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A Comparison between Random Hysteresis Current Control. Technique with Bipolar and Unipolar Modulations. Alireza Nami. Department of Power.
A Comparison between Random Hysteresis Current Control Technique with Bipolar and Unipolar Modulations Alireza Nami

Firuz Zare

Department of Power Mazandaran University Babol, Mazandaran, Iran [email protected]

School of Engineering Systems Queensland University of Technology Brisbane, GPO Box 2434,QLD 4001, Australia [email protected]

Abstract

phase inverter based on bipolar and unipolar modulations. This paper presents that the RBHCC based on unipolar modulation has less effect on distributing the spectrum contents of the load current compare to the bipolar modulation.

Current control methods play an important role in power electronic circuits. Among them, a hysteresis current control method has been used largely on AC motor drives due to a good dynamic response and simple operation. Random Pulse Width Modulation (RPWM) technique distributes the spectrum contents of load current without affecting the fundamental component and it may reduce the acoustic noise and mechanical vibration of an inverter-fed induction motor drive when the amplitude of harmonics around side bands is decreased. This paper presents a Random Band Hysteresis Current Control (RBHCC) technique with bipolar and unipolar modulations and compares spectrum contents of a load current. Mathematical analysis and simulation results show that the RBHCC with unipolar modulation has no considerable effect on distributing the spectrum contents of the load current compared to the bipolar modulation technique.

2-Random hysteresis current control A hysteresis current control generates proper switching signals for a power converter by comparing a load current ( i L ) and a reference current ( iref ). Fig.1 shows a block diagram of a single phase inverter with an inductive load and a hysteresis current controller. In bipolar modulation, when the error signal reaches to the higher (lower) band, switch pairs (Ta+ , Tb− ) are turned on (off) and (Ta− , Tb+ ) off (on) and as a result, two voltages level, + Vdc and − Vdc are generated by the controller at the output of the inverter.

1. Introduction The performance of the inverter system largely depends on the control system. Different current control techniques for a traditional inverter have been analysed by several authors [1-3]. The current controlled PWM inverters have some advantages compared to the conventional open-loop voltage source PWM inverters [2-6] such as: control of instantaneous current waveform with high accuracy, peak current protection, overload rejection, compensation of effects due to load parameter changes and semiconductor voltage drops of the inverters. By comparing a reference current and a load current, a current controller can generate switching signals for power electronic devices, which control the current error and provide the desired current waveform for the load. In AC motor drive systems the mechanical vibration and acoustic noise are important issues which may be affected by significant harmonic contents of the load current around the first switching side band. Recently different methods of RPWM which include Randomized Switching Frequency[2-6], Randomized Pulse Position [7,8] and Randomized Switching [9-11] are presented; in all of these methods harmonic spectrum of output voltage or current are distributed based on random variation of switching signal. In this paper, a RBHCC method has been analyzed for a single

Fig.1: A block diagram of a single phase inverter with hysteresis current control Fig.2 shows a hysteresis random band current control for a single phase inverter with bipolar modulation and the error signal is applied to a random band hysteresis controller. The output pulses and its complementary are produced by the controller to turns on and off the switches (Ta+ , Tb− ) and (Ta− , Tb+ ) . In unipolar modulation, as shown in Fig.3, two hysteresis band controllers are used to generate proper switching signals to turn on and off the +



+



switches (Ta , Ta ) and (Tb , Tb ) in order to control the load current. In this method, the switches are turned

on and off in such a way resulting the output voltage to be either Vdc or zero ( −Vdc or zero) over half a cycle.

3- Mathematical analysis of hysteresis band modulation

In this method, a random number generator is used to generate high and low random band heights according to HB = HB0 + HB1 N ( R) where N (R ) is a random number between zero and 1 and HBo, HB1 are constant numbers. As shown in Fig.3.a, two hysteresis controllers are used to generate random band heights.

In this section, a hysteresis band height of unipolar and bipolar modulation schemes are calculated in terms of system parameters using mathematical analysis and the results are compared.

( a)

3-1 Bipolar modulation Fig.4 shows the load current and voltage waveforms for the bipolar hysteresis band modulation. According to this figure, the rate of load and reference current changes can be expressed as follows: dia∗ 2 HB dia+ di + di * + = ⇒ 2 HB = a T1 − a T1 dt T1 dt dt dt

(1)

dia∗ 2 HB dia− di − di* − = ⇒ −2 HB = a T2 − a T2 dt T2 dt dt dt

(2)

dia+ = +Vdc dt di − L a = −Vdc dt

L

(3) ( 4)

( b) Fig.2: A random hysteresis current control with bipolar modulation (a) a block diagram of a random band controller (b) load current and voltage waveforms

(a)

Fig.4: Load current and voltage waveforms for the RBHCC with bipolar modulation. +



Where L is the load inductance and i a , ia are the rising and the falling segments of the load current, respectively. With regard to Fig.4:

T1 + T2 = Tc =

1 fc

(5)

Where T1 and T2 are switching intervals and the switching frequency. (b) Fig.3: A random hysteresis current control with unipolar modulation (a) a block diagram of random band controller (b) load current and voltage waveforms

 di * ( a )2 +  1 di a 4 HB =  − dt + f c  dt di ( a )  dt

by replacing Eq.3 in Eq.6 :

    

f c is

(6)

 Vdc 2 dia* 2  dia* 2 2 ( ) − ( ) ( V ) − ( L ) dc  1 dt  ⇒ HB= dt 4HB=  L (7) V fc  4LfcVdc  ( dc )   L It can be shown that in the bipolar modulation, the hysteresis band height is a function of the input dc voltage, the load parameters, the modulation frequency and the derivative of reference current.

3-2 Unipolar modulation Fig.5 shows the load current and voltage waveforms for the unipolar hysteresis band modulation. The rate of load and the reference current changes can be expressed as follows:

L

di a+ = +V dc dt

(8)

L

dia− =0 dt

(9)

di a* ) 1 dt T1 = f c di a+ ( ) dt Thus: (

(14)

di a* ) 1 1 T2 = − T1 = (1 − dt+ ) fc fc di ( a) dt (

(15)

Subtracting Eq.14 from Eq.15, (T1-T2) yields: di* di* di* ( a) ( a) 2( a ) 1 dt 1 1 T1 − T2 = − (1 − dt+ ) = ( dt+ − 1) f c dia+ fc f di di c ( ) ( a) ( a) dt dt dt

(16)

Replacing Eq.14 and Eq.16 in Eq.13:

According to Fig.5:

di a+ di * T1 − a T1 = 2 HB dt dt dia* − T2 = −2 HB dt

dia* dia+ di* ) ) −( a ) * ( 1 di 1 di 2 dia dt dt − 4HB = ( dt+ −1) = ( (17) f c dt f c dt dia f c dt dia+ ( ) ( ) dt dt And finally by replacing Eq.8 in Eq.17: * a

(10) (11)

* a

2(

di a* di * (Vdc − L a ) dt HB = dt 2 f cVdc

(18)

4- Mathematical analysis of random hysteresis current control modulations With regard to the main concept of the hysteresis current control and Eq.7 and Eq.18, variations of hysteresis band height have a direct effect on the switching frequency. In this study, by a random change of the hysteresis band height, the switching frequency is changed randomly which changes the frequency side bands around a determined frequency with distributed spectrum contents. In this section, a mathematical analysis for the unipolar and bipolar modulations is performed according to the equations discussed in the previous section. The effect of the random band height variation on each modulation is discussed. Fig.5: Load current and voltage waveforms for the RHBCC with unipolar modulation

4-1 Bipolar modulation With the assumption of sinusoidal reference current, Eq.7 is changed to the following form:

Adding Eq.10 and Eq.11:

T1

+ a

* a

di 1 di − =0 dt f c dt

(12)

di a+ di * − (T 1 − T 2 ) a dt dt

From Eq.12:

(19)

The minimum and maximum switching frequencies over one cycle can be obtained as follows:

Subtracting Eq.10 from Eq.11: 4 HB = T 1

(Vdc ) 2 − ( LωI m cos ωt ) 2 fc = 4LVdc HB

(13 )

V − ( L ωI m ) 2 cos ωt = 1 ⇒ f c min = dc 4 LV dc HB V cos ωt = 0 ⇒ f c max = dc 4 LHB 2

It is shown that the switching frequency varies between f c min and f c max over one cycle. Assuming (Vdc ) >> (LωI m ) , the switching frequency for the bipolar modulation can be defined as follows: 2

fc ≈

Vdc 4LHB

(20)

(Vdc ) 2 >> ( LωI m ) and Vdc >> ( LωI m ) The hysteresis band height is selected %1 and %3 of the reference current magnitude. Random band height is continuously generated with 1000 sampling in each cycle by a random generator. In this paper, we assumed HB1 = 1 and hysteresis band HB 0

4-2 Unipolar modulation The same analysis is applied for extracting the switching frequency of the unipolar modulation using Eq.18 and assuming a sinusoidal reference current. The switching frequency of the hysteresis current control with unipolar modulation is derived as follows: ωI cos ωt (V dc − LωI m cos ωt ) fc = m (21) 2V dc HB Considering Eq.21 and assuming that Vdc >> (ωLI m ) :

fc =

(ωI m cos ωt )(Vdc ) ωI m cos ωt = 2Vdc HB 2 HB

(22)

height is 0.2 A which is 1% of the reference current magnitude. In order to compare the RHBCC with the traditional hysteresis current control, the average band is defined as follows: Min of band + Max of band = Average band ( 23 ) 2

HB1 = HB 0 = 0.13 and ⇒

HB0 + 2 HB0 0.4 = 0.2 ⇒ HB0 = = 0.13 2 3

Thus:

HB = 0.13 + 0.13N(R ) and 0 < N ( R ) < 1

Thus, for a constant hysteresis band height, the switching frequency of the unipolar modulation varies much more than the bipolar modulation. It is shown that the load parameters, the DC voltage value and the reference current magnitude play important roles in determining the switching frequency. According to Eq.20, if the DC voltage value and the load inductance are kept constant, the switching frequency is inversely proportional to the hysteresis band height. According to Eq.22 the switching frequency of the hysteresis current control based on unipolar modulation is a function cos(wt), the reference current amplitude and hysteresis band height. Assuming that the hysteresis band height is kept constant, the switching frequency changes based on cos(wt). Thus, Eq.22 shows that the unipolar modulation inherently has variable frequency. Thus, the RBHCC based on unipolar modulation has less effect to spread the spectrum contents of the load current compare to the bipolar modulation.

5- Simulation results In this paper, a case study for a single phase inverter with hysteresis current control based on unipolar and bipolar modulation techniques has been carried out and the spectrum contents of the load currents have been compared. Table I shows the simulation parameters. Table I: Simulation parameters Hysteresis band

Reference current amplitude

Voltage source

Inductive load

Base frequency

0.2A

20A

100v

1mH

50Hz

0.6A

20A

100v

1mH

50Hz

These parameters satisfy the following conditions for bipolar and unipolar modulations:

Similarly, for HB=0.6 A (3% of reference current):



HB0 + 2 HB0 1 .2 = 0.6 ⇒ HB0 = = 0 .4 2 3

and

HB1 = HB0 = 0.4 Thus:

HB = 0.4 + 0.4 N ( R ) and 0 < N ( R ) < 1 The following figures show the simulation results associated with the hysteresis current control with constant and random band heights in the unipolar and bipolar modulations. Fig.6 and Fig.7 show that the RBHCC has a significant effect to distribute the spectrum contents of the load current in the hysteresis current control based on bipolar modulation whereas the peak of the harmonics around the side band has been decreased significantly. Fig.8 and Fig.9 show that the hysteresis current control based on unipolar modulation has a variable switching frequency and the RBHCC has less effect on spreading the spectrum contents of the load current. According to the simulation results and comparing the spectrum contents of two random band hysteresis modulations, the unipolar modulation has inherently distributed the spectrum contents of the load current and randomization of hysteresis band height has no considerable effect on the frequency spectrum n compare to bipolar modulation.

(a) (a)

(b) Fig.6: Spectrum contents of load current for the bipolar hysteresis current control modulation (a) fixed band with HB=0.2A (b) random band with average band height=0.2A

(b) Fig.8: Spectrum contents of load current for the unipolar hysteresis current control modulation (a) fixed band with HB=0.2A (b) random band with average band height=0.2A

(a) (a)

(b) Fig.7: Spectrum contents of load current for the bipolar hysteresis current control modulation (a) fixed band with HB=0.6A (b) random band with average band height=0.6A

(b) Fig.9: Spectrum contents of load current for the unipolar hysteresis current control modulation (a) fixed band with HB=0.6A (b) random band with average band height=0.6

6- Conclusions This paper presents a Random Band Hysteresis Current Control (RBHCC) technique with bipolar and unipolar modulations and compares spectrum contents of a load current. Mathematical analysis has been carried out to extract the switching frequency of the random hysteresis current control in term of systems parameter. Simulation have been carried out for the RBHCC with the bipolar and unipolar modulations and the results show that the random hysteresis current control based on bipolar modulation has a significant effect to distribute the spectrum contents of the load current whereas the peak of the harmonics around the side band has been decreased significantly. The hysteresis current control based on unipolar modulation has a variable switching frequency and the RBHCC has less effect on spreading the spectrum contents of the load current. According to the simulation results and comparing the spectrum contents of two random band hysteresis modulations, the unipolar modulation has inherently distributed the spectrum contents of the load current and randomization of hysteresis band height has no considerable effect on the frequency spectrum n compare to bipolar modulation.

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