Introduction to the basic principles of active filter circuits

1 Introduction

1.1 Classification of filters; Active filter is actually an amplifier with a specific frequency response. It is composed of adding some passive components such as R and C to the operational amplifier. Usually active filters are divided into:

Low pass filter

High Pass Filter

Bandpass filter

Bandstop filter

Their amplitude-frequency characteristic curves are shown in Figure 1.

Figure 1 Frequency response of active filter

Filters can also be made of passive reactive elements or crystals, which are called passive filters or crystal filters.

1.2 Purpose of filters
Filters are mainly used to filter out useless frequency components in signals. For example, there is a lower frequency signal that contains some interference from higher frequency components. The filtering process is shown in Figure 2.

Figure 2 Filtering process

2 Active low-pass filter (LPF)
2.1 Main technical indicators of low-pass filter
(1) Passband gain Avp
Passband gain refers to the voltage gain of the filter in the passband, as shown in Figure 3. The amplitude-frequency characteristic curve of a good LPF in the passband is flat, and the voltage gain in the stopband is basically zero.
(2) Passband cutoff frequency fp
Its definition is the same as the upper cutoff frequency of the amplifier circuit, as shown in Figure 3. The transition band between the passband and the stopband is called the transition band. The narrower the transition band, the better the selectivity of the filter.

Figure 3 Amplitude-frequency characteristic curve of LPF

2.2 Simple first-order low-pass active filter
The circuit of the first-order low-pass filter is shown in Figure 4, and its amplitude-frequency characteristic is shown in Figure 5. The dotted line in the figure is the ideal situation, and the solid line is the actual situation. The characteristics are simple circuit, too slow stopband attenuation, and poor selectivity.

Figure 4 First-order low-pass circuit (LPF) Figure 5 Amplitude-frequency characteristic curve of first-order LPF

When f = 0, the capacitor can be regarded as an open circuit, and the gain in the passband is

[page] The transfer function of the first-order low-pass filter is as follows
, where
The transfer function looks similar to the gain-frequency expression of the first RC low-pass link, except that the passband gain Avp is missing.

2.3 Simple second-order low-pass active filter
In order to make the output voltage drop at a faster rate in the high frequency band to improve the filtering effect, an RC low-pass filter link is added, which is called a second-order active filter circuit. It has better filtering effect than the first-order low-pass filter. The circuit diagram of the second-order LPF is shown in Figure 6, and the amplitude-frequency characteristic curve is shown in Figure 7.

Figure 6 Second-order low-pass circuit (LPF) Figure 7 Amplitude-frequency characteristic curve of the second-order low-pass circuit

(1) Passband gain
When f = 0, each capacitor can be regarded as an open circuit, and the gain in the passband is

(2) Second-order low-pass active filter transfer function
According to Figure 8-2.06, it can be written

(3) Passband cutoff frequency
Replace s with jω, let ω ₀ = 2πf ₀ = 1/(RC) to obtain

When f = fp, the modulus of the denominator in the above equation

Solve to get the cutoff frequency:

Compared with the ideal second-order Bode plot, after exceeding f0, the amplitude-frequency characteristic decreases at a rate of -40 dB/dec, which is faster than the first-order decrease. However, the amplitude-frequency characteristic does not decrease fast enough between the passband cutoff frequency fp→f0.

2.4 Second-order voltage-controlled low-pass active filter
(1) Second-order voltage-controlled LPF
The second-order voltage-controlled low-pass active filter is shown in Figure 8. One of the capacitors C1 was originally grounded, but is now connected to the output terminal. Obviously, the change of C1 does not affect the passband gain.

Figure 8 Second-order voltage-controlled LPF Figure 9 Amplitude-frequency characteristics of the second-order voltage-controlled LPF

[page] (2) Transfer function of second-order voltage-controlled LPF

For node N, the following equation can be listed

Solving the above three equations together, we can get the transfer function of LPF

The above equation shows that the passband gain of the filter should be less than 3 to ensure stable operation of the circuit.

(3) Frequency response
The frequency response expression can be written from the transfer function

When f=f0, the above formula can be simplified to

Define the quality factor Q value of the active filter as the ratio of the voltage gain modulus to the passband gain when f=f0 The above

two formulas show that when 21, the voltage gain at f=f0 will be greater than Avp, and the amplitude-frequency characteristic will be raised at f=f0, please refer to Figure 9 for details. When Avp≥3, Q=∞, the active filter is self-excited. Since C1 is connected to the output end, it is equivalent to adding a little positive feedback to the LPF at the high frequency end, so the gain at the high frequency end is raised, and it may even cause self-excitation.

2.5 Second-order inverting low-pass active filter
The second-order inverting LPF is shown in Figure 8-2.10. It is composed of adding an RC low-pass circuit to the input end of the inverting proportional integrator. The improved circuit of the second-order inverting LPF is shown in Figure 8-2.11.

Figure 10 Inverting second-order LPF Figure 11 Multi-feedback inverting second-order LPF
From Figure 11, we can see that

For node N, the following equations can be listed

[page]4 Active Bandpass Filter (BPF) and Bandstop Filter (BEF)
The circuit of active bandpass filter (BPF) is shown in Figure 14. The circuit of active bandstop filter (BEF) is shown in Figure 15.
The bandpass filter is composed of a low-pass RC link and a high-pass RC link. The lower cutoff frequency of the high-pass should be set to be less than the upper cutoff frequency of the low-pass. Otherwise, it is a bandstop filter.

Figure 14 Second-order voltage-controlled BPF Figure 15 Second-order voltage-controlled BEF

In order to obtain good filtering characteristics, a higher order is generally required. The design calculation of the filter is very troublesome. When necessary, you can use engineering calculation curves and related computer-aided design software.
Example 1: The second-order voltage-controlled LPF is required to have fc=400Hz and a Q value of 0.7. Try to find the resistance and capacitance values ​​in the circuit of Figure 16.

Figure 16 Second-order voltage-controlled LPF

Solution: According to fc, select C and then find R.
1. The capacity of C should not exceed 1uF. Because large-capacity capacitors are large in size and expensive, they should be avoided as much as possible.
Take C=0.1 uF, 1kΩ
to calculate R=3979Ω, and take R=3.9kΩ

2. Calculate R1 and Rf according to the Q value, because when f=fc, Q=1/(3-Avp)=0.7, then Avp=1.57. According to the relationship between Avp and R1 and Rf, the symmetry condition of the external resistors at the two input terminals of the integrated operational amplifier is 1+Rf/R1=Avp=1.57, R1//Rf=R+R=2R. Solution: