Butterworth Low-Pass Filter Design

When designing analog filters, the Butterworth low-pass filter is a popular choice due to its maximally flat frequency response in the passband. This RF filter design tutorial guide you through the process of using a low-pass prototype table to design a Butterworth low-pass filter. Consider the following circuit with Rs=50Ohm and RL=100Ohm.

Let's say the order of the butterworth filter is 4, that is $n = 4$

Now below is the guide to build the filter.

Understanding the Normalization

To begin, we need to normalize our terminations for $R L = 1 Ω$ . This normalization yields a value of $R s = 0.5$ . However, if we look at the Butterworth low-pass prototype table for $n=4$ , we notice that there is no $Rs/RL = 0.5$ ratio listed in the table below.

Finding the Right Values

Our next step is to choose the next best option: $R L / R s = 2$ . By reading up from the bottom of the table below, we can determine the correct values for the components in our low-pass prototype circuit.

Component Values for $n = 4$ and $R L / R s = 2$

Based on the Butterworth low-pass prototype table, the values are:

$L 1 = 0.218$
$C 1 = 2.452$
$L 2 = 0.883$
$C 2 = 3.187$

These values will form the basis of our Butterworth low-pass filter circuit.

Schematic Representation

To visualize the design, refer to the schematic that accompanies the table. This schematic will guide you on how to arrange the components based on the calculated values.

prototype Low Pass filter design circuit

Conclusion

Using the low-pass prototype approach, you can easily design a Butterworth low-pass filter that meets your specific needs. The key takeaway is to be flexible with your component ratios and to refer back to the table to find the closest matching values.

For more insights into filter design, check out my previous blog posts:

By understanding the process of using a low-pass prototype table, you'll be better equipped to tackle various filter designs in your projects.

Here’s the complete Butterworth Low-Pass Prototype Element Values table :

Butterworth Low-Pass Prototype Element Values

Order (n)	$\frac{Rs}{RL}$	C1	L2	C3	L4	C5	L6	L7
2	1.111	1.035	1.835
	1.250	0.849	2.121
	1.429	0.697	2.439
	1.667	0.566	2.828
	2.000	0.448	3.346
	2.500	0.342	4.095
	3.333	0.245	5.313
	5.000	0.156	7.707
	10.000	0.074	14.814
	$\infty$	1.414	0.707
3	0.900	0.808	1.633	1.599
	0.800	0.844	1.384	1.926
	0.700	0.915	1.165	2.277
	0.600	1.023	0.965	2.702
	0.500	1.181	0.779	3.261
	0.400	1.425	0.604	4.064
	0.300	1.838	0.440	5.363
	0.200	2.669	0.284	7.910
	0.100	5.167	0.138	15.455
	$\infty$	1.500	1.333	0.500
4	1.111	0.466	1.592	1.744	1.469
	1.250	0.388	1.695	1.511	1.811
	1.429	0.325	1.862	1.291	2.175
	1.667	0.269	2.103	1.082	2.613
	2.000	0.218	2.452	0.883	3.187
	2.500	0.169	2.986	0.691	4.009
	3.333	0.124	3.883	0.507	5.338
	5.000	0.080	5.684	0.331	7.940
	10.000	0.039	11.094	0.162	15.642
	$\infty$	1.531	1.577	1.082	0.383
5	0.900	0.442	1.027	1.910	1.756	1.389
	0.800	0.470	0.866	2.061	1.544	1.738
	0.700	0.517	0.731	2.285	1.333	2.108
	0.600	0.586	0.609	2.600	1.126	2.552
	0.500	0.686	0.496	3.051	0.924	3.133
	0.400	0.838	0.388	3.736	0.727	3.965
	0.300	1.094	0.285	4.884	0.537	5.307
	0.200	1.608	0.186	7.185	0.352	7.935
	0.100	3.512	0.091	14.095	0.173	15.710
	$\infty$	1.545	1.694	1.382	0.894	0.309
6	1.111	0.289	1.040	1.322	2.054	1.744	1.335
	1.250	0.245	1.116	1.126	2.239	1.550	1.688
	1.429	0.207	1.236	0.957	2.499	1.346	2.062
	1.667	0.173	1.407	0.801	2.858	1.143	2.509
	2.000	0.141	1.653	0.654	3.369	0.942	3.094
	2.500	0.111	2.028	0.514	4.141	0.745	3.931
	3.333	0.082	2.656	0.379	5.433	0.552	5.280
	5.000	0.054	3.917	0.248	8.020	0.366	7.360
	10.000	0.025	8.103	0.124	16.188	0.182	15.015
	$\infty$	1.553	1.759	1.553	1.202	0.758	0.259
7	0.900	0.299	0.711	1.404	1.489	2.125	1.727	1.296
	0.800	0.322	0.606	1.517	1.278	2.334	1.546	1.652
	0.700	0.357	0.515	1.688	1.091	2.618	1.350	2.028
	0.600	0.408	0.432	1.928	0.917	3.005	1.150	2.477
	0.500	0.480	0.354	2.273	0.751	3.553	0.951	3.064
	0.400	0.590	0.278	2.795	0.592	4.380	0.754	3.904
	0.300	0.775	0.206	3.671	0.437	5.761	0.560	5.258
	0.200	1.145	0.135	5.427	0.287	8.526	0.369	7.908
	0.100	2.257	0.067	10.700	0.142	16.822	0.182	15.748
	$\infty$	1.558	1.799	1.659	1.397	1.055	0.656	0.223
	$n$	$\frac{RL}{Rs}$	$L1$	$C2$	$L3$	$C4$	$L5$	$L6$

This table summarizes the Butterworth low-pass filter prototype element values, ensuring all requested information is included and clearly organized.