,est= 0.4Rs, Ω∗ = 942 rad/s,F =mf ≈9.3, ∆Tload = 0.6 Nm
(a) Time function of Ω andTe (b)isin steady state (c)Ψsin steady state
Figure 3.19: Simulated and measured responses for ramp like torque change applying DS SVM, Lm,est= 0.8Lm,Rs,est= 0.4Rs, Ω∗= 1162 rad/s, F =mf ≈7.6, ∆Tload = 0.6 Nm