Active Filters: Butterworth and Chebyshev

An analog filter passes a prescribed frequency band and attenuates frequencies outside it. An active filter combines resistors and capacitors with an amplifying device, normally an op-amp, to provide gain and buffering without inductors. The order $n$ is the highest power of $s$ in the denominator and, equivalently, the number of poles; larger $n$ gives a sharper transition.

Active filters offer voltage gain, high input impedance, low output impedance, easy cascading and practical low-frequency realization. Their limits are the op-amp GBW, slew rate, noise, output range and the tolerances of $R$ and $C$.

Response Classes and Basic Quantities

Idealized low-pass, high-pass, band-pass and band-stop responses.

The passband meets the allowed attenuation or ripple; the stopband meets a minimum attenuation; the region between them is the transition band. At a Butterworth cutoff, the magnitude is $1/\sqrt{2}$ of passband gain, or $-3.01\,\mathrm{dB}$. Each pole ultimately contributes $20\,\mathrm{dB/decade}$ ($6\,\mathrm{dB/octave}$) attenuation.

For a band-pass response,

$$\boxed{BW=f_H-f_L},\qquad \boxed{f_0=\sqrt{f_Lf_H}},\qquad \boxed{Q=\frac{f_0}{BW}}.$$

The geometric-center relation is exact for the standard second-order form.

First-Order Active Low-Pass and High-Pass Filters

Buffered first-order active filters with passband gain $K=1+R_f/R_g$.

The op-amp does not load the RC node. Applying the impedance-divider rule gives

$$\begin{aligned} H_{LP}(s)&=\boxed{\frac{K}{1+sRC}}, & H_{HP}(s)&=\boxed{\frac{KsRC}{1+sRC}},\\ \omega_c&=\frac{1}{RC}, & f_c&=\boxed{\frac{1}{2\pi RC}},\qquad K=1+\frac{R_f}{R_g}. \end{aligned}$$

At $f_c$, both magnitudes are $K/\sqrt2$. The low-pass phase changes from $0^\circ$ toward $-90^\circ$ and rolls off at $-20\,\mathrm{dB/decade}$; the high-pass rises at $+20\,\mathrm{dB/decade}$ below cutoff and approaches $K$ with phase tending to $0^\circ$ above cutoff.

Note

Example — First-order low-pass design

Design a unity-gain low-pass for $f_c=1\,\mathrm{kHz}$. Choose $C=10\,\mathrm{nF}$; then

$$R=\frac{1}{2\pi f_cC} =\frac{1}{2\pi(10^3)(10\times10^{-9})} =15.9\,\mathrm{k\Omega}.$$

The standard value $15.8\,\mathrm{k\Omega}$ gives $f_c\simeq1.007\,\mathrm{kHz}$. At $10f_c$, $|H|/K=1/\sqrt{101}=0.0995$, or approximately $-20.0\,\mathrm{dB}$.

Second-Order Sallen–Key Sections

A general second-order denominator is

$$D(s)=s^2+\frac{\omega_0}{Q}s+\omega_0^2.$$

Therefore

$$H_{LP}(s)=\frac{K\omega_0^2}{D(s)},\qquad H_{HP}(s)=\frac{Ks^2}{D(s)},\qquad H_{BP}(s)=\frac{K_B(\omega_0/Q)s}{D(s)}.$$

Low-pass and high-pass sections attenuate at $40\,\mathrm{dB/decade}$ in their stopbands. The band-pass peaks at $\omega_0$ and its selectivity is set by $Q$.

Second-order Sallen–Key low-pass and dual high-pass variants.

For the shown low-pass network,

$$\omega_0=\frac{1}{\sqrt{R_1R_2C_1C_2}},\qquad Q=\frac{\sqrt{R_1R_2C_1C_2}} {C_2(R_1+R_2)+C_1R_1(1-K)}.$$

With equal $R_1=R_2=R$, $C_1=C_2=C$, both dual circuits have

$$\boxed{\omega_0=\frac{1}{RC}},\qquad \boxed{Q=\frac{1}{3-K}}.$$

Thus an equal-component second-order Butterworth section needs $Q=1/\sqrt2$, hence $K=3-\sqrt2\simeq1.586$. A unity-gain equal-component section has $Q=0.5$, not the Butterworth value.

First- and second-order asymptotic behavior: $20$ and $40\,\mathrm{dB/decade}$.

Note

Example — Equal-component second-order Butterworth section

Design a low-pass with $f_c=2\,\mathrm{kHz}$ using $C_1=C_2=10\,\mathrm{nF}$. Then

$$R_1=R_2=\frac{1}{2\pi f_cC}=7.96\,\mathrm{k\Omega}.$$

Choose $R_g=10\,\mathrm{k\Omega}$ and $R_f=(K-1)R_g=5.86\,\mathrm{k\Omega}$ to obtain $K=1.586$ and $Q=1/\sqrt2$. The op-amp GBW should comfortably exceed $Kf_c$ and its slew rate must satisfy $SR\ge2\pi fV_p$.

Active Band-Pass Realization and Specifications

A practical wideband active band-pass may cascade a first-order high-pass at $f_L$ and a first-order low-pass at $f_H$; buffering prevents interaction:

$$H(s)=K\frac{s}{s+\omega_L}\frac{\omega_H}{s+\omega_H}, \qquad \omega_L<\omega_H.$$

This is second order overall. For a narrow, high-$Q$ band, use the standard second-order band-pass form above in a multiple-feedback or state-variable realization, because widely separated one-pole sections cannot produce high selectivity.

Buffered second-order wideband band-pass and its practical transmission specifications.

Note

Example — Wideband band-pass design

For $f_L=300\,\mathrm{Hz}$ and $f_H=3\,\mathrm{kHz}$, choose $C_L=C_H=10\,\mathrm{nF}$. Then

$$R_L=\frac{1}{2\pi f_LC_L}=53.1\,\mathrm{k\Omega},\qquad R_H=\frac{1}{2\pi f_HC_H}=5.31\,\mathrm{k\Omega}.$$

The nominal center is $f_0=\sqrt{300(3000)}=949\,\mathrm{Hz}$, $BW=2.70\,\mathrm{kHz}$ and $Q=0.351$, confirming a broad response.

Butterworth Approximation

The Butterworth response is maximally flat at DC and monotonic in both passband and stopband. For unity DC gain,

$$\boxed{|H_B(j\omega)|^2= \frac{1}{1+(\omega/\omega_c)^{2n}}}.$$

The first $2n-1$ derivatives of $|H|^2$ at $\omega=0$ vanish. At $\omega=\omega_c$, $|H|=1/\sqrt2$. Its ultimate stopband slope is $-20n\,\mathrm{dB/decade}$ and its phase is nonlinear.

The first two normalized low-pass forms are

$$H_1(s)=\frac{\omega_c}{s+\omega_c},\qquad \boxed{H_2(s)=\frac{\omega_c^2} {s^2+\sqrt2\,\omega_cs+\omega_c^2}},$$

with second-order damping $\zeta=1/\sqrt2$ and $Q=1/\sqrt2$. The stable poles lie equally spaced on the left half of a circle of radius $\omega_c$:

$$s_k=\omega_c\exp\!\left[j\frac{\pi(2k+n-1)}{2n}\right], \qquad k=1,2,\ldots,n.$$

Given maximum passband attenuation $A_p$ at $\omega_p$ and minimum stopband attenuation $A_s$ at $\omega_s>\omega_p$, the minimum Butterworth order is

$$\boxed{n\ge \frac{\log\!\left[(10^{A_s/10}-1)/(10^{A_p/10}-1)\right]} {2\log(\omega_s/\omega_p)}}.$$

Always choose the next integer. For the selected $n$, the permissible cutoff interval is

$$\frac{\omega_p}{(10^{A_p/10}-1)^{1/(2n)}}\le\omega_c\le \frac{\omega_s}{(10^{A_s/10}-1)^{1/(2n)}}.$$

Note

Example — Butterworth order and cutoff from specifications

Require $A_p=1\,\mathrm{dB}$ at $f_p=1\,\mathrm{kHz}$ and $A_s=40\,\mathrm{dB}$ at $f_s=4\,\mathrm{kHz}$:

$$n\ge\frac{\log_{10}[(10^4-1)/(10^{0.1}-1)]} {2\log_{10}4}=3.81\quad\Rightarrow\quad\boxed{n=4}.$$

The valid range is $1.184\,\mathrm{kHz}\le f_c\le1.265\,\mathrm{kHz}$. Choosing $f_c=1.20\,\mathrm{kHz}$ gives approximately $A(1\,\mathrm{kHz})=0.91\,\mathrm{dB}$ and $A(4\,\mathrm{kHz})=41.8\,\mathrm{dB}$. Implement the fourth order as two stable second-order sections.

Chebyshev Approximation

A Type-I Chebyshev filter accepts controlled equal ripple in the passband to obtain a sharper transition than a Butterworth filter of the same order. Its normalized low-pass magnitude is

$$\boxed{|H_C(j\omega)|^2= \frac{1}{1+\varepsilon^2T_n^2(\omega/\omega_p)}},\qquad \boxed{\varepsilon=\sqrt{10^{R_p/10}-1}},$$

where $R_p$ is passband ripple in dB and

$$T_n(x)=\cos\!\bigl(n\cos^{-1}x\bigr),\quad |x|\le1, \qquad T_{n+1}=2xT_n-T_{n-1}.$$

The first polynomials are $T_0=1$, $T_1=x$, $T_2=2x^2-1$ and $T_3=4x^3-3x$. Type I has a monotonic stopband and generally more nonlinear phase/group delay than Butterworth. Type II (inverse Chebyshev) instead has a flat passband and equiripple stopband.

For Type I specifications, the order is

$$\boxed{n\ge \frac{\cosh^{-1}\!\sqrt{(10^{A_s/10}-1)/(10^{R_p/10}-1)}} {\cosh^{-1}(\omega_s/\omega_p)}}.$$

Same-order Butterworth and Type-I Chebyshev responses: flatness versus selectivity.

For the preceding $1/40\,\mathrm{dB}$ and $4{:}1$ edge specifications,

$$n_C\ge \frac{\cosh^{-1}\sqrt{(10^4-1)/(10^{0.1}-1)}}{\cosh^{-1}(4)} =2.90,$$

so a third-order Chebyshev meets the selectivity for which Butterworth needed fourth order.

Property	Butterworth	Chebyshev Type I
Passband	Maximally flat, monotonic, no ripple	Equiripple within specified $R_p$
Passband edge	$-3.01\,\mathrm{dB}$ at $\omega_c$	Reaches $-R_p\,\mathrm{dB}$ at $\omega_p$
Transition/order	More gradual; usually higher order	Sharper; usually lower order
Stopband	Monotonic	Monotonic
Ultimate slope	$20n\,\mathrm{dB/decade}$	Also $20n\,\mathrm{dB/decade}$, reached sooner near edge
Phase/group delay	Nonlinear but generally smoother	More nonlinear and variable
Use	Audio, measurement, general filtering	Channel selection where ripple is acceptable

Butterworth and Type-I Chebyshev comparison.