A 6th-order Butterworth low-pass filter is an electronic circuit that allows low-frequency signals to pass while aggressively blocking high-frequency noise. It features a perfectly flat magnitude response in the passband (maximally flat) and rolls off at a steep rate of ) in the stopband.
Here is the comprehensive mathematical breakdown and electronic circuitry design behind this high-performance filter. 1. The Mathematical Foundation
The defining characteristic of a Butterworth filter is its maximally flat passband response. Magnitude Squared Response The power transfer function of an -th order Butterworth filter is expressed as:
|H(jω)|2=11+(ωωc)2nthe absolute value of cap H open paren j omega close paren end-absolute-value squared equals the fraction with numerator 1 and denominator 1 plus open paren the fraction with numerator omega and denominator omega sub c end-fraction close paren raised to the 2 n power end-fraction For a 6th-order filter ( ), this becomes:
|H(jω)|2=11+(ωωc)12the absolute value of cap H open paren j omega close paren end-absolute-value squared equals the fraction with numerator 1 and denominator 1 plus open paren the fraction with numerator omega and denominator omega sub c end-fraction close paren to the 12th power end-fraction is the operating angular frequency. ωcomega sub c
is the cutoff frequency (where the signal power drops to half, or Transfer Function and Pole Placement To find the transfer function in the Laplace domain (
), we must find the roots of the denominator polynomial, known as the Butterworth Polynomial. The poles lie evenly distributed on a left-half circle in the complex -plane with a radius equal to ωcomega sub c The angles of the 6 poles for an filter are calculated using:
θk=180∘+(2k−1)180∘2nfor k=1,2,…,6theta sub k equals 180 raised to the composed with power plus the fraction with numerator open paren 2 k minus 1 close paren 180 raised to the composed with power and denominator 2 n end-fraction space for k equals 1 comma 2 comma … comma 6
This yields three pairs of complex conjugate poles located at: Pair 1: Pair 2: Pair 3: Factored Transfer Function
Combining these poles into 2nd-order polynomial segments results in the normalized standard transfer function:
H(s)=1(s2+0.5176s+1)(s2+1.4142s+1)(s2+1.9319s+1)cap H open paren s close paren equals the fraction with numerator 1 and denominator open paren s squared plus 0.5176 s plus 1 close paren open paren s squared plus 1.4142 s plus 1 close paren open paren s squared plus 1.9319 s plus 1 close paren end-fraction
Each quadratic bracket represents a specific quality factor ( -factor) needed for implementation: Stage 1: (Highly resonant peak) Stage 2: (Critically damped / standard Butterworth) Stage 3: (Overdamped) 2. The Circuitry Design (Cascaded Architecture)
Building a 6th-order active filter requires cascading three individual 2nd-order filter stages in series. The most common topology for this is the Sallen-Key architecture using Operational Amplifiers (Op-Amps). The Sallen-Key 2nd-Order Stage
Each of the three stages utilizes the standard low-pass Sallen-Key configuration. A single stage consists of: Two resistors ( Two capacitors ( One Op-Amp configured as a voltage follower (unity gain) Component Selection Strategy
To make manufacturing simple, designers often keep the resistors equal ( ) and vary the capacitors to tune the specific -factor of each stage. The cutoff frequency ( ωcomega sub c ) and the quality factor ( ) for any given stage are dictated by:
ωc=1RC1C2omega sub c equals the fraction with numerator 1 and denominator cap R the square root of cap C sub 1 cap C sub 2 end-root end-fraction
Q=12C1C2cap Q equals one-half the square root of the fraction with numerator cap C sub 1 and denominator cap C sub 2 end-fraction end-root Calculation Steps for Tuning the Stages If you choose a global resistor value and target cutoff frequency ωcomega sub c , you calculate the capacitors for each stage ( ) using its distinct Qicap Q sub i Calculate capacitor ratio: Find Stage 1 Capacitors ( Find Stage 2 Capacitors ( Find Stage 3 Capacitors ( By cascading these stages from lowest to highest →right arrow →right arrow
Stage 1), you prevent the active components from clipping due to resonant peaking and ensure optimal signal headroom. Summary of Attributes Specification Passband Behavior Maximally flat (No ripple) Stopband Roll-Off Total Pole Count 6 poles (3 complex conjugate pairs) Hardware Required
3 Op-Amps, 6 Resistors, 6 Capacitors (Active implementation)
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