Please give me an analysis of a circuit consisting of an integrator and a bandpass filter

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Please give me an analysis of a circuit consisting of an integrator and a bandpass filter [Copy link]

The analysis of this circuit focuses on the center frequency of the filter

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2 Harmonic detection principle based on bandpass filter Taking the second-order bandpass filter as an example, the typical expression of the transfer function of the second-order bandpass filter is: ωo=2πfo, is the center angular frequency, fo is the center frequency; Q is the quality factor. When ω=ωo, H(iωo)=1. This shows that the amplitude of the bandpass filter at the center angular frequency ωo is particularly attenuated, and there is no phase delay, which is an important characteristic of the bandpass filter. This characteristic ensures the accuracy of the harmonic detection method based on the bandpass filter. In the active filter, we choose the center frequency fo of the bandpass filter to be 50Hz, then the bandpass filter has no attenuation of the fundamental amplitude, no phase delay, and other subharmonics are filtered out, which can detect the fundamental wave in real time. The load current ia, ib, ic passes through the bandpass filter to obtain the three-phase fundamental current ia1, ib1, ic1, and the three-phase harmonic current iah, ibh, ich can be obtained by subtracting the fundamental current from the load current. Based on this, the principle of harmonic current detection is shown in Figure 1. This detection method does not require coordinate transformation, but only needs to perform bandpass filtering on the three-phase currents respectively, which greatly reduces the amount of calculation. 3 Design and implementation of digital bandpass filter Digital filters can be divided into IIR type and FIR type according to their types. The PIR type has only zero points, and it is not easy to obtain better passband and stopband characteristics like the IIR type. Therefore, the IIR type is selected in general design. The IIR type can be divided into Butterworth type filter, Chebyshev I type filter, Chebyshev II type filter and elliptic filter. The digital filter design tool FDATool in the MATLAB toolbox can help you easily select and design the required digital filter. The main parameters of the digital bandpass filter include order, filter type, two cutoff frequencies, etc. The stopband attenuation characteristics of the high-order filter are very good, but it is difficult to achieve after the order is high. For active filters, the fundamental and main harmonic frequencies are relatively far apart, so the requirement for stopband attenuation is not very high, and a second-order filter can meet the requirement; and because the Buttermorth filter has a relatively flat characteristic in the passband and is relatively simple to implement, after comprehensive consideration, a second-order Butterworth bandpass filter is selected. The selection of the filter cutoff frequency is closely related to the quality factor Q. The larger the Q, the faster the harmonic attenuation, and the more accurate the fundamental component extracted by the bandpass filter; however, the larger the Q, the smaller the bandwidth, the slower the dynamic response speed, and the digital filter parameters will differ too much, which will increase the word length requirements. The passband width BW of the bandpass filter = ωo/(2πQ) = fo/Qofo is the center frequency of the system. Here we take Q at about 5, so that the bandwidth is about 10Hz. The two cutoff frequencies are selected as 45Hz and 55.6Hz respectively. It should be noted here that due to the asymmetry of the amplitude-frequency characteristics of the bandpass filter, the center frequency is not the average of the two cutoff frequencies. The selection criteria of the two cutoff frequencies are to ensure that the phase shift of the 50Hz center frequency is 0 and the amplitude is not attenuated. According to the above criteria, the filter transfer function is designed as the amplitude-frequency and phase-frequency characteristics of the filter are shown in Figures 2 and 3. The realization of the bandpass filter is to realize the transfer function of formula (2) in the DSP chip. In order to facilitate program implementation, formula (2) is changed into the form of a differential equation, as shown in formula (3). y(n)=0.003319x(n)-0.003319x(n-2)+1.9924y(n-1)-0.9934y(n-2) (3) To implement the above differential equation using DSP, three memory cells are used to store the values of x(n), x(n-1), and x(n-2), and three memory cells are used to store the values of y(n), y(n-1), and y(n-2). In each interrupt program, the values of these six memory cells are updated according to equation (3), and the final output y(n) is the fundamental wave value after filtering. If other forms of filters are used, the number of intermediate memory cells required may be different, and it should be determined according to the number of terms of x(n) and y(n) in the differential equation. If the bandpass filter program is implemented in a fixed-point DSP, attention should also be paid to the selection of the decimal point position of the filter coefficient. The digital filter coefficient has a great influence on the filter performance. Once the filter parameters differ even a little, the filter output may be far from the correct value, and sometimes it may make the system unstable. Therefore, the coefficients should be amplified as much as possible before calculation. Here, according to the characteristics of the three coefficients (0.003319, 1.9924, 0.9934) and DSP (16-bit fixed point), all coefficients are amplified by 214 times, and then reduced by 214 times after the filtering operation is completed, so that the calculated result is as accurate as possible. In the filter implementation, it is very important to select the appropriate amplification factor according to the filter coefficient. The principle is to use the full number of bits of the processor (here is 16 bits) as much as possible. 4 System simulation and test results The experimental system is a three-phase parallel active filter. The block diagram of the detection part is shown in Figure 4, where the dotted part is the DC side voltage control part. The principle of the system is: first, the load current passes through the bandpass filter to obtain the fundamental current ia1, ib1, ic1; then, the active current △iap, △ibp, △icp required to maintain the DC side voltage is superimposed, and then this part of the current is subtracted from the total load current to obtain the three-phase command current value; finally, the command current value is PI-regulated to control the output of the inverter, and the harmonic current is injected into the grid in reverse phase, so that the current of the grid is basically a sine wave. The system simulation uses the Simulink module in MATLAB, and the simulation results are shown in Figure 5. As can be seen from Figure 5, the grid current after compensation is greatly improved than the current waveform before compensation. The capacity of the experimental prototype is designed to be 6kW, the input voltage is three-phase 380V, and the load is a three-phase uncontrolled rectifier bridge. The control part is based on TI's TMS320LF2407 DSP, which is responsible for harmonic current calculation and PWM output control. The main part of the program is completed in the AD sampling interrupt. In the AO interrupt program, firstly, according to the voltage and current sampling values of the three phases, the current after filtering is calculated using formula (3), and then the command current value is calculated. Finally, after PI adjustment, it is sent to the PWM generation circuit to control the output of the inverter. Figure 6 is the intermediate calculation result of the program. In the figure, 1 is the grid voltage sampled by DSP, 2 is the load current sampled by DSP, and 3 is the fundamental component of the load current obtained by the bandpass filter. It can be seen from Figure 6 that the bandpass filter can well separate the fundamental component of the load current. Figure 7 is the experimental waveform of the system, where Figure 7 (a) is the grid voltage and current waveform before the active filter is put into use, and Figure 7 (b) is the grid voltage and current waveform after the active filter is put into use. It can be seen from Figure 7 (b) that the active filter based on the bandpass filter can play a good role in harmonic suppression. 5 Conclusion This paper proposes a harmonic detection method based on a bandpass filter, and verifies the feasibility of this method in the parallel active filter through simulation and experiment. The main conclusions are as follows: 1) The fundamental component in the load current can be detected relatively well by using a bandpass filter; 2) Since the filter load current generally has no even harmonics, if it is a three-phase symmetrical system, there are no third-order harmonics and multiples of 3, so as long as the center frequency of the bandpass filter is 50Hz, the bandwidth will not have a great impact on the system, but the phase-frequency characteristics of the bandpass filter will have a greater impact on the system; 3) Experiments have shown that parallel active filters based on bandpass filters can effectively suppress the harmonic currents of the power grid, but the disadvantage of this method is that it cannot compensate for reactive power at the same time.

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