Design and Implementation of a Sallen-­‐Kelley Low Pass Filter Shaun Eisenmenger 11/12/11 KEYWORDS Active Filter, Sallen-­‐Kelley, Low-­‐Pass, Electrical Noise, Aliasing ABSTRACT A Sallen-­‐Kelley Low Pass filter can be used to eliminate electronic noise in a circuit, but it can also be used to prevent aliasing by limiting the frequencies of signals to be processed. The Sallen-­‐Kelley LPF utilized in the MIT OpenCourseware Laptop Based Radar System is easy to design and implement – the filter parameters can be changed by changing a single resistor or capacitor value. INTRODUCTION An electronic filter is defined as an active or a passive electrical device or circuit that is used to attenuate undesirable electrical signals. In many cases, low-­‐pass filters are used to eliminate electrical noise. Electrical noise is any set of unwanted, random electrical signals that present themselves in electrical circuits. Nearly all electronic devices are susceptible to the negative effects noise, particularly those that utilize relatively small signals. Another use for electronic filtering is to prevent aliasing during signal processing applications. One can utilize a low-­‐pass filter to ensure that only low frequency signals will be passed through and that higher frequency signals (noise) will be eliminated. This document will serve as an introduction to active low-­‐pass filters, particularly the Sallen-­‐Kelley topography that has been utilized in the MIT Opencourseware Laptop Based Radar circuitry. OBJECTIVE This document will give the reader an introduction to the advantages of using an active low pass filter versus using a passive filter. Examples of how to design and implement the Sallen-­‐Kelly filter used in the Laptop Based Radar circuitry will also be given. ISSUES When presented with the choice between an active and a passive filter, one must consider the numerous advantages and disadvantages provided by each. A passive filter will consist of just resistors, capacitors and inductors, and will generally be cheaper and easier to implement than an active filter which utilizes an OP-­‐AMP. The passive filters do not require a separate power supply to operate, whereas an active filter will. Active filters do provide benefits over passive filters in that an active filter will not load the circuit that it is inserted to, while a passive filter will. An active filter will also have a higher quality factor, Q than a passive filter. The higher Q means that the filter can have a sharper peak at its operating frequency. This is shown in the graph and equation below: Figure 1: Energy of a system vs. frequency (B. Crowell) Q=
fo
Δf
Equation 1: Calculation of Q The Sallen-­‐Kelley €
Active filter topology offers an easy-­‐to-­‐implement filtering topology because the operating (cutoff) frequency can often be adjusted by changing a single resistor or capacitor value. A schematic is given below: Figure 2: Generic Sallen-­‐Kelley Active Low Pass Filter (eCircuit Center) STEPS In order to implement the Sallen-­‐Kelley Active Low Pass Filter, you must know what your desired cutoff frequency will be. From there you can determine what your component values should be. Formulas for calculating f0 and Q are given below: R1R2C1C2
Q=
C2 (R1 + R2 )
1
fo =
2π R1R2C1C2
€
EXAMPLE Figure 3: Active LPF used in the MIT OpenCourseware Laptop Based Radar In this particular case, the desired cutoff frequency, f0 is 15kHz. Since there are essentially 4 components that can be manipulated to change the cutoff frequency, it is best to pick easy numbers for 3 of the components and then solve for the 4th. In the filter used in the MIT OpenCourseware Laptop Based Radar, the first three component values are given and we can solve for the 4th. f o = 15kHz
C1 = 1nF
C2 = 1nF
R1 = 28kΩ
fo =
1
2π R1R2C1C2
Rearranging to solve for R2 yields the following equations: 1
R2 =
€
R1C1C2 (2πf o ) 2
‡
1
Quad
op-amp
MAX414
used in single-supply
R =
(28k)(1n)(1n)(2π ⋅ 15k)
configuration.
R = 4020.681Ω
‡
Gain
stage to amplify output of MXR1, adjust
Picking the nearest physical vbefore
alue of a resistor yields R2= 4.12kΩ. during FMCW mode
op-amp
clips
€
‡ Followed by 15 KHz 4th order LPF
±
prevents aliasing of 3&¶V input audio p
2
2
MIT Lincoln Labo
HARDWARE In the case of the MIT OpenCourseware Laptop Based Radar, the OP-­‐AMP used for the filter was from a MAX414. The MAX414 chip contains 4 OP-­‐AMPs and comes in a 14 pin DIP package. The OP-­‐AMPs require a 12V supply. Figure 4: MAX414 (Maxim IC) RESULTS The filter constructed in Lab was tested on an oscilloscope. The yellow waveform is the input signal and the green waveform is the signal that has been passed through the Sallen-­‐Kelley Low Pass Filter. Figure 5: Sallen-­‐Kelley LPF Results at 15kHz The filter is functioning as desired. A -­‐3dB gain factor was desired at 15kHz and the filter produced slightly higher attenuation than -­‐3dB. Calculations have been given below: Vin = 541.8mV
Vout(ACTUAL ) = 345.1mV
Vin 541.8mV
=
= 382.7mV
2
2
< Vout(IDEAL )
Vout(IDEAL ) =
Vout(ACTUAL )
CONCLUSIONS Since the filter is producing better than -­‐3dB attenuation at 15kHz, the Filter is working better than expected. This is likely due to parts tolerances that varied between €5%. 1% to A Sallen-­‐Kelley Low Pass Filter is an active filter topology that is very easy to use and implement. RECOMMENDATIONS Because of it’s ease of design and implementation, a Sallen-­‐Kelley filter is the perfect candidate for electronic filtering. It is ideal for eliminating electrical noise and since OP-­‐
AMPs draw a negligible amount of current, the filter will not load the circuit it is attached to. REFERENCES Maxim-­‐IC http://www.maxim-­‐ic.com/datasheet/index.mvp/id/1562 eCircuitCenter http://www.ecircuitcenter.com/circuits/opsalkey1/opsalkey1.htm MIT OpenCourseware http://ocw.mit.edu/resources/res-­‐ll-­‐003-­‐build-­‐a-­‐small-­‐radar-­‐
system-­‐capable-­‐of-­‐sensing-­‐range-­‐doppler-­‐and-­‐synthetic-­‐aperture-­‐radar-­‐imaging-­‐
january-­‐iap-­‐2011/index.htm TechLib http://www.techlib.com/reference/q.htm