What is the difference between Butterworth, Chebyshev, and elliptic responses?

These are three mathematical approximations for the ideal lowpass response, each trading different parameters. Butterworth (maximally flat): no passband ripple, smooth roll-off of 20N dB per decade (N = filter order). A 5th-order Butterworth at 1 GHz gives 100 dB per decade, reaching 40 dB rejection at 2.5 GHz. Chebyshev (equiripple): allows controlled passband ripple (0.1 to 3 dB) in exchange for steeper transition. A 5th-order Chebyshev with 0.5 dB ripple reaches 40 dB rejection at 1.8 GHz, much closer to the cutoff. Elliptic (Cauer): allows both passband ripple and stopband ripple in exchange for the steepest possible transition. A 5th-order elliptic reaches 40 dB rejection at 1.3 GHz but has finite rejection peaks that limit ultimate out-of-band attenuation. Choose Butterworth when flat passband is critical. Choose Chebyshev when transition sharpness matters and some ripple is tolerable. Choose elliptic when minimum order (size, cost) is the priority.

How many filter sections do I need to suppress harmonics?

The required filter order depends on the harmonic frequency relative to the cutoff and the required suppression. For a 900 MHz transmitter needing 60 dB suppression at 1800 MHz (second harmonic): the frequency ratio is 2:1. A Butterworth filter rolls off at 20N dB/decade; at an octave above cutoff, the attenuation is approximately 6N dB. For 60 dB: N = 10 (impractical). A Chebyshev with 0.5 dB ripple achieves roughly 10N dB at an octave, needing N = 6. An elliptic filter achieves 60 dB with N = 4 or 5. For the third harmonic at 2700 MHz (3:1 ratio), the attenuation is much higher and even a 3rd-order filter provides adequate rejection. The general rule: specify the filter to suppress the worst-case harmonic (usually the 2nd), and all higher harmonics will be suppressed by even more.

What are the implementation options for RF lowpass filters?

Lumped element (LC): inductors and capacitors in a ladder network. Size: 2 to 10 mm. Used below 6 GHz where chip inductors and capacitors have adequate Q. Performance limited by component parasitics above 3 to 4 GHz. Stepped impedance: alternating sections of high and low impedance microstrip or stripline on a PCB. The wide sections act as shunt capacitors and the narrow sections act as series inductors. No components needed. Used from 1 to 30 GHz. Size depends on wavelength. Coupled-line: parallel coupled microstrip lines implement bandpass behavior that can be adapted for lowpass with appropriate coupling. Cavity or waveguide: for very high power (kilowatts) and very high frequency (above 10 GHz) where insertion loss must be minimized. Size: centimeters. Cost: high.

Passive Components

Lowpass Filter

A 900 MHz cellular transmitter outputs +43 dBm at the fundamental and −20 dBc at the second harmonic (1800 MHz): that is +23 dBm of unwanted energy radiating directly into the DCS-1800 band used by another operator. Regulatory limits require −30 dBm: a 53 dB gap. A 5th-order elliptic lowpass filter with 950 MHz cutoff provides 0.3 dB passband loss and 60 dB rejection at 1800 MHz, solving the problem in a 4 × 3 mm footprint. The lowpass filter is the most common filter type in RF transmitter chains, and its primary job is simple: pass the fundamental, kill the harmonics.

Category: Passive Components

Primary Use: Harmonic suppression

Roll-off: 20N dB/decade (Butterworth)

Filter Approximation Comparison (5th Order, f_c = 1 GHz)

Type	Passband Ripple	Roll-off	40 dB Reached At	Group Delay Var.	Best For
Butterworth	0 dB (flat)	100 dB/decade	2.5 GHz	Low	Flat passband required
Chebyshev 0.1 dB	0.1 dB	~130 dB/decade	2.0 GHz	Moderate	Sharp transition, low ripple
Chebyshev 0.5 dB	0.5 dB	~150 dB/decade	1.8 GHz	Moderate-high	Compact harmonic filter
Elliptic	0.5 dB	Steepest	1.3 GHz	High	Minimum order / size
Bessel	0 dB	~60 dB/decade	4.0 GHz	Lowest	Pulse / digital signals

Butterworth attenuation:
A(f) = 10·log(1 + (f/f_c)^2N) dB
N = 5, f = 2f_c: A = 10·log(1 + 32) = 30.1 dB

Required order for target rejection:
N ≥ log(10^A/10 − 1) / (2·log(f_stop/f_c)) (Butterworth)

Harmonic rule of thumb:
Design for 2nd harmonic (worst case); higher harmonics get more rejection automatically

Common Questions

Frequently Asked Questions

Butterworth vs. Chebyshev vs. Elliptic?

Butterworth: zero ripple, smoothest roll-off. Chebyshev: allows passband ripple for steeper transition. Elliptic: ripple in both bands, steepest possible transition, minimum order. Choose based on: flatness, sharpness, or size priority.

How many sections for harmonics?

Depends on harmonic ratio and type. 2nd harmonic (2:1): Butterworth N=10 (impractical), Chebyshev N=6, Elliptic N=4. 3rd harmonic (3:1): N=3 sufficient. Design for worst case (2nd harmonic).

Implementation options?

Lumped LC: <6 GHz, 2 to 10 mm, limited by component Q. Stepped impedance microstrip: 1 to 30 GHz, no components needed. Cavity/waveguide: high power (kW), lowest IL, largest size. Digital FIR filters provide arbitrary sharp roll-off but only work after the ADC; an analog anti-alias LPF is always needed before sampling to prevent irreversible aliasing.

Related Terms

← LNA Microstrip →

Filter Design

Lowpass Filter Order Calculator

Enter cutoff frequency, stopband frequency, and required rejection. Get the minimum filter order for Butterworth, Chebyshev, and elliptic approximations with component values.

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