A Novel Random Hysteresis Current Control for a Single-Phase Inverter Alireza Nami
Hoda Ghoreishy
Department of Power Mazandaran University Babol, Mazandaran, Iran
Firuz Zare School of Engineering Systems Queensland University of technology Brisbane, GPO Box 2434, QLD 4001, Australia
[email protected]
Abstract Although hysteresis current control is a simple and a reliable modulation technique in switching control systems, its harmonic spectrum around a switching side band is not distributed broadly. The Random Pulse Width Modulation (RPWM) technique presented here is an effective scheme that distributes the harmonic spectrum of the load current and can reduce the acoustic noise and mechanical vibration of an inverter-fed motor drive. This paper presents a Random-Band Hysteresis Current Control (RBHCC) to spread more harmonic spectrum of the load current. In this paper, the effects of hysteresis band variations have been analysed in order to determine the best variation range for the hysteresis bands and the results have been compared with the traditional hysteresis current control. Theoretical analysis and simulations have been performed to describe the method.
1. Introduction Acoustic noise is a major problem associated with inverterfed AC drives. Two major sources of this problem are the motor design and the type of PWM scheme used in the inverters. A power inverter based on a standard sine weighted PWM scheme creates harmonic contents around switching side bands. These can produce acoustic noise, radio interference and mechanical vibration in electric machines depending on their frequencies and magnitude. Recently, new RPWM techniques based on different schemes have been proposed in order to spread the harmonics [1-4]. These techniques involve randomized pulse width, randomized switching frequency, randomized pulse position and the combination of these techniques. In this paper we propose a Random Band Hysteresis Current Control (RBHCC) technique with continuous and discontinuous sampling spreads the spectrum contents of load current. A traditional hysteresis current control technique is applied to a two-level inverter with inductive-resistive load and the results have been compared with RBHCC. The RBHCC is implemented with continuous and discontinuous sampling and with three different band ranges: small, medium and large variations.
Fig.1: Two level inverter with an inductive-resistive load
An error signal is made by comparing the reference current, Iref, and the output current, Io. If the output current crosses the upper hysteresis band, Ta- and Tb+ are turned on (Ta+ and Tb- are turned off). Similarly when it crosses the lower band, Ta- and Tb+ are turned off (Ta+ and Tb- are turned on) as shown in Fig.2 [7].
2. Traditional hysteresis current-control Among the conventional current-controlled schemes, the hysteresis current control provides a simple and robust current control performance with good stability; very fast response and an inherent ability to control peak current. Fig.1 shows a two-level inverter with an inductive-resistive load in which the traditional hysteresis control method can be applied.
Fig.2: Current and voltage waveforms based on a traditional hysteresis current control
In comparison with the other schemes, the hysteresis current control yields a continuously varying output
spectrum contents which is dependent on the derivative of the reference current, the variation of input voltage, back electromotive force (EMF) and the load parameters. However, the output harmonic spectrum is still gathered around the switching side band. Fig.3 shows the output current spectrum contents with high magnitude at 10 kHz and the annoying acoustic noise and the mechanical vibration caused by harmonics still exist [5]. One approach to eliminate this phenomenon is to reduce the hysteresis band and thus increase the dominant harmonic frequency range beyond the audible frequency range, but the switching frequency and losses will be increased significantly [6].
∆I ref 2B (3) ≈ = K1 ∆t ∆t Where 2B = hysteresis band height Thus, K K 1 fs = = 1 = 2 (4) B ∆t 2B Eq.4 shows the relationship between the switching frequency and the hysteresis band, around the peak of the reference current. According to this equation, with the random variation of the hysteresis band, the switching frequency as well as the output spectrum contents can be distributed broadly around the switching side band. The random band is defined as B = B 0 + B1 N( R ) where N ( R ) is a random number between 0 and 1 and B 0 and
B1 are constant numbers. According to Eq.4: fs =
Fig.3: Output current harmonic spectrum with traditional hysteresis band control R=0.5Ω, L=5mH, Vdc=100V, B=0.5A, f=10Hz
3. Hysteresis current control with random band Fig.4 shows the block diagram of a switching function based on random variations of hysteresis band. As can be seen, the random block generates a random number in an identified range. This random number defines the upper and the lower bands; and the error signal is compared with these bands. This comparison will lead to the production of proper control signals. These signals will be latched in an RS flip-flop and are applied to Ta+ and Tb- (the complementary is applied Ta- and Tb+) until the next control signal is generated.
k2 = B 0 + B1 × N(R )
(5) B1 × N(R ) B 0 1 + B0 B Eq.5 shows that the variations of 1 corresponds to the B0 variation of the random band height and the switching frequency. In order to analyse the effect of the band variation range on the switching frequency, we define three different band values, small ( B1 = 0.1 ), medium ( B1 = 1 ) B0 B0 and large (
B1 B0
0.1
1
10
fs
f s _ min
In order to study the effect of the hysteresis band variations on the harmonic characteristic, we have considered a sinusoidal reference current as follows: I ref = I m sin(ωt ) (1) The derivative of the reference current is:: dI ref = I m ω cos(ωt ) or ∆I ref ≈ I m f 2π cos(ωt ) (2) dt ∆t Due to the reduction of current variation around the peak of the reference current, hysteresis current control has the highest switching frequency and the hysteresis band can be defined in this region. According to Fig.2:
B1 = 10 ). The minimum and the maximum B0
switching frequency can be obtained by the variation of B1 and 0 < N (R ) < 1 . These frequencies are shown in B0 Table 1.
f s _ max
Fig.4: A block diagram of a random hysteresis current controller
k2
k B
2 0
k2 1.1B 0
k B
2 0
k2 2B 0
k2 B0
k2 11B 0
Table 1: Minimum and maximum switching frequencies in terms of
B1 B0
The variation of fs has to be taken into account in continuous and discontinuous sampling (the number of random variations of hysteresis band in each cycle). In this case, continuous sampling means changing the hysteresis band randomly based on a high sampling rate where as the hysteresis band is changed over a very short period of time. For discontinues sampling, the hysteresis band is changed randomly based on a few samples per each cycle.
3.1. Random hysteresis current control with small band variation
Fig.5 shows the switching frequency variations with respect to N (R ) for B1 = 0.1 . It can be seen that for B0
0 < N ( R ) < 1 , the switching frequency changes 10% which is a small variation.
Fig.7: Switching frequency variations in terms of N ( R ) and for B1 = 1
B0
Fig.5: Switching frequency variations in terms of
N ( R ) and for (a)
B1 = 0.1 B0
Fig.6.a and Fig.6.b show the switching frequencies within fs_min and fs_max for continuous and discontinuous sampling, respectively. As shown in Fig.6.a, due to the linear variations of the switching frequency and also more sampling, there are a lot of side bands gathering around a small frequency range. But in Fig.6.b, because of the discontinuity of sampling (a few samples per each cycle), the number of the side bands is few.
(b) Fig.8: Switching frequency side bands for B1 = 1 (a) continuous
B0
sampling (b) discontinuous sampling
3.3. Random hysteresis current control with large band variation Fig.9 shows the switching frequency variations in terms of N ( R ) for B1 = 10 . It can be seen that for 0 < N ( R ) < 1 , the B0
switching frequency changes 90% which has a significant variation.
(a)
(b) Fig.6: Switching frequency side bands for
B1 = 0.1 (a) continuous B0
sampling (b) discontinuous sampling
3.2. Random hysteresis current control with medium band variation Fig.7 shows the switching frequency variations with respect to N ( R ) for B1 = 1 . It can be seen that for B0 0 < N ( R ) < 1 , the switching frequency changes 50% which is a significant variation. In comparison with the previous case, it is apparent that the switching frequency has a wider range of variations. Fig.8.a and Fig.8.b show the switching frequency side bands within fs_min and fs_max and with the continuous and discontinuous sampling, respectively. Due to the linear and significant variations of the switching frequency, the side bands are distributed uniformly within fs_min and fs_max , as shown in Fig.8.
Fig.9: Switching frequency variations in terms of N (R ) and for
B1 = 10 B0
It can be seen that for 0 < N(R ) < 1 , the switching frequency has a significant nonlinear variations. If N(R ) has a uniform probability, the switching frequency is happened more around fs_min and fs_max. Fig.10.a and Fig.10.b show the switching frequency side bands within fs_min and fs_max with the continuous and discontinuous sampling, respectively.
(a)
(b) Fig.10: Switching frequency side bands for
B1 = 10 (a) continuous B0
sampling (b) discontinuous sampling
Fig.10.a and Fig.10.b show that the switching frequency range is large, but the side bands are not distributed uniformly within fs_min and fs_max and the switching frequency density is high around fs_min due to nonlinear variation of switching frequency.
4. Simulation results In this paper, a case study has been performed in order to analyse the output current harmonic spectrum of a single phase random hysteresis current-controlled inverter. The inverter parameters are as follows: Vdc = 100V R = 0.5 Ω (load resistance) L = 5 mH (load inductance) Iref = 10A f = 10Hz The average height of the random hysteresis band is 5% of the reference current magnitude. Simulations have been carried out to find the load current spectrum contents of a random hysteresis band currentcontrolled scheme based on small, medium and large variations of the hysteresis bands with continuous and discontinuous sampling. The results are analysed in the following sections.
Fig.11: Output current harmonic spectrum with small variation of the hysteresis band and with discontinuous sampling
Fig.12 shows the output current harmonic spectrum with the small variations of the random hysteresis band and with continuous sampling (1000 samples in each cycle). The simulation results show that the distribution of the load current harmonic spectrum is spread uniformly and the peak of harmonics around the switching side band is reduced compare to the traditional hysteresis current control.
4.1. Random hysteresis current control with small band variation In this case B1 = 0.1 , the average hysteresis band is 5% of B0 the reference current (0.5A); and according to Table 1 the average hysteresis band can be defined as follows:
B 0 + 1.1B 0 1 = 0.5 ⇒ B 0 = = 0.45 and B1 = 0.1B 0 = 0.045 2 2.1
Thus, B = 0.45 + 0.045N(R ) and 0 < N(R ) < 1 (6) Fig.11 shows the load current harmonic spectrum with the small variations of the hysteresis band and with discontinuous sampling (with 10 samples in each cycle). As it can be seen, the distribution of the load current is better than the traditional hysteresis current control technique shown in Fig.3 and the variation of the switching frequency around the switching side band is small.
Fig.12: Output current harmonic spectrum with small variation of the hysteresis band and with continuous sampling
4.2. Random hysteresis current control with medium band variation In this case B1 = 1 , the average hysteresis band is 0.5A B0 and according to Table 1, the average hysteresis band can be defined as follows:
B 0 + 2B 0 1 = 0.5 ⇒ B 0 = = 0.33 and B1 = B 0 = 0.33 2 3
Thus, B = 0.33 + 0.33N(R ) and 0 < N ( R ) < 1 (7) Fig.13 shows the output current harmonic spectrum with medium range variations of the random hysteresis band with discontinuous sampling (with 10 samples in each cycle).
As it can be seen, the distribution of the spectrum contents of the load current is broader than the traditional hysteresis control and the random hysteresis band with small variation. The peak of harmonics is reduced but the harmonic spectrum distribution is not uniform in the switching frequency range because of the discontinuity of the sampling. Fig.14 shows the output current harmonic spectrum with medium range variations of the random hysteresis band with continuous sampling (with 1000 samples in each cycle).
Thus, B = 0.083 + 0.83N(R ) and 0 < N(R ) < 1
(8)
Fig.15 shows the load current harmonic spectrum with large variation of the hysteresis band and with discontinuous sampling (with 10 samples in each cycle).
Fig.15: Output current harmonic spectrum with large variation of the hysteresis band and with discontinuous sampling
Fig.13: Output current harmonic spectrum with medium variation of the hysteresis band and with discontinuous sampling.
As it can be seen, the harmonic spectrum of the load current is distributed broadly but the switching frequency distribution is nonlinear, thus the switching frequency is centralized around f s _ min . Also, the harmonic spectrum is not uniform and has some peak due to the discontinuity of the sampling. Fig.16 shows the output current harmonic spectrum with large variation of the random hysteresis band and with continuous sampling (with 1000 samples in each cycle).
Fig.14: Output current harmonic spectrum with medium variation of the hysteresis band and with continuous sampling
Since the switching frequency range is high, the distribution of the load current harmonic is broader than the random hysteresis current control with small band variation and the peak of harmonic around the switching side band is reduced as shown in Fig.14.
4.3. Random hysteresis current control with large band variation B In this case 1 = 10 , the average hysteresis band is 0.5A B0 and according to Table 1 the average hysteresis band is as follows:
B 0 + 11B 0 1 = 0.5 ⇒ B 0 = = 0.083 and 2 12 B1 = 10B 0 = 0.83
Fig.16: Output current harmonic spectrum with large variation of the hysteresis band and with continuous sampling
As it can be seen, the harmonic spectrum of the load current is distributed widely with reduced peak of harmonic due to the large variation of the switching frequency. But, the switching frequency distribution is nonlinear and the side bands are centralized around f s _ min . According to simulation results, • the random hysteresis current control with different hysteresis band height variations distributes the spectrum contents of the load current around the switching frequency compare to the traditional hysteresis current control
•
•
increasing the range of the hysteresis band height will increase the variation of the switching frequency giving a wider distribution of harmonics in the random hysteresis band current control technique with medium variations; and with continuous sampling scheme, the harmonic spectrums are distributed uniformly and also the harmonic peak around the switching side band is reduced. Thus, by achieving 0.1 〈 B1 〈 1 , the B0
spectrum contents of the load current has a broad distribution and the peak of harmonics around the side band is reduced in amplitude with a low level of harmonic distortion. This is a better control method to reduce acoustic noise and mechanical vibration as the peak of spectrum contents is decreased
5. Conclusions This paper introduces a random hysteresis current control technique to distribute the spectrum contents of the load current around a switching sideband. Simulations have been carried out for traditional and random hysteresis current controllers using a two-level inverter. Simulation results show that the random hysteresis current-controlled scheme provides a broader distribution of the load current and reduces the magnitude of harmonics around the side band significantly. This paper also proposes a method for determining the best hysteresis band values to reduce the peak of spectrum contents in order to minimize the acoustic noise and the mechanical vibration in a motor drive system.
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