**Zhanqing Zhou 1, Xin Gu 2, Zhiqiang Wang 1, Guozheng Zhang 2,\* and Qiang Geng 2,\***


Received: 12 July 2019; Accepted: 30 July 2019; Published: 31 July 2019

**Abstract:** An improved direct torque control with space-vector modulation (DTC-SVM) scheme is presented in this paper. In the conventional DTC-SVM scheme, torque control performance is affected by the load conditions, due to the inappropriate linearization of the relationship between the flux angle and electromagnetic torque. Different from the conventional method, a torque controller with load angle estimation (TC-LAE) is proposed and the change rate of torque is regulated according to the variation of the load conditions, which could ensure the rapidity and consistency of torque performance at different load conditions. Meanwhile, an online permanent magnet synchronous motor and maximum torque per ampere (PMSM-MTPA) operation strategy based on the fitting solving method is proposed instead of the traditional two-dimensional look-up table, and the reference value of flux amplitude is calculated online to meet the MTPA requirement with the proposed method. The improved strategy is applied on a 6 kW PMSM, and the simulation and experimental results verified the effectiveness and the feasibility of the proposed strategy.

**Keywords:** direct torque control (DTC); permanent magnet synchronous motor (PMSM); maximum torque per ampere (MTPA) operation; DTC with space-vector modulation (DTC-SVM)

#### **1. Introduction**

A lot of work has been done to improve dynamic torque performance and to optimize the output efficiency of the torque of permanent magnet synchronous motors (PMSMs) in recent years [1–8]. Additionally, various optimal torque control strategies have been proposed, such as direct torque control (DTC) [4,5], predictive torque control [6,7] and nonlinear control strategies [8], etc. The DTC strategy combined with space vector pulse width modulation (PWM) [9], which used continuous rotated voltage vector to regulate the flux of the motor. The torque control performance was improved compared with conventional DTC [10–13].

The DTC-SVM scheme usually consists of two parts [14]: one is the selection of the flux reference based on the two-dimensional look-up table offline. In this part, the reference value of the current is achieved using the torque–current table based on the maximum torque per ampere (MTPA) criterion, and then, the reference value of flux is calculated according to the relationship between flux and current of the motor. Hence, the MTPA operation of the DTC-SVM scheme could be achieved [15]. The other is the calculation of the reference flux angle based on the proportional integral (PI) controller. The relationship between the electromagnetic torque and the flux angle is linearized approximately, which resulted in excellent torque performance which could be maintained at different load conditions.

The research of DTC-SVM scheme always focuses on two aspects: one is to reduce the impact on the MTPA operation brought about by the change of parameters of the motor. The other is to improve the control/precision of the torque against variations in the load conditions.

The validity of the data in the torque–current two-dimensional table depends on the accuracy of the motor's parameters in the selection mechanism of the reference flux amplitude. Slow variation of the parameters is inevitable because of copper loss and magnetic saturation. Hence, the operation's conditions might deviate from the MTPA [16]. Therefore, the online MTPA control is an ideal solution to these problems. At present, the online solution of MTPA could be classified into the direct solving method [17] and engineering optimization method [18,19]. The MTPA criterion is a fourth-order equation about the stator current. The direct solving method is to solve this fourth-order equation online using the Ferrari method. Then, the reference value of the current could be obtained. The engineering optimization method is to change the fourth-order equation to an online optimization problem. The direct solving relationship between the torque and MTPA criterion is established. The voltage limitations of the inverter, the extreme current of the motor and the operational conditions are used as the boundary criterions. The stator current, which meets the MTPA criterion, is the optimization object. Then, the solving of the current reference value is realized. After that, the reference value of the stator flux amplitude for the DTC-SVM scheme could be obtained with this reference value of the current [20]. It is worth illustrating that the impacts of parameter variation on the online MTPA operation can be eliminated using certain parameter identification or self-adaptive methods.

The linearization of the relationship between the flux angle and electromagnetic torque is used as the control object for the torque control of the conventional DTC-SVM scheme, thus, the torque loop can be regarded as a second-order system with a PI controller. However, the dampening of this torque loop will be affected by the load conditions, which could lead to different torque adjustable performances of the conventional DTC-SVM scheme. There are three kinds of control strategies to improve the performance of torque control caused by inappropriate linearization:


To improve the performance of the conventional DTC-SVM, a novel online MTPA method based on Lagrange interpolation and an improved torque controller with load angle estimation (TC-LAE) are proposed in this paper. Different from the existing MTPA scheme, the proposed MTPA scheme takes the stator flux linkage as a variable instead of the stator current. Furthermore, the direct selection of the reference flux amplitude satisfied with the MTPA criterion could be realized on-line by Lagrange interpolation. Besides, a P-type torque controller with load angle estimation is adopted instead of the inappropriate linearization PI controller, so that the parameters of torque controller could be adjusted online according to the actual load angle to improve the control performance of torque under different load conditions.

#### **2. Examination of Conventional DTC-SVM Scheme**

The structure diagram of the conventional DTC-SVM scheme is shown in Figure 1.

**Figure 1.** The conventional direct torque control with space-vector modulation (DTC-SVM) scheme.

#### *2.1. Selection of Reference Flux Amplitude Based on the MTPA Criterion*

When operating below the rated speed of a PMSM, the MTPA operation should be satisfied for high-efficiency operation, which means the relationship between the electromagnetic torque and the stator current might meet the requirements of the following equations [15].

$$\begin{cases} \dot{i}\_{\rm dn} = \frac{1}{2(L\_{\rm qn} - L\_{\rm dn})} - \sqrt{\frac{1}{4(L\_{\rm qn} - L\_{\rm dn})^2} + i\_{\rm qn}^2} \\\ T\_{\rm en} = \dot{i}\_{\rm qn} + (L\_{\rm dn} - L\_{\rm qn})\dot{i}\_{\rm dn}\dot{i}\_{\rm qn} \end{cases} \tag{1}$$

During mathematical derivation of this paper, the per-unit value is employed for generality. The base value is selected as follows: *i*<sup>b</sup> = *i*N; *L*<sup>b</sup> = ϕb/*i*b; ϕ<sup>b</sup> = ϕr; *T*eb = 1.5*p*ϕb*i*b; *T*sb = 1/ωN.

The reference value of *i*dn and *i*qn that meet the requirement of the MTPA criterion are obtained using look-up table, and the equation of the flux linkage after normalization can be obtained by:

$$\begin{cases} \begin{aligned} \varphi\_{\rm dn} &= L\_{\rm dn} i\_{\rm dn} + \varphi\_{\rm rn} = \varphi'\_{\rm dn} + 1\\ \varphi\_{\rm qn} &= L\_{\rm qn} i\_{\rm qn} \end{aligned} \tag{2} \end{cases} \tag{2}$$

where, ϕ*'*dn = *L*dn*i*dn. Furthermore, the reference value of stator flux amplitude is:

$$\left|q\rho\_{\rm sn}\right| = \sqrt{\left(L\_{\rm dn}i\_{\rm dn} + 1\right)^2 + \left(L\_{\rm qn}i\_{\rm qn}\right)^2} \tag{3}$$

It can seen from Equations (1)–(3) that the MTPA operation of the PMSM depends on the accuracy of the motor's parameters. Hence, the online MTPA operation method could be used to reduce the impact of the parameters on the operation of the MTPA. The essence is changing the offline look-up table which meets the requirement of Equation (1) to solve the fourth-order equation online, which takes *i*<sup>d</sup> as the independent variable and *T*en as the parameter. Finally, the reference amplitude could be obtained using the square root operation, as shown in Equation (3).

#### *2.2. Torque Control Based on the PI Controller*

The electromagnetic torque can be written as the expression of the flux amplitude and load angle, which is [14]:

$$T\_{\rm en} = \frac{|q\rho\_{\rm sn}|}{L\_{\rm qn}} \left| \rho \sin \delta - \frac{1}{2} |q\rho\_{\rm sn}| (\rho - 1) \sin 2\delta \right| \tag{4}$$

where, ρ = *L*qn/*L*dn, ρ > 1. δ represents the load angle, which is the angle between the stator voltage vector and the flux vector. The incremental quantity of the load angle Δδ is related to the phase angle of the motor's stator flux vector.

$$
\angle \mathfrak{op}\_{\text{ref}} = \angle \mathfrak{op}\_{s,k} + \Delta \delta \tag{5}
$$

The approximate linearization processing of Equation (4) is taken when the load angle is δ<sup>0</sup>

$$T\_{\rm en} = k\_{\rm T}(\delta - \delta\_0) + T\_0 \tag{6}$$

where,

$$k\_{\rm T} = \frac{|q\rho\_{\rm sn}|}{L\_{\rm qn}} [\rho \cos \delta\_k - |q\rho\_{\rm sn}| (\rho - 1) \cos 2\delta\_k] \tag{7}$$

$$T\_0 = T\_{\text{en}}|\_{\delta\_k = \delta\_0} \tag{8}$$

The PI controller is used to realize torque control for the conventional DTC-SVM scheme, according to Equation (6). The structure block diagram of the torque loop is shown in Figure 2. The close-loop transfer function of the torque control link could be obtained from this figure, and the damping ζ and natural characteristic frequency ω<sup>n</sup> can be derived as:

$$\zeta = \frac{K\_\mathrm{P}}{2} \sqrt{\frac{k\_\mathrm{T}}{K\_\mathrm{i} T\_\mathrm{s}}} \quad \omega\_\mathrm{n} \quad = \sqrt{\frac{K\_\mathrm{i} k\_\mathrm{T}}{T\_\mathrm{s}}} \tag{9}$$

**Figure 2.** The control block of the torque loop.

In the block diagram, as shown in Figure 2, the tuning processing of the parameters of the PI controller is as follows: firstly, the expectation value of ζ is 0.707 in engineering practice. Secondly, considering the regulation performance and the disturbance immunity of the control system, *T*<sup>i</sup> can usually be chosen as 15 *T*<sup>s</sup> to 25 *T*<sup>s</sup> in the digital control system, where *T*<sup>i</sup> = *K*p/*K*<sup>i</sup> [32]. At last, the change rate of torque *k*<sup>T</sup> corresponding certain load conditions is selected to calculate the parameters of the PI controller generally.

However, *k*<sup>T</sup> varies for different parameters, stator flux amplitudes and load angles of the motor, and because of the nonlinear characteristic of Equation (7), it is hard to select a particular *k*<sup>T</sup> to tune the PI parameters. For example, the change curve of *k*<sup>T</sup> is calculated according to (7), where the stator flux amplitude changes from 0.6 to 1.0 (per-unit value) and the load angle changes from 0.0 to 1.5 (per-unit value), as shown in Figure 3.

**Figure 3.** The relationship of the torque change rate *k*<sup>T</sup> with the flux amplitude and load angle.

As shown in Figure 3, *k*<sup>T</sup> is mainly affected by the load angle of the motor. The heavier the load is, the smaller the value of *k*<sup>T</sup> is, inversely. So if *k*<sup>T</sup> under the rated load condition is used to calculate the PI parameters, the *k*<sup>T</sup> would be increased under light load conditions. Thus, it can be known from equation (9), ζ will be larger than 0.707 for light load operations, so the adjustment time of the transiente torque will be longer. Conversely, if *k*<sup>T</sup> under the no-load condition is used, *k*<sup>T</sup> will be decreased when operating under rated load conditions. Then, ζ is smaller than 0.707, which is possible to cause an oscillation process of the torque regulation.

#### **3. An Improved DTC-SVM Scheme**

The block diagram of the proposed improved DTC-SVM scheme is presented in Figure 4. For the reference calculation of the flux amplitude, a novel MTPA criterion expressed by the stator flux linkage is constructed, and the reference value of flux amplitude can be obtained with Lagrange interpolation online directly. For the reference calculation of flux phase angle, this section puts forward a novel P-type torque controller with load angle estimation (TC-LTE), which could regulate the flux phase angle as load angle variation. Moreover, by adding the relevant correction, compensation and limitation blocks for the incremental quantity of the load angle Δδ, the impact of the voltage limitation circle, rotation of the permanent magnet and load angle stability on the torque control performance could be depressed.

**Figure 4.** The schematic diagram of the proposed DTC-SVM scheme.

#### *3.1. Novel Online MPTA Scheme*

By substituting (2) into (1), the following equation can be derived:

$$
\rho \rho \boldsymbol{\uprho}'^{4}\_{\mathrm{dn}} + b \boldsymbol{\uprho}'^{3}\_{\mathrm{dn}} + c \boldsymbol{\uprho}'^{2}\_{\mathrm{dn}} + d \boldsymbol{\uprho}'\_{\mathrm{dn}} + \mathfrak{e} T^{2}\_{\mathrm{en}} L^{2}\_{\mathrm{qn}} = 0 \tag{10}
$$

where, *a* = (1 − ρ) <sup>3</sup>ρ2; *<sup>b</sup>* <sup>=</sup> <sup>3</sup>ρ2(1 <sup>−</sup> <sup>ρ</sup>) 2; *<sup>c</sup>* <sup>=</sup> <sup>3</sup>ρ2(1 <sup>−</sup> <sup>ρ</sup>); *<sup>d</sup>* <sup>=</sup> <sup>ρ</sup>2; *<sup>e</sup>* <sup>=</sup> <sup>−</sup>(1 <sup>−</sup> <sup>ρ</sup>).

Equation (10) is a novel MTPA criterion expressed by flux linkage; it can determine the reference flux amplitude that satisfied MTPA criterion under different load conditions directly.

In order to solve (10), Lagrange interpolation is adopted to fit the left polynomial of (10). Then, the feasible solution of (10) is equivalent to the zero point of the fitting polynomial. At last, the reference value of the flux amplitude can be determined with simple calculation. Assuming:

$$f\_1(\varphi'\_{\rm dn}) = a\varphi'^4\_{\rm dn} + b\varphi'^3\_{\rm dn} + c\varphi'^2\_{\rm dn} + d\varphi'\_{\rm dn} + eT^2\_{\rm em}L^2\_{\rm qn} \tag{11}$$

According to the theory of Lagrange interpolation [33], there are two necessary steps to fit the above polynomial. The first step is to confirm the solution region of (10). The second step is to select the samples and calculate the remainder of the interpolation.

Calculating the derivative of ϕ dn in (11), and making this derivation equal to zero, that is:

$$4(1 - \rho)^3 \rho'^3{}\_{\rm dn} + 9(1 - \rho)^2 \rho'^2{}\_{\rm dn} + 6(1 - \rho)\rho'{}\_{\rm dn} + 1 = 0\tag{12}$$

The above equation has a single real root <sup>1</sup> <sup>4</sup>(ρ−1) and a double real root <sup>1</sup> <sup>ρ</sup>−<sup>1</sup> . Taking the second order derivative at these two points, we can get d2 *<sup>f</sup>*<sup>1</sup> dϕ<sup>2</sup> dn ϕ dn<sup>=</sup> <sup>1</sup> 4(ρ−1) <sup>&</sup>lt; 0 and d2 *<sup>f</sup>*<sup>1</sup> dϕ<sup>2</sup> dn ϕ dn<sup>=</sup> <sup>1</sup> (ρ−1) < 0. Obviously, ! <sup>1</sup> <sup>4</sup>(ρ−1), *<sup>f</sup>*1( <sup>1</sup> <sup>4</sup>(ρ−1))] and <sup>1</sup> <sup>ρ</sup>−<sup>1</sup> , *<sup>f</sup>*1( <sup>1</sup> <sup>ρ</sup>−<sup>1</sup> )] are the maximum points of *<sup>f</sup>* 1. Besides, considering the intercept of the ϕ dn <sup>−</sup> *<sup>f</sup>* <sup>1</sup> plot, *<sup>f</sup>* 1(0) <sup>=</sup> *eT*en2*L*qn<sup>2</sup> <sup>&</sup>gt; 0, it can be concluded that (10) has one positive solution and one negative solution. Furthermore, as shown in (1), *i*dn < 0. Hence, the negative solution could be the unique feasible solution of (10). The schematic diagram of the interpolation trajectory and its remainder are drawn in Figure 5.

**Figure 5.** The schematic diagram of the interpolation trajectory and remainder. (**a**) Interpolation; (**b**) remainder of interpolation.

To avoid the irreversible demagnetization of the permanent magnet, always keep ϕ dn > −0.5 [34]. Therefore, the solution region of (10) can be determined as (−0.5, 0).

In general, the value of ϕ dn is always small under the MTPA operation, then the interplotion samples could be selected in [−0.25, 0]. This section adopted Lagrange parabolic interpolation to fit the curve of *f* 1, and the interplotion samples are chosen as (0, *C*0), (−0.15, *C*1) and (−0.25, *C*2), where *C*<sup>0</sup> = *f*1(0) = *eT*<sup>2</sup> en*L*<sup>2</sup> qn, *C*<sup>1</sup> = *f*1(ϕ dn) ϕ dn=−0.15 and *C*<sup>2</sup> = *f*1(ϕ dn) ϕ dn=−0.25 , respectively. Based on the interpolation formula [33], The final interpolation polynomial can be expressed as follows:

$$f\_2(\boldsymbol{\varphi}'\_{\rm dn}) = (26.67\text{C}\_0 - 66.67\text{C}\_1 + 26.67\text{C}\_2)\boldsymbol{\varphi}'\_{\rm dn}^2 + (10.67\text{C}\_0 - 16.67\text{C}\_1 + 4\text{C}\_2)\boldsymbol{\varphi}'\_{\rm dn} + \text{C}\_0 \tag{13}$$

and the interpolation remainder for (13) is:

$$R\_{\rm n}(\varphi'\_{\rm dn}) = \frac{\varphi'\_{\rm dn} f\_1^{\prime\prime\prime}(\varphi'\_{\rm dn})}{6} (\varphi'\_{\rm dn} + 0.15)(\varphi'\_{\rm dn} + 0.25) \tag{14}$$

For ϕ dn ∈ [−0.25, 0] and ρ ∈ [1.0, 2.0], the numerical analysis results of *R*n(ϕ dn) is shown in Figure 5b. It can be seen from this figure that −0.01 < *R*n(ϕ dn) < 0.01, thus the solution of *f* 2(ϕ dn) = 0 could be regarded as the solution of (10), approximately. This means that we can determine the reference flux amplitude with the solution of *f* 2(ϕ dn) = 0 online, and the complex process for solving (10) directly can be avoided. Particularly, the following extra conditions must be satisfied during the determination of the reference flux amplitude.

1. To ensure stable operation of the PMSM, the reference value of the flux amplitude [20],

$$\left| \left. q \rho\_{\mathrm{sn,ref}} \right| < \frac{\rho}{\rho - 1} \right. \tag{15}$$

2. If the PMSM works under no-load conditions, the electromagnetic torque and stator current are almost zero. According to (3),

$$\left| q \rho\_{\text{sn,ref}} \right| = 1 \tag{16}$$

### *3.2. Torque Controller with Load Angle Estimation (TC-LAE)*

The control block of the proposed TC-LAE is shown Figure 4. The estimation equation of the load angle can be expressed with stator current and flux, which is:

$$\delta\_k = \arctan[\frac{1.5 p L\_{\rm qn} T\_{\rm en}}{\left| q \rho\_{\rm sn,k} \right|^2 - L\_{\rm qn} (\varphi\_{\rm \alpha} i\_{\rm \alpha} + \varphi\_{\rm \beta} i\_{\rm \beta})}] \tag{17}$$

by substituting the estimated load angle into (7), we can predict the value of *k*<sup>T</sup> in real-time. Furthermore, the incremental quantity of the load angle at the next control instant, which is denoted as Δδ, can be obtained by taking the difference operation on both sides of (4), that is:

$$
\Delta \delta = \frac{1}{k\_{\rm T}} \Delta T\_{\rm em} \tag{18}
$$

In addition, a P-type controller is employed in the TC-LAE for Δδ trimming, to depress the impact of several disturbance factors, such as sampling error and parameter mismatches, on torque performance. The control parameter of this P-type controller is *K*c, and *K*<sup>c</sup> > 0.

The conventional DTC-SVM takes a constant *k*<sup>T</sup> for the parameter tuning of the PI controller. Differening from the conventional method, the proposed TC-LAE adjusts Δδ with the appropriate *k*T, which is calculated based on the actual load angle. With the aid of TC-LAE, the improved DTC-SVM can achieve a fast and consistent torque response under different load conditions.

#### *3.3. Correction, Compensation and Limitations of* Δδ

#### 3.3.1. Correction

In DTC-SVM strategies, during large torque demands, the torque controller will give an output that demands the selection of voltage vectors to increase the torque. However, once the reference voltage vector tip point lies outside the hexagon, the space-vector PWM yields a negative time length, resulting in an inevitable volt-seconds error [35,36]. A voltage vector on the hexagon boundary (the modified reference voltage vector) must be selected and at least one back step has to be taken to recalculate the vector time lengths that generate the modified reference voltage vector. Shown in Figure 6, the two popular modified reference vector choices are the minimum magnitude error PWM (MMEPWM) method (also called the one-step-optimal method), and the minimum phase error PWM (MPEPWM) method. However, the MMEPWM and MPEPWM could not ensure the stable output of the load angle and flux amplitude simultaneously at the transient instant [37]. Hence, a special voltage vector correction algorithm is proposed in this section, which is shown in Figure 6.

**Figure 6.** The correction algorithm of the incremental value of the load angle.

Assuming the reference torque is increased at the instant *kT*s, and ϕsn,*<sup>k</sup>* donotes the current stator flux vector; ϕref denotes the reference stator flux vector; *V*ref denotes the reference vetor of the stator voltage obtained by (17), which hopes to make the stator flux vector (ϕsn,*k*<sup>+</sup>1) equal to ϕref at the instant (*k* + 1)*T*s. However, it can be seen from Figure 6 that the actual voltage vector is *V*act due to the existance of the voltage limitation circle. Consequently, ϕsn,*k*+<sup>1</sup> could not follow ϕref under the effects of *V*act, it will lead to an ampltide error for flux control. To make sure |ϕsn,*k*+1|=|ϕref |, and fullly ulitize the voltage capablity of the VSI at the same time, we should revise Δδ when the amplitude of *V*ref is beyond the voltage limitation circle. On the basis of the vector raltionship in Figure 6, and with the help of cosine theorem, the revised Δδ\*\* can be derived as:

$$
\Delta \delta^{\*\*} = \arccos(\frac{|V\_{\text{ref}} T\_{\text{sn}}|^2 - \left| q \rho\_{\text{ref}} \right|^2 - \left| q \rho\_{\text{sn},k} \right|^2}{-2 \left| q \rho\_{\text{ref}} \right| \left| q \rho\_{\text{sn},k} \right|} \tag{19}
$$

where, *T*sn denotes the per-unit value of the control period, and the base value of time is selected as 1/ωN, ω<sup>N</sup> is the rated electrical angular frequency of the PMSM; *V*ref denotes the voltage vector with the Δδ revising algorithm, its amplitude equals the maximum value of the output voltage of the VSI.

#### 3.3.2. Compensation

The rotor permanent magnet of the PMSM keeps rotating during normal operation. Assuming the rotor rotates counterclockwise, and ω denotes the rotor electrical angular frequency, it can be seen from Figure 6 that the value of Δδ obtained by controller is ω*T*<sup>s</sup> less than the actual required value because of the rotation of the permanent magnet. This angular deviation will result in offsets for torque control during high speed operations, hence it is necessary to compensate the angular deviation, that is:

$$
\Delta \delta^\* = \Delta \delta + a \, T\_s \tag{20}
$$

#### 3.3.3. Limitation

Load angle stability must be ensured when the PMSM is operating under heavy load conditions, hence, the limitation block should be utilized for Δδ adjustment. Taking the derivative of δ in (7), and making this derivation equal to zero, the maximum load angle δ<sup>m</sup> can be obtained:

$$\delta\_{\rm m} = \arccos\left[\frac{\rho - \sqrt{\rho^2 + 8\left|\rho \mathbf{e}\_{\rm sn}\right|^2 (\rho - 1)^2}}{4\left|\rho \mathbf{e}\_{\rm sn}\right| (\rho - 1)}\right] \tag{21}$$

Therefore, the maximum allowable vaule of Δδ (denoted as Δδm) can be obtained:

$$
\Delta \delta\_{\rm m} = \delta\_{\rm m} - \delta\_{\rm k} \tag{22}
$$

#### **4. Simulation Results**

In order to study the control performance of the proposed TC-LAE, numerical simulations have been carried out using Matlab/Simulink. The parameters of the control system are presented in Table 1. It should be illustrated that the speed loop consisting of a PI controller is added out of the torque loop. The maximum value of the output torque, which is restricted by the PI controller of the speed loop, is 1.2*T*N, which is 230 Nm.


**Table 1.** Parameters of the control system.

#### *4.1. The Correction and Compensation for* Δδ

The simulation waveforms with/without the correction algorithm for Δδ are shown in Figure 7. The motor operates at 100 r/min with no load. When *t* = 1.5 s, the reference value of speed *n*ref is set to 200 r/min, the speed PI controller reaches the positive limitation. Without the correction algorithm, Δδ increased rapidly, resulting from the sudden increase of *T*ref. The amplitude of the stator voltage vector is increased correspondingly according to the analysis in Section 3.3. As can be seen from Figure 7, taking the voltage amplitude limitation of the SVM into consideration, when the reference voltage (*v*α, *v*β)is beyond the range of the voltage limitation circle, the amplitude of the stator flux linkage |ϕs| slides for a short time and a dynamic deviation will appear between the reference value and the actual value of the stator flux linkage. So, the stator current will increase rapidly(*i*max = 23 A). With the correction algorithm, the tracking ability of the flux control is improved, and the dynamic current is reduced effectively when the reference torque is changed suddenly(*i*a,max = 21 A).

**Figure 7.** The simulation waveforms for the proposed DTC-SVM Scheme. (**a**) Waveforms with Δδ correction; (**b**) waveforms with Δδ correction.
