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ANewDesignProcedureforOutputLCFilterofSinglePhaseInvertersCONFERENCEPAPERJANUARY2010

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AhmadAleAhmdBabolNoshirvaniUniversityofTechnology14PUBLICATIONS59CITATIONS

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2010 3rd International Conference on Power Electronics and Intelligent Transportation System

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Abstract- This paper presents a new design procedure

for output LC filter of single phase inverter. Two main goals of the procedure are to meet the IEEE Std. 1547 requirements for attenuating of harmonics distortion and to limit the high frequency current of switches in acceptable value. The design steps and their considerations are discussed comprehensively. This procedure is verified with simulation results for a 220V, 5KVA inverter. The simulations run either linear or nonlinear full loads.

Keyword- Inverter, LC filter, cut off frequency, THD, inductor current ripple,

I. INTRODUCTION

Today, inverter with an output LC filter has an especial application such as distributed generation, active filter, stand-alone application based on renewable energy, uninterruptible power supply (UPS) and dynamic voltage restorers [1-3]. Two main duties of the output LC filter are to attenuate the output voltage ripple and to limit the high frequency ripple current of inverter switches. The attenuation of switching frequency voltage at the output node is depended on the cut off frequency of filter. Also, bandwidth of inverter controller is limited by the cut off frequency of the filter [3]. The cut off frequency of the filter have to be selected small for perfect voltage ripple attenuation, and the bandwidth of the controller have to be wide for fast response to step or nonlinear load. So, there is a trade off between the bandwidth of the controller and filter attenuation. After selecting the cut off frequency of filter, determining the L and C values is very important issue, because they affect on ripple current of inverter switches, the inverter output impedance [4], efficiency [4-5], transient response [3] and also the cost of the inverter. In [4-6], the cut off frequency of LC filter is designed based on the Fourier series of the inverter output voltage. Then by using the relation between the filter capacitor and the system time constant, the capacitor and inductor value are designed [4]. In [5], the L and C are selected to minimize the filter reactive power. Authors of [6] defined a cost function based on reactive power where the reactive power of inductor is weighted two times higher than the reactive power of capacitor, then calculating the L and C value to minimize this cost function. Other ones used the same method too, but the ripple current of inductor,

switching frequency and its relation to the cut off frequency of filter is not well considered. Also, none of them were presented a straight forward method or relation to calculate the L and C values. This paper analyses the characteristics of the output LC filter for PWM inverter. The cut off frequency of filter and its relation to the modulation factor and switching frequency are determined to meet the IEEE Std. 1547 requirements for attenuating the harmonics distortion. Considering the switches current ripple, the inductance and capacitance value are calculated. The specifications and design criterias are illustrated in this paper. This procedure is verified with simulation results.

II. LC FILTER ANALYSIS

Design of an LC filter for the PWM based inverter is very important issue. This filter should reduce the high frequency distortion of output voltage and control the switching current. The switching devices generate this distortion. The IEEE Std. 1547 requirement for maximum harmonic voltage distortion is shown in table 1. For the medium power inverters whose PWM frequency is higher than 3 KHz, the low frequency harmonics (2nd, 3rd, 5th, and 7th) are usually rejected by controller perfectly. So, the high frequency distortion is only included switching or PWM frequency. According to the standard this distortion should be limited under 0.3%. TABLE 1. IEEE STD. 1547 REQUIREMENTS FOR MAXIMUM HARMONIC

VOLTAGE DISTORTION Individual Harmonic

order h


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Fig 1. a) Output LC filer connected to a load, b) input voltage and c) output voltage of the filter.

A. Minimum Load condition

In minimum load condition (RL=), (1) can be summarized as:

2r

NL 2 2r

H ()=

- (2)

2r

1 =

LC (3)

where r=2fr and fr is the cut off frequency of the LC filter. The amplitude of input voltage at the switching frequency depends on DC input voltage and duty ratio of PWM signal. It will be maximum when the duty ratio is 50%. So:

DCi s max

V4V ( ) =

2 (4)

where s=2fs and fs is the switching frequency. On the other hand, the amplitude of input voltage at fundamental frequency depends on DC voltage and modulation factor. It can obtain as:

i 1 DCV ( ) =mV (5)

where 1=2f1 and f1 and m are fundamental frequency and modulation factor, respectively. Because, at the fundamental frequency, the voltage drop across the filter can not be estimated before the L and C values are specified, we neglect this voltage drop by the first assumption. It means that the attenuation of the LC filter at the fundamental frequency is approximately 0dB, so according to the standard limitation for output voltage distortion, we can write:

o s i s NL smax maxNL s

o 1 i 1 NL 1

V ( ) V ( ) H ( ) 2 0.3= H ( )

V ( ) V ( ) H ( ) m 100

(6) So to meet the IEEE Std. 1547 requirements, the attenuation of the LC filter at the switching frequency should satisfy following inequality:

NL s

3mH ( )

2000 (7)

If the inequality (7) is solved for m=0.95, the cut off frequency of the LC filter (fr) must be less than fs/15 to satisfy the standard limitation. So:

r s

1 and k 15

k (8)

Really, the attenuation due to the filter at the fundamental frequency obtains as follow:

2s

NL 1 2 2s 1

H ( ) =

-(k ) (9)

Fig 2 shows the |HNL(1)| as a function of fs and k where the k factor and switching frequency have been changed from 15 to 20 and 3KHz to 10KHz, respectively. These curves reveal that our first assumption about the attenuation of the filter at the fundamental frequency is not far from our expectation.

Fig 2. Filter attenuation at the fundamental frequency in minimum load situation.

B. Maximum Load condition

At the maximum load condition (RL= RLm) the filter attenuation at the switching frequency is more than minimum load condition. But the filter attenuation at the fundamental frequency should be studied. The transfer function of the LC filter at the maximum load condition is:

2r

FL2 2r

Lm

H ()=

- +j

R C

(10)

Then: 2r

FL

2 2 2 2r

Lm

H () =

( - ) +( )

R C

(11)

(11) shows that the attenuation amplitude depends on the cut off frequency, load value and also capacitor or inductor value. There are several methods to determine the inductor and capacitor value. In the first approach, we calculated the L and C value to minimize the filter reactive power. This is suitable to increase the inverter efficiency. The reactive power of filter at the

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fundamental frequency, Q, can be obtained as a function of inductor value, cut off frequency of filter, output voltage and the load, as follow:

321 1 1O2 2 4

L r r

L Q= - + V

R L L

(12)

To minimize the filter reactive power at the maximum load, the inductor value can be determined as follow: Q=0 (13) So:

2 2Lmr 12

r

RL=

(14)

If the r is at least 3 times larger than 1 then the inductor value can be approximated by:

Lm

r

RL

(15)

Also, the reactive power at the minimum load is calculated as:

1L Lm

r

Q(R = ) P

(16)

where the PLm is the output power at the maximum load. The equation (16) shows that to decrease the filter reactive power at the minimum load, the cut off frequency of filter should be selected as large as possible (greater than the fundamental frequency of the inverter). Although, this approach improves the inverter efficiency, it provides a very large inductor value. So, it increases the cost and the size of filter. The large inductor value increases the output impedance of inverter too. It also causes a large over or under-shoot voltage in the step load condition. All of these evidences indicate that other criteria should be selected to calculate the inductor value. The main duty of inductor is the control of the switching frequency of inverter ripple current. So, the maximum acceptable ripple current and the switching frequency of the inverter can determine the minimum inductor value. The ripple can be estimated as:

LL

VI = t

L (17)

According to Fig 1, when the inverter switches are on:

L DC o DC om 1V =V -V =V -V sin( t) (18)

Where Vom is the amplitude of output voltage. When the inverter switches are on, the t is obtained as follow:

1

s s

m sin( t)Dt= =

f f

(19)

Where m is modulation index and 0


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42Lm 1

41 r

R L= -

(25)

21

2 4 4Lm r 1

1

R -C (26)

To obtain real answer for the equations (25) and (26), the following condition must be satisfied:

1r

>

(27)

or

s 1

kf > f

(28)

Now at the full load condition, the magnitude of transfer function from Vi to Vo can be written as:

2r

FL

2 2 2 2 2 4 4r r 1

1

H () =

( - ) +( ) ( - )

(29)

And at fundamental frequency:

rFL 1

2 2 2r 1

H ( ) =

(1+ ) -2 (30)

Fig 6 and 7 show the |HFL(1)| as function of r and for fs equal to 3KHz and 20KHz, respectively.

Fig 5. The filter attenuation at the fundamental frequency in

maximum load for fs=3KHz.

The figures 5 and 6 imply that if is varied from 0.02 to 0.2, the maximum attenuation of filter will only be 3%. So, this aims to optimize the size of inductor and capacitor . Another important result of these figures is that, if the cut off frequency of the filter is decreased below the 200Hz, the filter can amplify the fundamental frequency. This is sometimes useful, but the filter size will be increased. Also, the stability of the system will be very critical, because the phase margin of system decreases when the cut off frequency is decreased.

Fig 6. The filter attenuation at the fundamental frequency in maximum load for fs=20KHz.

III. PROPOSED LC FILTER DESIGN PROCEDURE

The LC filter of an inverter can be designed in the following mentioned 4 steps:

1) Selecting the switching frequency. High switching frequency aims to reduce the filter size, but the maximum frequency of solid state switches and their dynamic losses limit the switching frequency. It is usually chosen between 3KHz to 15KHz for IGBT based inverter and 10KHz to 100KHz for MOSFET based inverter.

2) Selecting k factor. The standard requirement will be meet, if k=15, but lager k factor causes more attenuation at switching frequency and little amplification at the fundamental frequency. If the modulation factor is less than 0.95, the minimum of k should be calculated by equations (7) and (8).

3) Selecting factor. completely depends on the switching frequency and acceptable inductor ripple current. 20% to 40% is an acceptable range for ripple current. So using the equation (20) or Fig 4, this factor can be selected. The inequalities (27) and (28) should be satisfied. If it is not maintained, then the k and factor should be renewed and selected again. 4) Now using equations (8), (25) and (26), the necessary L and C can be calculated.

IV. A DESIGN EXAMPLE

To verify the algorithm, an LC filter is designed for an inverter whose main characteristics are mentioned in table 2.

TABLE 2. INVERTER CHARACTERISTICS.

VDC 360V VO 220VRMS Sout 5KVA fs 20KHz f1 50Hz

According to DC input and AC output voltage of the inverter, the 0.95 is appropriate for modulation factor, so k=15. To limit the inductor ripple current below the


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40%, factor should be more than 0.025 at fs=20KHz. This aims us the filter specification which is listed in table3.

TABLE 3. THE LC FILTER CHARACTERISTICS.

L 770H C 18F fr 1330Hz

The control loop is illustrated in Fig 7. To reject the main low frequency harmonics (2nd, 3rd, 5th, 7th) of output voltage the bandwidth of control loop should be at least 350Hz. The capacitor current loop is employed to guarantee the stability of the inverter. Also, the inner loop make the inverter to present a fast response to nonlinear loads. Using the sliding mode control theory, the gains of voltage and current loop are easily obtained.

Fig 7. The controller of single phase inverter.

The inverter with designed output filter has been simulated with Simulink Toolbox in MATLAB. Fig 8 a and b show the output voltage and its spectrum at maximum linear load, respectively. The THD of the output voltage is below 0.5% and also the switching frequency distortion is below 0.1%. So, it meets the standard requirements for harmonics distortion.

a)

b)

Fig 8. a) The output voltage and current and b) The spectrum of output voltage in maximum linear load.

The fig 9 (a) and (b) show the output voltage and its spectrum at maximum nonlinear load. The current crest factor is 3. The THD is below the standard limitation. a) b)

Fig 9. a) The output voltage and current and b) The spectrum of output voltage in maximum nonlinear load with CF=3.

Fig 10 a) shows the inductor current. Also, fig 10 b) shows the ripple of inductor current in one-fourth of a period. As we have designed, the ripple is below 40% and the maximum ripple is occured about 45 degree. a) b)

Fig 10. a) The inductor current and its ripple.

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V. CONCLUSION

A complete algorithm to design output LC filter of a single phase inverter is developed in this paper. To meet the IEEE Std. 1547 requirements for attenuating of harmonics distortion, a relation between cut off frequency of the filter and switching frequency is calculated. The inductor value is designed to limit the high frequency ripple of switches current. This algorithm is verified with simulation results for a 220V, 5KVA inverter. The THD of output voltage is less than 0.4% and 1.1% at linear and nonlinear full load, respectively. In both simulations, the HD of switching frequency is lower than 0.15%.

VI. REFERENCES

[1] 1. Patricio Corts, M., IEEE, Gabriel Ortiz, Juan I. Yuz, Member, IEEE, Jos Rodrguez, Senior Member, IEEE, and M. Sergio Vazquez, IEEE, and Leopoldo G. Franquelo, Fellow, IEEE, Model Predictive Control of an Inverter With Output LC Filter for UPS Applications. IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, 2009. 56(6): p. 14. [2] 2. JOSEP M. GUERRERO, L.G.D.V., and JAVIER UCEDA, Uninterruptible power supply systems provide protection. IEEE Industrial Electronics Magazine, 2007. 1(1). [3] 3. Hyosung Kim , S.K.S., Analysis on Output LC Filter for PWM Inveter. IPEMC2009, 2009: p. 6. [4] 4. J. Kim, S.M., IEEE, J. Choi, Member, IEEE, H. Hong, Student Member, IEEE, Output LC Filter Design of Voltage Source Inverter Considering the Performance of Controller. 2000: p. 6. [5] 5. Pekik A. Dahono, A.P., Qamaruzzaman, An LC Filter Degign Method for single Phase PWM Inverter. 1995: p. 6. [6] 6. S. B. Dewan, P.D.Z., Optimum Filter Design for a Single Phase Solid State UPS System. IEEE Transaction on Industrial Application, 1975. IA-21(3): p. 6.

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