In this paper, a new method for the online estimation of equivalent series resistances (ESR) of the DClink capacitors in induction machine (IM) drive systems with a frontend diode rectifier is proposed, where the ESR estimation is conducted during the regenerative operating mode of the induction machine. In the first place, a regulated AC current component is injected into the
q
axis current component of the induction machine, which induces the current and voltage ripple components in the DClink. By processing these AC signals through digital filters, the ESR can be estimated by a recursive least squares (RLS) algorithm. To acquire the AC voltage across the ESR, the DClink voltage needs to be measured at a double sampling frequency. In addition, the ESR current is simply reconstructed from the stator currents and switching states of the inverter. Experimental results have shown that the estimation error of the ESR is about 1.2%, which is quite acceptable for condition monitoring of the capacitor.
I. INTRODUCTION
Power electronic systems play an increasingly important role in industrial applications, especially in adjustable speed machine drives, where PWM inverters have been commonly used
[1]

[4]
. The electrolytic capacitors in the DC link have been employed for filtering DC voltage ripples and for storing energy as a buffer
[5]

[7]
. One of challenging issues in the power electronic systems is the reliability of the overall system, where the power semiconductor devices and electrolytic capacitors are the most vulnerable components. According to the surveys in
[8]
and
[9]
, the failure rates of the semiconductors and capacitors in power electronic converters are 21% and 30%, respectively. Hence, the electrolytic capacitor is the key component, which determines the life time of a whole system.
In order to avoid unexpected faults in a system, a reliable diagnosing technique for the DClink capacitor in IM drive systems is essential. Many of the datasheets from capacitor manufacturers specify that the capacitor is considered to be faulty when the ESR of the capacitor increases to more than twice its initial value and the capacitance decreases to below 80% of its initial value
[10]

[12]
. The increase in the ESR is usually more pronounced than the decrease in the capacitance
[13]
,
[14]
. Therefore, it is known that the condition monitoring of electrolytic capacitors based on variations of the ESR rather than the capacitance is more reliable.
So far, many studies have been presented for monitoring the condition of capacitors. A simple method, called as an offline method, is to disconnect the capacitor and measure its capacitance and ESR with an LCR meter
[12]
,
[15]
. However, it is inconvenient to interrupt the system operation for this measurement. On the other hand, an online monitoring technique for the lifecycle condition of capacitors was proposed
[13]
, where the ESR of the capacitor is estimated from the capacitor losses which are calculated from the capacitor voltage and current. This method requires an additional circuit for acquiring the voltage and current. Another technique that applies to adjustablespeed machine drives has been presented in
[16]
, where the condition of the DClink electrolytic capacitors is monitored based on the estimated values of the ESR and capacitance, which are calculated from the inverter side when the motor stops. In this method, complicated signal processing for the measured voltage and current is needed. On the other hand, an ESR estimation method for the DClink capacitor of an AC/DC PWM converter was introduced by utilizing current injection
[14]
, where the ESR is calculated from the instantaneous values of the voltage and current in the ESR at the midpoint instant of the sampling period under the no load condition. Recently, an estimation method of the capacitance of DClink capacitors has been proposed for AC machine drive systems with a frontend diode rectifier, where it is possible to achieve the regenerative operating mode
[17]
,
[18]
.
In this paper, a new method for the online estimation of ESR in the DClink capacitors of AC machine drives is proposed, by which the deterioration condition of a DClink capacitor can be diagnosed. An IM drive fed by a VSI (voltagesource inverter) through a coupling of the DClink capacitor with a frontend diode rectifier is shown in
Fig. 1
. The DClink capacitor is usually modeled as an ESR in series with a capacitance. The ESR is estimated from the AC components of the voltage and the current ripples in the DC link, which are produced by the AC current injected into the
q
axis current component in the machine side. For the controller, PR (proportional resonant) controllers are used to regulate the AC component of the
q
axis current, whereas PI (proportional integral) controllers are used for the DC component of the
q
axis current generated from the speed controller and the
d
axis current. The capacitor current can be calculated from only the stator currents since the diode rectifier is blocked while the induction motor is in the regenerative operating mode. An experimental test on a 3kW IM drive is carried out to verify the validity of the proposed method. The experimental results show that the proposed technique gives a reliable result with an ESR estimation error of 1.2%.
Induction machine drive system with a frontend diode rectifier.
II. RELATIONSHIP OF THE VOLTAGES IN THE DC LINK FOR THE PWM SWITCHING PERIOD
The relationship of the voltages and current in the DC link of a voltage source inverter for the PWM switching interval is briefly described below. It is similar to the concept in the AC/DC PWM converter in
[14]
.
With the switching states of the VSI, the instantaneous DClink current is reproduced from the motor phase currents which also defines the relationship between the DClink voltage and the capacitor voltage, which is shown in
Fig. 2
. The switching states and gating times for the upper switches of each inverter leg are denoted as
S_{a}
,
S_{b}
, and
S_{c}
, and
T_{ga}
,
T_{gb}
, and
T_{gc}
, as shown in
Fig. 2
, and
T_{s}
is the sampling time. The instantaneous DClink current can be calculated from the stator phase currents and the switching states of the inverter as
[14]
:
Details in the DC link. (a) Switching states. (b) Stator currents and instantaneous DClink current. (c) Relationship of voltages and current in the DC link.
where
i
_{2}
is the instantaneous DClink current, and
i_{as}
,
i_{bs}
, and
i_{cs}
are the stator currents. It is illustrated in
Fig. 2
(a)  (b) and (1) that the instantaneous DClink current is zero during the interval of the zero voltage vectors, where all of the upper or lower switches turn on or off simultaneously. During the regenerative mode of the machine, the capacitor current is identical to the negative of the DClink current due to the blocking sate of the diode rectifier (
i
_{1}
= 0), which is shown in
Fig. 2
(b) (c) and expressed as:
where
i_{dc}
is the capacitor current.
As shown in
Fig. 1
, the DClink voltage,
v_{dc}
, is a summation of the capacitance voltage,
v_{cap}
, and the ESR voltage,
v_{ESR}
, which is expressed as:
where
R_{esr}
is the ESR of the DClink capacitor.
It is known that the DClink voltage in the VSI is usually measured at the interval of the zero voltage vectors for every sampling period. In this period, from (1) and (2), (3) can be rewritten as:
It can also be seen in
Fig. 2
that for the interval of the active voltage vectors, there is a current flowing through the DC capacitor, which results in a voltage drop across the ESR, as shown in the bottom waveform in
Fig. 2
.
III. CONTROL OF IM DRIVES FOR ESR ESTIMATION OF DCLINK CAPACITORS
As previously mentioned, the ESR of the DClink capacitors in IM drives is estimated during the regenerative operation. Here, the regenerative operating principle and the control of the induction motor are shortly described
[17]
.
The power balance for an IM drive system is expressed as
[19]
:
where
p
_{1}
and
p
_{2}
are the DClink input and output powers,
p_{dc}
is the DClink power,
C
is the capacitance of the DClink capacitor,
J
is the inertia of the drive system,
ω_{m}
is the machine speed,
P_{load}
is the load power applied to the machine, and
P_{loss}
is the loss in both the VSI and machine.
Fig. 3
shows a control block diagram of the induction machine
[20]
. The
q
axis current reference is produced from the outer speed control loop. Then, the inner
dq
axis current loops are employed to control the rotor flux and the machine torque, where the
d
axis current is controlled at a constant value of 5 A. To acquire AC signals from the voltage and current of the DC link, an AC current component,
is injected into the
q
axis stator current during the estimation process, of which the waveform is sinusoidal beginning at a zero in order to avoid a large transient for the current control, which is expressed as:
Control block diagram of IM for ESR estimation.
where
I_{inj}
and
f_{inj}
are the magnitude and frequency of the injected current, respectively. By this injection, the motor torque ripple and the speed pulsation are induced. They are limited to an allowable range with an appropriate selection of the magnitude of the injected current. It is noted that the estimation duration is very short, which has a slight effect on the operation of the motor. The selection criteria for the frequency and magnitude of the AC injected current has been described in detail in
[17]
. According to this criteria, the values 30 Hz and 3 A are used in this work.
As shown in
Fig. 3
, PR regulators are used to control the injected AC component of the
q
axis current, whereas PI controllers are employed to regulate the DC components of the
q
axis currents and the
d
axis current. For the feedback control, a bandpass filter (BPF) and bandstop filters (BSF) with a cutoff frequency of 30 Hz are employed to extract the AC and DC components of the
q
axis current as well as the machine speed.
IV. PROPOSED SCHEME FOR ESR ESTIMATION
The ESR of the DClink capacitor is calculated from the voltage across the ESR and the current flowing through the ESR, which is expressed as:
 A. Calculation of Voltage across the ESR
A process to calculate the ESR voltage in the DClink capacitor of the AC/DC PWM converter can be applied to the VSI of AC machine drive systems, which is briefly explained as follows.
As previously mentioned, the DC voltage is usually sensed for the interval of the zero voltage vector at point
as shown in
Fig. 2
, where no current flows through the ESR, resulting in a zero voltage drop across the ESR. However, it is shown in
Fig. 2
(c) that there is a difference between the actual DClink and capacitor voltages for the interval of the active voltage vectors, which is the ESR voltage. Therefore, to acquire the ESR voltage, the DC voltage needs to be sampled for the interval of the active voltage vectors
[14]
. In this study, the capacitor voltage is measured at the instant of the zero voltage vectors, whereas the midpoint values of the DClink voltage, consisting of the capacitor voltage and the ESR voltage, can be sensed by doubling the sampling frequency,
v
_{dc_mid}
_{(2}
_{fs}
_{)}
, at point
as shown in
Fig. 2
.
The midpoint value of the capacitor voltage,
v
_{c_mid}
, is calculated as the average value of two consecutive DClink voltages, which is depicted in
Fig. 2
(c) and expressed as
[14]
:
where
v
_{c_1}
and
v
_{c_2}
are the DClink voltages sensed for the interval of the zero voltage vectors.
Then, the voltage drop across the ESR can be calculated as:
 B. Reconstruction of Capacitor Current
The instantaneous DClink current in (1) is a pulsewise waveform, where the switching functions (
S_{a}
,
S_{b}
, and
S_{c}
) are required to reconstruct the DClink current. For easy implementation in practical applications, the DClink current is reconstructed from the phase currents and the gating times of the VSI instead of the switching functions. For ESR calculation, the mean value of the capacitor current,
i
_{2,}
_{cal}
, during the active voltage vectors is required, which is expressed as
[21]
:
where
T
_{max}
and
T
_{min}
are the maximum and minimum gating times of (
T_{ga}
,
T_{ab}
, and
T_{gc}
), respectively.
Then, the average value of the capacitor current,
i_{dc,cal}
, in a sampling period is obtained as:
 C. Estimation of ESR in the DCLink Capacitor
Eq. (8) can be rewritten with (11) and (12) as:
In this study, the mean value of the reconstructed capacitor current in a sampling period is used for the estimation process, instead of the midpoint values of the capacitor current which require complicated signal processing
[14]
.
Then, by applying bandpass filters with a cutoff frequency of 30 Hz on both sides of (13), only the AC components of the ESR voltage and current only processed, resulting in:
where
BPF
[ · ] represents the bandpass filtered signal.
Since the ESR estimated from the voltage and current usually contains a ripple, an RLS algorithm is applied for a reliable estimation. It is well known that the RLS algorithm is used to minimize the least square cost function in each sampling period whenever a data is updated. An error cost function for (14) is defined as
[21]
,
[22]
:
where
is the estimated value of the ESR.
From (15), the gradient of the error with respect to
is given as:
To minimize the error cost function,
should be updated as illustrated in (16). Therefore, the updated ESR value by the RLS algorithm is given as:
where
γ
(
k
) is an adjustable gain that was selected, by the trial and error method, as a constant of 1.2 × 10
^{6}
. The procedure to estimate the ESR is illustrated in
Fig. 3
(in the top left side).
V. EXPERIMENTAL RESULTS
To confirm the effectiveness of the proposed scheme, an experimental test is carried out on a prototype of an IM drive system in the laboratory, which is illustrated in
Fig. 4
. The hardware setup consists of a 3kW SCIM (squirrelcage induction motor) coupled with a PMSG (permanentmagnet synchronous generator) to apply a load. A frontend diode rectifier supplied by a 3phase AC source of 215V/60Hz is used to provide the input voltage for the VSI. A digital controller based on a DSP TMS320VC33 chip was used to control the IGBT inverter, where the switching frequency is 3.33 kHz. The sampling periods of the current and speed control loops are 150 µs and 1.5 ms, respectively. The parameters of the IM drive are listed in
Table I
. Four sound capacitors were used in the DC link, where the capacitors are connected in series and parallel. Connection 1 consists of three capacitors connected in series, whereas connection 2 is composed of two capacitors connected in parallel, which are in series with the other two capacitors. The parameters of the effective DClink capacitor and ESR for the two connections are listed in
Table II
.
Experimental setup.
PARAMETERS OF INDUCTION MACHINE
PARAMETERS OF INDUCTION MACHINE
PARAMETERS OF DCLINK CAPACITOR
PARAMETERS OF DCLINK CAPACITOR
The performance of the IM control is shown in
Fig. 5
for the ESR estimation process with connection 1 of the DClink capacitor, in which the machine speed is reduced from 1,500 rpm with a deceleration rate of 204 rad/s
^{2}
from 0.6 s for the regenerative operation as shown in
Fig. 5
(a).
Fig. 5
(b) shows the
d
axis stator currents, which are regulated at the rated value of 5 A for excitation. The
q
axis stator current consists of the DC and AC injected components as shown in
Fig. 5
(c) and (d), respectively. It can be seen in
Fig. 5
(c) that the DC component of the
q
axis stator current is negative, which implies that the power from the machine flows back to the DClink capacitor due to the deceleration operation. Under the no load condition, the
q
axis stator current also exists as shown in
Fig. 5
(c). The AC injected component of the
q
axis stator current is well regulated by the PR controller as shown in
Fig. 5
(d).
Control of IM for the estimation process. (a) Machine speeds. (b) daxis components of stator current. (c) DC components of qaxis stator current. (d) AC components of qaxis stator current.
The response of the DClink and input AC sides during the deceleration of the machine is shown in
Fig. 6
.
Fig. 6
(a) shows the DClink voltage. It can be seen that during the regenerative operation, the DClink voltage is increased higher than that of in the motoring operation. With this DClink voltage, the diode rectifier is blocked, resulting in no current flowing from the AC source to the DClink side in this interval as shown in
Fig. 6
(b).
Fig. 6
(c) shows the reconstructed capacitor current. Through the signal processing of the DClink voltage and current, the AC components of the ESR voltage and current are acquired. Then, the ESR of the DClink capacitor is calculated according to (16) as shown in
Fig. 6
(d), in which the estimation error is about 1.2%. Then, the deterioration condition of the capacitor can be monitored with a consideration of the effect of temperature on the ESR, where the estimated ESR value is compared with the referred initial value with regard to the operating temperature. Even though its effect has not been investigated in this study, it can be simply implemented, and has been described in detail in
[14]
.
Behavior of DClink and AC sides during deceleration mode. (a) DClink voltage. (b) Source phase current. (c) Reconstructed capacitor current. (d) Estimated ESR value.
Fig. 7
shows magnified waveforms from
Fig. 5
, where the sensed speed and its reference are shown in
Fig. 7
(a). It can be seen from
Fig. 7
(a) that the actual machine speed fluctuates a little, which is only about 2.5% compared with the operating speed. The speed fluctuation depends on the frequency of the injected current and the machine inertia. Due to this ripple component of the machine speed, the BSF needs to acquire the feedback signal for the speed controller as shown in
Fig. 3
. By virtue of the BSF, the speed controller works well. The daxis stator currents are shown in
Fig. 7
(b).
Fig. 7
(c) and (d) show the DC and AC components of the
q
axis stator current, respectively, which are kept close to their references.
Control of IM during the estimation process (magnified waveforms from Fig. 5). (a) Speeds of machine. (b) daxis components of stator current. (c) DC components of qaxis stator current. (d) AC components of qaxis stator current.
Fig. 8
shows the signals of the processed DClink voltage and current. The DClink currents are shown in
Fig. 8
(a), where the average value of the reconstructed current,
i_{dc,cal}
, is almost equal to that of the quantity measured directly by the probe,
i_{dc,mea}
.
Fig. 8
(b) shows the ESR voltage, which is acquired from the sensed and calculated values of the DClink voltage as shown in (10). It can also be seen that the ESR voltage contains a30Hz ripple component with the same frequency as that of the injected current. The ESR voltage and capacitor current are filtered out by the BPF with a 30Hz cutoff frequency, as shown in
Fig. 8
(c) and (d), respectively.
Signal processing in the DC link. (a) Capacitor currents. (b) ESR voltage. (c) Bandpass filtered ESR voltage. (d) Bandpass filtered capacitor current.
Fig. 9
shows the estimations of the ESR values in the extreme case of an abrupt variation under the conditions of
Table II
. First, connection 2 of the DClink capacitor is applied, and then it is changed to connection 1 by disconnecting the parallelconnected capacitor with a circuit breaker at 6.8 s.
Fig. 9
(a) shows the machine speed, where the machine is operated under the regenerating mode twice for 2 s ~ 2.7 s and 6.75 s ~ 7.45 s. During this operation, the current controllers of the
dq
axis stator currents still work well as shown in
Fig. 9
(b) and (c), respectively.
Fig. 9
(d) shows the DClink voltage, which is greatly increased during the deceleration operations of the machine. It can be seen in
Fig. 9
(d) that the DClink voltage during the second regenerative operation is higher than that in the first one due to a reduction of the DClink capacitance in the second operation.
Fig. 9
(e) shows the phaseA current of the source, where no current flows into the DClink side from the source during deceleration. The capacitor current is shown in
Fig. 9
(f).
Fig. 9
(g) shows the voltage drop across the ESR, where the ESR voltage during the second deceleration is higher than that in the first deceleration due to an increase in the ESR value of the DClink capacitor. The ESR value is calculated as shown in
Fig. 9
(h). It can be seen that the estimation errors for the two cases are less than 1.2%. In addition, it can be seen in
Fig. 9
(h) that the estimation is fast enough for the variation of the ESR of the capacitor.
Estimation in an abrupt variation of ESR of DClink capacitor. (a) Motor speeds. (b) qaxis stator currents. (c) daxis stator currents. (d) DClink voltage. (e) PhaseA current of source. (f) DClink current. (g) ESR voltage. (h) Estimated ESR.
VI. CONCLUSIONS
In this paper, a novel online estimation scheme for the ESR of the DClink capacitors in AC machine drive systems has been proposed. The regenerative operation of the machine is utilized, which increases the DClink voltage, causing a blocking of the diode rectifier. Then, the DClink current can be directly calculated by the stator currents and the switching times of the VSI. Injecting the AC regulated current component into the
q
axis stator current induces AC current and voltage components in the DC link as well as in the ESR of the capacitor, where the RLS algorithm is applied to achieve a reliable estimation of the ESR. In this proposed method, no additional hardware is required. It can be implemented simply in practical machine drive systems. However, the proposed scheme has a limitation in that it can only estimate the total capacitance of the DClink capacitors, but not individually for a capacitor bank with multiple units. Experimental results obtained witha3kW IM drive system in the laboratory have shown that the estimation error of the proposed method is less than 1.2%, by which the fault diagnosis of the DClink capacitor is performed.
Acknowledgements
This research was supported by the Yeungnam University Research Grants in 2012.
BIO
Thanh Hai Nguyen was born in Dong Thap, Vietnam. He received his B.S. degree in Electrical Engineering from the Ho Chi Minh City University of Technology, Ho Chi Minh City, Vietnam, in 2003, and his M.S. and Ph.D. degrees in Electrical Engineering from Yeungnam University, Gyeongbuk, Korea, in 2010 and 2013, respectively. He is presently working as a Research Professor at Yeungnam University. From May 2003 to February 2008, he was an Assistant Lecturer in the College of Technology, Can Tho University, Can Tho, Vietnam. His current research interests include power converters, machine drives, HVDC transmission systems, and wind power generation.
Quoc Anh Le was born in Can Tho, Vietnam, in 1988. He received his B.S. degree in Electrical Engineering from Can Tho University, Can Tho, Vietnam, in 2010, and his M.S. degree from the Ho Chi Minh City University of Technology, Ho Chi Minh City, Vietnam, in 2013. He is presently working toward his Ph.D. degree in the Department of Electrical Engineering, Yeungnam University, Gyeongbuk, Korea. In 2011, he was an Assistant Lecturer in the College of Technology, Can Tho University. His current research interests include high power converters and multilevel converters.
DongChoon Lee received his B.S., M.S., and Ph.D. degrees in Electrical Engineering from Seoul National University, Seoul, Korea, in 1985, 1987, and 1993, respectively. He was a Research Engineer with Daewoo Heavy Industry, Korea, from 1987 to 1988. Since 1994, he has been a faculty member in the Department of Electrical Engineering, Yeungnam University, Gyeongbuk, Korea. As a Visiting Scholar, he joined the Power Quality Laboratory, Texas A&M University, College Station, TX, USA, in 1998; the Electrical Drive Center, University of Nottingham, Nottingham, UK, in 2001; the Wisconsin Electric Machines and Power Electronic Consortium, University of Wisconsin, Madison, WI, USA, in 2004; and the FREEDM Systems Center, North Carolina State University, Raleigh, NC, USA, from September 2011 to August 2012. His current research interests include ac machine drives, the control of power converters, wind power generation, and power quality. Professor Lee is currently the EditorinChief for the Journal of Power Electronics of the Korean Institute of Power Electronics.
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