A scheme of simple band-pass RC- and LR- filters on operational amplifiers containing only one capacitor or inductor and 3 resistors is proposed. A comparison is made of the amplitude-frequency characteristics of the proposed filters, as well as the RC filter of Robert Allen Pease and its modified LR- variant.

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From the whole set of simple low-frequency filters, one can highlight the Sallen-Key filters [1, 2]. Despite their attractive external simplicity, these filters are far from easy to set up and require the use of coordinated parts.

The RC filter, proposed in 1971 by an engineer of George A. Philbrick Research—Robert Pease—**Figure** **1** [3, 4], has several unique properties. It is extremely simple, and its resonant frequency is controlled by only one potentiometer R2, and the transmission coefficient of the filter almost does not depend on the resistance value of this potentiometer. The amplitude-frequency characteristics of this filter when adjusting the potentiometer R2 are shown in Figure 1 [5].

**Figure** **1** Electrical diagram of the Pease RC-filter and its amplitude-frequency characteristics when R2: 1) 10.0 kΩ; 2) 3.0 kΩ; 3) 1.0 kΩ; 4) 0.3 kΩ; 5) 0.1 kΩ; 6) 0.03 kΩ.

By slightly modifying Pease’s circuit, namely, by replacing capacitors with inductors, we get a modified filter circuit. The amplitude-frequency characteristics of the modified LR-filter during the adjustment of the R2 potentiometer are shown in **Figure** **2** [5].

**Figure** **2** Electrical diagram of the modified LR-filter and its amplitude-frequency characteristics when R2: 1) 0.03 kΩ; 2) 0.1 kΩ; 3) 0.3 kΩ; 4) 1.0 kΩ; 5) 3.0 kΩ; 6) 10.0 kΩ. L1=L2=20 mH.

In addition to the op-amp, the filters discussed above contain 5 components each. However, it is possible to offer even simpler filters that contain only 4 where the elements R3 + R4 can be replaced with one potentiometer.

The “resonant” frequency of the RC filter, **Figure** **3**, is determined from the expression:

where *f*_{0} is in Hz, R is in Ω, C is in F, *a* is a constant depending on the model of the op-amp.

So, for example, for LM324 *a* ≈ 426. The equivalent *Q-*factor of the filter *Q* is proportional to the expression:

where *b* is a constant (*b* ≈ 110).

In the calculations: C = C1; R = R3 + R4. Thus, the “resonant” frequency of the filter depends only on the nominal values of the elements R = R3 + R4 and C = C1. The ratio R2/R1 does not affect the frequency of the “resonance”, but affects only the value of the equivalent quality factor of the filter and the transmission coefficient of the filter at the frequency of the “resonance”.

**Figure** **3** Electrical diagram of the RC-filter with the adjustment of the “resonance” position by the potentiometer R4.

The amplitude-frequency characteristics of the RC-filter are shown in **Figure** **4**.

**Figure** **4** Amplitude-frequency characteristics of the RC-filter with the adjustment of the “resonance” position when the resistance value R = R3 + R4 varies.

Replacing the capacitor C1 with the inductor L1 and swapping the frequency-determining components R and L, we get the LR-version of the filter, **Figure** **5**. Its amplitude-frequency characteristics with varying values of R are shown in **Figure** **6**.

The “resonant” frequency of the LR-filter, Figure 3, is determined from the expression:

where *f*_{0} is in Hz, R is in Ω, L is in H, and *a* is a constant. The ratio R2/R1 affects the same parameters as before.

**Figure** **5** Electrical diagram of the LR-filter with the adjustment of the “resonance” position by the potentiometer R4.

**Figure 6** Amplitude-frequency characteristics of the LR-filter with the adjustment of the “resonance” position when the resistance value R = R3 + R4 varies.

*Michael A. Shustov is a doctor of technical sciences, candidate of chemical sciences and the author of over 850 printed works in the field of electronics, chemistry, physics, geology, medicine, and history.*

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- Painless reduction of analog filter noise
- Toward better behaved Sallen-Key low pass filters
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**References**

- Sallen R.P., Key E.L. “A Practical Method of Designing RC Active Filters”. IRE Transactions on Circuit Theory, 1955, Vol. 2, № 1 (March), pp. 74–85.
- Tietze U., Schenk Ch. “Halbleiter-Schaltungstechnik”, 12. Auflage, Berlin-Heidelberg, Springer Verlag, 2002, 1606 S.
- Pease R. “An easily tunable notch-pass filter”. Electronic Engineering, December 1971, p. 50.
- Hickman I. “Notches, Top”. Electronics World Incorporating Wireless World, 2000, V. 106, No. 2 (1766), pp. 120–125.
- Shustov M.A. “Circuit Engineering. 500 devices on analog chips”. St. Petersburg: Science and Technology, 2013, 352 p.

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