Indirect Space Vector Modeling of Asynchronous Motor for High-Speed Electric Vehicle Propulsion

Vehicle propulsion system now-a -days mainly concern to the eco-friendly and green transportation. The depletion of fossil fuels, carbon dioxide (CO₂) emission and ever- growing energy demand are the leading contributors of climatic change and environmental pollution [1], which paid attention to the EV. EV’s are the promising technology for the energy efficient and pollution free environment compared to internal combustion engine vehicle (ICEV) [2, 3].

Advances in power converter design for renewable energy sources and charging electric vehicles made EV’s more reliable [4]. In EV’s, numerous types of machines are employed, out of them synchronous motors with permanent magnet (PMSM) and ASMs are used for propelling. ASM is widely preferred because it gives best compliance in manufacturing cost and power density. It is also simple in construction, high reliable, minimum torque ripple, easy in control, low noise and a robust machine. Moreover, it is highly efficient at medium and high frequencies with minimum iron losses [5, 6]. The control schemes mainly applied to ASM are V/F control, adaptive control, direct torque control and fuzzy logic control [7-10]. The IDSVC of ASM is the basic closed loop control scheme with less complexity in design and operation and could achieve the requirements of high-performance speed regulation.

In this paper, the indirect space vector control scheme implementation for asynchronous motor involves the conversion of the actual stator and rotor three phase windings by an equivalent set of orthogonal di-phasic windings and further to d-q orthogonal windings, which produces same MMF in the air gap, without direct consideration of Park’s transformation matrix [11]. Nevertheless, transformation matrix is derived by physical approach. The indirect space vector control strategy for torque and speed regulation is exercised and then finally the SVPWM inverter in association with SOLP RLC filter with Butterworth approximation is employed to achieve harmonic free speed and torque curve in high speed traction.

For dynamic analysis of ASM, we need two orthogonal windings by which the flux and torque can alone be governed. The stator and rotor have equal number of turns (N_s). Assume the stator windings be wye-connected where neutral not accessible and the number of turns of orthogonal windings as $\sqrt{3 / 2}$N_s such that d-q windings have same stator resistance and leakage reactance similar to three phase windings.

2.png

Figure 2. d-q winding representation of stator and rotor currents

$\overrightarrow{u_{s}^{a}}(t)=u_{a}(t) e^{j 0}+u_{b}(t) e^{\frac{j 2 \pi}{3}}+u_{c}(t) e^{\frac{j 4 \pi}{3}}=u^{\wedge}(t) e^{j \theta_{u_{s}}(t)}$    (1)

$\overrightarrow{i_{s}^{a}}(t)=i_{a}(t) e^{j 0}+i_{b}(t) e^{\frac{j 2 \pi}{3}}+i_{c}(t) e^{\frac{j 4 \pi}{3}}=i^{\wedge}(t) e^{j \theta_{i_{s}}(t)}$    (2)

$\overrightarrow{\psi_{s}^{a}}(t)=\psi_{a}(t) e^{j 0}+\psi_{b}(t) e^{\frac{j 2 \pi}{3}}+\psi_{c}(t) e^{\frac{j 4 \pi}{3}}=\psi^{\wedge}(t) e^{j \theta_{\psi_{s}}(t)}$    (3)

where, $\overrightarrow{u_{s}^{a}}(t), \overrightarrow{i_{s}^{d}}(t)$ and $\overrightarrow{\psi_{s}^{a}}(t)$ are stator three phase space vectors and $u^{\wedge}(t) e^{j \theta_{u_{s}}(t)}, i^{\wedge}(t) e^{j \theta_{i_{s}}(t)}$ and $\psi^{\wedge}(t) e^{j \theta_{\psi_{s}}(t)}$ are stator three phase phasors referred to stator a-axis. The MMF distribution produced by the hypothetical d-q winding in the air gap is same, similar to three phase stator windings [Appendix 1]. The stator current space vector with reference d-axis is given in Eq. (4), (5) [13].

$\overrightarrow{l_{s}^{d}}(t)=i_{a}(t) e^{-j \theta_{d s a}+i_{b}}(t) e^{-j\left(\theta_{d s a}-\frac{2 \pi}{3}\right)}+i_{c}(t) e^{-j\left(\theta_{d s a}-\frac{4 \pi}{3}\right)}$    (4)

$\overrightarrow{i_{s}^{d}}(t)=\sqrt{\frac{3}{2}}\left(i_{s d}(t)+j i_{s q}(t)\right)$    (5)

From Figure 2 it is observed

$i_{s d}(t)=\sqrt{\frac{2}{3}}\left(\right.$ Projection of $i_{s}(t)$ along $d$-axis $)$

$=\sqrt{\frac{2}{3}} \cdot\left(\frac{3}{2} \hat{i}_{\mathrm{s}}\right) \cos \theta_{\mathrm{dsa}}$    (6)

$\mathrm{i}_{\mathrm{sq}}(\mathrm{t})=\sqrt{\frac{2}{3}}\left(\right.$ Projection of $\mathrm{i}_{\mathrm{s}}(\mathrm{t})$ along $\mathrm{q}$-axis $)$

$=\sqrt{\frac{2}{3}} \cdot\left(\frac{3}{2} \hat{\mathrm{i}}_{\mathrm{s}}\right) \sin \theta_{\mathrm{dsa}}$    (7)

Equating the real and imaginary parts of Eq. (4) and Eq. (5). In the situation of an isolated neutral, the sum of all 3-phase currents is always zero. The dq0 winding variables can be determined in terms of abc winding variables. The bottom third row of ones in the matrix shows the situation where the total of all 3-phase currents is zero, $i_{s 0}(t)=i_{a}(t)+i_{b}(t)+i_{c}(t)=0$. Therefore, we omit the third row and consider a 2x3 matrix instead of 3x3 matrix as shown in Eq. (8.2).

$\left(\begin{array}{c}i_{s d}(t) \\ i_{s q}(t) \\ i_{s 0}(t)\end{array}\right)=$

$\sqrt{\frac{2}{3}}\left(\begin{array}{ccc}\cos \left(\theta_{d s a}\right) & \cos \left(\theta_{d s a^{-}} \frac{2 \pi}{3}\right) & \cos \left(\theta_{d s a^{-}} \frac{4 \pi}{3}\right) \\ -\sin \left(\theta_{d s a}\right) & -\sin \left(\theta_{d s a^{-}} \frac{2 \pi}{3}\right) & -\sin \left(\theta_{d s a^{-}} \frac{4 \pi}{3}\right) \\ 1 & 1 & 1\end{array}\right)$

$\left(\begin{array}{l}i_{a}(t) \\ i_{b}(t) \\ i_{c}(t)\end{array}\right)$    (8.1)

$\left(\begin{array}{l}i_{s d}(t) \\ i_{s q}(t)\end{array}\right)=$

$\sqrt{\frac{2}{3}}\left(\begin{array}{ccc}\cos \left(\theta_{d s a}\right) & \cos \left(\theta_{d s a^{-}} \frac{2 \pi}{3}\right) & \cos \left(\theta_{d s a^{-}} \frac{4 \pi}{3}\right) \\ -\sin \left(\theta_{d s a}\right) & -\sin \left(\theta_{d s a^{-}} \frac{2 \pi}{3}\right) & -\sin \left(\theta_{d s a^{-}} \frac{4 \pi}{3}\right)\end{array}\right)$

$\left(\begin{array}{l}i_{a}(t) \\ i_{b}(t) \\ i_{c}(t)\end{array}\right)$    (8.2)

The 2×3 matrix is called state transformation matrix [Z_s]_abc-dq0 given in Eq. (8.2). Similarly, we apply the same transformation matrix for stator voltages and flux linkages. Consider Eq. (8.1) and solve for abc phase currents in terms of dq0 winding currents. The inverse 3x3 matrix in Eq. (8.1) is given in Eq. (9.1). We get the necessary relationship by removing the third column of ones which has no contribution as shown in Eq. (9.2).

$\left(\begin{array}{l}i_{a}(t) \\ i_{b}(t) \\ i_{c}(t)\end{array}\right)=\sqrt{\frac{2}{3}}\left(\begin{array}{ccc}\cos \left(\theta_{d s a}\right) & -\sin \left(\theta_{d s a}\right) & 1 \\ \cos \left(\theta_{d s a}+\frac{4 \pi}{3}\right) & -\sin \left(\theta_{d s a}+\frac{4 \pi}{3}\right) & 1 \\ \cos \left(\theta_{d s a}+\frac{2 \pi}{3}\right) & -\sin \left(\theta_{d s a}+\frac{2 \pi}{3}\right) & 1\end{array}\right)$

$\left(\begin{array}{c}i_{s d}(t) \\ i_{s q}(t) \\ i_{s 0}(t)\end{array}\right)$    (9.1)

$\left(\begin{array}{c}i_{a}(t) \\ i_{b}(t) \\ i_{c}(t)\end{array}\right)=\sqrt{\frac{2}{3}}\left(\begin{array}{cc}\cos \left(\theta_{d s a}\right) & -\sin \left(\theta_{d s a}\right) \\ \cos \left(\theta_{d s a}+\frac{4 \pi}{3}\right) & -\sin \left(\theta_{d s a}+\frac{4 \pi}{3}\right) \\ \cos \left(\theta_{d s a}+\frac{2 \pi}{3}\right) & -\sin \left(\theta_{d s a}+\frac{2 \pi}{3}\right)\end{array}\right)$

$\left(\begin{array}{l}i_{s d}(t) \\ i_{s q}(t)\end{array}\right)$    (9.2)

Here, the 3×2 matrix is also called state transformation matrix [Z_s]_dq-abc in reverse direction. The same procedure is adopted to derive the rotor d-q quantities, assuming the rotor is wound with three phase winding similar to stator three phase winding and replacing θ_dsa with θ_dra. To derive the Stator winding equations, consider a pair of orthogonal α-β windings lined up to stator which is shown in Figure 3. That is α-axis lined up to stator a-axis. Similarly, we can derive rotor winding equations by considering Figure 4.

3.png

Figure 3. α-β and d-q equivalent windings of stator

4.png

Figure 4. α-β and d-q equivalent windings of rotor

u_sα = R_si_sα + $\frac{d}{d t} \psi$_sα    (10)

u_sβ = R_si_sβ + $\frac{d}{d t} \psi$_sβ    (11)

Since α and β components are orthogonal combine Eq. (10) and Eq. (11) with j operator for space vector representation.

$\overrightarrow{u_{s_{-} \alpha \beta}^{\alpha}}=R_{s} \overrightarrow{i_{s_{-} \alpha \beta}^{\alpha}}+\frac{d}{d t} \overrightarrow{\psi_{s  \alpha \beta}^{\alpha}}$    (12)

where, $\overrightarrow{u_{s_{-} \alpha \beta}^{\alpha}}$ =u_α+ju_β.

The α-axis reference orthogonal α and β space vector components can be related to d-axis reference orthogonal d-q components. The function of this representation is to convert orthogonal α- β components in stationary frame of reference into d-q components in rotating frame of reference. This is how the Park’s transformation can be achieved.

$\overrightarrow{u_{s \quad \alpha \beta}^{\alpha }}=\overrightarrow{u_{s \quad d q}^{\alpha} } e^{j \theta_{d s a}}$    (13)

$\overrightarrow{i_{s \quad \alpha \beta}^{\alpha }}=\overrightarrow{i_{s \quad d q}^{\alpha} } e^{j \theta_{d s a}}$    (14)

Substituting Eq. (13), (14) in Eq. (12),

$\overrightarrow{u_{s_{-} d q}^{\alpha}}$ = R_s $\overrightarrow{i_{s_{d q}}^{\alpha} }$ + $\frac{d}{d t} \overrightarrow{\psi_{s \quad d q }^{\alpha}}$ +jω_d $\overrightarrow{\psi_{s\quad {d q}}^{\alpha} }$     (15)

where, $\frac{\mathrm{d}}{\mathrm{dt}}$θ_dsa= ω_dsa, instantaneous speed (rad.s^-1).

u_sd = R_si_sd + $\frac{d}{d t} \psi$_sd - ψ_sqω_dsa    (16)

u_sq = R_si_sq +$\frac{d}{d t} \psi$ _sq + ψ_sdω_dsa    (17)

$\left[\begin{array}{l}u_{s d} \\ u_{s q}\end{array}\right]=R_{s}\left[\begin{array}{l}i_{s d} \\ i_{s q}\end{array}\right]+\frac{d}{d t}\left[\begin{array}{l}\psi_{s d} \\ \psi_{s q}\end{array}\right]+\omega_{d s a}\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]\left[\begin{array}{l}\psi_{s d} \\ \psi_{s q}\end{array}\right]$    (18)

To obtain the rotor equations, we assume the rotor was wound with 3-phase windings and follow the same approach as was used to obtain the stator windings earlier. Simply replace the suffix s (stator) with r (rotor) and dsa with dra in Eq. (18) instead of going through the overall process.

$\left[\begin{array}{l}u_{r d} \\ u_{r q}\end{array}\right]=R_{r}\left[\begin{array}{l}i_{r d} \\ i_{r q}\end{array}\right]+\frac{d}{d t}\left[\begin{array}{l}\psi_{r d} \\ \psi_{r q}\end{array}\right]+\omega_{d r a}\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]\left[\begin{array}{l}\psi_{r d} \\ \psi_{r q}\end{array}\right]$    (19)

Note that for squirrel cage rotor with short circuited rotor bars $\left[\begin{array}{l}u_{r d} \\ u_{r q}\end{array}\right]=\left[\begin{array}{l}0 \\ 0\end{array}\right]$. Flux linkages in terms of input voltages and currents are obtained from Eq. (18), (19).

$\frac{d}{d t}\left[\begin{array}{l}\psi_{s d} \\ \psi_{s q}\end{array}\right]=\left[\begin{array}{l}u_{s d} \\ u_{s q}\end{array}\right]-R_{s}\left[\begin{array}{l}i_{s d} \\ i_{s q}\end{array}\right]-\omega_{d s a}\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]\left[\begin{array}{l}\psi_{s d} \\ \psi_{s q}\end{array}\right]$    (20)

$\frac{d}{d t}\left[\begin{array}{l}\psi_{r d} \\ \psi_{r q}\end{array}\right]=\left[\begin{array}{l}u_{r d} \\ u_{r q}\end{array}\right]-R_{r}\left[\begin{array}{l}i_{r d} \\ i_{r q}\end{array}\right]-\omega_{d r a}\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]\left[\begin{array}{l}\psi_{s d} \\ \psi_{s q}\end{array}\right]$    (21)

Eq. (20), (21) can be rewritten as

$\frac{d}{d t}\left[\overrightarrow{\psi_{s_{-} d q}}\right]=\left[u_{s_{-}} d q\right]-R_{s}\left[i_{s_{-} d q}\right]-\omega_{d s a}\left[A_{\text {rotate }}\right]\left[\psi_{s_{-} d q}\right]$    (22)

$\frac{d}{d t}\left[\overrightarrow{\psi_{r_{-} d q}}\right]=\left[u_{r_{-} d q}\right]-R_{r}\left[i_{r_{-}} d q\right]-\omega_{\text {dra }}\left[A_{\text {rotate }}\right]\left[\psi_{r_{-} d q}\right]$    (23)

Space vector $\overrightarrow{\psi_{s_{-} d q}}$ is rotated by an angle of π/2 with the matrix $\left[A_{\text {rotate }}\right]$ =$\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]$_. On rotor d-axis and q-axis winding, electromagnetic torque is given in Eq. (24), (25) [13].

$T_{d(\text { rotor })}=\frac{p}{2}\left(L_{m} i_{s q}+L_{r} i_{r q}\right) i_{r q}=\frac{p}{2}\left(\psi_{r q}\right) i_{r d}$    (24)

$T_{q(\text { rotor })}=-\frac{p}{2}\left(L_{m} i_{s d}+L_{r} i_{r d}\right)=-\frac{p}{2}\left(\psi_{r d}\right) i_{r q}$    (25)

The constraints for the establishment of flux linkages are (1) The d-axis windings and q-axis windings have zero mutual coupling. (2) The flux that links any winding is caused by its self current as well as the current of the other winding on the same axis. One could write flux expressions for all four windings using this logic. The flux linkages in terms of their direct and quadrature currents in stator and rotor are given in Eq. (26), (27), (28) and (29).

$\psi_{s d}=L_{s} i_{s d}+L_{m} i_{r d}$    (26)

$\psi_{s q}=L_{s} i_{s q}+L_{m} i_{r q}$    (27)

$\psi_{r d}=L_{r} i_{r d}+L_{m} i_{s d}$    (28)

$\psi_{r q}=L_{r} i_{r q}+L_{m} i_{s q}$    (29)

where, R_s and R_r are per phase stator and rotor resistances in ohm respectively. L_s, L_r and L_m are per phase stator, rotor and magnetizing inductances in Henry respectively. Total electromagnetic torque $T_{E}=T_{d(\text { rotor })}+T_{q \text { (rotor })}$ (N-m).

$T_{E}=\frac{p}{2}\left(\psi_{r q} i_{r d}-\psi_{r d} i_{r q}\right)$    (30)

$T_{E}=\frac{p}{2} L_{m}\left(i_{s q} i_{r d}-i_{s d} i_{r q}\right)$    (31)

$\omega_{M e c h}=\frac{2}{p}\left(\omega_{m}\right)$    (32)

$\frac{d}{d t} \omega_{M e c h}=\frac{\left(T_{E^{-}} T_{L}\right)}{J}$    (33)

where, $\omega_{M e c h}$ is mechanical speed of motor (rad-s^-1), T_L is the load torque (N-m), Representing the Eq. (26, 27, 28, 29) in matrix form. Let the M stands for 4x4 matrix [Appendix 2].

$\left[\begin{array}{l}\psi_{s d} \\ \psi_{s q} \\ \psi_{r d} \\ \psi_{r q}\end{array}\right]=\left[\begin{array}{cccc}L_{s} & 0 & L_{m} & 0 \\ 0 & L_{s} & 0 & L_{m} \\ L_{m} & 0 & L_{r} & 0 \\ 0 & L_{m} & 0 & L_{r}\end{array}\right]\left[\begin{array}{l}i_{s d} \\ i_{s q} \\ i_{r d} \\ i_{r q}\end{array}\right]$    (34)

$\left[\begin{array}{llll}i_{s d} & i_{s q} & i_{r d} & i_{r q}\end{array}\right]^{T}=M^{-1}\left[\begin{array}{llll}\psi_{s d} & \psi_{s q} & \psi_{r d} & \psi_{r q}\end{array}\right]^{T}$    (35)

In Figure 2 each alone stator and rotor d-axis windings are aligned to rotor flux ($\overrightarrow{\psi_{r}}$) to draw simplified equations. We will represent $\boldsymbol{\omega}_{d r a}$ and electromagnetic torque in terms of $i_{s q}$ and $\psi_{r d}$. Under vector-controlled operation, $\psi_{r d}$ is kept invariable and $i_{s q}$ steer the electromagnetic torque generation. The dynamics for $\psi_{r d}$ are built. In the alignment of d-axis of both stator and rotor to rotor flux ($\overrightarrow{\psi_{r}}$), which that means that the q-axis of both stator and rotor windings will be orthogonal to the rotor flux. So, q-axis rotor winding will not have any flux linking with the rotor flux. Hence q-axis rotor flux linkages are zero as given in Eq. (36). The absence of q- axis rotor flux linkages indicates that the rate of change of their flux linkage will definitely be equals to zero.

$\psi_{r q}=0, \Rightarrow \frac{d}{d t}\left(\psi_{r q}\right)=0$    (36)

For a squirrel cage rotor $\left[\begin{array}{l}u_{r d} \\ u_{r q}\end{array}\right]=\left[\begin{array}{l}0 \\ 0\end{array}\right]$    (37)

Substituting Eq. (36) in Eq. (29).

$i_{r q}=-\left(\frac{L_{m}}{L_{r}}\right) i_{s q}$    (38)

On substitution of Eq. (36), (37) in Eq. (19) for $\mathrm{\omega}_{d r a}$.

$\omega_{d r a}=-R_{r}\left(\frac{i_{r q}}{\psi_{r d}}\right)$    (39)

Let rotor time constant from Figure $5, \tau_{r}=\left(\frac{L_{r}}{R_{r}}\right)$    (40)

Substituting Eq. (38), (40) in Eq. (39).

$\omega_{d r a}=\left(\frac{L_{m}}{\tau_{r} \psi_{r d}}\right) i_{s q}$    (41)

Then substituting Eq. (36) in Eq. (30) for $T_{E}$.

$T_{E}=-\frac{p}{2}\left(\psi_{r d} i_{r q}\right)$    (42)

Substitute Eq. (38) in Eq. (42),

$T_{E}=\frac{p}{2} \psi_{r d}\left(\frac{L_{m}}{L_{r}} i_{s q}\right)$    (43)

5.png

Figure 5. A simplified d-axis circuitry with current excitation

Figure 5 is the d-axis circuit for short-circuited squirrel cage rotor with current excitation. Where $L_{l r}$ is per phase rotor leakage inductance and $L_{r}=L_{l r}+L_{m}$. Solve the circuit for $i_{r d}$ in Laplace domain.

$i_{r d}(s)=-\frac{s L_{m}}{\left(R_{r}+s L_{r}\right)} i_{s d}(s)$    (44)

Substituting Eq. (44) in Eq. (28) and using Eq. (40)

$\psi_{r d}(s)=\frac{L_{m}}{\left(1+s \tau_{r}\right)} i_{s d}(s)$    (45)

Figure 6 presents the block diagram of actual ASM, which was developed employing Eq. (41), (43, (45). Here stator direct and quadrature currents along with mechanical speed of rotation are given as input and electromagnetic torque, d-axis rotor flux linkages and rotor position are output quantities. Where $\omega_{d s a}$ is the instantaneous speed of stator d-q winding pair with reference to stator a-axis and $\omega_{d s a}=\omega_{d r a}+\omega_{m}$. The Simulink model ASM corresponding to Figure 6 is shown in Figure 7.

In Figure 8, utilizing the computed values of $i_{s d}, i_{s q}$ and $\psi_{r d}$ for an any value of $\omega_{d s a}$ measured, the reference voltages $u_{s d}^{r e f}$ and $u_{s q}^{r e f}$ are generated. The direct and quadrature stator voltages with the estimated value of $\theta_{d s a}$ undergo dq to abc transformation and reference voltage signals $u_{a}^{r e f}, u_{b}^{r e f}$ and $u_{c}^{r e f}$ are produced. The reference voltage signals are fed to SVPWM converter to the operating three phase voltages $u_{a}, u_{b}$ and $u_{c}$ to ASM to acquire desired operation. The three phase stator currents of actual motor undergo abc to dq and are driven to estimated motor model to achieve the estimated values of torque $T_{E \text { (est) }}$, $\psi_{r d \text { (est) }}$ and $\theta_{d s a(\text { est })}$. The direct and quadrature stator currents of actual motor model are compared with their corresponding reference values to generate an error signal. Each alone d-axis and q-axis error signals are given respective PI torque controllers as shown in Figure 9. To design PI torque controller the following procedure is implemented.

6.png

Figure 6. Actual ASM model d-axis oriented with ($\overrightarrow{\psi_{r}}$)

7.png

Figure 7. Simulink model of actual ASM

8.png

Figure 8. A block diagram of indirect space vector control of ASM accompanied by applied voltages

Firstly, we have to compute the required reference stator voltages that SVPWM power processing unit that have to supply ASM as shown in Figure 8. Let leakage factor (σ) of ASM is given in Eq. (46) [13].

$\sigma=\frac{L_{s} L_{r}-L_{s}^{2}}{L_{s} L_{r}}$    (46)

Solve for $i_{r d}$ in Eq. (28) and then substituting in Eq. (26).

$\psi_{s d}=\sigma L_{s} i_{s d}+\frac{L_{m}}{L_{r}} \psi_{r d}$    (47)

Substituting Eq. (38) in Eq. (27) for $\psi_{s q}$.

$\psi_{s q}=\sigma L_{s} i_{s q}$    (48)

Substituting Eq. (46), (47) in Eq. (16), (17) respectively.

$u_{s d}=\left(R_{s} i_{s d}+\sigma L_{s} \frac{d}{d t} i_{s d}\right)+\left(\frac{L_{m}}{L_{r}} \frac{d}{d t} \psi_{r d}-\omega_{d s a} \sigma L_{s} i_{s q}\right)$     (49)

$u_{s q}=\left(R_{s} i_{s q}+\sigma L_{s} \frac{d}{d t} i_{s q}\right)+\left(\omega_{d s a} \frac{L_{m}}{L_{r}} \frac{d}{d t} \psi_{r d}+\omega_{d s a} \sigma L_{s} i_{s d}\right)$    (50)

In Eq. (49) for d-axis stator voltage, on RHS only the initial two terms in the first set of brackets rely on d-axis stator current, while the other two terms are treated as disturbances and can be ignored. Similarly, disturbances for q-axis stator voltage in Eq. (50) are omitted. The error caused by the disturbances in stator d-axis and q-axis voltages is only about two to three percent of the sum of initial two terms in the first set of brackets in Eq. (49) and Eq. (50) respectively. These disturbance terms are treated as compensated terms. They could be compensated by considering these compensation terms. In this analysis, they have been ignored.

$u_{s d}^{\prime}=\left(R_{s} i_{s d}+\sigma L_{s} \frac{d}{d t} i_{s d}\right)$    (51)

$u_{s q}^{\prime}=\left(R_{s} i_{s q}+\sigma L_{s} \frac{d}{d t} i_{s q}\right)$    (52)

Applying Laplace transformation to Eq. (51), (52).

$u_{s d}^{\prime}(s)=\left(R_{s}+s \sigma L_{s}\right) i_{s d}(s)$    (53)

$u_{s q}^{\prime}(s)=\left(R_{s}+s \sigma L_{s}\right) i_{s q}(s)$    (54)

Figure 9 show the block diagram current control loop on d-axis winding, designed from Eq. (53). Similarly speed control loop on q-axis winding is designed from Eq. (54).

9.png

Figure 9. A d-axis current control loop schematic

In steady state under vector control $\psi_{r d}$ is constant, which implies $\frac{d \psi_{r d}}{d t}=0$, from Eq. (36), $\psi_{r q}=0$, $\frac{d \psi_{r q}}{d t^{2}}=0$, and for squirrel-cage rotor $u_{r d}=0$. Substituting above values in Eq. (19).

$i_{r d}=0$    (55)

On substitution of Eq. (55) in Eq. (28)

$\psi_{r d}=L_{m} i_{s d}$    (56)

On substitution of Eq. (56) in Eq. (43)

$T_{E}=\frac{p}{2}\left(\frac{L_{m}^{2}}{L_{r}}\right) i_{s d}{ }^{r e f} i_{s q}$    (57)

where, $K_{T}=\frac{p}{2}\left(\frac{L_{m}^{2}}{L_{r}}\right) i_{s d}{ }^{r e f}$    (58)

$\omega_{M e c h}=\frac{1}{J} \int\left(T_{E}-T_{L}\right)$    (59)

10.png

Figure 10. A speed loop controller schematic

Figure 10 is the block diagram of speed control loop, where $\omega_{M e c h}^{r e f}$cc is reference speed and $\omega_{M e c h}$ actual mechanical speed of ASM. The error in the position is fed to PI controller. The proportionality constants for current and speed loop controllers for particular values of crossover frequency and phase margin are derived. In the control scheme of ASM, at initial start-up before zero speed the electromagnetic torque is zero, where $\psi_{r d}$ builds up to its rated value which is dependent on $i_{s d}$ evident from Eq. (56). Once the core gets full magnetized, $\psi_{r d}$ remain unchanged and then the torque varies according to $i_{s q}$ from Eq. (57). There after the drive follows the speed, torque and position commands.

To illustrate the design of low pass filter, Butterworth filter approximation was considered because it advantages maximum flat response in magnitude, good roll-off to high frequencies, minimum phase and perfect bandwidth limitation [15]. Here second-order low-pass passive filter was designed. SOLP’s can also be structured by cascading of two first-order low- pass filters, but the cut-off frequency doesn’t remain same as cut-off frequency of first-order filter. For n^th order filter the cut-off frequency varies as $\omega_{n c o}=\omega_{c o} \sqrt{2^{1 / n}-1}$. In Butterworth approximation the cascading of any order filters doesn’t affect the cut-off frequency. The circuit of SOLP passive filter is shown in Figure 16. Substitute s=jω and observe the behaviour of the filter for zero and infinite frequencies.

16.png

Figure 16. RLC second-order low-pass filter

For $\omega$= 0, the inductor will be short-circuited, while capacitor is open-circuited. The output voltage is nearly equal to the input voltage, and the voltage gain is close to unity. For $\omega=\infty$, inductor will be open-circuited while capacitor is short-circuited and voltage gain is zero. The filter admits all the signals for lower frequencies beneath cut-off frequency and attenuates for higher frequencies upon cut-off frequency. Thus, it acts as low-pass filter. The transfer function of the above filter circuit is shown in Figure 16 is given in Eq. (71).

$\frac{U_{o u t}(s)}{U_{i n}(s)}=\frac{1 / L C}{\left(s^{2}+\frac{R}{L} s+\frac{1}{L C}\right)}$    (71)

The transfer function of SOLP filter is given in Eq. (72).

$\frac{U_{o u t}(s)}{U_{i n}(s)}=\frac{\omega_{c o}^{2}}{\left(s^{2}+\frac{\omega_{c} s}{Q}+\omega_{c o}^{2}\right)}$    (72)

where, $\omega_{c o}$ is cut-off or resonant frequency in rad.s^-1 and Q is quality factor. $\omega_{c o}=\frac{1}{2 \pi \sqrt{L C}}$, and $Q=\frac{1}{\omega_{c o} R C}=\frac{\omega_{c o} L}{R}$. In Figure 17, we observe the roll-off rate is 40 dB per decade and the amount of peaking was decided by the value of quality factor (Q). Maximally flat response was observed for Q = 0.707, which is Butterworth response. The Butterworth SOLP filter transfer function is given in Eq. (73). Similarly, one can design a SOLP with the quality factor (Q) equal to 2 and the transfer function of such SOLP filter with quality factor is given in Eq. (74) [15].

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Figure 17. A Magnitude Vs frequency plot for SOLP filter

$\frac{U_{o u t}(s)}{U_{i n}(s)}=\frac{\omega_{c}^{2}}{s^{2}+1.414 \omega_{c} s+\omega_{c}^{2}}$    (73)

$\frac{U_{o u t}(s)}{U_{i n}(s)}=\frac{\omega_{c}^{2}}{s^{2}+0.5 \omega_{c} s+\omega_{c}^{2}}$    (74)

The switching frequency ($f_{s w}$) of SVPWM inverter is selected 10 kHz and it would be the cut-off frequency (f_co). Substituting the value of $\omega_{c o}$, in Eq. (73), for $\omega_{c o}$=2πf_co, and compared with Eq. (71). For a selected value capacitor (C) = 0.01µF, the values of resistance (R) and inductance (L) was determined. The corresponding values of R and L were found to be 2.25kΩ and 25.3mH respectively. Thus, the passive filter was developed by Butterworth approximation. Finally, the filtering is implemented by placing a transfer function per phase in the proposed Simulink model.

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Figure 20. Torque and Speed plot when torque is reduced to half of its value at 0.1 sec of actual ASM

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Figure 21. 3-Փ Phase voltages generated by SVPWM inverter

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Figure 22. Unfiltered stator voltages

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Figure 23. Filtered stator voltages

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Figure 24. Torque and speed plots for IDSVC of ASM without SOLP RLC filter

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Figure 25. Torque and speed plots for IDSVC of ASM with both SOLP RLC filters wih Q = 0.707 and Q = 2

The block diagram shown in Figure 8 is modelled in Matlab / simulink as shown in Figure 19. A reference speed drive curve is considered over a time period of 3s and system is simulated. The stator voltage waveforms of SVPWM inverter connected without and with SOLP filter is shown in Figure 22 and Figure 23 respectively. In Figure 22, non-uniformity in waveforms is observed at higher amplitudes of stator voltages generated by SVPWM inverter in each of the three phases when operated without SOLP filter. Figure 24 and Figure 25 shows electromechanical torque and output mechanical speed plots when SVPWM inverter is connected without and with filter respectively. Actually, the SVPWM inverter injects required phase voltages into the stator windings. At high–speed (≥126 rad.s^-1) operation of vehicle, the switching frequency ($f_{s w}$) of SVPWM inverter is inconsistent, which injects distorted voltages into stator windings of ASM. Spikes are observed in torque and speed plots during this high-speed range as shown in Figure 24. The adversity was mastered by placing a SOLP RLC filter with Butterworth approximation in conjunction with SVPWM inverter as shown in Figure 25.

The speed profile of the vehicle model with Butterworth approximation with quality factor equal to 0.707 is compared with another SOLP filter with quality factor equal to 2 as depicted in Figure 25. It was noticed that the speed profile with Butterworth approximation tracks the reference speed with in short period of time compared to other filter. This is because; in Butterworth approximation the magnitude of the speed has flat response without peaking. As the quality factor increases the peaking increases which require more time to settle and does not track the speed response perfectly.