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Butterworth Filter Prototype Element Values

Butterworth poles lie along a circle and are spaced at equal angular distances around a circle. It is designed to have a frequency response which is as flat as mathematically possible in the passband, and is often referred to as a 'maximally flat magnitude' filter. Prototype value real and imaginary pole locations (ω=1 at the 3 dB cutoff point) for Butterworth filters are presented in the table below.

The Butterworth type filter was first described by the British engineer Stephen Butterworth in his paper "On the Theory of Filter Amplifiers", Wireless Engineer (also called Experimental Wireless and the Wireless Engineer), vol. 7, 1930, pp. 536-541.

The table below lists prototype element values for the normalized lowpass function, which assumes a cutoff frequency of 1 rad/sec and source and load impedances of 1 Ω. Either an input capacitor (top title line in table) or an input inductor (bottom title line in table) can be used.

Convert Butterworth prototype values to other cutoff frequencies, impedances, and to highpass, bandpass or bandstop using the equations here.

See my online Butterworth filter calculators and plotters here.

Complex poles are here.


RF Cafe - Prototype filter schematic - capacitor input           RF Cafe - Prototype filter schematic - inductor input

Capacitor Input                                                     Inductor Input

Order C1 L2 C3 L4 C5 L6 C7 L8 C9 L10
1 2.000                  
2 1.41421 1.41421                
3 1.00000 2.00000 1.00000              
4 0.76537 1.84776 1.84776 0.76537            
5 0.61803 1.61803 2.00000 1.61803 0.61803          
6 0.51764 1.41421 1.93185 1.93185 1.41421 0.51764        
7 0.44504 1.24698 1.80194 2.00000 1.80194 1.24698 0.44504      
8 0.39018 1.11114 1.66294 1.96157 1.96157 1.66294 1.11114 0.39018    
9 0.34730 1.00000 1.53209 1.87938 2.00000 1.87938 1.53209 1.00000 0.34730  
10 0.31287 0.90798 1.41421 1.78201 1.97538 1.97538 1.78201 1.41421 0.90798 0.31287
  L1 C2 L3 C4 L5 C6 L7 C8 L9 C10
Related Pages on RF Cafe
- Filter Transfer Functions
- Filter Equivalent Noise Bandwidth
- Filter Prototype Denormalization
- Filter Design Resources
- Bessel Filter Poles
- Bessel Filter Prototype Element Values
- Butterworth Lowpass Filter Poles
- Butterworth Filter Prototype Element Values
- Chebyshev Lowpass Filter Poles
- Chebyshev Filter Prototype Element Values
- Monolithic Ceramic Block Combline Bandpass
  Filters Design
- Coupled Microstrip Filters: Simple Methodologies for
  Improved Characteristics

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