The article presents the results of simulating the operation of the first-order active filters, designed on the basis of operational amplifiers with the help of simulation MULTISIM. The system allows the simulation studies and other electronic devices.
Using MULTISIM for the design of active filters of the first order
Yury Ponomarev, Olga Dyachuk, Zhitomir
Currently, design engineers are looking for new products (tools) to enable them to facilitate the work on the creation of new electronic devices. Recently increasingly popular system of circuit simulation MULTISIM company National Instruments , which successfully replaces a known system Electronics Workbench. This system allows you to design and simulate circuits in a variety of conditions.
The main advantages of the system:
• user-friendly and intuitive interface that allows you to quickly learn how to operate the system;
• the presence of many (more than 15) of different modes of analysis;
• choice of models of real (not virtual) components;
• the possibility of «3D» design;
• the ability to use virtual instrumentation, whose characteristics are the best examples of digital storage devices, and others.
All of this can significantly reduce the overall design time.
To demonstrate some of these features on the example of specific schemes. As an example, take the well-known circuits in the electrical active filters of the first order, built on the basis of operational amplifiers (op amps).
Brief theoretical information
The main characteristic of the filter is considered to be its amplitude-frequency characteristic (AFC). By referring response filters are divided into low pass filters (LPF), a high pass filter (HPF), bandpass filters, notch filters, etc.
Active filters can be first, second and larger orders. The procedure sets the slope of the asymptotic ACHH. For filters of the first order, he is 20 dB / decade. For the second — 40 dB / decade, etc.
Active filter design technique generally involves three steps :
• job specifications (raw data, such as frequency response, phase-frequency characteristic (PFC), and others.);
• determination of the transfer function;
• active filter design.
At the moment before the last stage is expedient to add a separate stage simulation, for example, using a system MULTISIM.
The transfer function can be expressed in the operator R (P) and forms in the frequency where the dimensionless (normalized) variable,
— complex variable, i
— the angular frequency,
— the cutoff frequency,
— filter gain in the frequency band,
— relative frequency,
— frequency signal.
The cutoff frequency of the filter — a signal frequency at which the power reduction occurs twice, i.e. filter coefficient transmission V2 decreases in time compared with the transmission coefficient K0 to zero (for the LPF) or infinite (for HPF) frequency.
When designing active filters of different orders, it is advisable to use the already calculated coefficients  of the corresponding polynomials, which specifies a particular type of filter.
According to , if the roots of the polynomial are complex, the transfer function of the low-pass filter can be written as a product of factors first and second order:
K (P) = R0 / ((1 + a1 + b1 • P • P2) + (1 + a2 + b2 • P • P2)) (1) where, a1 and b1 — positive real coefficients.
Filters Bessel, Butterworth and Chebyshev just different values of these coefficients.
The transfer function of the HPF can be obtained from the LPF replacing operator P (1) 1 / R:
K (P) = R0 / ((1 + a1 • 1 / R + b1 • 1 / P 2) + (1 + a2 • 1 / R + b2 • 1 / P2), …). (2)
Methods for the optimization and calculation of the coefficients of polynomials approximating HPF remain unchanged.
We use MULTISIM system for the design and analysis of active filters of the first order, namely, LPF, HPF, band. The simulation results it is advisable to obtain different frequencies (below cutoff frequency, at the cutoff frequency and at the frequency f0 greater
The procedure for calculating the filter
To calculate the parameters of the designed filters is advisable to use the following sequence:
1. Set the cutoff frequency of the filter design.
2. Set the gain of the filter.
3. Set the capacitance of the capacitor.
4. Calculate for the corresponding formulas resistors.
5. Replace the calculated values of resistors standardized.
6. Select an operational amplifier.
Capacitance value of the capacitor often selected from the E12 and resistors — number of E24.
Selecting a specific OU requires consideration of several factors. For simplified modeling MULTISIM all filters Select the standard schemes (virtual) OC type «ORAMR_3T_VIRTUAL».
The low pass filter of the first order
The active low-pass filter of the first order can be built as a scheme to the inverting and non-inverting input of op-amp. Next, we will consider only filters with the inverting input of op-amp. The scheme of the filter and logarithmic frequency response (LACHH)
Considering the expression (1), we can write the transfer function of the LPF represented in operator form:
K (P) = Co / (1 + a1 • P), (3) where Co = -R2 / R1 — the coefficient of the filter at zero frequency, a1 = 2¶ • Fo • R2 • C1 (4)
— coefficient in the transfer function (3).
Proizvedёm calculation of the filter.
As an example, we choose the initial data: the cutoff frequency Fo = 15 kHz; gain filter to = 4. The capacity of the condenser will choose from a number of E12, eg 1.8 nF. Resistor R2 found from expressions (4) with the proviso that a1 = 1. This is true no matter what type of filter used (Chebyshev, Butterworth, Bessel).
R2 = 1 / (2¶ • Fo • C1) = 5.9 kW.
From the expression Co = | R2 / RYA1 | find the resistance of the resistor R1 = 1,48 kW. Given the number of E24 choose resistors:
R2 = 6,2 kilohms, R1 = 1.5 kOhm. The gain at the cutoff frequency is equal to
The simulation results are calculated for a low-pass filter frequency signal shown on the oscilloscope and Bode plotter system MULTISIM (Figure 2).
Figure 2 shows:
1. The function generator — the parameters specified by the harmonic signal (frequency of 1.5 kHz, amplitude 1).
2. In a two-channel oscilloscope — the curves of the input (1, Channel_A) and output (2 Channel_V) signals. Estimated value of the gain lowpass filter can be calculated as the ratio of the maximum amplitude.
R0 (0,1 • Fo) INTS = 4,110 V / 999.577 = 4.1 mV, which approximately corresponds to the declared value R0 = 4.
Inaccuracy due to selection of the specific value of the resistor circuit of ryadya E24, not calculated.
From the waveforms it shows that the output signal actually 4 times greater than the input and is in antiphase, since we use a circuit with an inverting op-amp.
In decibels the transmission coefficient of the filter at frequencies close to zero, is:
K0 (0.1 • Fo) INTS (dB) = 20lg (R2 / R1) = 20lg (6,2 / 1,5) = 20lg 4,1 = 12,26 dB.
3. Bode plotter 1 (XBP1) — filter response. At a frequency of 1.48 kHz (close to 0,1 • Fo = 1.5 kHz — red vertical line) or rather it is simply impossible to install the program with the arrow keys «left», «right», the transmission coefficient LPF is 12.28 dB, practically corresponds to the calculated.
4. Bode plotter 2 (XBP2) — PFC filter. The phase shift is changed from 180 ° (at low frequencies) to 90 ° (at high frequencies). At a frequency of 1.48 kHz it is approximately 174 °, ie, substantially in antiphase.
Figure 3 shows the results of simulation of the filter cutoff frequency (15 kHz).
The function generator sets the frequency of the harmonic signal of 15 kHz.
The output signal on the oscilloscope is already not 4 as in Figure 2, and 2,846 B (blue vertical line window Channel_B). Gain is
R0 (Fo) INTS = 2,846 V / 999.757 = 2.85 mV, which roughly corresponds to R0, 7 = 2.83.
On Bode plotter 1 (XBP1) at a frequency of 15.058 kHz (red vertical line) calculated higher the gain is set to gain filter 9.073 dB. The difference between Figure 2 and Figure 3 is 12.28 dB — 9.073 dB = 3.2 dB, which corresponds to approximately 3 dB in Figure 1.
On Bode plotter 2 (HVR2) at a frequency of 15.058 kHz phase shift of approximately 133 °, ie, not in antiphase, as seen on an oscilloscope.
The function generator sets the frequency of the harmonic signal of 150 kHz.
On the oscilloscope the output of 390.904 mV (blue vertical line window Channel_V). Gain is
R0 (1M0) INTS = 390.904 mV / 999.446 mV «0.39, which indicates a significant weakening of the signal. On Bode plotter 1 (HVR1) at a frequency of 149.497 kHz (approximately 150 kHz) of the filter coefficient of -8.122 dB. The difference between Figure 4 and Figure 3 (on the horizontal axis of frequencies — a decade) is 9.073 dB — (- 8.122 dB) «17.2 dB, which corresponds approximately to reduce the frequency response 20 dB per decade
On Bode plotter 2 (HVR2) at a frequency of 149.497 kHz phase shift is about 95 °, which can be clearly seen on the oscilloscope.
Thus, the results of the simulation system designed MULTISIM first order LPF indicate correctly calculate and options selected circuit components of the filter at the specified procedure and that it works correctly.
In one discussed above procedure it is advisable to analyze the work of any filter (HPF, band, etc.), but to further simplify the analysis, we present the results of only one particular frequency.
The high pass filter of the first order
Driving active first-order high-pass filter to the inverting input of op-amp and its LACHH shown in Figure 5.
Based on the expression (2), the transmission coefficient of the first order HPF in operator form can be written
Proizvedёm calculation of the filter.
As an example, we choose the same data as for the LPF: cutoff frequency Fo = 15 kHz; filter gain Ko = 4; resistors as in the circuit LPF: R1 = 1.5 ohms, R2 = 6,2 kohm. Since, in this case, all parameters are set HPF than capacitance of the capacitor C2, as opposed to the low pass filter, we are not going to calculate the resistance (R1, R2), namely C2.
We calculate C2
Choose from a number of E12 capacitance value of the capacitor C2 closest to the calculated C2 = 6.8 nF.
1. The function generator — the parameters specified by the harmonic signal (frequency of 150 kHz, amplitude 1).
2. On the oscilloscope — the curves of the input (1 black color) and output (2, red) signals. Estimated value of the gain of the HPF is:
Co. (1o • Fo) INTS = 4,110 V / 999.493 mV ≈ 4,11, which corresponds approximately to = 4.
From the waveforms shows that the output signal of the filter actually 4 times larger than at the inlet, and in antiphase.
In decibels transfer coefficient HPF same as for the LPF, but on a frequency tending towards infinity, is Co (1o • Fo) INTS (dB) = 20lg (R2 / R1) = 12,26 dB.
3. Bode plotter 1 (HVR1) — filter response. At a frequency of 14.7 kHz (near Fo = 15 kHz — the red vertical line) HPF transfer coefficient is about 9.1 dB, ie 12,3-9,1 at «less than 3.2 dB, which roughly corresponds to 3 dB in Figure 5.
4. Bode plotter 2 (HVR2) — PFC filter. The phase shift varies from -9o ° (at low frequencies) to -18o ° (high). At a frequency of 14.8 kHz (close to the cutoff frequency), the phase shift is approximately -133 °, i.e. not in antiphase as on an oscilloscope (Fig.6) for the frequency 15o kHz.
It can be concluded that the filter works correctly.
The band-pass filter of the first order
Bandpass filter and LACHH shown in Figure 7.
Construct a bandpass filter by combining LPF and HPF of the first order, which have been discussed above, ie, with the same parameters. Accordingly, the bandpass filter parameters are as follows: Fo = 15 kHz, Ko = 4, R1 = 1.5 kOhm, R2 = 6,2 kohm C1 = 1.8 nF, C2 = 6.8 nF.
Where Fo is not cut-off frequency and the average frequency of the operating range of the filter (the frequency at which a maximum amplitude).
In this filter conditions must be satisfied: R2 • S1≥R1 • C2, C2 / C1 = R2 / R1.
In this case, these conditions are performed approximately (6.2 1.8 • • 11≥1,5 = 6.8 and 10 = 6.8 / 1.8 = 3.7 = 6.2 / 1.5 = 4, 1, respectively).
The gain at the resonance frequency is Fo = Kfo Co., 7 / √2 = (Co / √2) / √2 = Co / (√2) 2 = Co / 2 = 4/2 = 2. Kfo (DB) = 20lg (Vout / Vin) = 20lg2≈6 dB.
The gain at the cutoff frequency f1, f2 is Kf1 = Kf2 = Kf0 / √2 = 2 / 1.41 = 1.41. Kf1 (dB) = Kf2 (dB) = 20lg1,41≈3 dB.
1. The function generator — the parameters specified by the harmonic signal (frequency 15 kHz, amplitude 1).
2. On the oscilloscope — the curves of the input (1, Channel_A) and output (2 Channel_V) signals. Estimated value of the gain is equal to
Kf0rasch = 2,022 V / 999.724 mV ≈2, that the statement Kf0 = 2.
From the waveforms it shows that the output is valid two times larger than the inlet, and in antiphase (Scheme inverting input of OA).
3. Bode plotter 1 (XBP1) — filter response. At a frequency of 14.8 kHz (close to the fo = 15 kHz — the red vertical line) of the filter coefficient is 5.9 dB, which corresponds approximately to the calculated value (6 dB).
4. Bode plotter 2 (XBP2) — PFC filter. The phase shift varies from -90 ° to -180 ° (at frequencies from zero to fo) and from 180 ° to 90 ° (at frequencies fo to infinity). At a frequency of 14.8 kHz, it is -167 °, ie, It is approximately in antiphase.
Figure 9 shows the simulation results of the filter at points where the gain is reduced to a level of about 1.41, which corresponds Kf1, Kf2. On the left is represented by the lower cutoff frequency f |, right — the upper frequency limit f2.
Analyzing Figure 9, we can determine the frequency of the bandpass filter cutoff points: f1≈6,2 kHz, f2≈35,4 kHz. The average frequency of the filter is associated with the expression of the boundary frequencies f0 = √ (f1 • f2) = √ (6,2 kHz • 35,4 kHz) = 14.8 kHz. The same frequency is determined by the parameters of the components of the formula: f0 = 1 / 2√ (R1C2R2C1) = 1 / 6,28√ (1,5 kW • 6,8 nF • 6,2 kW • 1,8 nF) = 14, 9 kHz.
As you can see, in the first and in the second case, the calculated frequency of about 15 kHz relevant statements.
The results show the correct operation of the filter, although it should be noted that the slope of the rise and fall of his response simulation results proved to be somewhat worse than in Figure 7, and is approximately 17 dB per decade.
Summing up the analysis of all the above filters can note the following:
1. The system allows MULTISIM comfortable enough to hold a full and fast analysis of these filters.
2. The simulation results show the correctness of the chosen methods of calculating the parameters of the filter circuit components and accuracy of their work.
3. Existing discrepancies between the model results and the calculated values can be partly explained by the fact that the parameters of the filter circuit selected on the basis of real components that are slightly different from those calculated.
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4. Vodovozov AM, AS Elyukov Design of active filters. — Vologda: VSTU 2009.