


  
TRANSACTIONS ON ENVIRONMENT AND ELECTRICAL ENGINEERING ISSN 2450-5730 Vol 1, No 4 (2016) 

© Jiao Jiao and R. M. Nelms 

 

Abstract— Explored in this paper is the grid impedance effect 

on the stability of a single-phase grid connected inverter with an 

LC filter based on an analysis of the inverter output impedance. 

For a single-phase grid connected inverter, a PI controller is 

often used to regulate the current injected into the grid. However, 

the control performance can be influenced when the inverter is 

connected to a weak grid. Also, the utility grid has background 

harmonic noise, which can make the injected current distorted. 

Therefore, analysis of the output impedance of a single-phase 

grid connected inverter is important for the robustness and 

stability of the system. By modeling the output impedance of 

inverter, it can be determined that the proportional gain and 

integral gain of the controller have an effect on the output 

impedance. Analytical results show that by adjusting the PI 

controller parameters, the ability for harmonic reduction and 

stability of the system can be improved. Simulation and 

experiments using a 1 kW single-phase grid connected inverter 

verify the effectiveness of the theoretical analysis.  

 
Index Terms— Grid impedance, inverter output impedance, 

LC filter, PI controller, single-phase inverter 

 

I. INTRODUCTION 

ISTRIBUTED generation technologies such as solar 

panels and wind turbines are being investigated because 

they are environmentally friendly. The voltage source inverter, 

which is the connection interface between distributed 

generation and the utility grid, plays an important role and has 

received more and more attention. An LC filter or an LCL 

filter is commonly used to reduce the pulse width modulation 

(PWM) switching harmonics. Generally, the LCL filter has 

better performance in attenuating higher order harmonics and 

smaller component size and weight compared to an LC filter. 

However, the LCL filter is third order, which can introduce a 

resonant peak into the system that will cause an oscillation. 

For a small power inverter (a few kW), an LC filter is a better 

choice for the harmonic attenuation. In this paper, we employ 

an LC filter for the single-phase grid connected inverter. 

In the case of long distribution lines and lower power 

 
Submitted on: 27 Sep. 2016.  Accepted on: 30 Oct. 2016.  

Jiao Jiao is with the Electrical and Computer Engineering Department, 
Auburn University, Auburn, AL 36849 USA. (e-mail: jzj0034@auburn.edu). 

R. M. Nelms is with the Electrical and Computer Engineering Department, 

Auburn University, Auburn, AL 36849 USA. (e-mail: nelmsrm@auburn.edu). 

transformers in a distribution system, the grid can have a large 

impedance, which is a typical weak grid. However, the 

controller design for a grid-connected inverter usually doesn’t 

take the grid impedance into consideration. When a voltage 

source inverter is connected into a weak grid, the inverter 

control performance can be influenced [1]. The variation of 

the grid impedance may decrease the current controller 

bandwidth. In order to find the effect of grid impedance on the 

stability of the system, the external characteristics of the 

inverter need to be explored. Generally, it can be modeled by 

the output impedance of the inverter. The relationship between 

the output impedance of the inverter and the grid impedance 

has been studied by many researchers [2-4]. For stability 

analysis, the impedance-based stability criterion can be used to 

examine the ratio of the inverter output impedance to the grid 

impedance [2]. However, it doesn’t mention a way to improve 

the stability of the system. In order to reduce the effect of grid 

impedance, the output impedance can be regulated 

accordingly. The output impedance depends on the design of 

the LC filter and the control structure and parameters. To 

improve the stability of the system and the ability for 

harmonic rejection, the output impedance can be changed by 

adjusting the controller parameters. Therefore, the output 

impedance based method is explored under a weak and 

distorted grid.  

In this paper, a 1 kW single-phase grid connected inverter is 

used to investigate the output impedance of the system. By 

analyzing the output impedance in the frequency domain, the 

controller parameters can be adjusted to regulate the output 

impedance. By increasing the proportional gain of the PI 

controller, the magnitude of the output impedance can be 

increased to improve the ability for harmonic rejection. By 

increasing the integral gain of the PI controller, the phase of 

the output impedance can be increased to improve the stability 

of system. Experimental results are in agreement with 

simulation analysis. 

II. STABILITY ANALYSIS OF A SINGLE-PHASE INVERTER 
CONSIDERING GRID IMPEDANCE VARIATION 

A. Modeling of the single-phase inverter 

The structure of a single-phase grid connected inverter with 

an LC filter is presented in Fig. 1. Vdc is the input DC link 

voltage, vinv is the output voltage of the H-bridge inverter and 

Adjusting output impedance using a PI 

controller to improve the stability 

of a single-phase inverter under weak grid 

D 

Jiao Jiao and R. M. Nelms 



 

vg is the utility grid voltage. The LC filter consists of inverter 

side inductances L1 and L2 and a filter capacitance C. The 

inductances L1 and L2 are two equal inductances used for 

attenuating the common mode noise current in the circuit. In 

addition, the electromagnetic interference (EMI) effects are 

alleviated. Lg is the grid impedance. The inductor current iL, is 

used for current control to regulate the injected current with 

lower harmonics and unity power factor. The utility grid is 

sensed using a phase loop lock (PLL) to make the single-phase 

inverter synchronize with the grid voltage. 

 

PLL

PISPWM


dc
V

1
L

2
L

C





ref
I

ref
i

L
i

g
v

inv
v

g
L





L
i

g
i

1
r

2
r





c
v





sin

 
 

Fig. 1.  Single-phase grid connected inverter with LC filter 

 

According to Kirchhoff’s voltage law, we can get the circuit 

equation shown as follows: 

 

                         
cinvL

L vvirr
dt

di
LL 

2121
 (1) 

 

Where r1 and r2 are the parasitic resistances of the 

inductors. By transforming to the frequency domain, the 

transfer function between inductor current and the difference 

between the inverter output voltage and the capacitor voltage 

can be derived. 

 

                           
 

    RLssVsV
sI

sG
cinv

L
p







1  (2) 

 

Where 1 2L L L  , 1 2R r r  . 

B. Output impedance of a single-phase inverter  

Many power converters are digitally controlled which will 

introduce sampling and computational delays into the system. 

Also, the PWM process will introduce delay due to a zero-

order hold [5]. Therefore, the total delay can be approximated 

by one and a half sampling period, which can be expressed by 

Gd (s) shown below. 

 

                             
Ts

esG
Ts

d
5.11

15.1




  (3) 

 

The variable T is the sampling period. A PI controller is 

applied for current control in this inverter. The transfer 

function of a PI controller is shown as follows: 

 

                                       









s

ksG
i

pc


1
1  (4) 

 

Here kp is the proportional gain and τi is the integral gain. The 

system control block diagram is shown in Fig. 2 [6]. Fig. 2(a) 

can be simplified to Fig. 2(b) by control block equivalent 

transformations. The resulting transfer function of G1(s) and 

G2(s) can be expressed by: 

 

                       
   

   sGsGKRLs

sGsGK
sG

dcpwm

dcpwm




1
 (5) 

              
   

    122 




CssGsGKRCsLCs

sGsGKRsL
sG

dcpwm

dcpwm  (6) 

 

The output impedance of the inverter can be defined by: 

 

                               
 
 

 sG
si

sv
sZ

refi
g

c
o 2

0








 (7) 

 

 

 1
Ls R

pwm
K cG s

L
i 1

sC



1

g
sL

g
v

g
i


c
v  invv

 dG s
ref
i

 
(a) 



 Li 1
sC



1

g
sL

g
v

g
i


c
v 



1

c d pwm
G G K

 1G s
ref
i

     



L
i 

1

g
sL

g
v

g
i


c
v 

 1G s  2G s
ref
i

 
 

(b) 
Fig. 2.  Control block diagram for a single-phase grid connected inverter (a) and its equivalent transformation (b) 



 

Therefore, the single-phase grid connected inverter with an 

LC filter can be modeled by the Norton equivalent circuit, as 

shown in Fig. 3. The inverter is represented by a current 

source and a parallel output impedance, and the utility grid is 

modeled by a voltage source and a grid impedance [2]. In this 

effort, only the grid inductance is considered in the analysis as 

a worst case scenario. In reality, the resistance in the grid will 

help to stabilize the system. 

 









 gZ s gI s

 gV s oZ s sI s  oV s

 
 

Fig. 3.  Norton equivalent circuit 
 

An expression for the inverter output current can be derived 

by: 

 

                  sV
ZZ

Z
sI

ZZ
sI

g

og

o
s

og

g






1

1

1

1  (8) 

 

From (8), it can be found that in order to mitigate the effect 

caused by grid voltage and grid impedance variation, the 

output impedance should be designed as high as possible to 

operate stably [2]. It can be found from (7) that the output 

impedance depends on the design of the LC filter and the 

controller structure and parameters. Therefore, the inverter 

output impedance can be shaped by adjusting the controller 

parameters to improve the system stability. 

C. Relationship between inverter output impedance and grid 
impedance  

A current controller is designed by assuming that the 

inverter is connected into an ideal grid (Lg = 0). The bode plot 

of the uncompensated system (without a current controller) 

and the bode plot of the compensated system (with current 

controller) are shown in Fig. 4. When the inverter is connected 

into a utility grid, the grid impedance can influence the 

inverter control performance. Fig. 5 shows the relationship 

between inverter output impedance and grid impedances in the 

frequency domain. According to [3-4], the stability of the 

inverter depends on the inverter output impedance’s phase at 

the intersection point of Zo and Zg. With an increase in the grid 

impedance, the phase of the output impedance at the 

intersection frequency point is decreasing, which implies the 

system is less stable. 

In order to enhance the stability of the system under a wide 

range of grid impedance, the phase of the inverter output 

impedance at the intersection point should be increased by 

shaping the current controller parameters. The magnitude of 

inverter output impedance also needs to be designed higher to 

achieve better harmonic rejection ability. 

 
 

Fig. 4.  Bode plot of the compensated and uncompensated system 

 

 
 

Fig. 5.  Bode plot of the inverter output impedance and different grid 

impedances 
 

D. Shaping PI controller parameters to improve the stability 
of system under weak grid 

By analyzing (7), it can be determined that the output 

impedance of the inverter can be changed by adjusting the 

parameters of the PI controller. As seen in Fig. 6, the lower 

frequency part of the bode diagram shows that the magnitude 

of the output impedance increases with increasing proportional 

gain kp, which improves the ability for harmonic reduction. 

However, the phase of the inverter output impedance is 

decreasing, which means the system might be becoming less 

stable. 

As for increasing the integral gain of the PI controller, the 

phase of the inverter output impedance at the intersection 

frequency is increased, but the magnitude of the output 

impedance is decreased in the low frequency range. In the 

high frequency range, the magnitude and phase of inverter 

output impedance does not change much. It can be concluded 

that increasing the integral gain has the opposite effect on the 

output impedance as compared to increasing the proportional 



 

gain of the controller. Therefore, it is possible to increase the 

magnitude and phase of the output impedance by adjusting the 

PI controller parameters without affecting the performance of 

the inverter. 

 

 
 

Fig. 6.  Bode plot of inverter output impedance and grid impedance (a) 
proportional gain changes (top) (b) integral gain changes (bottom) 

 

E. Output impedance sensitivity to circuit parameters 

From (7), it can be found that the inverter output impedance 

depends on the circuit parameters and controller parameters. 

In reality, the circuit parameters can change due to the 

temperature variation of the surrounding environment. 

Therefore, it is necessary to explore the effect of circuit 

parameter variations on the inverter output impedance. Since 

the parasitic resistance of the filter inductors has little effect 

on the inverter output impedance, the resistance variation is 

not investigated here. 

 
 

Fig. 7.  The inverter output impedance sensitivity to inductance variation 

 

 
 

Fig. 8.  The inverter output impedance sensitivity to capacitance variation 

 

From Fig. 7, it can be seen that the frequency response of 

the inverter output impedance shifts to the left slightly as the 

inductance increases. It mainly affects the magnitude and 

phase of inverter output impedance around the peak. As 

shown in Fig. 8, with the increase of filter capacitance, the 

magnitude of the inverter output impedance remains 

unchanged in low frequency range, but it decreases in the high 

frequency range. The phase margin of the inverter output 

impedance is also decreased. 

III. SIMULATION RESULTS 

The system shown in Fig. 1 is simulated by MATLAB to 

validate the theoretical analysis. The system parameters are 

given in Table I. For the simulation, the utility grid voltage (vg 

in Fig. 1) is modeled by 7% third harmonics and 5% fifth 

harmonics and 3% seventh harmonics with phase 30°, 90° and 

270°, respectively. 

 



 

 
Fig. 9 to Fig. 10 show the simulation results when the 

proportional gain was changed from kp = 2 to kp = 3; the grid 

impedance was 19.5 mH. The grid current total harmonic 

distortion (THD) is 4.30% and 3.76%, respectively. The 

harmonic reduction ability is improved, which validates the 

analysis in Section II. Fig. 11 to Fig. 12 show the simulation 

results when the integral gain was changed from τi = 0.0005 to 

τi = 0.01; the grid impedance was 19.5 mH. The grid current 

total harmonic distortion (THD) is 4.51% and 4.83%, 

respectively. Table II and Table III show the harmonic 

analysis of the inverter output current. 

 

 
 

Fig. 9.  Output voltage and output current when Lg = 19.5 mH (kp = 2) 
 

 
 

Fig. 10.  Output voltage and output current when Lg = 19.5 mH (kp = 3) 

 

 

 
 

Fig. 11.  Output voltage and output current when Lg = 19.5 mH (τi = 0.0005) 
 

 
 

Fig. 12.  Output voltage and output current when Lg = 19.5 mH (τi = 0.01) 

 

 

IV. EXPERIMENTAL RESULTS 

A 1 kW Texas Instruments single-phase grid connected 

inverter with an LC filter was utilized for the experimental 

TABLE I 

CIRCUIT PARAMETERS 

Circuit Parameter Symbol Value 

DC-link Voltage Vdc 380 V 

Utility Grid Voltage vg 120 V 

Fundamental Frequency f0 60 Hz 

Inverter Inductance L 7 mH 

Inductance parasite resistance R 0.4 Ω 

Filter Capacitance C 1 μF 

Switching Frequency fs 19.2 kHz 

 

 

TABLE II 

OUTPUT CURRENT HARMONIC ANALYSIS 

Harmonic 

Order 

Lg = 6.5 mH Lg = 19.5 mH 

kp = 2 kp = 3 kp = 2 kp = 3 

3rd 0.71% 0.73% 1.12% 0.88% 

5th 0.34% 0.42% 0.53% 0.17% 

7th 0.45% 0.33% 0.53% 0.32% 

9th 0.16% 0.17% 0.24% 0.25% 

11th 0.13% 0.15% 0.19% 0.18% 

13th 0.16% 0.10% 0.12% 0.18% 

THD 3.76% 3.51% 4.30% 3.76% 

 

 

TABLE III 
OUTPUT CURRENT HARMONIC ANALYSIS 

Harmonic 
Order 

Lg = 6.5 mH Lg = 19.5 mH 

τi = 0.0005 τi = 0.01 τi = 0.0005 τi = 0.01 

3rd 0.52% 0.74% 0.74% 0.98% 

5th 0.34% 0.29% 0.14% 0.49% 

7th 0.37% 0.46% 0.28% 0.45% 

9th 0.14% 0.25% 0.28% 0.40% 

11th 0.05% 0.01% 0.20% 0.12% 

13th 0.15% 0.10% 0.11% 0.04% 

THD 3.51% 3.77% 4.51% 4.83% 

 

 



 

investigation. An ideal and a distorted grid voltage were 

simulated using a programmable AC source. In order to 

examine the effect of the grid impedance, an adjustable 

impedance is inserted between the inverter and the AC source. 

The measured single-phase inverter output voltage and output 

current are presented in Fig.13- Fig. 22. The inverter output 

current harmonics measured by Tektronix oscilloscope are 

given in Table IV and Table VII. Under the ideal grid, the 

THD of inverter output current was reduced from 4.94% to 

4.65% by increasing the proportional gain of current controller 

when the grid impedance is 19.5 mH. The output voltage and 

output current without PI controller is also measured for 

comparison, which demonstrate the current distortion can be 

improved by shaping PI controller parameters. The THD of 

the inverter output current was decreased from 5.09% to 

4.73% by decreasing the integral gain of the current controller 

when Lg is 19.5 mH. Under the distorted grid, the THD of 

inverter output current was reduced from 5.21% to 4.69% by 

increasing the proportional gain of current controller when the 

grid impedance is 19.5 mH. The THD of the inverter output 

current was decreased from 5.28% to 4.74% by decreasing the 

integral gain of the current controller. As can be seen from 

these results, the THD of the inverter output current can be 

reduced by adjusting the PI controller gains. Because the 

single-phase inverter is connected to the AC source through a 

transformer and relay, which makes the grid side inductance 

larger than the grid impedance in the simulation, the THD of 

experimental results are higher than the THD of simulation 

results. But it’s still in agreement with the theoretical analysis.  

 

 
 

Fig. 13.  Output voltage and output current when Lg = 19.5 mH (without PI) 

 

 
 

Fig. 14.  Output voltage and output current when Lg = 19.5 mH (kp = 2) 
 

 
 

Fig. 15.  Output voltage and output current when Lg = 19.5 mH (kp = 3) 

 

 
 

 
 

Fig. 16.  Output voltage and output current when Lg = 19.5 mH (τi = 0.0005) 
 

 

 
 

Fig. 17.  Output voltage and output current when Lg = 19.5 mH (τi = 0.01) 

 

TABLE IV 
OUTPUT CURRENT HARMONIC ANALYSIS 

Harmonic 

Order 

Lg = 6.5 mH Lg = 19.5 mH 

No PI kp = 2 kp = 3 No PI kp = 2 kp = 3 

3rd 6.87% 2.34% 1.54% 7.29% 2.70% 1.55% 

5th 5.53% 1.52% 0.91% 5.71% 1.63% 0.72% 

7th 2.83% 1.77% 1.37% 2.64% 1.91% 1.67% 

9th 1.70% 1.97% 1.61% 1.61% 2.20% 1.61% 

11th 1.09% 1.00% 0.73% 1.09% 1.18% 0.86% 

13th 1.03% 0.69% 0.77% 0.89% 0.91% 0.68% 

THD 6.83% 4.85% 4.64% 6.61% 4.94% 4.65% 

 

Current (2A/div) 

Voltage (50V/div) 

Voltage (50V/div) 

Voltage (50V/div) 

Voltage (50V/div) 

Current (2A/div) 

Voltage (50V/div) 

Current (2A/div) 

Current (2A/div) 

Current (2A/div) 



 

 
 

 
 

Fig. 18.  Output voltage and output current when Lg = 19.5 mH (without PI) 

 

 

 
 

Fig. 19.  Output voltage and output current when Lg = 19.5 mH (kp = 2) 

 

 
 

Fig. 20.  Output voltage and output current when Lg = 19.5 mH (kp = 3) 

 

 
 

 
 

Fig. 21.  Output voltage and output current when Lg = 19.5 mH (τi = 0.0005) 

 

 

 
 

Fig. 22.  Output voltage and output current when Lg = 19.5 mH (τi = 0.01) 

 

 

 

TABLE V 

OUTPUT CURRENT HARMONIC ANALYSIS 

Harmonic 

Order 

Lg = 6.5 mH Lg = 19.5 mH 

τi = 0.0005 τi = 0.01 τi = 0.0005 τi = 0.01 

3rd 1.82% 2.29% 2.33% 2.05% 

5th 0.98% 1.80% 0.92% 2.01% 

7th 1.57% 2.02% 1.54% 1.77% 

9th 2.13% 1.96% 1.58% 1.93% 

11th 1.31% 0.72% 0.77% 0.63% 

13th 0.94% 0.54% 0.54% 0.40% 

THD 4.78% 5.28% 4.73% 5.09% 

 

 

TABLE VI 

OUTPUT CURRENT HARMONIC ANALYSIS 

Harmonic 

Order 

Lg = 6.5 mH Lg = 19.5 mH 

No PI kp = 2 kp = 3 No PI kp = 2 kp = 3 

3rd 7.62% 3.80% 2.71% 8.10% 3.47% 2.39% 

5th 3.56% 1.68% 1.52% 4.43% 1.92% 0.76% 

7th 4.51% 1.90% 1.28% 3.90% 2.02% 1.24% 

9th 1.64% 1.57% 1.14% 1.61% 1.95% 1.45% 

11th 1.13% 0.85% 0.55% 1.09% 1.22% 0.94% 

13th 1.05% 0.92% 0.52% 1.08% 0.74% 0.74% 

THD 6.91% 5.09% 4.77% 6.86% 5.21% 4.69% 

 

 

TABLE VII 

OUTPUT CURRENT HARMONIC ANALYSIS 

Harmonic 

Order 

Lg = 6.5 mH Lg = 19.5 mH 

τi = 0.0005 τi = 0.01 τi = 0.0005 τi = 0.01 

3rd 2.78% 3.65% 2.75% 2.89% 

5th 1.59% 1.51% 1.62% 1.06% 

7th 1.36% 2.64% 1.59% 2.32% 

9th 1.59% 1.67% 1.89% 1.71% 

11th 1.06% 0.86% 0.85% 0.60% 

13th 0.95% 0.89% 0.78% 0.58% 

THD 4.85% 5.51% 4.74% 5.28% 

 

 

Voltage (50V/div) 

Voltage (50V/div) 

Voltage (50V/div) 

Voltage (50V/div) 

Current (2A/div) 

Current (2A/div) 

Current (2A/div) 

Current (2A/div) 

Voltage (50V/div) 

Current (2A/div) 



 

V. CONCLUSION 

Introduced in this paper is a method to increase the output 

impedance of a single-phase grid connected inverter with an 

LC filter to improve the stability and harmonic reduction 

ability of the system when the inverter is connected to a weak 

distorted grid. By modeling the output impedance of the 

inverter, the relationship between output impedance and grid 

impedance can be investigated. The grid impedance can 

degrade the control performance of the inverter and make the 

system less stable. In order to mitigate this effect, PI controller 

parameters are adjusted to increase the output impedance, 

which can improve the ability for the harmonic reduction and 

the stability of the system. Finally, simulation and experiment 

results for a 1 kW single-phase grid connected inverter with an 

LC filter verify the effectiveness of the proposed method. 

 

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 Jiao Jiao received her B.S. and M.S. 

degree in Electrical Engineering from 

China Agricultural University, Beijing, 

China, in 2011 and 2013, respectively. 

She is currently pursuing her Ph.D. degree 

in Electrical Engineering in the Electrical 

and Computer Engineering Department at 

Auburn University. Her current research 

interests include grid-connected inverter 

modeling, simulation and control. 

 

 

 

 

R. M. Nelms (F’04) received the B.E.E. 

and M.S. degrees in electrical engineering 

from Auburn University, Auburn, AL, 

USA, in 1980 and 1982, respectively, and 

the Ph.D. degree in electrical engineering  

from Virginia Polytechnic Institute and 

State University, Blacksburg, VA, USA, 

in 1987. He is currently Professor and 

Chair of the Department of Electrical and Computer 

Engineering, Auburn University. His research interests are in 

power electronics, power systems, and electric machinery. Dr. 

Nelms is a registered Professional Engineer in Alabama. In 

2004, he was named an IEEE Fellow “for technical leadership 

and contributions to applied power electronics.”