Common-Mode and Differential-Mode Active Damping for PWM ...€¦ · 3188 IEEE TRANSACTIONS ON...

3188 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 6, JUNE 2014

Common-Mode and Differential-Mode ActiveDamping for PWM Rectifiers

Mohammad H. Hedayati, Anirudh Acharya B., Member, IEEE, and Vinod John, Senior Member, IEEE

Abstract—Modern pulse-width-modulated (PWM) rectifiers useLCL filters that can be applied in both the common mode and dif-ferential mode to obtain high-performance filtering. Interactionbetween the passive L and C components in the filter leads toresonance oscillations. These oscillations need to be damped eitherby the passive damping or active damping. The passive dampingincreases power loss and can reduce the effectiveness of the fil-ter. Methods of active damping, using control strategy, are losslesswhile maintaining the effectiveness of the filters. In this paper, anactive damping strategy is proposed to damp the oscillations inboth line-to-line and line-to-ground. An approach based on poleplacement by the state feedback is used to actively damp both thedifferential- and common-mode filter oscillations. Analytical ex-pressions for the state-feedback controller gains are derived forboth continuous and discrete-time model of the filter. Tradeoffin selection of the active damping gain on the lower order powerconverter harmonics is analyzed using a weighted admittance func-tion. Experimental results on a10-kVA laboratory prototype PWMrectifier are presented. The results validate the effectiveness of theactive damping method, and the tradeoff in the settings of thedamping gain.

Index Terms—Active damping (AD), admittance measurement,discrete-time systems, harmonic distortion, LCL filters, passive fil-ters, pulse-width-modulated (PWM) power converters, state feed-back, state-space methods.

NOMENCLATURE

PWM Pulse width modulation.CM Common mode.DM Differential mode.PD Passive damping.AD Active damping.CSVPWM Conventional space vector PWM.ADC Analog-to-digital converters.DT Discrete time.CT Continuous time.DMAD Differential mode active damping.CMAD Common mode active damping.

Manuscript received February 26, 2013; revised June 5, 2013; accepted July11, 2013. Date of current version January 29, 2014. This work was supported byIndian Space Research Organization-IISc Space Technology Cell under ProjectISTC244. Recommended for publication by Associate Editor A. Lindemann.

M. H. Hedayati and V. John are with the Department of Electrical Engineer-ing, Indian Institute of Science, Bangalore 560012, India (e-mail: [email protected]; [email protected]).

A. Acharya B. is with ABB GISL, Grid System R&D, Chennai 600089, India(e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPEL.2013.2274102

THD Total harmonic distortion.DFT Discrete Fourier transform.WY Weighted admittance.EC S Energy contained in signal.VS , Vi, VC Grid-side voltage, inverter-side voltage, and

LCL filter capacitor voltage.L1 , L2 , CL Grid-side inductance, inverter-side induc-

tance, and capacitance of LCL filter.Cy1 , Cy2 , CM g Common mode filter capacitances.Cdc Dc bus capacitance.Cd,Rd Damping branch capacitance, and damping

branch resistance.fres , fresc m LCL filter resonance frequency, and com-

mon mode filter resonance frequency.Pb, Vb , Rb Base power, base voltage, and base

impedance.Icomg , Icom i Grid-side common mode current, and

inverter-side common mode current.Vcom Common mode voltage of inverter.iL1 , iL2 Grid-side current, and inverter-side current.K1 ,K2 ,K3 DMAD gain in continuous time.K1D ,K2D ,K3D DMAD gain in discrete time.K1c m ,K2c m CMAD gain in continuous timeK1D c m ,K2D c m CMAD gain in discrete time.T ADC sampling time.n Harmonic number.Yn Input admittance of the power converter of-

fered to nth harmonic.Φ,Φcm DM state matrix in DT, and CM state matrix

in DT.Γ,Γcm DM input matrix in DT, and CM input matrix

in DT.A,Acm DM state matrix in CT, and CM state matrix

in CT.B,Bcm DM input matrix in CT, and CM input matrix

in CT.x, x[kT ] State vector in CT, and state vector in DT.X(f) Continuous Fourier transform of x(t).X[k] Discrete Fourier transform of x[n].Fsw Switching frequency.

I. INTRODUCTION

IN applications like motor drives where regenerative actionis necessary, grid-connected converters for photovoltaic and

wind power generation, etc., a pulse-width-modulated (PWM)rectifier is a preferred topology [1]–[3]. The fast transition timesand possibility of higher switching frequency in insulated-gatebipolar transistor results in lower switching ripple, elimination

0885-8993 © 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

HEDAYATI et al.: COMMON-MODE AND DIFFERENTIAL-MODE ACTIVE DAMPING FOR PWM RECTIFIERS 3189

Fig. 1. PWM rectifier with combined CM and DM filters. Utilizing split capacitor Rd Cd Passive Damping.

of lower order harmonics, and lower power loss. However, thefast transition of voltage, i.e., high dv/dt, at the output ter-minals of the power converter causes several problems, suchas excitation of parasitic capacitors in the power circuit, high-frequency ripple current injection into the grid, high groundcurrent, etc., [4]. To minimize these effects and decrease theparasitic influence, the common-mode (CM) choke and linefilters with Y-capacitor are used in [5] and [6]. Issues such asswitching ripple, ground current, etc., are addressed either usingpassive or active filtering or combination of these filtering tech-niques [2], [7]–[9]. These passive filters are connected betweenthe grid and the PWM rectifier; the filters used for this purposeare L or LCL [2], [9], [10]. Higher order filters, LCL filters, arepreferred over lower order L filters for their performance suchas higher attenuation rate and smaller filter parameter values.The higher order filters required to meet the standards IEEE519 [11], CISPR 11 [12], and IEC 61000 [13] are in generalless bulky compared to the lower order filters. However, thehigher order filters have a tendency to oscillate at a particularfrequency, referred to as the resonance frequency, due to the dis-turbances around these frequencies. Even a small disturbanceproduces large amplitude of oscillations if the damping in thefilter is poor. The damping in the filter can be enhanced by ap-propriately introducing the resistive elements; this is referredto as passive damping (PD). The passive damping is simple toimplement but has drawbacks. It increases the power loss anddecreases the effectiveness of the filter. In order to decrease thepower loss in the filters and to maintain their effectiveness, it isdesirable to damp the oscillations using a suitable control strat-egy, referred to as active damping (AD). AD emulates the actionof the resistors and damps the oscillations, and it is a losslessdamping method. However, the AD methods need additionalsensors and increase the complexity of the control algorithmwhich makes the system less reliable compared to the passivedamping. Various methods of implementing the PD and ADare discussed in [10] and [14]–[20]. In the passive technique, aresistive, resistive-capacitive branch, inductive-resistive branch,or combinations of these networks are introduced in the filterstructure [10], [20]. The inclusion of resistive elements dampsthe oscillations and the resistance is chosen as a tradeoff be-tween the power loss and quality factor. The combination of a

resistive element with a capacitor or an inductor helps reducingthe power loss and improves the performance of the filter [10].The AD methods based on virtual resistor, lead-lag compen-sation, notch filter, state-space method, etc., have been widelystudied in the literature [14]–[19]. The primary attraction of theAD method is that, it is lossless while maintaining the filteringeffectiveness.

The CM filtering requirements of the power circuit have sofar been considered to be independent of the need to filter theline-to-line PWM output of the converter. Hence, the main fo-cus of the damping design in the literature has been on theinvestigation of AD on a line to line basis or differential mode(DM) alone. However, if the DM and CM filtering can be inte-grated into a single filter structure, then it becomes importantto simultaneously address the concern of both the DM and CMresonances of the filter. The higher order filter configurationshown in Fig. 1 is proposed in [21]. It integrates the filtering forline-to-line or DM, and line-to-ground or CM. The passive filterused in this manner reduces the overall number of the passiveelements required to mitigate the DM and CM electrical noises.Fig. 1 shows the filter configuration where addition of the threecapacitors, Cy1 , Cy2 , and CM g to an LCL filter adapts it forboth DM and CM filtering.

In this paper, differential mode active damping (DMAD)and common mode active damping (CMAD) are proposed anddemonstrated for this topology and also compared with PD. Thestate-space method is used to analyze the filter oscillations andits damping. Pole placement using state feedback is adopted toobtain the desired level of the damping. Closed-form analyticalexpressions are derived for the DM and CM damping controllergains in both the continuous time (CT) and discrete-time (DT)domain. The damping of resonance with both the passive andactive methods is analyzed in terms of the damping factor. Acomparison between the effectiveness of the two techniques isexperimentally validated in terms of a measure called energycontained in a signal (EC S ). The performances of DMAD andCMAD are evaluated and compared with the PD. The methodfor obtaining the input admittance of the converter at the har-monic frequencies in [22] has been used to study the effectof the AD gains on the low-frequency harmonic distortion ofthe PWM rectifier. This has been used to obtain guidelines for


Fig. 2. Passive Damping of the oscillation in the LCL filter using a splitcapacitor parallel Rd Cd damping branch.

setting the AD gains. This tradeoff between the AD gain andlow-frequency distortion seen in the PWM rectifier input cur-rent is validated by experimental results on a 10-kVA laboratoryprototype.

II. HIGHER ORDER FILTERS WITH THE PD

The structure of the filter and the values of the componentsare as discussed in [21]. The analysis and discussion below isfrom the perspective of the damping, harmonic performance,and power loss in the filter.

A. DM Filter

Fig. 1 shows the LCL filter interface between the grid andPWM rectifier, which reduces switching current ripple injec-tion into the grid. The design of the LCL filter follows theguidelines suggested in [10]. The advantage of the LCL filtersover an L filter is that, they have an attenuation of −60 dB/decfor frequencies higher than the resonance frequency fres . How-ever, they suffer from oscillations between the passive L and Ccomponents. These oscillations are due to a pair of undampedcomplex-conjugate poles on the imaginary axis. To damp theseoscillations, the undamped poles should be shifted to the lefthand side of the imaginary axis. This is done by introducing adamping resistor to the filters. Fig. 2 shows the LCL filter witha damping branch consisting of a capacitor Cd and a resistorRd on a DM basis. In this configuration, the current is sharedbetween the capacitor CL and the damping branch. This resultsin a lower power loss compared to having a damping resistor inseries with a capacitor CL . The transfer function Vc (s)

Vi (s) is givenas follows:

Vc(s)Vi(s)

=L1(1 + sRdCd)

(L1 + L2)(s3LP CLCdRd + s2LP CP + sCdRd + 1)(1)

where LP = (L1L2)/(L1 + L2) and CP = CL + Cd . Thebode plot of (1) is shown in Fig. 3 for different values of thedamping resistance. The resonance peak of the Vc (s)

Vi (s) transferfunction decreases with increase in the resistance Rd . However,beyond a certain value of the resistance, the resonance peakstarts increasing again. This indicates that there exist an opti-mum value of the damping resistance that can be chosen to get abetter damping. A wide range of alternate PD circuit topologiesare available in the literature [20]. In this paper, the PD shownin Fig. 2 is adopted due to its low power loss and quality factor,while maintaining an adequate stability with the grid [23], [24].

Fig. 3. Frequency response plot of transfer function V c (s)V i (s) given in (1) for

different values of damping resistance, where Rb =3V 2

bP b

= 17.28 Ω.

In this paper, the effects of the PD circuit in terms of DM andCM damping of the filter are investigated and the correspondingpower loss is evaluated.

B. CM Filter

Fig. 1 shows the filter capacitors Cy1 , Cy2 , and CM g intro-duced at the dc bus. The neutral point of the capacitors Cy1

and Cy2 is connected to the LCL filter neutral “M”. The ca-pacitor CM g acts as a link between “M” and ground. Theaddition of Cy1 , Cy2 , and CM g provides CM filtering of thedc-bus output of the PWM rectifier. The filter constraints anddesign procedure on the CM-based circuit of a power converteris based on the procedure suggested in [21]. The filter config-uration restricts the injection of lower order harmonics due toan advanced PWM technique such as conventional space vectorPWM into the grid. Additionally, by the appropriate selectionof the filter circuit parameters, it minimizes the third harmoniccurrents that circulates within the filter through the dc bus sothat the increase in the filter inductor current rating is less than2.5%. Also, the high-frequency ground currents due to para-sitic capacitances are reduced. The CM equivalent circuit ofthe filter is shown in Fig. 4(a) with a damping branch, whereLa = L1

3 , Lb = L23 , Rdc m = Rd

3 , Ca = 3CL,Cdc m = 3Cd , andCb = Cy1 + Cy2 .

Due to the small value of the capacitor, CM g , the CM circuitfor low-frequency components can be approximated as shownin Fig. 4(b). The inductance Lb of the inverter side LCL filteralong with the effective filter capacitance Ccm acts as a low-pass filter for the CM components. This eliminates the needfor additional passive elements to realize a low-pass CM fil-ter. By suitably selecting the corner frequency of the filter, thelower order harmonics due to the advance PWM techniques andswitching frequency component will circulate within the sys-tem, with a small magnitude. The resonance frequency of theCM filter with zero damping resistance is given as follows:

fresc m =1

2π√

LbCcm(2)


(a)

(b)

Fig. 4. (a) CM equivalent circuit for the PWM rectifier shown in Fig. 1.(b) Low-frequency approximation of CM equivalent circuit of the PWM rectifier.

Fig. 5. Frequency response plot of the transfer function Ic o m i (s)V c o m (s) given in (3)

for different values of the damping resistance.

where Ccm = ((Ca + Cdc m )Cb)/(Ca + Cdc m + Cb). The filteradmittance transfer function for the circulating inverter CM cur-rent Ic o m i (s)

Vc o m (s) is given in (3) as shown at the bottom of this page.The bode plot of (3) is shown in Fig. 5. It can be observed that,by increasing the damping resistance, the CM current resonanceis reduced and it reaches its minimum value at an optimum valueof the damping resistance. If the damping resistance is further in-creased, the CM resonance peak starts increasing again. Hence,the value of Rd selection in the split capacitor RdCd dampingnetwork needs to cater to both the DM and CM damping of thefilter.

C. Selection of the Damping Resistance Rd

To select the value of the damping resistance, it is increasedfrom zero in small steps. At each step, the magnitude of the

(a)

(b)

Fig. 6. Effect of the damping resistance on the filter magnitude response andpower loss. (a) Peak magnitude of response at resonance frequency versus the

damping resistance. For the transfer function | V c (s)V i (s) | in dB, and the transfer

function | ic o m i (s)V c o m (s) | in dB(mho). (b) Power loss in the damping resistors versus

damping resistance values due to both the DM and CM excitation.

resonance in the LCL capacitor voltage and in the CM currentis calculated. Fig. 6(a) plots the peak of the resonance of themagnitude transfer functions (1) and (3) as a function of thedamping resistance value. From Fig. 6(a), it can be seen that theminimum resonance in the capacitor voltage, as well as in theCM current, occurs while having the damping resistance close to1 p.u. The minimum numerically is observed at 1.1 p.u., whichis chosen as the PD resistance used in the damping network.

D. Power Loss Calculation in the Damping Resistor Rd

To calculate the power loss in the damping resistor Rd , thecircuit shown in Fig. 2 is evaluated at different frequencies. Thelosses in Rd are mainly due to fundamental (50 Hz), triplenharmonics (150 Hz, . . .), and at switching frequency (10 kHz).The third harmonic losses are due to the space vector PWMmethod and the switching losses are due to the switching action

Icom i(s)Vcom(s)

=CaCbCdc m Rdc m s2 + (CaCb + CbCdc m )s

CaCbCdc m LbRdc m s3 + (CaCbLb + CbCdc m Lb)s2 + (CaCdc m Rdc m + CbCdc m Rdc m )s + (Ca + Cb + Cdc m )(3)


of the PWM rectifier. All the calculated losses are added up andthe result is plotted in Fig. 6(b). The power loss for the valueof Rd calculated in the previous section is marked to be 39.4 Wfor a power converter rated for 10 kVA.

III. AD OF THE HIGHER ORDER FILTERS

PD increases the power loss in the inverter. Additionally, insome damping configuration, it decreases the performance ofthe filter. To damp the oscillations without increasing the powerloss and without reduction in the attenuation performance ofthe filter, the AD is used [14]–[19]. Also, if the configurationof the PD RdCd branches are connected in Delta, on a line-to-line basis, then it would address only the resonance of the DMand not the CM. In such cases, the AD is necessary to dampthe oscillations in the CM circuit, while the DM oscillationsare passively damped. To model the LCL filter circuit for AD,the PD resistor is shorted. The circuit is analyzed in terms ofDM and CM variables separately [25]. The DM and CM circuitmodels have been used here to analyze the AD of the filterusing the state-space approach. The CM voltage and current aresubtracted from the measured states of the LCL filter in orderto obtain the DM filter equations.

A. DM Filter Model

The state variables used to model the DM LCL filter are x1 =VCL

, x2 = iL2 , and x3 = iL1 . The input vector u = [Vi Vs ]′,with the grid voltage Vs being considered a disturbance in-put. The state-space representation without the PD branch is asfollows:

x = Ax + Bu

y = Cx (4)

where

A=

⎡⎢⎢⎢⎢⎢⎢⎣

0 − 1CP

1CP

1L2

0 0

− 1L1

0 0

⎤⎥⎥⎥⎥⎥⎥⎦, B =

⎡⎣b1 b2

⎤⎦ =

⎡⎢⎢⎢⎢⎣

0 0

− 1L2

0

01L1

⎤⎥⎥⎥⎥⎦

.

The Matrix C is the identity as the inductor currents and capac-itor voltages are sensed. The eigenvalues of A are determinedby setting

|sI − A| = 0 ⇒ s3 + s1

LpCP= 0

s1 = 0, s2,3 = ±jωres (5)where ωres = 1√

Lp CP

.

B. CM Filter Model

The dynamic equations describing the filter shown in Fig. 4(b)are as follows:

Lbd(Icom i)

dt= −Vcom + Vcm (6)

CcmdVcm

dt= −Icom (7)

where IL2 a, IL2 b

, and IL2 care the converter side phase currents,

Vcm = VCa− VCb

and Icom i = IL2 a+ IL2 b

+ IL2 c. The state

variables are x4 = Vcm , x5 = Icom i , xcm = [x4 x5 ]′, and theinput is ucm = Vcom . The state-space representation is given asfollows:

xcm = Acmxcm + Bcmucm (8)

where

Acm =

⎡⎢⎢⎣

0 − 1Ccm

1Lb

0

⎤⎥⎥⎦, Bcm =

⎡⎣

0

− 1Lb

⎤⎦.

The eigenvalues of matrix Acm are located at s1,2 =±jωresc m = ± j√

Lb C c m.

IV. STATE-SPACE CONTROL LAW FOR THE AD

The purpose of the state-feedback control law is to locatepoles of the system such that the marginally stable poles arebrought to stable region with the satisfactory dynamic response.It is advantageous to use the state-space method when more thanone input are involved or more than one output are sensed [26],[27]. The control law used is a linear combination of the statevariables for the DM and CMAD. The analysis below providesthe expressions for the state-feedback gain in the CT. This isthen used to obtain expressions for the damping gain in the DTfor implementation in a digital controller.

A. DM Control Law

State feedback can be used to shift the poles of the systemto obtain an adequate damping. The control and the gain matrixcan be expressed as u=−(K1x1 + K2x2 + K3x3)=−K × x.The state-space representation of the modified system is x =(A − b1K)x. The eigenvalues of the system after state-spacefeedback are located at

|sI−A+b1K|=det

⎡⎢⎢⎢⎢⎢⎢⎢⎣

s1

CP− 1

CP

− (1 + K1)L2

(s − K2

L2

)−K3

L2

1L1

0 s

⎤⎥⎥⎥⎥⎥⎥⎥⎦=0.

Solving the aforementioned determinant leads to characteristicequation given as

s3 − K2

L2s2 +

{(1 + K1)L2CP

+1

L1CP

}s −

{(K2 + K3)L1L2CP

}= 0.

(9)The original poles of the system are given by (5). Let the desiredpole locations be at 0 and −σ ± jωd . where σ = ζωres andωd = ωres

√1 − ζ2 . Then, the characteristics equation with new

pole locations would be

s{(s + σ − jωd)(s + σ + jωd)} = 0

s3 + 2σs2 + ω2ress = 0. (10)


Equating (9) and (10), the coefficient of gain matrix K to obtainthe DM damping is

K1 = 0

K2 = −2ζωresL2

K3 = +2ζωresL2 . (11)

B. CM Control Law

A similar analysis is extended for the CM filter. The con-trol input for the CM resonance damping is given by ucm =−(K1c m x4 + K2c m x5). The state-space representation is mod-ified as xcm = (Acm − BcmKcm)xcm . The state-feedback gaincoefficients for CMAD are given by

K1c m = 0

K2c m = −2ζ

√Lb

Ccm. (12)

V. ANALYSIS IN DT DOMAIN

Analog information from the sensors is converted into digitalform using the analog-to-digital converters (ADC). The digitalcontrol has to act upon the samples of the sensed plant output.The data are sampled by ADC at a sampling time T , whereT = 1

2Fsw. Since each sampled value is held constant until the

next value is available, compared to continuous signal, the aver-age value of the sampled data lags by T/2 [28]. Further the statematrix coefficients vary based on the sampling rate. The ADanalysis from Section IV is presented in DT domain to facilitateimplementation in a digital controller. CT to DT transformationbased on impulse invariant transform approach is adopted [29].This approach helps match the frequency response characteris-tics of the continuous system and the DT domain. Closed-formexpressions are derived to understand the dependences of thestate matrices on the filter parameters and the sampling rate andto relate the filter damping factor (ξ) to the state-feedback gain.

A. DT Representation

The CT state-space model can be represented in DT domainwith a good match of the poles and zeros of the transfer functionusing the impulse invariant transformation [30]. This approachis used for transforming the CT representation to DT and canbe expressed as

x[(k + 1)T ] = Φx[kT ] + Γu[kT ]

y[kT ] = Hx[kT ]. (13)

1) Expression for Φ and Γ for the DM Filter Modeling: Theclosed-form expression of Φ and Γ helps in designing the ADcontroller which is analytically calculated as follows:

Φ = I + α1A + α2A2 (14)

where α1 and α2 can be obtained using (31) derived inAppendix A

α1 = k1T =sin(ωresT )

ωres

α2 = k2T2 =

1 − cos(ωresT )ω2

res.

Substituting for A from (4) into (14)

Φ =

⎡⎢⎢⎢⎢⎢⎢⎣

1 − α2

CP

(1L1

+1L2

)− α1

CP

α1

CP

α1

L21 − α2

L2CP

α2

L2CP

−α1

L1

α2

L1CP1 − α2

L1CP

⎤⎥⎥⎥⎥⎥⎥⎦

.

(15)Φ and Γ can be related to following equation as [29], and isderived in Appendix A in (33):

Γ = A−1(Φ − I)B

= α1B + α2AB

Γ =

⎡⎣ γ1 γ2

⎤⎦ =

⎡⎢⎢⎢⎢⎢⎣

α2

L2CP

α2

L1CP

−α1

L20

0α1

L1

⎤⎥⎥⎥⎥⎥⎦

. (16)

2) Expression for Φcm and Γcm for the CM Filter Modeling:The closed-form expression of Φcm and Γcm for the CM modelcan be calculated in a similar method. The steps for derivationof Φcm and Γcm are provided in Section B of Appendix A

Φcm =

⎡⎢⎢⎣

cos(ωresc m T ) − sin(ωresc m T )ωresc m Ccm

sin(ωresc m T )ωresc m Lb

cos(ωresc m T )

⎤⎥⎥⎦ (17)

Γcm =

⎡⎣

1 − cos(ωresc m T )

− sin(ωresc m T )ωresc m Lb

⎤⎦. (18)

B. DT State-Feedback Control Law

1) DM Control Law in the DT: The pole location in z-domaincan be related to s-domain as z = esT . The desired pole loca-tions are s1 = 0, s2,3 = −σ ± jωd their values in z-domain areas follows:

z1 = e0T = 1

z2 = e(−σ+jωd )T = e−σT (cos(ωdT ) + j sin(ωdT ))

z3 = e(−σ−jωd )T = e−σT (cos(ωdT ) − j sin(ωdT )). (19)

The DT model of the DM filter with the AD is given by

x = (Φ − γ1KD )x (20)

where KD = [K1D K2D K3D ]. The eigenvalues of the systemafter the DT state feedback are given by |zI − Φ + γ1KD | = 0.Equating the new characteristics with the desired characteristics,the gain matrix KD can be calculated as

|zI − Φ + γ1KD | = (z − z1)(z − z2)(z − z3). (21)

Solving (21) for K1D ,K2D , and K3D using (34) inAppendix A, we get


K1D =2 cos(ωresT ) − 2e−σT cos(ωdT ) + e−2σT − 1

2α2CP L2

K2D = −2 cos(ωresT ) − 2e−σT cos(ωdT ) − e−2σT + 12α1L2

K3D = +2 cos(ωresT ) − 2e−σT cos(ωdT ) − e−2σT + 1

2α1L2

. (22)

The state-feedback gains in (22) are used for the DMAD in thedigital controller.

2) CM Control Law in the DT: The desired pole loca-tions are s1,2 = −σcm ± jωdc m their values in z-domain are asfollows:

z1c m = e(−σc m +jωd c m )T

= e−σc m T (cos(ωdc m T ) + j sin(ωdc m T ))

z2c m = e(−σc m −jωd c m )T

= e−σc m T (cos(ωdc m T ) − j sin(ωdc m T )). (23)

The gain matrix KD c m = [K1D c m K2D c m ] can be calculated us-ing the following equation:

|zI − Φcm + ΓcmKD c m | = (z − z1c m )(z − z2c m ). (24)

Solving (24) for K1D c m and K2D c m using (39) fromAppendix A, we get

K1D c m =−2 cos(ωresc m T ) + e−2σ c m T − 2e−σ c m T cos(ωd c m T ) − 12 cos(ωresc m T ) − 2

K2D c m =−2 cos(ωresc m T ) − e−2σ c m T − 2e−σ c m T cos(ωd c m T ) + 1

2 sin(ω r e s c m T )L b ω r e s c m

.

(25)

The state-feedback gains in (25) are used to implement theCMAD in the digital controller.

VI. DISTORTIONS AT THE LOW FREQUENCY

AND RESONANCE FREQUENCY

A. Effects of the AD Gain on Grid Current Lower OrderHarmonics

Due to the presence of lower harmonics in the line voltage,the input current to the PWM rectifier contains lower harmonics.However, introduction of the AD reduces the resonance oscilla-tions; it is expected to increase the lower order harmonics in theinput current as reported in the literature [31]. That is due to theAD state feedback which feeds the lower harmonic, as well, tothe controller from the sensed input as seen from state-feedbacklaw given in Section IV-A. Higher gain amplifies the harmonicswhich is fed back. Hence, increasing the low-frequency totalharmonic distortion (THD) observed in the input current to theconverter.

To analyze the effect of damping on the line current distortion,the current controller and AD control loops have been modeledin CT domain. Simulation using MATLAB is used to calculatethe input admittance of the power converter at the harmonicfrequencies. The admittance Yn = In

Vnfor 5th, 7th, 11th, and

Fig. 7. Admittance offered by the PWM rectifier to different harmonics andweighted admittance, evaluated for the different DMAD gain given in (11).

13th harmonics was calculated and is shown in Fig. 7 alongwith the weighted admittance using the following equation:

WY =

√∑n

|Yn |2n2 (26)

where In is the input current harmonics drawn by the PWMrectifier, Vn is the line voltage harmonics, and n = 5, 7, 11, and13. The admittance of the power converter is evaluated usingthe method from [22] by superimposing the harmonic excita-tion over the fundamental nominal operation of the converterfor the different AD factors. It is assumed that the amount ofline voltage even harmonics is small and negligible. The thirdharmonic and its multiples cannot flow in the line current ofthe three-phase converter. So, the system has been evaluatedfor 5th, 7th, 11th, and 13th harmonics. It is also assumed thatthe magnitude of harmonic components in the line voltage isinversely proportional to the harmonic number. As can be seenfrom Fig. 7, by increasing the AD gain KD , which is propor-tional to damping factor ξ, the admittance offered by the PWMrectifier is increased. The input current harmonic to the powerconverter depends on the input admittance offered to each har-monic by the power converter. Hence, the THD of line currentdepends on the AD gain when there are harmonics in the inverterand the grid voltage.

B. Energy Contained in the Signal.

The waveforms of the power converter contain many signalfrequencies due to fundamental excitation, low-frequency dis-tortion, and triplens due to space-vector modulation in additionto the switching frequencies and its side bands. The strength ofthe signal in the LCL filter is evaluated at the resonance fre-quency using (27). This is used to evaluate the filter dampingperformance for different damping coefficients. The objective ofthe EC S definition is to quantify the damping provided near theresonance frequency. Based on Parseval’s theorem in AppendixB, the total energy contained in the waveforms around the res-onance frequency is calculated using (27). The expression for


Fig. 8. Control block diagram of a vector-controlled PWM rectifier with the DM and CMAD loops.

EC S between the frequency range F1 and F2 is

EC S =1

F2 − F1

F2∑F =F1

|x(F )|2 (27)

where F1 < F2 . The EC S value is used to compare the energycontained in the LCL filter capacitor line-to-line voltage for theDM resonance, and dc-bus midpoint to the ground voltage forthe CM resonance for different cases of the AD and PD.

VII. EXPERIMENTAL RESULTS

A. Experimental Setup

A three-phase, 10-kVA, 415-V PWM rectifier prototype hasbeen built in the laboratory with the filter topology as shown inFig. 1. The base and the designed values of the filter parametersare given in Table. I. The PWM rectifier is loaded using a mo-tor drive inverter. The experimental setup consists of the PWMrectifier, a motor drive inverter, and a 6.5-kW induction motorwhose shaft is coupled to another induction generator. This testconfiguration, which is explained in [32], allows circulation ofpower and draws only losses from the grid. The synchronousreference frame controller [33] is used to independently controlthe active and reactive power input to the power converter andregulate the dc-bus voltage of the PWM rectifier using state vec-tor PWM [34]. The control block diagram of the PWM rectifierwith the DM and CMAD loops is shown in Fig. 8. The parts ofthe block diagram within the dashed lines show the AD parts.Note that K2D and K3D are equal in magnitude but with oppo-site signs. Also, K1D and K1D c m are numerically much smallerthan K3D and K2D c m , respectively, in the practical design. Thesetup has been tested with different damping methods underloaded condition. The control is implemented in a Cyclone IIFPGA using 16-bit fixed-point arithmetic.

The performances of DM and CM damping are evaluatedfor five cases: case (i) when damping is absent; case (ii) withPD; case (iii) with DMAD; case (iv) with CMAD, and case

TABLE IDESIGN VALUES FOR THE FILTER

(v) with both the DMAD and CMAD. The waveforms are mea-sured using a LeCroy 6050, 500-MHz oscilloscope.

B. Passive Damping

The setup has been tested under the loaded condition (6 kW)with and without the PD. The values of the damping resis-tance calculated in Section II-C are used for the PD branches.The results are shown in Fig. 9(a) and (b). The power lossin the damping resistors was measured, using a YOKOGAWAWT1600 digital power meter, to be 40 W. It is observed fromFig. 9 that the resonance oscillations in the LCL capacitor line-to-line voltage, grid side current, and in the dc-bus capacitor


(a) (b) (c)

Fig. 9. (Top) Filter capacitor line-to-line voltage and grid-side phase current. (Middle) CM dc-bus capacitor midpoint to the ground voltage. (Bottom) FFT ofthe line-to-line voltage (F1), FFT of the grid current (F2), and FFT of the CM voltage (F3). (a) Case (i)—no damping, (b) case (ii)—PD and (c) case (v)—withAD. With load of 6 kW and AD gain corresponding to ξ = 0.7.

midpoint to the ground voltage are reduced as compared to thecase of no damping. It is seen that the performance differencebetween the cases is better captured by the spectra of the wave-forms, which is obtained using fast Fourier transform (FFT) ofthe captured time-domain waveforms. The PD network is effec-tive in attenuating both the DM and CM resonance oscillations.

C. Active Damping

To study the effectiveness of the AD, the PD resistors areshorted and the AD loop is activated. Fig. 9 shows the filtercapacitor line-to-line voltage, the grid phase current, the dc-buscapacitor midpoint to the ground voltage, and their spectra, cor-responding to cases (i), (ii), and (v) mentioned in Section VII-A.From Fig. 9, the reduction in the resonance oscillations due tothe AD case can be observed. However, it can also be seenthat the lower frequency distortion in this case is increased.The LCL capacitor line-to-line voltage and dc-bus midpoint tothe ground voltage are also measured to study the effects ofseparately enabling DMAD and CMAD. The spectra of these

TABLE IISTATE-SPACE GAIN FOR THE DAMPING COEFFICIENT ξ = 0.7

voltages are shown in Fig. 10. The three cases considered hereare: case (iii) activating only DMAD; case (iv) activating onlyCMAD; and case (v) activating both the DMAD and CMAD.The damping gains are given in Table II. The actual values of theAD gains using (11) and (12) are in column 4. Column 3 repre-sents the gain normalized by Rb . Column 5 represents the DTgains used in the experimental setup, using (22) and (25). It canbe observed from Fig. 10 that simultaneous activation of both


(a) (b) (c)

Fig. 10. Spectrum of the LCL capacitor line-to-line voltage (F1), spectrum of the dc-bus capacitor midpoint to the ground voltage (F2). (a) Case (iii)—withonly DMAD, (b) case (iv)—with only CMAD, and (c) case (v)—with both DMAD and CMAD. With AD gain of ξ = 0.7.

TABLE IIIENERGY CONTAINED SIGNALS (V 2 s) NEAR THE FILTER

RESONANCE FREQUENCY

the DMAD and CMAD effectively suppresses the oscillationsat the corresponding resonance frequencies.

D. Analysis of the Experimental Results

A set of experiments consisting of testing the setup with nodamping, with the PD, and with the AD activated are performedunder the loaded conditions. The AD tests are repeated for dif-ferent values of the damping factors of ξ = 0.5, 0.7, 0.9, and1.0. The calculated EC S is used to evaluate the performancechange with gain settings. The EC S for the DM resonance isevaluated between frequencies 0.75 and 2 kHz in the filter ca-pacitor line-to-line voltage signal. CM resonance performanceis evaluated by calculating the EC S between 1.0 and 3 kHz forthe dc-bus midpoint to the ground voltage. Five measurementswere made for each operating condition and the average valuesare tabulated in Table III.

The comparison between different tests are as follows:1) No Damping and the PD: The energy contained signals

of DM and CM resonance with no damping are 1101.7 and685 V2s, respectively. By introducing the damping resistors tothe LCL filter, they have been reduced to 711.2 V2s (65%)and 402.9 V2s (59%), respectively. It is clearly seen that, withPD, the DM and CM resonance are damped up to someextent.

2) Activating Only the DMAD: By activating only theDMAD, the DM resonance is reduced. By increasing the damp-ing factor, the DM resonance is effectively damped as can be

seen from Table III. However, it is not so effective to damp theCM resonance.

3) Activating Only the CMAD: By activating only theCMAD, the CM resonance is reduced. By increasing the damp-ing factor, the CM resonance is effectively damped. The CMADhas almost no damping effect on the measured DM resonance.

4) Activating the DMAD and CMAD: In comparison to thePD with ξ = 0.53, the AD with ξ = 0.5 has a slightly higherDM resonance, whereas, the CM resonance is better damped inthe AD. As the damping factor is increased, both the DM andCM resonances are damped more effectively. Activating boththe DMAD and CMAD simultaneously, in case (v), has betterdamping compared to cases (iii) and (iv) at the same dampingfactor.

E. Transient Performance of the Damping Methods

To test the effects of the AD on transient response, the setuphas been tested under step load change from 1 to 6 kW on the in-duction motor load. The filter capacitor line-to-line voltage andphase current corresponding to cases (i), (ii), and (v) are shownin Fig. 11. From Fig. 11, it can be seen that the introduction ofthe AD does not affect the transient response of the system andit remains stable during the transients of load change.

F. AD Effects on Low-Frequency THD

To evaluate the effects of the DMAD on low-frequency dis-tortion in the grid current, the grid current spectrum was evalu-ated between the 3rd harmonic (150 Hz) and the 13th harmonic(650 Hz), while varying the damping factor from ξ = 0.1 toξ = 1.0. Frequencies up to the 13th harmonic were evaluatedto check the grid current distortion away from the DM andCM resonance of the filter in the power converter. It can beseen from Fig. 12 that the introduction of the AD increasesthe low-frequency distortion in the grid current. This is be-cause the AD feedback effectively multiplies the capacitive cur-rent using the state-feedback law given in Section IV-A. Thisis included in the modulation signal of the power converter.Based on the simulation in Section VI.A, this effectively causesan increased admittance of the converter at harmonic frequen-cies for higher values of the damping gain. If any lower orderharmonics are presented in the grid, lower order currents will


(a) (b) (c)

Fig. 11. Filter capacitor line-to-line voltage and grid side phase current for step load change from 1 to 6 kW at t = 0.1 s. (a) Case (i) with no damping, (b) case(ii) with PD, and (c) case (v) with the AD and gain corresponding to ξ = 0.7.

Fig. 12. Grid side current low-frequency THD for different settings of theDMAD gains.

flow through the filter. This increases the low-frequency THDof the line current. Hence, the damping gain of the AD cannotbe chosen with the resonance damping as the sole objective, andthe amount of grid lower order harmonics must be taken intoconsideration.

The measured results and cubic curve fit of the measured dataare shown in Fig. 12. Comparing the analysis of the converterinput admittance, shown in Fig. 7, with the experimental results,shown in Fig. 12, it can be seen that both are following a similartrend. The increase DMAD gain increases the admittance of theconverter to the harmonics in the grid voltage and the distortionsin the inverter output voltage. Hence, increase in the DMADgain increases the low-frequency THD of the grid current of thePWM rectifier under the practical operating conditions.

G. Recommended Damping Settings

Based on the analysis and the experimental evaluation ofexperimental performance, we have the following.

1) The AD is preferred over the PD due to savings in lossand retaining the effectiveness of the LCL filter.

2) It has been found that a DM damping factor of 0.5 ispreferable, as it reduces the energy content near the res-onance frequency by around 34%. Damping factor of 0.5is recommended, while the line current THD is not exces-

sive. If the line current THD is small, the damping factorcan be increased further more.

3) CM damping factor of 0.9 provides acceptable perfor-mance, as it reduces the energy content near the resonancefrequency by around 55%.

4) Simultaneously activation of both DMAD and CMADwith these gains provides good overall performance.

VIII. CONCLUSION

The comparison of the PD along with the need for the ADfor the CM and DM LCL filter is presented. The importanceof the PD circuit topology used for damping the resonance isillustrated. The power loss measured in the PD resistors is foundto match the analytical calculations. A state-space approach isused to calculate the AD gains for the CT and the DT model ofthe filter and to obtain state-feedback controller gains in both theCT and DT. The analytically derived controller gains are usedto implement the AD for different damping factors on a digitalcontroller. The laboratory PWM rectifier with the LCL DM andCM filter is tested with different damping methods and differ-ent damping conditions. The damping performance is correlatedwith measurement in terms of the energy contained signal forboth DM and CM damping. The comparison of the results showsthe effectiveness of the AD of oscillations in the dc-bus mid-point to the ground voltage, and the LCL capacitor line-to-linevoltage. However, it is also shown that the state-feedback-basedapproach for AD increases the lower order harmonics in theline currents, for high values of DMAD gain. This is shown bycorrelating the admittance of the power converter at harmonicfrequencies and using a weighted admittance function WY andthe measured line harmonics in the line current. This is due tothe presence of the lower order harmonics during the practicaloperation of the power converter and in the grid voltage. Theanalysis of the DMAD and CMAD performance is validated bythe results from a 10-kVA laboratory power converter.

APPENDIX A

A. DM Model of the Filter in the DT

The impulse invariant model of the filter in the DT state-space form is obtained as follows. To derive the closed-formexpressions for Φ and Γ as defined by (13), consider the function


defined as f(AT ) = eAT and g(AT ) to be a polynomial suchthat

g(AT ) = k0I + k1(AT ) + k2(AT )2 . (28)

The poles of the CT characteristic equation are λ = 0,±ωres

f(λT ) = eλT (29)

g(λT ) = k0I + k1(λT ) + k2(λT )2 (30)

when λ = 0, f(0) = 1 ⇒ k0 = 1when λ = jωres , e

jω r e s T = 1 + k1(jωresT ) + k2(jωresT )2 .Equating the real and imaginary parts, k1 and k2 can be ex-pressed as

k1 =sin(ωresT )

ωresT

k2 =1 − cos(ωresT )

(ωresT )2 . (31)

Substituting the value of k1 and k2

Φ = eAT = k0I + k1(AT ) + k2(AT )2

= I +sin(ωresT )

ωresA +

1 − cos(ωresT )(ωres)2 A2 . (32)

This is used to obtain expression (15). Γ is calculated as

Γ =∫ T

0eAηdηB

=(

T +AT 2

2+

A2T 3

3!+ · · ·

)B =

∞∑K =0

AK TK +1

(K + 1)!

= A−1(Φ − I)B. (33)

The values of the resulting Φ and Γ are given in (15) and (16),respectively. Using (21) and equating the coefficients of z in(21), the following expressions are obtained:

2 cos(ωresT ) − 2e−σT cos(ωdT )

− K1Dα2

L2C+ K2D

α1

L2= 0

1 − 2e−σT cos(ωdT ) − e−2σT + 2 cos(ωresT )

− 2(K2D + K3D )α1

L2

α2

L1C+ 2K2D

α1

L2= 0

e−2σT − K1Dα2

L2C− K2D

α1

L2− 1 = 0. (34)

Solving for K1D ,K2D , and K3D , the final expressions for theDT AD controller gain are obtained in (22).

B. CM Model of the Filter in the DT

Consider the following function defined as f(AT ) = eAT

and g(AT ) to be a polynomial such that

g(AT ) = k0I + k1(AT ). (35)

The poles of the CT characteristic equation are λ = ± j√Lb C c m

=±jωresc m

f(λT ) = eλT (36)

g(λT ) = k0I + k1(λT ) (37)

when λ = jωresc m ⇒ ejω r e s c m T = k0 + k1(jωresc m T ). Equat-ing the real and imaginary parts, k0 and k1 can be expressedas

k0 = cos(ωresc m T )

k1 =sin(ωresc m T )

ωresc m T. (38)

This is used to obtain Φcm and Γcm given in (17) and (18),respectively. Using (24) and equating the coefficients of z in(24), the following expressions are obtained:

2 cos(ωresc m T ) − K1D c m − 2e−σc m T cos(ωdc m T )

+ K1D c m cos(ωresc m T ) +K2D c m sin(ωresc m T )

Lbωresc m

= 0

1 − e−2σc m T − K1D c m cos(ωresc m T ) + K1D c m

+K2D c m sin(ωresc m T )

Lbωresc m

= 0. (39)

Solving for K1D c m and K2D c m , the final expressions (25) forthe digital AD controller gain are obtained.

APPENDIX B

Parseval’s theorem: Total energy contained in a waveformx(t) summed across all t is equal to the total energy of thewaveform’s Fourier transform X(f) summed across all of itsfrequency components f

∫ ∞

−∞|x(t)|2dt =

∫ ∞

−∞|X(f)|2df (40)

where X(f) = F [x(t)] represents the continuous Fourier trans-form of x(t) and f represents the frequency component of x.Alternatively, for the discrete Fourier transform (DFT), the re-lation becomes

N −1∑n=0

|x(n)|2 =1N

N −1∑k=0

|x(k)|2 (41)

where X[k] is the DFT of x[n].

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Mohammad H. Hedayati was born in Kazeroon,Iran, in 1984. He received the B.Tech. degree inelectrical engineering from Islamic Azad University,Kazeroon, Iran, in 2006 and the M.E. degree fromthe Indian Institute of Science, Bengaluru, India, in2010, where he is currently working toward the Ph.D.degree.

His research interests include power electronics,motor drives, active damping, common-mode filters,and high-power converters.

Anirudh Acharya B. (S’09–M’12) received the B.E.degree from Visvesvaraya Technological University,Belgaum, India, in 2007, and the M.Sc. (Eng.) degreein electrical engineering from the Indian Institute ofScience, Bengaluru, India, in 2011.

He is currently with ABB GISL, Grid SystemR&D, Chennai, India. His research interests includegrid-connected converters, high-power dc converters,motor drives, power filters, EMI/EMC, and controlsystems.

Vinod John (S’92–M’00–SM’09) received theB.Tech. degree in electrical engineering from theIndian Institute of Technology, Chennai, India, theM.S.E.E. degree from the University of Minnesota,Minneapolis, MN, USA, and the Ph.D. degree fromthe University of Wisconsin–Madison, Madison, WI,USA.

He has worked in research and development posi-tions at GE Global Research, Niskayuna, NY, USA,and Northern Power, Barre, VT, USA. He is currentlyworking as an Assistant Professor at the Indian Insti-

tute of Science, Bengaluru, India. His research interests include power electron-ics and distributed generation, power quality, high-power converters, and motordrives.

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