Chebyshev Lowpass Filter Poles

Chebyshev poles lie along an ellipse, rather than a circle like the Butterworth and Bessel. Prototype value real and imaginary pole locations (ω=1 at the ripple attenuation cutoff point) for Chebyshev filters are presented in the table below.

See Espresso Engineering Workbook (free) for a calculator.

Pole locations are calculated as follows, where K=1,2,...,n. n is the filter order.

where Chebyshev pole equation sinh arguement - RF Cafe Website

0.01 dB Ripple
Order (N)	Re Part (-σ)	Im Part (±jω)
1	20.82774
2	2.22776	2.33729
3	0.79469 1.58937	1.62621
4	0.41087 0.99192	1.35553 0.56148
5	0.25251 0.66109 0.81715	1.22820 0.75907
6	0.17147 0.46845 0.63992	1.15867 0.84820 0.31046
7	0.12426 0.34818 0.50313 0.55844	1.11664 0.89548 0.49695
8	0.09429 0.26852 0.40187 0.47404	1.08934 0.92350 0.61706 0.21668
9	0.07405 0.21321 0.32665 0.40070 0.42641	1.07060 0.94147 0.69879 0.37182
10	0.05971 0.17329 0.26991 0.34011 0.37701	1.05720 0.95371 0.75687 0.48594 0.16744

0.10 dB Ripple
Order (N)	Re Part (-σ)	Im Part (±jω)
1	6.55220
2	1.18618	1.38095
3	0.48470 0.96941	1.20616
4	0.26416 0.63773	1.12261 0.46500
5	0.16653 0.43599 0.53891	1.08037 0.66771
6	0.11469 0.31335 0.42804	1.05652 0.77343 0.28309
7	0.08384 0.23492 0.33947 0.37678	1.04183 0.83549 0.46366
8	0.06398 0.18220 0.27268 0.32165	1.03218 0.87504 0.58468 0.20531
9	0.05044 0.14523 0.22251 0.27294 0.29046	1.02551 0.90182 0.66935 0.35616
10	0.04079 0.11837 0.18437 0.23232 0.25753	1.02071 0.92080 0.73075 0.46917 0.16166

0.25 dB Ripple
Order (N)	Re Part (-σ)	Im Part (±jω)
1	4.10811
2	0.89834	1.14325
3	0.38361 0.76722	1.09155
4	0.21252 0.51306	1.05678 0.43773
5	0.13503 0.35350 0.43695	1.03788 0.64145
6	0.09339 0.25515 0.34854	1.02689 0.75173 0.27515
7	0.06845 0.19178 0.27714 0.30760	1.02001 0.81798 0.45395
8	0.05232 0.14900 0.22299 0.26304	1.01545 0.86085 0.57520 0.20198
9	0.04130 0.11890 0.18217 0.22347 0.23781	1.01227 0.89018 0.66071 0.35156
10	0.03342 0.09700 0.15108 0.19037 0.21102	1.00998 0.91112 0.72307 0.46424 0.15997

0.50 dB Ripple
Order (N)	Re Part (-σ)	Im Part (±jω)
1	2.86278
2	0.71281	1.00404
3	0.31323 0.62646	1.02193
4	0.17535 0.42334	1.01625 0.42095
5	0.11196 0.29312 0.36232	1.01156 0.62518
6	0.07765 0.21214 0.28979	1.00846 0.73824 0.27022
7	0.05700 0.15972 0.23080 0.25617	1.00641 0.80708 0.44789
8	0.04362 0.12422 0.18591 0.21929	1.00500 0.85200 0.56929 0.19991
9	0.03445 0.09920 0.15199 0.18644 0.19841	1.00400 0.88291 0.65532 0.34869
10	0.02790 0.08097 0.12611 0.15891 0.17615	1.00327 0.90507 0.71826 0.46115 0.15890

1.0 dB Ripple
Order (N)	Re Part (-σ)	Im Part (±jω)
1	1.96523
2	0.54887	0.89513
3	0.24709 0.49417	0.96600
4	0.13954 0.33687	0.98338 0.40733
5	0.08946 0.23421 0.28949	0.99011 0.61192
6	0.06218 0.16988 0.23206	0.99341 0.72723 0.26618
7	0.04571 0.12807 0.18507 0.20541	0.99528 0.79816 0.44294
8	0.03501 0.09970 0.14920 0.17600	0.99645 0.84475 0.56444 0.19821
9	0.02767 0.07967 0.12205 0.14972 0.15933	0.99723 0.87695 0.65090 0.34633
10	0.02241 0.06505 0.10132 0.12767 0.14152	0.99778 0.90011 0.71433 0.45863 0.15803

Data taken from "Filter Design," by Steve Winder, Newnes Press, 1998. This is a great filter design book, and I recommend you purchase a copy of it.

Related Pages on RF Cafe

- Butterworth Filter Equations for Magnitude, Phase, and Group Delay

- Chebyshev Filter Equations for Magnitude, Phase, and Group Delay

- Butterworth Lowpass Filter Gain, Phase, and Group Delay Equations

- Butterworth Highpass, Bandpass, & Bandstop Filter Gain, Phase, and Group Delay Equations

- 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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