Deadtime element must be set into space vector pulsed width modulation signals to avoid short circuits of the inverter. However, the deadtime element distorts the output voltage vector, which deteriorates the performance of electrical machine drive system. In this paper, deadtime effect and its compensation control strategy are analyzed. Based on the analysis, the voltage distortion caused by deadtime 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 deadtime, a control scheme using two linear extended state observers is proposed. This method provides a strong ability to suppress deadtime 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 maincurrent switching device of VSI is the insulated gate bipolar transistor (IGBT). Given that IGBT has finite turnon time and turnoff time, a proper length of time delay must be imposed between the turnon signal of one IGBT and the corresponding turnoff signal of the other IGBT within the same leg to avoid possible short circuit. This time delay, known as deadtime, 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 deadtime 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 deadtime compensation is to reject the disturbance caused by the deadtime 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 offline calculation or additional hardware circuits. However, obtaining accurate current polarity at zerocrossing 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 deadtime. The compensating voltage can be mainly obtained in two ways. One way calculates the candidate compensating voltages offline 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 deadtime 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 deadtime effect in the implementation of SVPWM is presented, which scrutinizes the disturbance caused by deadtime and its adverse effect on electrical machine drive system. A control scheme using extended state observer (ESO) is proposed to compensate the deadtime effect. This scheme does not require precise motor parameters or any additional hardware. The deadtime 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 deadtime. 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 DEADTIME 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
U_{c}
, two contiguous basic voltage vectors
U_{b1}
and
U_{b2}
are used to feed the motor for
T_{b1}
and
T_{b2}
respectively to achieve the equivalent effect of
U_{c}
feeding the motor alone for each PWM period
T_{s}
.
Six basic nonzero voltage vectors.
As an example,
Fig. 2
shows the scenario when the commanded voltage vector
U_{c}
is in the sector bounded by
U_{4}
and
U_{6}
.
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
T_{on}
and
T_{off}
respectively. For the sake of security, a small length of time that neither one of the two IGBTs conducts during the deadtime inevitably exists. This time is called nonconducting 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 nonconducting time in this situation. Similarly, when the current direction of phase A is negative, the output voltage is positive during the nonconducting 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
T_{d}
+
T_{on}

T_{off}
shorter (longer) than the commanded conducting time when the direction of the phase current is positive (negative).
Configuration of phase A in an inverter.
Gate signals and corresponding output voltages. (a) Commanded gate signal. (b) Actual gate signal including deadtime. (c) Commanded output voltage. (d) Actual output voltage for i_{A} > 0. (e) Actual output voltage for i_{A} < 0.
The actual output voltage of each phase is altered from the commanded output voltage because of the deadtime. Analyzing the deadtime 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
U_{c}
is located in Sector 1, which is bounded by
U_{4}
and
U_{6}
. Given that the currents of phases B and C are negative, both of their positive output voltages are elongated. Therefore, actual
T_{6}
is equal to commanded
T_{6}
. Meanwhile, given that the direction of the current of phase A is positive, its positive output voltage is contracted. Thus, actual
T_{4}
is 2(
T_{d}
+
T_{on}

T_{off}
) shorter than commanded
T_{4}
. Here, ∆
T
is introduced to denote 2 (
T_{d}
+
T_{on}

T_{off}
). In this case, the adverse effect of the deadtime is that
T_{4}
is contracted by ∆
T
. Therefore, the difference between actual voltage vector and commanded voltage vector is −∆
T
⋅
U_{4}
(
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 deadtime, 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
U_{s}
is the stator voltage vector,
R_{s}
is the stator resistance,
I_{s}
is the stator current vector,
Ψ_{ss}
is the stator selfinduced flux linkage vector with a modulus of
Ψ_{ss}
,
θ_{ss}
is the vector angle of
Ψ_{ss}
,
j
is the complex operator,
ω
is the electric angular speed, and
Ψ_{f}
is the flux linkage vector established by magnets. The vector angle of
U_{s}
can be different from that of
I_{s}
because of the variation of system state and control strategy. In most industrial cases, especially the ones with surfacemounted PMSM, the goal is to control the current of
d
axis to zero, which makes
I_{s}
π
/2 phase angle ahead of
Ψ_{f}
. refore, under such kind of control strategy, the difference between the vector angle of
U_{s}
and that of
I_{s}
can be small in the steady state, given that
is much smaller than
R_{s}
I_{s}
+
jω
Ψ_{f}
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
I_{s}
locates (
Table I
).
Commanded voltages and actual voltages.
Relationship between commanded voltage vector and actual voltage vector only considering deadtime effect.
Six parts according to the different combinations of phase currents directions.
PARTS AND CORRESPONDING DIFFERENCE VECTORS CAUSED BY DIFFERENT COMBINATIONS OF PHASE CURRENT DIRECTIONS
PARTS AND CORRESPONDING DIFFERENCE VECTORS CAUSED BY DIFFERENT COMBINATIONS OF PHASE CURRENT DIRECTIONS
According to the principle of fieldoriented control strategy,
dq
rotor reference frame is rotating at the same speed with the rotor, and the
d
axis is aligned with
Ψ_{f}
. 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
d_{d}
and
d_{q}
imposed on the
dq
axes respectively(
Fig. 8
). To obtain detailed information about
d_{d}
and
d_{q}
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
t_{1}
, the difference vector switches to −∆
T
⋅
U_{4}
, whose modulus is denoted as d. As long as the current vector is in Part 1,
d_{d}
and
d_{q}
, which are the modulus of
d_{d}
and
d_{q}
respectively, can be calculated from the following equations:
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,
d_{d}
and
d_{q}
are periodic variables, whose angular frequency is six times of
ω
. This kind of disturbance with speedvarying frequency deteriorates the performance of current tracking, which eventually causes the torque ripple.
III. COMPENSATION OF DEADTIME EFFECT
In this section, a control scheme using two linear ESOs is proposed to compensate for the deadtime 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 deadtime). 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 DeadTime Effect
Based on the previous analysis in this paper, the mathematical model of PMSM considering the difference vector caused by the deadtime can be expressed as follows:
Decomposing Eq. (6) into the
dq
rotor reference frame leads to the following equations:
where
U_{cd}
and
U_{cq}
are the commanded voltages,
L_{d}
and
L_{q}
are the inductances, and
i_{d}
,
i_{q}
are the actual currents of
dq
axes.
Ψ_{f}
is the modulus of
Ψ_{f}
. In practice,
U_{cd}
and
U_{cq}
are determined by the current controllers of
dq
axes. Meanwhile,
d_{d}
and
d_{q}
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
L_{d}
by measurement is difficult. Furthermore, the value of
L_{d}
varies with operation conditions. Here, the measured value of
L_{d}
is denoted as
L_{m}
, 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
D_{d}
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,
z_{1}
is the estimate of
i_{d}
, and
z_{2}
is the estimate of
D_{d}
. When −
p
is set large enough,
z_{2}
can track
D_{d}
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
U_{cd}
can be designed as follows:
where
K_{d}
is the adjustable coefficient, and
i_{rd}
is the reference current of
d
axis. Substituting
U_{cd}
in Eq. (11) with Eq. (14) yields the following equation:
Notably, the lumped disturbance
D_{d}
will be compensated remarkably when
z_{2}
tracks
D_{d}
fast enough, which makes the whole system a firstorder integralchain system as follows:
The transfer function between
i_{rd}
and
i_{d}
can be written as follows:
Consequently, the system between
i_{rd}
and
i_{d}
is equivalent to an inertial system with time constant of
L_{m}
/
K_{d}
when
D_{d}
is compensated. Therefore, the relatively high frequency disturbance caused by the deadtime along with other uncertainties, which deteriorate the performance of current tracking, is inhibited by the compensation of
D_{d}
.
A similar result can be acquired by introducing a new variable
D_{q}
for
q
axis, which is also estimated through the ESO. The block diagram of the proposed current control scheme is shown in
Fig. 9
.
Block diagram of the PMSM driving system.
IV. SIMULATIONS AND EXPERIMENTS
In this section, simulations and experiments were conducted on an eightpole 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 deadtime, whereas the phase current with the proposed control scheme is much closer to the normal sinusoid.
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 deadtime is the sixthorder harmonics, whose angular frequency is six times of
ω
, which matches the former analysis [
Figs. 11
(b) and
11
(d)]. By contrast, the sixthorder harmonics caused by the deadtime considerably decreased with the proposed control scheme [
Figs. 11
(a) and
11
(c)].
Simulated spectra of dq axes currents at 100 rpm. (a) Spectrum of i_{q} with the proposed method. (b) Spectrum of i_{q} with the PI controller. (c) Spectrum of i_{d} with the proposed method. (d) Spectrum of i_{d} 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 fixedpoint DSP (TMS3202812) with a clock frequency of 150 MHz. The DC link voltage is 310 V, and phase currents are measured by Halleffect 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 deadtime is set as 3.2
μs
. The parameters of the tested motor are shown in
Table II
.
Tested PMSM drive system.
PARAMETERS OF TESTED MOTOR
PARAMETERS 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 surfacemounted 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 deadtime. 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 sixthorder 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 deadtime 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 deadtime 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 secondorder 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 2
nd
order harmonics by tracking the reference current. Significant secondorder 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
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.
Comparison of current spectrum with operation speed at 50 rpm. (a) Spectrum of i_{d} with the proposed control scheme. (b) Spectrum of i_{d} with the PI controller. (c) Spectrum of i_{q} with the proposed control scheme. (d) Spectrum of i_{q} with the PI controller.
Comparison of current spectrum with operation speed at 100rpm. (a) Spectrum of i_{d} with proposed control scheme. (b) Spectrum of i_{d} with the PI controller. (c) Spectrum of i_{q} with the proposed control scheme. (d) Spectrum of i_{q} with the PI controller.
Meanwhile, the adverse effects caused by the deadtime 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 lowpass filters, whose cutoff frequency is determined by the inductance and resistance of the stator. According to the analysis, the disturbances
d_{d}
and
d_{q}
are six times the frequency of
ω
, which means their frequency can increase dramatically when the motor speeds up. Thereby,
d_{d}
and
d_{q}
are filtered by the PMSM itself at relatively high operation speed, where the frequency of
d_{d}
and
d_{q}
exceeds the cutoff frequency. This is the reason that deadtime effects mainly occur at relatively low operation speed range.
Fig. 16
shows the spectra of
dq
currents at 500 rpm. The sixthorder harmonics (200
Hz
at 500 rpm) in each subgraph 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.
Comparison of current spectrum with operation speed at 500 rpm. (a) Spectrum of i_{d} with the proposed control scheme. (b) Spectrum of i_{d} with the PI controller. (c) Spectrum of i_{q} with the proposed control scheme. (d) Spectrum of i_{q} with the PI controller.
As generally accepted, three normally sinusoidal phase currents with 2
π
/3 phase angle difference among each other are the prerequisite for outputting constant electromagnetic torque for a perfectly designed threephase 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 deadtime effect can cause the sixthorder 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
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 deadtime at high speed range. All of these experimental results accord with the analysis provided in this paper.
V. CONCLUSION
A detailed analysis of deadtime effect is presented in this paper. Based on the analysis, the deadtime causes two periodic disturbances
d_{d}
and
d_{q}
with six times the frequency of electric angle on
dq
axes respectively. These relatively highfrequency 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 8pole PMSM to verify the effectiveness of the proposed current control scheme and the analysis in this paper.
Acknowledgements
This work was supported by the Science Foundation for Distinguished Young Scholars of Jiangsu Province under Grant BK20130018 and the Highlevel Talents Program in Six Industries of Jiangsu Province under Grant DZXX30
BIO
Jie Shi was born in 1990 in Wuhan, China. He received his B.S. degree in Automation in 2012 from Shenyang University of Technology, Shenyang, China. He is currently working toward his M.S. degree at the School of Automation, Southeast University, Nanjing, China. His research interests include control of electrical machines and power electronics.
Shihua Li was born in 1975 in Pingxiang, China. He received his B.S., M.S., and Ph.D. degrees all in Automatic Control in 1995, 1998, and 2001, respectively, from Southeast University, Nanjing, China. Since 2001, he has been with the School of Automation, Southeast University, where he is presently working as a Professor. His main research interests include nonlinear control theory and its application to robots, spacecrafts, AC motors, and other mechanical systems.
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