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Butterworth Filter Lowpass Prototype Element Values

Simulations of Normalized and Denormalized LP, HP, BP, and BS Filters

Prototype Lowpass Filter Schematics (Butterworth, Chebyshev, Bessel) - RF Cafe

Lowpass Filters
(above)

Highpass Filters
(above)

Bandpass and Bandstop Filters
(above)

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, 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 denormalization equations.  Complex poles are here.

Prototype filter schematic - capacitor input - RF Cafe

Capacitor Input

Prototype filter schematic - inductor input - RF Cafe

Inductor Input

 Capacitor Input, RS=RL=1 Ω, f=1 rad/sec
OrderC1L2C3L4C5L6C7L8C9L10
12.000         
21.414211.41421        
31.000002.000001.00000       
40.765371.847761.847760.76537      
50.618031.618032.000001.618030.61803     
60.517641.414211.931851.931851.414210.51764    
70.445041.246981.801942.000001.801941.246980.44504   
80.390181.111141.662941.961571.961571.662941.111140.39018  
90.347301.000001.532091.879382.000001.879381.532091.000000.34730 
100.312870.907981.414211.782011.975381.975381.782011.414210.907980.31287
 L1C2L3C4L5C6L7C8L9C10
 Inductor Input, RS=RL=1 Ω, f=1 rad/sec
Related Pages on RF Cafe
- How to Use Filter Equations in a Spreadsheet
- Filter Transfer Functions
- Filter Equivalent Noise Bandwidth
- Filter Prototype Denormalization
- 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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