Stability Analysis and Comprehensive Design ofGrid-tied Inverter with Active Damped LCL Filter
Jun-ichi Itoh
Dept. Science of Technology
Innovation
Nagaoka University ofTechnology
Nagaoka, Japan
itoh@vos.nagaokaut.ac.jp

Tetsunori Kinoshita
Dept. Electrical, Electronics, andInformation Engineering
Nagaoka University ofTechnology
Nagaoka, Japankinoshita_t@stn.nagaokaut.ac.jp

Abstract— This paper proposes a comprehensive design
strategy for an LCL filter and controller parameters of an active
damped LCL filter which has low inverter-side inductance in agrid-tied inverter. Design methods of a current controller and an
active damping gain become more challenging because grid
impedance affects the stability when the inverter-side inductanceis low. First, a maximum damping gain range is derived to achievea virtual positive damping resistor with cutoff frequency of highpass-filter (HPF) and the LCL resonance frequency. Next, thepractical implementation solutions are discussed based on the
proposed design procedure. The detailed design strategy is
exhibited with the design procedure and the design example. The
validity of the theoretical analysis is demonstrated by a 1-kW
prototype with or without the designed active damping. The gridcurrent THD below 1.2% is maintained over the grid impedance
range from 0% to 10%.
Keywords— Active damping, LCL filter design, Grid current
feedback control

I. INTRODUCTION
An output filters in single-phase grid-tied inverters are
required to suppress current harmonics and meet the grid current
harmonics standards such as IEEE 1547 [1] with small filter
parameters. An LCL filter is widely used to obtain an effect of
the switching harmonic attenuation with the lower inductancecompared to the L filters. However, the current control loop
becomes unstable due to the LCL filter resonance even if the
proportional gain is small in the grid-tied inverter with the LCL
filters [2]-[5].
On the other hand, the computation delay and the PWM
delay due to zero-order-hold (ZOH) occur when digital control
is applied, i.e. DSP and FPGA. Generally, in the grid-tied systemwith the LCL filter, the grid current feedback control system iswell-known unstable when the resonance frequency is less thanquarter of sampling frequency fsamp [6]-[12]. According to [13],in the grid current feedback control system, an appropriate delay
is required for stable operation. However, the excessive delaycauses worse control performance, i.e. reducing control
bandwidth. If the resonance frequency is less than fsamp/4, thecurrent control system requires some attenuation, i.e. the passive
damping or the active damping.
Therefore, the active damping methods based on concepts ofa virtual resistor has been proposed in order to suppress the LCLXXX-X-XXXX-XXXX-X/XX/$XX.00 ©20XX IEEE

Akio Toba
Fuji Electric Co., Ltd.
Tokyo, Japan
toba-akio@fujielectric.com

Satoru Fujita
Fuji Electric Co., Ltd.
Tokyo, Japanfujita-satoru@fujielectric.com

resonance and avoid the power loss of the passive resistor [14][16]. In general, an active damping method uses the filter
capacitor current feedback through a proportional gain or a highpass filter (HPF). In these active damping methods, a virtual
resistor or a virtual RC damper are respectively provided inparallel with the filter capacitor. However, these damping
methods is not effective when the virtual resistor seem to anegative resistance depending on the ratio of the LCL filter
resonance frequency to the sampling frequency underconsidering the digital sampling and computational delays [10].
The current control block diagram contains the virtual negative
damping resistor by the open-loop [14], [15]. Note that the step
response includes the unfavorable inverse response due to thevirtual negative damping resistor. The conventional design
strategy focuses on the design of the LCL filter parameters andthe parameters of the active damping controller, which are theinsufficient design strategy for minimized the LCL filter for highpower density. In particular, the current controller and the active
damping gain designs considering the resonance in the LCL
filter become more challenging due to a wide variation range ofthe grid impedance when the inverter-side inductor is lowinductance. Nevertheless, the design strategy of a filter, thecurrent controller, and active damping gain considered the LCL
filter with low inductance has not been reported.
This paper proposes a comprehensive design strategy for the
LCL filter, active damping controller parameters, and current
controller parameters of the active damped LCL filter systemwith the low inverter-side inductance in a grid-tied inverter. The
new contribution of this paper is that the stability range isprovided by the diagram easily in unrelated to the rating of theoutput power and voltage. The practical implementation
solutions are discussed based on the proposed design procedure.
The detailed design strategy is exhibited with the design
procedure and the design example. The validity of thetheoretical analysis is demonstrated by a 1-kW prototype withor without active damping.
This paper is organized as follows; first, the issue of the
proportional active damping is explained, and the HPF active
damping method is employed to extend the range of the positive
virtual resistance. In addition, the practical implementation
solutions are discussed based on the proposed design procedurein Section II. Second, the design procedure of the LCL filter, theactive damping controller parameters, and the current controllerparameters are presented, and the detailed design strategy is
exhibited in Section III. Third, in Section IV, the design example
is introduced for the experiments. Fourth, the experiment results
are provided in order to validate the theoretical analysis in
Section V. Finally, Section VI concludes this paper.
II. CAPACITOR CURRENT FEEDBACK ACTTIVE DAMPING
Fig. 1 indicates the circuit configuration of the single-phasegrid-tied inverter with an LCL filter. In this paper, a single-phaseH-bridge two-level inverter is applied due to its simplicity. In
this paper, the grid current, capacitor current, and capacitor
voltage are detected for the current control, active damping, andphase-locked-loop.

resonance frequency range of the positive resistance is less than
0.39fsamp when the cutoff frequency of HPF in (1) is set to 0.5fsamp.
B. Discretized model
The modified pulse transfer functions of the inverter-outputvoltage-to-grid-current Gvo→ig[z,m] and the inverter-outputvoltage-to-filter-capacitor-current Gvo→ic[z,m] from Fig. 2 areexpressed as follows to evaluate the current control stability,
respectively
GvO  ig [ z , m ] 

hpf

L 
sin Tsamp  cos Tsamp 
Kt C f (1),

where L is the inverter-side inductance, Cf is the filter
capacitance, Tsamp is the sampling period, Kt is the active
damping gain, and fhpf (= hpf/2) is the cutoff frequency of HPF.
According to (1), the resonance frequency range of the positive
resistance is less than fsamp/4 when fhpf = 0, i.e. the proportional
active damping is employed [17]. On the other hand, theresonance frequency range of the negative resistance is avoided
by increasing the cutoff frequency of HPF. However, if thecutoff frequency of HPF is set to higher than the Nyquist
frequency, the effect of observation noise increases. Thus, the
SW1

SW2
L

LCL Filter

iL
vO

Vdc
SW2

iC

Grid
Lg

Lf
ig

vg

Cf

vC



vO

z 1
1

z L  L f  Lg

 Tsamp  m  z  1  1 z sin m r Tsamp  sin 1  m   r Tsamp 




2
 r  z 2  2 z cos  r Tsamp  1 z  1
(2)
GvO  iC [ z , m ]A. Negative resistance issue of active damping
Fig. 2 shows the block diagram of the grid current feedback
control system and the equivalent circuit. As shown in Fig. 2, byemploying the active damping, the virtual impedance ZAD(j) is
connected in parallel with the filter capacitor. The virtual
impedance is constructed with the virtual resistance RAD and thevirtual reactance XAD. The virtual resistance can be the negativevalue due to the change of the resonance frequency byconsidering the delay time [15]. Thus, the virtual resistance with
the HPF-active-damping is expressed as follows;
RAD ( j ) igiz  1 1 z sin m r Tsamp  sin 1  m  r Tsamp C vOz L rz 2  2 z cos  r Tsamp  1
fr 

r
1

2
2

L  L f  Lg

(4),

L  L f  Lg  C f

where fr is the resonance frequency of the LCL filter, Lf is thegrid-side filter inductance, and Lg is the gird inductance. Notethat m is 1-, and  is the variable that depends on the duty
update timing [6], [7].  is 0.5 when sampling at the top of thecarrier and updating at the bottom of the carrier. In this paper, is discussed at 0.5. HPF for the virtual RC damper is discretizedby Tustin transformation, which is expressed by
G AD [ z ] 

2 K t  z  1

2   T  z   T
hpf

samp

hpf

samp

(5).

2

C. Inner loop stable region
Fig. 3 shows the block diagram of the discretized grid current
feedback control system with the active damping. As shown in
Fig. 3, the two types loop are introduced: the outer loop is thegrid current feedback control loop, and the inner loop is the filter
capacitor current feedback loop for the active damping. The
stability of the inner loop depends on (3) and (5). The closedloop pulse transfer function of the inner loop is expressed as

SW1

Gclosed _ inner [ z ] 

iC vC ig

1
1  GAD [ z ]GvO iC [ z , m]

(6).

Fig. 1. Single-phase grid-tied inverter with LCL filter.
   0.5Tssamp
Gd ( s )  e
 0    1

ig*

GC(s)

GAD(s)vg
vO

vL

GAD ( s ) Kt ss  hpf

Gd(s)

1
sL

iL

iC

vC

1
sCf
1
ZAD(s) ZAD( j ) 

LCL Filter1s(Lf+Lg)L
GAD ( j )Gd ( j )C f

ig

Equivalent circuit
ZAD=RAD+jXAD(a) Grid current feedback control system with active damping.
(b) Equivalent circuit.
Fig. 2. Block diagram of grid current feedback control system and equivalent circuit. Active damping is equivalent to virtual
impedance Z A D connected in parallel with filter capacitor C f .
Identify applicable funding agency here. If none, delete this text box.

(3),

vO

GC[z]

GvO ig [ z , m]ig

Phase [deg] Magnitude [dB]ig*

Inner loop
GAD[z]

GvO iC [ z , m]Outer loop

Fig. 3. Discretized grid current feedback system with active damping.

Damping gain is limited by
LCL resonance frequency andcutoff frequency of HPF.

0.8
0.6

0.0
0.0

0.1
0.2
0.3
0.4
0.5
LCL resonance frequency ratio fr/fsamp
Fig. 4. Maximum active damping gain range to achieve inner
loop stability with cutoff frequency of HPF and LCL resonancefrequency.

The stability criterion of the closed-loop pulse transfer
function in the z-dominant is determined from the roots of thedenominator characteristic equation. The system is stable when
all poles are in the unit circle. Note that the poles of (6) have acomplex conjugate pole, and the condition for arranging thepoles on the unit circle is expressed by2

 2 cos  T
r

samp

Lr

sin 1  m  r Tsamp h  sin 1  m  r Tsamp  sin mr Tsamp

PM2

-180
-270
0.02fsamp

0.6

m = 0.5

2  hpf TsampPM1

0

0.2fsamp 0.35fsamp fsamp/2
Frequency
Fig. 5. Definition of gain and phase margins of inner loop at different
resonance frequencies.

0.2

Kt 

180

Outside of line introduces
open-loop right-half-plane(RHP) pole

0.4

GM

0

(7)

sin mr Tsamp   2  h  sin 1  m  r Tsamp

where h = (2-hpfTsamp) / (2+hpfTsamp).
Fig. 4 depicts the maximum active damping gain range toachieve the inner loop stability for the cutoff frequency of HPF
and the LCL resonance frequency. As shown in Fig. 4, the highcutoff frequency of HPF is required for the active damping
under the high LCL resonance frequency to achieve the inner
loop stability with the same damping gain, which is consistent
with (1). Fig. 4 and (7) show the relationship of each parameterto achieve the stable inner loop. However, (7) shows only thestability limit. The next step is to evaluate based on the stability
margin, i.e. the gain margin and the phase margin.
Fig. 5 depicts the definition of gain and phase margins of theinner loop. As shown in Fig. 5, a gain margin GM and two phasemargins PM1, PM2 are defined. The gain margin GM and thephase margin PM2 of the inner loop are improved by decreasing
the LCL resonance frequency, whereas the phase margin PM1 of
the inner loop worse due to decreasing the LCL resonancefrequency. Note that the phase margin PM2 decreases due to theunknown delay of the real system, i.e. detection delay. Generally,
the gain margin and phase margin are recommended to be notless than 3dB and 30 degrees, respectively [16]. If the LCL
resonance frequency is fsamp/3 or more, the phase margin of 30

Damping gain Kt/(2fsampL)

Damping gain Kt/(2fsampL)

1.0

At different resonancefrequencies fr

Limited by PM1

Outside the line, phase margin
is less than 30 degrees

0.5
Limited by PM2

0.4

Eq.(8)
0.3
0.2
0.1
0.0
0.0

0.1
0.2
0.3
LCL resonance frequency ratio fr/fsamp
Fig. 6. Relationship between cutoff frequency and damping gain toachieve a phase margin of 30 degrees.

degrees or more is not achieved when the  is 0.5. Furthermore,in the worst case, which is 0.35fsamp or more, the inner loop might
be unstable. It is a limit of the HPF active damping method.
Fig. 6 depicts the relationship between the cutoff frequencyof HPF and the LCL resonance frequency considering the active
damping gain to achieve PM = 30 degrees. As shown in Fig. 6,
The range of the LCL resonance frequency is limited to 0.3fsamp
when the cutoff frequency of HPF is set to 0.5fsamp. As shown in
Fig. 6, the stable range is decreased as compared with the stable
range in Fig. 6 by considering the phase margin. The inside ofthese triangles in Fig. 6 are the stable region of the operation
with the active damping. Note that the increasing part of thetriangle is decided from PM1, and the decreasing part of thetriangle is decided from PM2. As a design guide, the active
damping controller parameters design to satisfy the phasemargin PM2 at the maximum LCL resonance frequency, whichis the stiff grid. The equation by polynomial curve fitting in Fig.
6 is expressed byy  a2 x 2  a1 x  a0
 a2  2278.3w6  4064.8 w5  2905.54 w 4 1075.1w3  225.49 w 2  24.62 w  6.3423

 a1  0.77674 w  0.045246
 a  0.98265 w  0.19033
 0

(8)

where y is Kt/(2fsampL), x is fr/fsamp, and w is fhpf/fsamp.
Therefore, the active damping gain is determined by the LCL
resonance frequency and the cutoff frequency of HPF. Next, thecutoff frequency of HPF is determined to maximize the current
controller gain based on the root locus. The open-loop transfer

function of the discrete grid current feedback control system inFig. 3 is expressed by
GC [ z ]GvO  ig [ z , m]

Fig. 8 depicts the gain range of the current controller where
the current control system is stabilized by the cutoff frequencyof HPF. As shown in Fig. 8, there is the cutoff frequency of HPF
to maximize the current controller gain by the LCL resonancefrequency. The current controller gain with the HPF active
damping method is lower than that with the proportional active
damping method, which is clearly a weakness of the HPF active
damping method. The maximum Kp at the resonance frequency
when the cutoff frequency of HPF to maximize the current
controller gain is selected is given by the polynomial curve
fitting as follows
y  437398 x  640416 x  378886 x
87

116724 x  19892 x  1805.3 x
5

4

6

3

(10)

75.692 x 2  1.7 x  0.195452

where y is Kp/(2fsamp(L+Lf)), x is fr/fsamp.
Fig. 9 depicts the characteristics of the LCL resonancefrequency and the cutoff frequency of HPF to achieve themaximum current controller gain. Note that the maximum
current controller gain means the stability limit. The equation bypolynomial curve fitting in Fig. 9 is expressed as follows
0for  f r f samp  0.1
32

134.08 x  48.747 x  6.9042 x  0.3342
yfor  0.1  f r f samp  0.26 for  0.26  f r f samp 

 0.5

(11)

where y is fhpf/fsamp, x is fr/fsamp. The proportional active damping
method is effective when the LCL resonance frequency is lessthan 0.1fsmap. Otherwise, the HPF active damping method iseffective and the cutoff frequency of HPF is set to 0.5fsamp whenthe LCL resonance frequency is 0.26fsamp or more.
III. DESIGN OF MULTIPLE OBJECT
Fig. 10 describes the LCL filter and the controller design
procedure base on the LCL resonance frequency. The LCL filter

0.6

fhpf/fsamp=0.5

0.4
Imaginary Axis

Fig. 7 depicts the pole trajectory of the closed-loop transfer
function of the discrete grid current feedback control system atthe stiff grid, i.e. the grid inductance of 0 mH. The cutoff
frequency of HPF is swept from 0 to the Nyquist frequency. The
current controller is only proportional gain Kp for simplicity. As
shown in Fig. 7, the poles move into the unit circle, i.e. the gridcurrent feedback control system becomes stable by increasing
the cutoff frequency of HPF. The instability at the low cutoff
frequency of HPF is insufficient damping due to thedestabilization of the inner loop. On the other hand, theinstability at the high cutoff frequency of HPF is due to thedecrease in the outer loop gain margin. As shown in Fig. 7, thecutoff frequency of HPF is determined to maximize the current
controller gain. If the current controller gain is small, the control
performance decrease. Thus, maximizing the current controller
gain is an important factor.

0.2

every 0.01

fhpf/fsamp=0

0
-0.2
-0.4
-0.6
-0.8
-1

0
0.5
1
Real Axis
Fig. 7. Root locus of grid current feedback system when cutoff
frequency of HPF is swept.

Proportinal gain Kp/(2fsamp(L+Lf))

1  G AD [ z ]GvO iC [ z , m]

fr/fsamp = 0.15 Minimum absolute value offc/fsamp = 1/15 dominant characteristic root

0.8

(9)

-1

-0.5

0.20

fhpf/fsamp=0

0.15

every 0.01

Eq.(10)0.10
0.05
fhpf/fsamp=0.5

0.00
0.0

0.1
0.2
LCL resonance frequency ratio fr/fsamp

0.3

Fig. 8. Gain range of current controller where current control
system is stabilized by HPF cutoff frequency.
HPF cutoff frequency ratio fhpf/fsamp

Gop [ z ] 1

0.5

y = 134.08x3 – 48.747x2+ 6.9042x – 0.3342(0.1 < x < 0.26)
Eq. (11)

0.4
0.3
0

0.2

0.5

0.1
0.0
0.0

0.1
0.2
0.26
LCL resonance frequency ratio fr/fsamp

0.3

Fig. 9. Relationship between resonant frequency and cutoff
frequency of HPF to maximize gain of grid current feedback
system.

and the controller design starts with the initialization of thefollowing parameters; the nominal power Pn, the dc-link voltage
Vdc, the grid voltage Vg, the grid frequency fg, the switchingfrequency fsw, the sampling frequency fsamp, the fraction ofsampling period m, the number of output voltage levels N. First,
the desired LCL resonance frequency at the stiff grid is selectedbased on the ratio to the sampling frequency. The inverter sideinductance is determined by the current ripple ratio (=

ΔiL/ILrms). Next, the relationship between the LCL resonancefrequency and minimum k (= Lf/L) satisfying design
requirements is calculated considering the five design
requirements as follows: the reactive power restriction andharmonic suppression, the crossover frequency restriction at
weak grid, the phase margin restriction at weak grid, and thepower factor at stiff grid. Then, the minimum k and maximum frthat satisfy these conditions are determined by drawing in the
figure. Finally, the cutoff frequency of HPF is determined by the
LCL resonance frequency at the stiff grid. The active damping
gain is determined by the LCL resonance frequency and thecutoff frequency of HPF. The important definition of eachparameter in this design is expressed as follows, respectively
Zb 

Vg 2

%ZL 

2 f g L

L f  kL  k % Z L

f r _ max

f r _ min1
2 f g Z b

Zb
2 f g% Z L 1  k   % Z Lg



%YC f % Z L k % Z L  % Z Lg

% Z Lg  0% Z Lg (%YC f  5%)(14),(0  k  1)

(15),



(16),

Crossover frequency
restriction at weak-gridPhase margin
restriction at weak-gridPower factor
restriction at stiff-grid

 fg

1 k
k %YC f % Z L

(17),

 fg

1
%YC f % Z L(18),In this paper, the grid-side feedback current control applies
a PI controller. Note that the controller parameter design without
the filter capacitor voltage feedforward and with employed the
low-inverter-side inductance is required to be more carefully
determined for the system with the PI controller than for thesystem with a PR controller. The discretized transfer function of
the PI controller is expressed byK p Ti  Tsamp  z  Ti
Ti

Drawing Kmin4 by Eq.(30), Eq.(10)Step 5:
Drawing Kmin5 by Eq.(32), Eq.(10)Step 6:
Minimize k, and then maximize fr
based on Kmin1, Kmin2, Kmin3,
Kmin4, Kmin5, fr_max/fsamp

Determinationof k and fr

z 1

Design Cf based on fr (17)Design Lf based on k (15)Calc. of fhpf, Kt : Eq.(11),Eq.(8)Calc. of Kp, Ti : Eq.(22),Eq.(21)Output: LCL Filter value
Active damping parameter
Current controller parameter

Fig. 10 Design flowchart of LCL filter and controller parameters

GvO  ig [ z , m ] where Zb is the base impedance, %ZL is the inverter-side
percentage impedance, Cb is the base capacitance, %YCf is thepercentage capacitance, k is the ratio of the inverter sideinductance and the grid-side inductance, %ZLg is the grid
impedance, fr_max is the maximum LCL resonance frequency as
stiff grid, and fr_min is the minimum LCL resonance frequency as
weak grid. Generally, worst case of grid impedance %ZLg is 10%[18]. Note that the filter capacitor is limited by the decrease of
the power factor at the rated power (generally less than 5%) [19].

GC [ z ] Harmonic constraints

(13),

Zb

C f  %YC f Cb  %YC f

fr  f g

Reactive power
restriction

(12),

Pn

Input: Pn, Vdc, Vg, fg, fsw, fsamp, m, N
Step 1:
Design L(%ZL) based on  (24)
Step 2:
Drawing Kmin1 by Eq.(25), Eq.(10)Step 3:
Drawing Kmin2 by Eq.(27), Eq.(10)Step 4:
Drawing Kmin3 by Eq.(28), Eq.(10)(19)

where Kp is the proportional gain, and Ti is the integral period.
Basically, The LCL resonance makes no phase contribution until
the resonance frequency is actually reached [16], [17]. In thiscase, the phase is approximated by the ideal L filter-based
system, which transfer function is written as g Tsamp  m  z  1  1

Z b % Z L 1  k   % Z Lg z  z  1(20).

The integral period and the proportional gain are calculated
from the crossover frequency fc and the phase margin PM basedon (19) and (20), which the integral period and the proportional
gain are expressed by, respectively
Ti 

Tsamp  tan  PM   cTsamp   1
2  tan 0.5cTsamp

K p  Ti

% Z L 1  k  Z b

g

 2

tan 0.5cTsamp  Tsamp

(21),
2

1  tan 2  PM  cTsamp (22).

Note that the current controller design is applied byassuming Lg = 0. According to (21), the crossover frequency islimited as follows
fc
f samp





2

 PM
2

(23).

Thus, the maximum crossover frequency is limited below(10) and (23). Within these ranges, the PI controller parameters
to achieve the desired power factor PF, the phase margin PMweak,
and the crossover frequency fc_weak at weak grid are determined.
Step 1: Design inverter side inductance L (%ZL)
Based on the current ripple ratio  (= ΔiL/ILrms) of theinverter-side inductor, the impedance of the inverter-side
inductor is expressed by%Z L 

2 f gVdc
4  N  1 f swVg 
2

(24).

Step 2: Derivation of Kmin1 satisfying reactive power
restriction
The relationship between the LCL resonance frequency andk considering the reactive power restriction is expressed asfollows with (17),2 f samp f r 
K min1   % Z L %YC f max  1

 f



 g f samp 



1

(25).

Step 3: Derivation of Kmin2 satisfying harmonic constraints
Based on the regulation of IEEE standard 1547, therelationship between percentage capacitance and the ratio of theinverter-side inductance and the grid-side filter inductance forsatisfying current harmonics suppression is expressed by,
2

k %YC f 

fg
 
1 
1
4 


 1 
% Z L   N  1 f sw  
0.3%  2 (26).

Moreover, the relationship between the LCL resonancefrequency and k considering the harmonic suppression
regulation is expressed as follows with (17) and (26),

TABLE I System Parameters
Circuit Parameter
VDC
DC-link Voltage
Vg
Grid Voltage
Pn
Nominal Powerfg
Grid Frequency
Zb
Base Impedance
Cb
Base Capacitance
Switching Device (SiC-MOSFET)
fsw
Switching Frequency
Current Ripple Ratio
L
Inductor (%ZL = 0.44%)
Cf
Filter Capacitor (%YCf = 1.25%)
Lf
Filter Inductor (k = 0.42)
fr_stiff
LCL Resonance Frequency
Controller Parameter
fsamp
Sampling Frequencym
Fraction of sampling period
Td
Dead time
fc
Crossover Frequency
PM
Phase Margin
Kp
Proportional Gain
Ti
Integral Period
Kt
Active Damping Gain
fhpf
Cutoff Frequency of HPF

2

 f
 
f samp1
4 
K min 2   r  1  1  f0.3%  2 
 samp  N  1 f sw  (27).

Step 4: Derivation of Kmin3 and Kmin4 satisfying crossover
frequency and phase margin at weak grid
The relationship between the LCL resonance frequency andk considering the crossover frequency and phase margin at weakgird is expressed as follows with (10) and (19)- (22),K min 3 

% Z Lg1
1% Z L F1  1

tan 0.5cTsamp

(28),

2

F1 

tan 0.5c _ weak Tsamp
2

1



tan 2 0.5c _ weak Tsamptan 2 0.5cTsamptan 2  PM  cTsamp (29),

1  tan 2  PM  cTsamp K min 4 F2 

% Z Lg1
1% Z L F2  1

tan 2  PM  c Tsamp tan 2  PM weak  c _ weak Tsamp (30),

K min 5 

350 V
200 Vrms
1 kW
50 Hz
40 W
79.6 F
SCT3030AL
100 kHz31.25%
560 H1 F
235 H12.5 kHz
50 kHz0.5
200 ns
3.2 kHz45 deg13.8 W
111.7 s25.9 W
22 kHz

AVg 2

B 


% Z L Z b Pout  tan  cos 1 PF  tan 1  B
1 A 




1

2

1  tan 2  PM   c Tsamp  A    g Tsamp 2
tan 2 0.5c Tsamp g Tsamp  tan  PM  c Tsamp  


 1
B 
2
 tan 0.5c Tsamp




(32).
Step 6: Select k and fr and Design Cf, Lf, fhpf, Kt, Ti, and Kp
The recommended values of k and fr within the range
satisfying the requirements are desired to be small k and thenlarge fr in order to minimize the LCL filter. Based on k and fr
selected, each parameter is designed as follows; Cf is designedby (17), Lf is designed by (15), fhpf is designed by (11), Kt isdesigned by (8), Ti is designed by (21), and Kp is designed by(22).
IV. DESIGN EXAMPLE

1  tan 2  PM weak  c _ weak Tsamp (31).

1  tan 2  PM  c Tsamp Step 5: Derivation of Kmin5 satisfying power factor at stiff
grid
The power factor is derived by the continuous system model,
assuming the influence of the computation delay on the gridfrequency is small enough. The relationship between the LCL
resonance frequency and k considering the power factor at stiff
gird is expressed by

Table I shows the system parameters for the experiments.
The robustness is examined with Lg varying by more than %ZLg
= 10%, which corresponds to 12.7 mH.
According to the design flowchart in Fig. 10, first,
substituting the current ripple ratio  = 31.25% into (19) and(13), %ZL = 0.44 and L = 560 H are calculated.
Fig. 11 depicts the relationship between fr and k considering
reactive power restriction in (25), current harmonics suppression
regulation in (27), the crossover frequency restriction at weakgird in (28), the phase margin restriction at weak grid in (30),and the power factor restriction at stiff grid in (32). Finally, the
LCL filter parameters and controller parameters are determinedby considering the control performance, i.e. the power factor for
stiff grid, the crossover frequency and the phase margin for weakKmin5: (32)0.8
Lf/L = k

fr_max/fsamp

Range to meetrequirements

Phase [deg] Magnitude [dB]

1.0

Design point
fr/fsamp = 0.25
k = 0.44%YCf = 1.25%

0.6
0.4
0.2

0.00

0.05

Kmin2: (27)Kmin1: (25)

0.10

0.15

0.20

0.25

0.30

0.35

0.40

LCL resonance frequency ratio fr/fsamp

Fig. 11. Relationship between fr and k considering current harmonics
suppression regulation, resonance frequency, crossover frequency at
weak-grid, phase margin at weak-grid and power factor at stiff-grid.

grid. As a result, fc = 3.2 kHz, Kp = 13.8 W, Ti = 111.7 s are
decided from (25)-(32) when the condition is required asfollows; PM = 45 degrees, PMweak = 17 degrees, fc_weak = 500 Hz,
PF = 0.995.
Fig. 12 depicts the inner and outer loop characteristics of thecontroller. As shown in Fig. 12 (a), the stability margin of theinner loop is robust to the grid impedance variation. The phasemargin of the inner loop at %ZLg = 0% is PM = 30 degrees
according to the design strategy. The design values of thecrossover frequency and the phase margin are 3.2 kHz and 45
degrees, although note that these actual values have errors dueto the approximation of the circuit model as shown in Fig. 12 (b).

VI. CONCLUSION
This paper proposed the comprehensive design strategy for
the LCL filter and the controller parameters of the active damped
LCL filter system with the low inverter-side inductance in thegrid-tied inverter.

PM1=63 deg

0
-180
-270
1

PM2=30 deg
fsamp/2
10

-50
-90
-180
-270
-360
-450
0.1

PM=17 deg
(f = 500 Hz)

1: w/o active damping: with active damping

0.8

PM=43.4 deg(f = 3.55 kHz)%ZLg=0%0.6

every 0.1%

Imaginary Axis

0.4

V. EXPERIMENTAL RESULTS

Fig. 15 shows the experimental waveforms to verify thevalidity of the theoretical analysis, with or without active
damping. As shown in Fig. 15 (a), the system stably operates
without resonant current when the active damping is enabled,
whereas the system stably operates even without the active
damping as shown in Fig.15 (b). The reason for the stable
operation in Fig. 15 (b) is the nonlinearity of the discontinuous
current mode is induced by the dead-time.

-50
180

1
10
fsamp/2
Frequency [kHz](b) Outer loop due to grid current feedback for current control.
Fig. 12. Inner and outer loop characteristics.

Fig. 13 depicts the pole trajectory of the current control
system. As the grid impedance increases, the pole moves toward
the boundary of the circle, i.e. the response of the system
becomes poor. However, the system is stable even with grid
impedance variation. Thus, the proposed design methodenhances the system robustness to the grid impedance variation.
Fig. 14 shows the experimental waveforms to verify thevalidity of the theoretical analysis, with the different grid
impedances. As shown in Fig. 14 (a) and (b), the system stably
operates without the resonant current. In other words, the good
grid-tied operation is achieved regardless of the grid impedance.
The grid current THD is maintained below 1.2% over the grid
impedance range from 0% to 10%.

GM1=6.4 dB%ZLg=10% %ZLg=0%0

Frequency [kHz](a) Inner loop due to filter capacitor current feedback for active
damping.
50%ZLg=0%GM=4.1 dB(f = 8.81 kHz)0
%ZLg=10%Phase [deg] Magnitude [dB]0.0

Kmin3: (28)Kmin4: (30)50

%ZLg=10%

0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1

-0.5

0
0.5
1
Real Axis
Fig. 13. Root locus of grid current feedback system when grid
inductance is swept.

The influence of the active damping gain and the grid
impedance on the stability of the active damped LCL filter in the
single-phase grid-tied inverter was clarified. Then, the practicalimplementation solutions were discussed based on the proposed
design procedure. The detailed design strategy was exhibitedwith the design procedure and the design example. The
experimental results were provided to verify the effectiveness ofthe design strategy. The grid current THD is maintained below
1.2% over the grid impedance range from 0% to 10%.
REFERENCES[1]IEEE Application Guide for IEEE Std 1547(TM), IEEE Standard forInterconnecting Distributed Resources with Electric Power Systems,"
in IEEE Std 1547.2-2008 , vol., no., pp.1-217, 15 April 2009

Filter capacitor voltage vC (100 V/div)
Grid current ig (5 A/div)0Filter capacitor voltage vC (100 V/div)
Grid current ig (5 A/div)0

PF = 0.99 or more

THDig = 1.0%

PF = 0.99 or moreTime (10 ms/div)(a) Stiff grid at %Zg = 0% (Lg = 0 mH).
Fig. 14. Inverter operation waveforms with active damping.

Time (10 ms/div)(b) Weak grid at %Zg = 10% (Lg = 12.7 mH).

Filter capacitor voltage vC (100 V/div)
Grid current ig (5 A/div)0Filter capacitor voltage vC (100 V/div)
Grid current ig (5 A/div)0With active damping

W/o active damping

With active damping

Time (20 ms/div)(a) Stiff grid at %Zg = 0% (Lg = 0 mH).
Fig. 15. Inverter operation waveforms when active damping is disabled from enabled.

[2]

THDig = 1.2%

W. Wu, Y. He, T. Tang and F. Blaabjerg, "A New Design Method for the
Passive Damped LCL and LLCL Filter-Based Single-Phase Grid-Tied
Inverter," in IEEE Transactions on Industrial Electronics, vol. 60, no. 10,
pp. 4339-4350, Oct. 2013.
[3] C.T. Lee, A. Kikuchi, and T. Ito,”An Auto-Tuning Damper for the
Harmonic Resonance of Grid-Connected Converters,”IEEJ J. Industry
Applications, vol. 8, no. 6, pp. 884–892, 2019.
[4] Tuomas Messo, Roni Luhtala, Aapo Aapro, and Tomi Roinila”Accurate
Impedance Model of a Grid-Connected Inverter for Small-Signal Stability
Assessment in High-Impedance Grids,”IEEJ J. Industry Applications, vol.
8, no. 3, pp. 488-496, 2019.
[5] M. Semasa, T. Kato, K. Inoue, “Simple and Effective Time Delay
Compensation Method for Active Damping Control of Grid-Connected
Inverter with an LCL Filter”, in IEEJ Journal of Industry
Applications,.vol. 7, no. 6, pp. 454-461, 2018[6] P. Mattavelli, F. Polo, F. Dal Lago and S. Saggini, "Analysis of ControlDelay Reduction for the Improvement of UPS Voltage-Loop Bandwidth,"in IEEE Trans. on Ind. Electron., vol. 55, no. 8, pp. 2903-2911, Aug. 2008.
[7] D. Pan, X. Ruan, X. Wang, F. Blaabjerg, X. Wang and Q. Zhou, "A
Highly Robust Single-Loop Current Control Scheme for Grid-Connected
Inverter With an Improved LCCL Filter Configuration," in IEEE Trans.
on Power Electronics, vol. 33, no. 10, pp. 8474-8487, Oct. 2018.
[8] S. G. Parker, B. P. McGrath and D. G. Holmes, "Regions of Active
Damping Control for LCL Filters," in IEEE Transactions on Industry
Applications, vol. 50, no. 1, pp. 424-432, Jan.-Feb. 2014.
[9] J. Wang, J. D. Yan, L. Jiang and J. Zou, "Delay-Dependent Stability ofSingle-Loop Controlled Grid-Connected Inverters with LCL Filters,"
in IEEE Trans. on Power Electron., vol. 31, no. 1, pp. 743-757, Jan. 2016.
[10] C. Zou, B. Liu, S. Duan and R. Li, "Influence of Delay on System Stability
and Delay Optimization of Grid-Connected Inverters With LCL Filter,"in IEEE Trans. on Ind. Inform., vol. 10, no. 3, pp. 1775-1784, Aug. 2014.
[11] X. Li, J. Fang, Y. Tang, X. Wu and Y. Geng, "Capacitor-Voltage
Feedforward With Full Delay Compensation to Improve Weak Grids
Adaptability of LCL-Filtered Grid-Connected Converters for Distributed

W/o active damping

Time (20 ms/div)(b) Weak grid at %Zg = 10% (Lg = 12.7 mH).

Generation Systems," in IEEE Transactions on Power Electronics, vol.
33, no. 1, pp. 749-764, Jan. 2018.
[12] M. Lu, A. Al-Durra, S. M. Muyeen, S. Leng, P. C. Loh and F. Blaabjerg,"Benchmarking of Stability and Robustness Against Grid Impedance
Variation for LCL -Filtered Grid-Interfacing Inverters," in IEEE Trans.
on Power Electronics, vol. 33, no. 10, pp. 9033-9046, Oct. 2018.
[13] W. Yao, Y. Yang, Y. Xu, F. Blaabjerg, S. Liu and G. Wilson, "Phase
Reshaping via All-Pass Filters for Robust LCL-Filter Active Damping,"
in IEEE Transactions on Power Electronics, vol. 35, no. 3, pp. 3114-3126,
March 2020.
[14] D. Pan, X. Ruan, C. Bao, W. Li and X. Wang, "Capacitor-CurrentFeedback Active Damping With Reduced Computation Delay for
Improving Robustness of LCL-Type Grid-Connected Inverter," in IEEE
Trans. on Power Electronics, vol. 29, no. 7, pp. 3414-3427, July 2014.
[15] X. Wang, F. Blaabjerg and P. C. Loh, "Virtual RC Damping of LCLFiltered Voltage Source Converters With Extended Selective Harmonic
Compensation," in IEEE Transactions on Power Electronics, vol. 30, no.
9, pp. 4726-4737, Sept. 2015.
[16] Y. He, X. Wang, X. Ruan, D. Pan, X. Xu and F. Liu, "Capacitor-Current
Proportional-Integral Positive Feedback Active Damping for LCL-Type
Grid-Connected Inverter to Achieve High Robustness Against Grid
Impedance Variation," in IEEE Transactions on Power Electronics, vol.
34, no. 12, pp. 12423-12436, Dec. 2019.
[17] D. Pan, X. Ruan, C. Bao, W. Li and X. Wang, "Optimized Controller
Design for LCL-Type Grid-Connected Inverter to Achieve High
Robustness Against Grid-Impedance Variation," in IEEE Transactions on
Industrial Electronics, vol. 62, no. 3, pp. 1537-1547, March 2015.
[18] M. Liserre, R. Teodorescu and F. Blaabjerg, "Stability of photovoltaic andwind turbine grid-connected inverters for a large set of grid impedancevalues," in IEEE Transactions on Power Electronics, vol. 21, no. 1, pp.
263-272, Jan. 2006.
[19] S. Jayalath and M. Hanif, "Generalized LCL-Filter Design Algorithm for
Grid-Connected Voltage-Source Inverter," in IEEE Transactions on
Industrial Electronics, vol. 64, no. 3, pp. 1905-1915, March 2017.