Analysis and Compensation Control of Dead-Time Effect on Space Vector PWM

Abstract

Dead-time element must be set into space vector pulsed width modulation signals to avoid short circuits of the inverter. However, the dead-time element distorts the output voltage vector, which deteriorates the performance of electrical machine drive system. In this paper, dead-time effect and its compensation control strategy are analyzed. Based on the analysis, the voltage distortion caused by dead-time is regarded as two disturbances imposed on dq axes in the rotor reference frame, which degenerates the current tracking performance. To inhibit the adverse effect caused by the dead-time, a control scheme using two linear extended state observers is proposed. This method provides a strong ability to suppress dead-time effects. Simulations and experiments are conducted on a permanent magnet synchronous motor drive system to demonstrate the effectiveness of the proposed method.

I. INTRODUCTION

Space vector pulse width modulation (SVPWM), which is based on voltage source inverter (VSI), is one of the most commonly applied modulation techniques to drive electrical machines. The main-current switching device of VSI is the insulated gate bipolar transistor (IGBT). Given that IGBT has finite turn-on time and turn-off time, a proper length of time delay must be imposed between the turn-on signal of one IGBT and the corresponding turn-off signal of the other IGBT within the same leg to avoid possible short circuit. This time delay, known as dead-time, distorts output voltages [1]-[3], [5]-[14]. Notably, for AC motor vector control systems, the commanded voltage vector is given by the current controller, whose objective is to force the actual current vector of the motor to track the reference current vector. The difference between actual voltage vector and commanded voltage vector caused by the dead-time can disturb this objective of current tracking. And this deterioration of current tracking can result in the distortion of phase currents, which leads to the ripples in the electromagnetic torque. Therefore, from the perspective of system control, the purpose of dead-time compensation is to reject the disturbance caused by the dead-time and improve the performance of current tracking.

Different approaches have been discussed to overcome this problem. These methods can be divided into two main groups. The first group is based on the modification of pulse width modulation (PWM) signals [6]-[8]. Most methods in this group are dependent on the precise detection of current polarity. The modified PWM signals are obtained by off-line calculation or additional hardware circuits. However, obtaining accurate current polarity at zero-crossing instants, where the current amplitude is rather small, is difficult [9], [10]. The other group is based on a feed forward control approach. In this group, a compensating voltage is added to the commanded voltage to counteract the voltage error caused by the dead-time. The compensating voltage can be mainly obtained in two ways. One way calculates the candidate compensating voltages off-line and employs the corresponding one according to the directions of phase currents, such as in [10] and [11]. Therefore, methods belonging to this group also rely on the precise detection of current polarity, which is difficult to achieve. Another way is based on the disturbance observer (DOB). Strategies belonging to this way do not require any additional hardware or accurate detection of current polarity. They consider dead-time as the disturbance voltage vector and obtain the estimate of disturbance via DOB, from which the compensating voltage is acquired [12]-[14]. However, the design of DOBs is based on the precise parameters of motors. The variations of the parameters, such as resistance varying with temperature and inductance varying with flux density, can affect the accuracy of estimation to some extent.

In this paper, detailed analysis of dead-time effect in the implementation of SVPWM is presented, which scrutinizes the disturbance caused by dead-time and its adverse effect on electrical machine drive system. A control scheme using extended state observer (ESO) is proposed to compensate the dead-time effect. This scheme does not require precise motor parameters or any additional hardware. The dead-time effect is regarded as two disturbance voltages added to the commanded voltages of dq axes in the rotor reference frame, which undermines the performance of current tracking. To reject the disturbance, two linear ESOs are employed to obtain the estimates of the disturbances. The estimates are then delivered to the commanded voltages of dq axes to counteract the disturbances caused by the dead-time. To verify the effectiveness of the proposed control scheme, experiments are conducted on a permanent magnet synchronous motor (PMSM) drive system, and all the algorithms are achieved through digital signal processor (DSP) TMS320F2812.

 

II. ANALYSIS ON DEAD-TIME EFFECT

According to the different combinations of switching states of the inverter, eight basic voltage vectors, including six nonzero vectors (100, 110, 010, 011, 001, and 101) (Fig. 1) and two zero vectors (000 and 111), can be obtained [15]. The whole plane is evenly divided into six sectors by the nonzero voltage vectors. When given any commanded voltage vector Uc, two contiguous basic voltage vectors Ub1 and Ub2 are used to feed the motor for Tb1 and Tb2 respectively to achieve the equivalent effect of Uc feeding the motor alone for each PWM period Ts.

Fig. 1.Six basic nonzero voltage vectors.

As an example, Fig. 2 shows the scenario when the commanded voltage vector Uc is in the sector bounded by U4 and U6.

Fig. 2.Case of commanded voltage located in sector 1.

In this section, only the analysis on phase A is presented, and the results can be extended to the other two phases. The basic configuration of phase A in an inverter is shown in Fig. 3. Fig. 4 gives the commanded gate signal and actual gate signal of phase A as well as the corresponding output voltages they produce. Figs. 4(a) and 4(b) show the commanded gate signal and actual gate signal respectively. Fig. 4(c) shows the commanded output voltage. In practice, IGBTs require short period of time to be turned on and off completely after the switching signals occur, which are denoted as Ton and Toff respectively. For the sake of security, a small length of time that neither one of the two IGBTs conducts during the dead-time inevitably exists. This time is called non-conducting time. During this time, the output voltage is only dependent on the direction of the phase current. For example, when the direction of phase A current is positive, which moves from left to right (Fig. 3), the bottom diode will conduct at the time when both IGBTs are turned off. This phenomenon is caused by the fact that the current direction cannot be changed in such a short time, given the relatively strong inductance of PMSM. Therefore, the output voltage will be negative during the non-conducting time in this situation. Similarly, when the current direction of phase A is negative, the output voltage is positive during the non-conducting time. Thus, two scenarios of actual output voltage according to the actual gate signal and the direction of the phase current can be determined [Figs. 4(d) and 4(e)]. As a consequence, the actual conducting time of positive output voltage is Td + Ton - Toff shorter (longer) than the commanded conducting time when the direction of the phase current is positive (negative).

Fig. 3.Configuration of phase A in an inverter.

Fig. 4.Gate signals and corresponding output voltages. (a) Commanded gate signal. (b) Actual gate signal including dead-time. (c) Commanded output voltage. (d) Actual output voltage for iA > 0. (e) Actual output voltage for iA < 0.

The actual output voltage of each phase is altered from the commanded output voltage because of the dead-time. Analyzing the dead-time effect caused by the three phases becomes more complicated. As an example, Fig. 5 shows a comparison between commanded output voltages and actual output voltages, where the current of phase A is positive and the currents of phases B and C are negative. The commanded voltage vector Uc is located in Sector 1, which is bounded by U4 and U6. Given that the currents of phases B and C are negative, both of their positive output voltages are elongated. Therefore, actual T6 is equal to commanded T6. Meanwhile, given that the direction of the current of phase A is positive, its positive output voltage is contracted. Thus, actual T4 is 2(Td+Ton-Toff) shorter than commanded T4. Here, ∆T is introduced to denote 2 (Td+Ton-Toff). In this case, the adverse effect of the dead-time is that T4 is contracted by ∆T. Therefore, the difference between actual voltage vector and commanded voltage vector is −∆T ⋅ U4 (Fig. 6). However, the commanded voltage vector and the current directions vary with time. Thus, the adverse effect caused by the dead time changes frequently. In this paper, difference vector is defined as the difference between actual voltage vector and commanded voltage vector. To further analyze the difference vector caused by the dead-time, the plane in Fig. 1 is again evenly divided into six parts according to the different combinations of phase current directions (Fig. 7 and Table I). The vector equation of the PMSM can be expressed as follows:

where

where Us is the stator voltage vector, Rs is the stator resistance, Is is the stator current vector, Ψss is the stator self-induced flux linkage vector with a modulus of Ψss, θss is the vector angle of Ψss, j is the complex operator, ω is the electric angular speed, and Ψf is the flux linkage vector established by magnets. The vector angle of Us can be different from that of Is because of the variation of system state and control strategy. In most industrial cases, especially the ones with surface-mounted PMSM, the goal is to control the current of d-axis to zero, which makes Is π/2 phase angle ahead of Ψf. refore, under such kind of control strategy, the difference between the vector angle of Us and that of Is can be small in the steady state, given that is much smaller than RsIs + jωΨf n Eq. (2). As a result, when the motor is operating at steady state, based on the similar analysis of the example in Fig. 5, the difference vectors can be sorted into six different cases due to the six different parts in which Is locates (Table I).

Fig. 5.Commanded voltages and actual voltages.

Fig. 6.Relationship between commanded voltage vector and actual voltage vector only considering dead-time effect.

Fig. 7.Six parts according to the different combinations of phase currents directions.

TABLE IPARTS AND CORRESPONDING DIFFERENCE VECTORS CAUSED BY DIFFERENT COMBINATIONS OF PHASE CURRENT DIRECTIONS

According to the principle of field-oriented control strategy, dq rotor reference frame is rotating at the same speed with the rotor, and the d-axis is aligned with Ψf. Within one rotation cycle, the difference vector changes six times because of the six different parts that the current vector passes. For each part where the current vector locates in, the corresponding difference vector d can be decomposed into two separate vectors dd and dq imposed on the dq axes respectively(Fig. 8). To obtain detailed information about dd and dq during one entire rotation cycle, the case of the current vector locating in Part 1, is analyzed in this section as an example. The results can be extended to other scenarios. Here, the rotating direction is assumed to be counterclockwise (Fig. 8). When the current vector first passes the lower boundary of Part 1 at time t1, the difference vector switches to −∆T ⋅ U4, whose modulus is denoted as d. As long as the current vector is in Part 1, dd and dq, which are the modulus of dd and dq respectively, can be calculated from the following equations:

Fig. 8.Case of current vector locating in Part 1 and its corresponding difference vector.

Given that all difference vectors have identical modulus d and the six parts as well as their corresponding difference vectors are distributed evenly and symmetrically in the plane, the same equations as that of Part 1 can be deduced for the other five parts. Therefore, when the system is in the steady state, dd and dq are periodic variables, whose angular frequency is six times of ω. This kind of disturbance with speed-varying frequency deteriorates the performance of current tracking, which eventually causes the torque ripple.

 

III. COMPENSATION OF DEAD-TIME EFFECT

In this section, a control scheme using two linear ESOs is proposed to compensate for the dead-time effect. The proposed control scheme is based on the active disturbance rejection control strategy. Its objective is to make the output (actual current vector) behave as desired by designing a proper input (commanded voltage vector), which contains the negative estimate of disturbance (here, caused by the dead-time). Therefore, the adverse effect brought by the disturbance is counteracted by the negative estimate of disturbance in the designed input. Details of the proposed method are provided below.

A. Mathematic Model of PMSM Considering Dead-Time Effect

Based on the previous analysis in this paper, the mathematical model of PMSM considering the difference vector caused by the dead-time can be expressed as follows:

Decomposing Eq. (6) into the dq rotor reference frame leads to the following equations:

where Ucd and Ucq are the commanded voltages, Ld and Lq are the inductances, and id, iq are the actual currents of dq axes. Ψf is the modulus of Ψf. In practice, Ucd and Ucq are determined by the current controllers of dq axes. Meanwhile, dd and dq are periodic disturbances caused by the dead time.

B. Design of Current Controller Using ESO

In this section, only the details of designing d-axis current controller are presented. The controller of q-axis can also be achieved through a similar method. To design the ESO, Eq. (7) is rearranged as follows:

In practice, obtaining an accurate Ld by measurement is difficult. Furthermore, the value of Ld varies with operation conditions. Here, the measured value of Ld is denoted as Lm, and the difference between is denoted as Δ. Then, Eq. (9) can be rewritten as the following equation by replacing the fourth element on the right side with

Then, a new variable Dd is introduced to denote which is regarded as the lumped disturbance on d-axis. Thus, Eq. (10) can be simplified into the following equation:

Considering an extended state system can then be constructed as follows:

Therefore, a linear ESO, which is easy to implement and only needs a small source of computation, can be designed as follows [16]–[17]:

where − p is the desired double poles of ESO, z1 is the estimate of id, and z2 is the estimate of Dd. When − p is set large enough, z2 can track Dd fast enough [18]–[19]. However, the noise involved in the detected currents can be amplified dramatically through the ESO when the pole is set too far from the origin. Therefore, − p is required to be set in an appropriate range in practical implementation.

According to the disturbance rejection control strategy, the input Ucd can be designed as follows:

where Kd is the adjustable coefficient, and ird is the reference current of d-axis. Substituting Ucd in Eq. (11) with Eq. (14) yields the following equation:

Notably, the lumped disturbance Dd will be compensated remarkably when z2 tracks Dd fast enough, which makes the whole system a first-order integral-chain system as follows:

The transfer function between ird and id can be written as follows:

Consequently, the system between ird and id is equivalent to an inertial system with time constant of Lm/Kd when Dd is compensated. Therefore, the relatively high frequency disturbance caused by the dead-time along with other uncertainties, which deteriorate the performance of current tracking, is inhibited by the compensation of Dd .

A similar result can be acquired by introducing a new variable Dq for q-axis, which is also estimated through the ESO. The block diagram of the proposed current control scheme is shown in Fig. 9.

Fig. 9.Block diagram of the PMSM driving system.

 

IV. SIMULATIONS AND EXPERIMENTS

In this section, simulations and experiments were conducted on an eight-pole PMSM to verify the effectiveness of the proposed compensation method.

A. Simulations

Figs. 10(a) and 10(b) show the tracking performance of dq axes currents with the proposed control scheme and PI controller respectively. The reference of q-axis current is given by the speed PI controller and the reference of d-axis is set to zero for both methods. Given the disturbance caused by the dead time, harmonics significantly exist in the dq axes currents with PI controller, which deteriorate the tracking performance. When compared with the PI controller, the harmonics of dq axes currents are reduced remarkably with the proposed control scheme based on its stronger ability of disturbance rejection, which improves the tracking performance. In addition, Figs. 10(c) and 10(d) provide waveforms of phase current with the proposed control scheme and the PI controller respectively. The phase current with the PI controller is severely distorted by the dead-time, whereas the phase current with the proposed control scheme is much closer to the normal sinusoid.

Fig. 10.Simulated waveforms of dq axes and phase currents at 100 rpm. (a) Waveforms of dq axes currents with proposed method. (b) Waveforms of dq axes currents with PI controller. (c) Waveform of phase current with proposed method. (d) Waveform of phase current with PI controller.

Fig. 11 shows the spectra of dq axes currents with the proposed control scheme and PI controller. The main component of harmonic caused by dead-time is the sixth-order harmonics, whose angular frequency is six times of ω, which matches the former analysis [Figs. 11(b) and 11(d)]. By contrast, the sixth-order harmonics caused by the dead-time considerably decreased with the proposed control scheme [Figs. 11(a) and 11(c)].

Fig. 11.Simulated spectra of dq axes currents at 100 rpm. (a) Spectrum of iq with the proposed method. (b) Spectrum of iq with the PI controller. (c) Spectrum of id with the proposed method. (d) Spectrum of id with the PI controller.

B. Experiments

Fig. 12 shows the structure of the tested PMSM drive system. The processor of the tested drive system is a fixed-point DSP (TMS3202812) with a clock frequency of 150 MHz. The DC link voltage is 310 V, and phase currents are measured by Hall-effect devices. An absolute encoder is employed to obtain the rotating speed and the position information of the rotor. The PWM period is set as 100 μs, and the dead-time is set as 3.2 μs. The parameters of the tested motor are shown in Table II.

Fig. 12.Tested PMSM drive system.

TABLE IIPARAMETERS OF TESTED MOTOR

The performance of the proposed control scheme is compared with that of the commonly used PI controller. In industrial applications, the reference current of q-axis is usually given by the speed controller, whereas the reference current of d-axis is set to zero in most cases, especially for surface-mounted PMSM. Hence, in the experiments of this paper, a speed PI controller is employed to obtain the reference current of q-axis for both the proposed control scheme and the PI controller. The reference current of d-axis is set to zero for all the cases in this paper. The motor operates without load to avoid the possible uncertainties introduced by the load actuator. Fig. 13 presents the performances of current tracking in the steady state under different operation speeds. The solid lines are the reference currents of dq axes, and the dashed lines are the actual currents of dq axes. When the system operates at a relatively low speed such as 50 rpm or 100 rpm [Figs. 13(a), 13(b), 13(c), and 13(d)], the currents of both dq axes fail to follow the reference currents close with the current PI controller because of the disturbance caused by the dead-time. By contrast, the performance of current tracking is remarkably improved with the proposed control scheme because of its stronger ability of disturbance rejection. The following comparison can be acquired by analyzing the spectra of dq currents in the steady state. Figs. 14 and 15 show the spectra of dq currents with operation speeds at 50 and 100 rpm respectively. Here, the electric angle frequency is defined as fundamental frequency. The sixth-order harmonics of dq currents with PI controller are significant at relatively low operation speed, whereas that of dq currents are much smaller with the proposed control scheme. This phenomenon matches the analysis that the disturbances caused by the dead-time on dq axes are six times the frequency of the electrical angle. Moreover, this comparison also shows the proposed control scheme has good ability to compensate for the dead-time effect. It should be pointed out that the tested motor exhibits being disturbed eight times by mechanical resistance during each rotation cycle of the rotor, which results in the second-order harmonics of the rotor speed. Then this signal of two times the fundamental frequency is passed to the reference current of q-axis by the speed PI controller. As a consequence, the current of q-axis acquires the 2nd order harmonics by tracking the reference current. Significant second-order harmonics are observed in the q-axis current for both the proposed control scheme and the PI controller [Figs. 14(c), 14(d), 15(c), and 15(d).]undefined

Fig. 13.Current tracking performance at different operation speeds. (a) Case of 50 rpm with the proposed control scheme. (b) Case of 50 rpm with PI controller. (c) Case of 100 rpm with the proposed control scheme. (d) Case of 100 rpm with PI controller. (e) Case of 500 rpm with the proposed control scheme. (f) Case of 500 rpm with PI controller.

Fig. 14.Comparison of current spectrum with operation speed at 50 rpm. (a) Spectrum of id with the proposed control scheme. (b) Spectrum of id with the PI controller. (c) Spectrum of iq with the proposed control scheme. (d) Spectrum of iq with the PI controller.

Fig. 15.Comparison of current spectrum with operation speed at 100rpm. (a) Spectrum of id with proposed control scheme. (b) Spectrum of id with the PI controller. (c) Spectrum of iq with the proposed control scheme. (d) Spectrum of iq with the PI controller.

Meanwhile, the adverse effects caused by the dead-time become much smaller when the operation speed increases to a certain level. As shown in Fig. 13(f), the performance of the current tracking with PI controller at 500 rpm is much better than that of Figs. 13(b) and 13(d). This phenomenon is based on Eqs. (7) and (8), where the PMSM behaves as two inertia elements of dq axes. Therefore, the PMSM can be regarded as two low-pass filters, whose cut-off frequency is determined by the inductance and resistance of the stator. According to the analysis, the disturbances dd and dq are six times the frequency of ω, which means their frequency can increase dramatically when the motor speeds up. Thereby, dd and dq are filtered by the PMSM itself at relatively high operation speed, where the frequency of dd and dq exceeds the cut-off frequency. This is the reason that dead-time effects mainly occur at relatively low operation speed range. Fig. 16 shows the spectra of dq currents at 500 rpm. The sixth-order harmonics (200 Hz at 500 rpm) in each sub-graph is negligibly small at this speed, which matches the analysis that the disturbances caused by dead time are filtered by the PMSM itself at relatively high speed. The harmonics with frequencies of 130, 270, and 400 Hz, which exist for both the proposed controller and the PI controller at this operation speed, result from the noise introduced by the hardware and sampling.

Fig. 16.Comparison of current spectrum with operation speed at 500 rpm. (a) Spectrum of id with the proposed control scheme. (b) Spectrum of id with the PI controller. (c) Spectrum of iq with the proposed control scheme. (d) Spectrum of iq with the PI controller.

As generally accepted, three normally sinusoidal phase currents with 2π/3 phase angle difference among each other are the prerequisite for outputting constant electromagnetic torque for a perfectly designed three-phase PMSM. In addition, the distortion of the phase currents, which results in the ripples of electromagnetic torque, is caused by their harmonics. Based on the previous analysis, the dead-time effect can cause the sixth-order harmonics of dq currents, which change into the corresponding harmonics of phase currents after coordinate transform and distort the waveforms of phase currents. Thus, the waveforms of phase currents are displayed in Fig. 17 to show a straightforward comparison

Fig. 17.Phase currents at different operation speeds. (a) Phase currents at 50 rpm with the proposed control scheme. (b) Phase currents at 50 rpm with the PI controller. (c) Phase currents at 100 rpm with the proposed control scheme. (d) Phase currents at 100 rpm with the PI controller. (e) Phase currents at 500 rpm with the proposed control scheme. (f) Phase currents at 500 rpm with the PI controller.

between the proposed control scheme and the PI controller. In the cases of low speed range [Figs. 17(a), 17(b), 17(c), and 17(d)], the waveforms of phase current with the PI controller are severely distorted because of the harmonics, whereas the waveforms with the proposed control scheme are much closer to normal sinusoid. However, these waveforms are incompletely sinusoidal because of the existence of harmonics introduced by the mechanical resistance. In the cases of relatively high operation speed [Figs. 17(e) and 17(f)], the phase current waveforms with both of the proposed method and the PI controller are close to normal sinusoid. This phenomenon is because that the PMSM itself inhibits the adverse effects caused by the dead-time at high speed range. All of these experimental results accord with the analysis provided in this paper.

 

V. CONCLUSION

A detailed analysis of dead-time effect is presented in this paper. Based on the analysis, the dead-time causes two periodic disturbances dd and dq with six times the frequency of electric angle on dq axes respectively. These relatively high-frequency disturbances result in the degeneration of current tracking performance and the distortion of phase currents. To compensate for these adverse effects, a current control scheme using two linear ESOs are proposed. The ESOs can track the disturbances fast enough when the poles are set into proper range. Therefore, the commanded voltages given by the proposed current control scheme contain the negative estimates of the disturbances, which counteract the real disturbances and improve the performance of current tracking. Simulations and experiments are carried out on a 8-pole PMSM to verify the effectiveness of the proposed current control scheme and the analysis in this paper.