A Study on the Speed Control of Induction Motor Using Modified SVPWM Control Method

1. Space vector voltage modulation

Space vector voltage modulation is a technique of modulating the three-phase command voltage as a space vector in complex space.

This method can linearly generate a voltage that is 15.5% larger than the SPWM technique, and has the advantage that the harmonics in the current and torque are less than other techniques when the modulated voltage is applied to the motor. In this technique, since the command voltage is given as a space vector, the output voltage of the inverter needs to be expressed as a space voltage vector. As shown in [Fig. 1], a three-phase inverter has a total of eight different switching states: different three phase voltages. The space voltage vectors for three phase voltages corresponding to eight different switching states of these inverters are shown in <Table 1>.

[Fig. 1]

Space vector expressed in a complex number plane.

<Table 1>

Space vector according to switching state.

Of these, six voltage vectors V₁ ~ V₆ are an active voltage vector for applying a valid voltage to a load, all of which are the same as 2 V_dc/3 and only different in phase. In contrast, the two voltage vectors V₀ and V₇ are called zero voltage vectors because they do not apply a valid voltage to the load. The use of this zero voltage vector has a great impact on the modulation performance.

If the three-phase command phase voltage changes over time, the command voltage vector rotates counterclockwise in a complex space plane or d-q stationary coordinate system, as shown in [Fig. 2].

[Fig. 2]

Movement of the reference voltage vector.

In the space vector modulation method, the command voltage given as the voltage vector is generated by using eight voltage vectors that can be generated by the inverter.

Two effective voltage vectors (V_n, V_n+1) adjacent to the reference voltage vector V^*and a zero voltage vector (V₀, V₇) are used to make an average voltage equal to the reference voltage vector for a predetermined modulation period.

That is, two effective voltage vectors V₁ and V₂ closest to the command voltage vector V^* among the effective voltage vectors are properly synthesized to form a voltage vector having the same magnitude and phase as the command voltage vector V^*. The process of synthesizing the voltage is generated in three stages, as shown in [Fig. 3], which is repeated for each voltage modulation period T_s determined by the switching frequency. First, it is assumed that the magnitude and phase of the command voltage vector V^* is constant during the modulation period. In the first step of synthesis, one vector V₁ of two effective voltage vectors adjacent to the command voltage vector V^* is applied for T₁ and next, apply the remaining adjacent vector V₂ for T₂ to match the phase and magnitude with V^*. The same voltage as V^* can then be generated. Finally, if the sum of the application times T₁ and T₂ of the two vectors is less than the voltage modulation period T_s, the zero voltage vector V₀ or V₇ is applied for the remaining T₀(=T_s - T₁ - T₂) so that no further voltage is applied.

[Fig. 3]

Process of voltage modulation.

The application times T₁, T₂ and T₀ of two effective voltage vectors and zero voltage vectors required in this modulation process are shown in equation (1).

Assuming that the command voltage vector V^* and the direct current input voltage V_dc are constant during the voltage modulation period T_s, equation (2) is as follows.

The formula of two complex axis components when the command voltage vector V^* is located in the region ①(0 ≤ θ 60°) is expressed as follows.

In Eq. (3), |V^*| and θ are the magnitude and phase of the voltage vector, respectively. The application times of the effective voltage vector and the zero voltage vector are calculated from this equation as follows.

here, $$ a=\left|V^{\mathrm{*}}\right|/\left(\frac{2}{3}{V}_{dc}\right)$$

The application times of the effective voltage vector and the zero voltage vector in the remaining areas are also the same. In the space vector voltage modulation technique, the sum of the application times of the effective voltage vectors should not be greater than the modulation period T_s. That is, T₁ + T₂ ≤ T_s. The magnitude of the command voltage that satisfies this condition can be obtained as shown in equation (7).

From Eq. (7), the range of modulated command voltage vectors is the inner region of the hexagon, which consists of six effective voltage vectors, as shown in [Fig. 4]. The maximum magnitude of the command voltage is 2 V_dc/3.

[Fig. 4]

Controlled voltage areas in SVPWM.

The inside of the circle inscribed in the hexagon in this area is the area of the command voltage capable of linear modulation. The radius of the inscribed circle is $$ V_{dc}/\mathrm{ }\sqrt{3}$$, which is the maximum value of fundamental phase voltage that can be output by the space vector voltage modulation technique, which is about 15.5% larger than the SPWM method. In addition, it corresponds to 90.7% of the voltage during 6-step operation.(Kim, 2016)

A comparison of the voltage utilizations for the various PWM techniques is shown in <Table 2>.

<Table 2>

Comparison of voltage utilization by modulation technique.

2. Modified Space Vector Modulation

Conventional space vector modulation is difficult to implement compared to other types of PWM because of the approach based on the position of the reference voltage vector in two-dimensional vector space. Since the calculation is performed based on the spatial region to which the reference voltage vector belongs, the problem of double calculation that requires the calculation of the relationship with the actual voltage of the three phases to be output again is the biggest disadvantage of the existing SVPWM.

The modified SVPWM technique applied in this paper can implement SVPWM relatively simply while maintaining the existing performance by constructing SVPWM in terms of the valid time of applying the effective voltage. By the simple relocation algorithm of the newly developed zero-voltage vector, the applied PWM method shows the same performance as the existing SVPWM, furthermore it can be implemented more easily because the amount of calculation similar to the triangular wave comparison current control technique is required.

2.1 Space vector modulation pattern

In general SVPWM, voltage is treated as vector and analyzed in vector space. However, in real 3-phase inverter system, the time-base analysis can reduce the computational complexity because switching time determines three voltage states. Based on the d-q conversion theory, the reference phase voltage of an induction motor is as follows.

$\[ \left[\begin{array}{c}{V}_{as}^{\mathrm{*}}\\ V_{bs}^{\mathrm{*}}\\ V_{cs}^{\mathrm{*}}\end{array}\right]=\left[\begin{array}{cc}1& 0\\ -\frac{1}{2}& +\frac{\sqrt{3}}{2}\\ -\frac{1}{2}& -\frac{\sqrt{3}}{2}\end{array}\right]\cdot \left[\begin{array}{c}{V}_d^{s\text{*}}\\ V_q^{s\text{*}}\end{array}\right] \]$

(8)

Using the reference phase voltage, switching time can be obtained directly from the basic concept of SVPWM without the hassle of checking and reassembling the region as in conventional SVPWM. To do this, check the relationship between the time T₁, T₂ at which the effective vector is applied and the reference phase voltage. Assuming the case where the reference voltage is located in the region ①(0 ≤ θ 60°)and |V^*| ∙ cosθ is $$ V_d^{s\text{*}}$$ and |V^*| ∙ sinθ is $$ V_q^{s\text{*}}$$ in equation (3), it can be expressed as follows.

$\[ \begin{array}{c}\begin{array}{c}{T}_s\cdot V_d^{s\text{*}}=T_1\cdot \left(\frac{2}{3}{V}_{dc}\right)\hfill \\ +T_2\cdot \left(\frac{2}{3}{V}_{dc}\right)\cos 60° \end{array}\hfill \\ T_s\cdot V_q^{s\text{*}}=T_2\cdot \left(\frac{2}{3}{V}_{dc}\right)\sin 60°\hfill \end{array} \]$

(9)

Equation (9) is expressed based on T₁ and T₂ as follows.

$\[ \begin{array}{c}{T}_1=T_s\cdot \frac{3}{2V_{dc}}\left(V_d^{s\text{*}}-\frac{1}{\sqrt{3}}{V}_q^{s\text{*}}\right)\hfill \\ =\frac{T_s}{V_{dc}}\left(\frac{3}{2}{V}_d^{s\text{*}}-\frac{\sqrt{3}}{2}{V}_q^{s\text{*}}\right)\hfill \\ =\frac{T_s}{V_{dc}}\left(V_d^{s\text{*}}+\frac{1}{2}{V}_d^{s\text{*}}-\frac{\sqrt{3}}{2}{V}_q^{s\text{*}}\right)\hfill \\ =\frac{T_s}{V_{dc}}\left(V_d^{s\text{*}}\right)+\frac{T_s}{V_{dc}}\left(\frac{1}{2}{V}_d^{s\text{*}}-\frac{\sqrt{3}}{2}{V}_q^{s\text{*}}\right)\hfill \\ =\frac{T_s}{V_{dc}}\left(V_{as}^{\text{*}}\right)-\frac{T_s}{V_{dc}}\left(V_{bs}^{\text{*}}\right)=T_{as}-T_{bs}\hfill \end{array} \]$

(10)

$\[ \begin{array}{c}\begin{array}{c}{T}_2=T_s\cdot \frac{3}{2V_{dc}}\left(\frac{2}{\sqrt{3}}{V}_q^{s\text{*}}\right)\hfill \\ =\frac{T_s}{V_{dc}}\left(0\cdot V_d^{s\text{*}}+\sqrt{3}{V}_q^{s\text{*}}\right)\hfill \end{array}\hfill \\ \begin{array}{c}=\frac{T_s}{V_{dc}}\left(\frac{1}{2}{V}_d^{s\text{*}}+\frac{\sqrt{3}}{2}{V}_q^{s\text{*}}-\frac{1}{2}{V}_d^{s\text{*}}\right.\hfill \\ +\left.\frac{\sqrt{3}}{2}{V}_q^{s\text{*}}\right)\end{array}\hfill \\ =\frac{T_s}{V_{dc}}\left(\frac{1}{2}{V}_d^{s\text{*}}+\frac{\sqrt{3}}{2}{V}_q^{s\text{*}}\right)\hfill \\ -\frac{T_s}{V_{dc}}\left(\frac{1}{2}{V}_d^{s\text{*}}-\frac{\sqrt{3}}{2}{V}_q^{s\text{*}}\right)\\ =\frac{T_s}{V_{dc}}\left(V_{bs}^{\text{*}}\right)-\frac{T_s}{V_{dc}}\left(V_{cs}^{\text{*}}\right)=T_{bs}-T_{cs}\hfill \end{array} \]$

(11)

From the equations (10), (11), the time difference at which the switching state change of each phase voltage of the induction motor is represented by the effective vectors T₁ and T₂. In the other five regions, we can obtain relations similar to equations (10), (11). At this time, the time T_as, T_bs, T_cs at which the phase voltage transitions is a ratio between the reference phase voltage and the DC voltage for a predetermined period as shown in equation (12).

$\[ \begin{array}{c}{T}_{as}=T_s\cdot \frac{V_{as}^{\text{*}}}{V_{dc}},\hfill \\ T_{bs}=T_s\cdot \frac{V_{bs}^{\text{*}}}{V_{dc}},\hfill \\ T_{cs}=T_s\cdot \frac{V_{cs}^{\text{*}}}{V_{dc}}\hfill \end{array} \]$

(12)

With the proviso that $$ V_{as}^{\text{*}}+V_{bs}^{\text{*}}+V_{cs}^{\text{*}}=0$$ is T_as + T_bs + T_cs = 0.

The time when the effective vector is applied in the SVPWM is actually the time of applying the line voltage as the difference of each phase voltage of the induction motor. Assuming that each phase voltage transitions from V_dc to 0 [V] within one period, when the reference voltage is given to the region① as shown in the above formulas, we can see that the voltage as shown in [Fig. 5](a) is applied to the induction motor. In other words, a DC link voltage is applied between the a-b lines during T₁, between the b-c lines during T₂, and between the a-c lines during T₁ + T₂. Thus, the valid time T₁, T₂ in the conventional SVPWM can be defined as the time difference between the time when the switching state of each phase of the induction motor transition. After calculating each region separately in the existing SVPWM, these times must be recalculated to the actual switching times, which requires a lot of computation time and inefficient efficiency. If SVPWM is analyzed in terms of the effective time that zero voltage is applied to each line voltage rather than the idea that SVPWM is a space vector, it can be found that the times required to apply voltage on each phase of the induction motor are automatically calculated without distinguishing between sectors. When the reference phase voltages $$ V_{as}^{\text{*}}$$, $$ V_{bs}^{\text{*}}$$ and $$ V_{cs}^{\text{*}}$$ are obtained from the reference voltages given to the vector, and the time periods T_as, T_bs and T_cs are applied to the phase voltages are calculated, the times when the phase voltages are applied already include the information on the voltages applied to the line voltages. This means that there is no need to recombine the effective time for each region or to know the reference phase voltage position in the vector space.

[Fig. 5]

Switching pattern generation method of the applied SVPWM.

Of course, T_as, T_bs and T_cs times are virtual points at which the voltage applied to each phase voltage is changed from 0 to V_dc or V_dc to 0. Only the difference in each time has a physical meaning. These are referred to as virtual switching times as variables for obtaining the actual switching time. [Fig. 5] is a conceptual diagram of the modified SVPWM. [Fig. 5](a) shows the virtual switching time that produces the calculated phase voltage of the induction motor when the reference phase voltage is located in the region➀. Since the neutral point potential of the induction motor is not fixed, from the recognition that the line voltage of the induction motor is the only valid voltage, by adding an arbitrary delay time to the virtual switching time in which the (-) time value exists, the actual switching time for the inverter can be selected without any difference. That is, if all switching operations are completed in one period, the general switching pattern as shown in [Fig. 5](b) can be obtained by shifting the valid time portion of [Fig. 5](a). The effective time T_eff in [Fig. 5](a) is the voltage between T_max and T_min. There is a way to move T_eff to any part of the period, but as T_eff is centered, as shown in [Fig. 5](b), the ripple of the current can be minimized. First, the actual switching time is obtained as shown in equation (13) by adding a specific deviation to the virtual switching time.

$\[ \begin{array}{c}{T}_{ga}=T_{as}+T_{offset}\hfill \\ T_{gb}=T_{bs}+T_{offset}\hfill \\ T_{gc}=T_{cs}+T_{offset}\hfill \end{array} \]$

(13)

Equation(14) can be obtained from the fact that T_eff should be located at the center of one sampling period.

$\[ \begin{array}{c}{T}_{eff}=T_{\text{max}}-T_{\text{min}}\hfill \\ T_{zero}=T_s-T_{eff}\hfill \\ \begin{array}{c}{T}_{\text{min}}+T_{offset}=T_{zero}/2\hfill \\ T_{offset}=T_{zero}/2-T_{\text{min}}\hfill \end{array}\hfill \end{array} \]$

(14)

Among the virtual switching points T_as, T_bs and T_cs the largest and smallest values are T_max and T_min. Therefore, three virtual switching points should be arranged by time. Since what has been described so far is related to the case where the switching signal is OFF, the case of ON should also be considered. However, since the switching pattern is symmetric every two periods of sampling, using the fact that the voltage applied to the induction motor does not change no matter where the T_eff is placed, you can generate the pattern simply when the switching signal is turned on as in equation(15).

$\[ \begin{array}{c}{T}_{ga}\left(ON\right)=T_s-T_{ga}\left(OFF\right)\hfill \\ T_{gb}\left(ON\right)=T_s-T_{gb}\left(OFF\right)\hfill \\ T_{gc}\left(ON\right)=T_s-T_{gc}\left(OFF\right)\hfill \end{array} \]$

(15)

This can be obtained simply by adding the switching time in which the switch is turned ON to the switching time at the switch OFF.

Using the concept of the effective time (T_eff), SVPWM can be easily implemented in the phase voltage formula of the reference voltage. Unlike the conventional SVPWM, there is no need to check the region where the reference voltage vector is located, to select the applied vector, or to calculate the valid application time and recombine it. SVPWM can be easily implemented by simply moving the calculated virtual point in time using a simple three-element sorting algorithm. Therefore, the modified SVPWM method, which is applied to this paper, is very simple to implement in the low cost controller with low performance because the computational structure is very simple and the computation time is shorter than that of general SVPWM.

The switching pattern of SVPWM applied in this paper is shown in [Fig. 6]. After the valid time is obtained from the reference phase voltage, the actual switching time is obtained by shifting the valid time over time. If the switch is changed to the ON state, it is moved symmetrically based on time 0.

[Fig. 6]

Implementation of the PWM method.

<Table 3> shows the difference in computation between general SVPWM and applied SVPWM. Compared to the complicated method of implementing conventional SVPWM, the amount of computation is greatly reduced, so it can be seen that SVPWM can be simply implemented(Kim and Sul, 1995)

<Table 3>

Comparision of the calculation burden between the Conventional SVPWM and the Modified SVPWM

1. Simulation

In order to prove the effectiveness of the modified SVPWM technique for the induction motor speed control algorithm, computer simulations were performed using the PSIM program in the high and low speed regions.

In the low speed region(300[rpm]), step load torque was applied and in the high speed region(1500[rpm]), the load was proportional to the speed square.

<Table 4> shows the parameters and system constants of the induction motor used in the simulation.

<Table 4>

Parameters of induction motor and system constants used for computer simulation.

[Fig. 7] shows the PSIM block diagram of induction motor driving simulation.

[Fig. 7]

PSIM diagram of induction motor driver.

1.1 Induction motor high speed range

[Fig. 8] & [Fig. 9] shows the speed response characteristics of the induction motor when the speed command is applied from 0 [rpm] to 1,500 [rpm].

[Fig. 8]

Simulation responses for step change of speed setting of DTC control(0→1,500[rpm]).

[Fig. 9]

Simulation responses for step change of speed setting of modified SVPWM control(0→1,500[rpm]).

1.2 Induction motor low speed range

[Fig. 10] & [Fig. 11] shows the response state when load torque of 10 [N·m] is applied during steady state operation at 300 [rpm].

[Fig. 10]

Simulation responses for step change of load torque of DTC control(300[rpm], 10[N·m]).

[Fig. 11]

Simulation responses for step change of load torque of modified SVPWM control(300[rpm], 10[N·m]).

2. Review simulation results

When comparing [Fig. 8](a) and [Fig. 9](a), [Fig. 8](b) and [Fig. 9](b), the direct torque control method and the modified SVPWM technique are applied when the speed command is applied from 0[rpm] to 1,500[rpm], which is the high speed range. They were found that the speed and torque are stably controlled.

When comparing [Fig. 8](c) and [Fig. 9](c), the THD included in the input stage current is 3.41[%] in the direct torque control method and 2.71[%] in the modified SVPWM method, showing a current waveform close to the sine wave. It has been confirmed that the input current is very good quality.

 When comparing [Fig. 8](d) and [Fig. 9](d), the maximum value of the load current is not much different between 4.6[A] and 4.7[A] in the steady state section, respectively in two methods. It can be seen that the current is controlled stably in the transient period.

When comparing [Fig. 10](a) and [Fig. 11](a), [Fig. 10](b) and [[Fig. 11](b), when the speed command is applied from 0[rpm] to 300[rpm], which is a low speed range, in the direct torque control method it is confirmed that the fine oscillation of speed and torque occurs in the steady state section, and in the modified SVPWM method, the speed and torque are stably controlled.

When comparing [Fig. 10](c) and [Fig. 11](c), THD of 5.32[%] included in the input torque of direct torque control method is shown in the steady state section where load is applied. The input current waveform of the modified SVPWM is 3.34[%], which shows that the input current of better quality is supplied and the change of current is faster according to the load variation.

In comparison between [Fig. 10](d) and [Fig. 11](d), the maximum load current values of the direct torque control method and the modified SVPWM method are 6.8 ~ 12.8[A] and 9.1[A] in the steady state section after applying the load torque, respectively. It was found that the modified SVPWM method was controlled very stably, and it was confirmed that the SVPWM method was able to control stably even when the load was applied, even between the tools.

In comparison between [Fig. 8](e) and [Fig. 9](e), both of two methods, the power factor is 0.99, which is very good as the AFE rectifier is applied.