Designing a fourth order Butterworth BPF



The perfect BPF would have little or no loss within the passband and infinite attenuation at all frequencies outside the passband. Such filters don't exist in the real world. The designer of a practical filter will always have to make some compromises. These compromises involve various trade-offs between filter bandwidth, passband loss, stopband attenuation, passband ripple, I/O matching and phase/delay characteristics.

Butterworth band-pass filters have a maximally flat response (minimum ripple) at frequencies near the centre of the passband. The cost of this flat response is inferior stopband attenuation when compared to some other types of filter. Chebyshev filters offer high stopband attenuation at a cost of increased passband ripple. The image below shows transmission loss through a 4th order filter for the impossible case of a perfect lossless filter (green) Butterworth (red) and Chebyshev with passband ripple of 0.5dB (blue).

Filter TL plots
Transmission through Ideal, Chebyshev and Butterworth filter.


The filter design procedure used here is based on work by Anatol Zverev[ref 1] and the many excellent articles about BPF filter design by Wes Hayward W7ZOI [ref 2].

The filter described here is one of a set of band pass filters for my new HF transceiver. One of the design goals for this project is to keep spurious signals to an absolute minimum. To this end, I have decided to use 4th order (four LC resonators) filters instead of the more commonly used 2nd or 3rd order filters. A generic example is shown below.

BPF schematic
4th order BPF


The optimum input/output termination resistance for this type of filter is usually higher than the industry standard value of 50 ohms. Typical values range from a few hundred ohms to several thousand ohms. There are a few different options for matching the filter I/O to 50 ohms. I have decided to use inductive coupling. This means L1 and L4 are transformers with a link coupling winding which is shown as 'Lk' in the schematic. This arrangement works well with either PIN diode or relay switching. The Lk to L turns ratio is sqrt(Rp/50) where Rp is the optimum termination resistance for the filter.

Filter design formula
Component value calculation



The filter is based on four parallel resonant LC circuits with capacitive top-coupling between each resonator. The first task is to choose a value of inductance for L1/2/3/4. An inductor with XL (inductive reactance) somewhere in the 50-150 ohms region would be a reasonable choice. To keep the design as simple as possible, all four inductors (L1-L4) will have the same value.

A coil wound on a T50-2 toroid will have an inductance of approximately N^2*5 nH, where N is the number of turns. 30 turns on a T50-2 toroid will have an inductance of 4500nH or 4.5 microhenries. This is a suitable value for use in a filter for the 80m amateur band. The European 80m band extends from 3.5MHz to 3.8MHz, with a centre frequency of 3.65MHz. 2*PI*f*L = 2*PI*3.65*4.5 gives us the inductive reactance (XL) value of 103 ohms. This is right in the middle of the required range.

To make a filter for the 80m band, we should start by specifying a centre frequency and filter bandwidth.
Centre frequency: f = 3.65MHz
Bandwidth: B = 0.4MHz

Most calculations define bandwidth as the -3dB BW (half power bandwidth). Many applications call for a much smaller variation in filter loss within the filter passband. I will use a figure of -1dB to specify the bandwidth of the 80m filter. Graphs from Zverev (Ref 1) show that the 1dB BW of a N=4 Butterworth filter is about 0.84 times the 3dB BW. 0.4MHz * 0.84 = 0.336MHz. This is just enough to cover the 300kHz amateur band.

The next task is to find the value of capacitance that will resonate with L at the filter centre frequency. From the formula above:
1000000/((2*PI*f)²×4.5) = 422.5pF
This is the nodal capacitance or the total value of capacitance in each resonator.

The value for C12 and C34 is:
422.5×((0.8409×0.4)/3.65) = 39pF

A different k (coupling) value is used for C23:
422.5×((0.5412×0.4)/3.65) = 25pF
I used a 27pF ceramic disc capacitor.

C1 and C4 is simply 422.5-39 = 383.5pF

C2 and C3 = 422.5-39-27 = 356.5pF

I used 330pF in parallel with a 65pF trimmer capacitor. Take care if you are using low tolerance capacitors. Errors of 10% or more would place resonance beyond the tuning range of the trimmers. If possible, measure the actual value of the 330pF capacitors before using them.

The filter end section Q is used to calculate the I/O (source/load) resistance. A resonator Q of 200 was used for the calculation. k and q values for a 4th order Butterworth are shown above.

(0.7654×3.65×200)/(0.4×200-0.7654×3.65) = 7.237

The optimum I/O resistance is:
2×PI×3.65×4.5×7.237 = 747 ohms

The I/O coupling link turns ratio is:
sqrt(747÷50) = 3.865. The number of turns required is 30/3.865=7.762. As we can only use whole turns on a toroid core, I used 8 turns. The final filter design is shown below.

80m filter
80m filter


I used the scientific calculator for the Gnome Desktop (gcalctool) for calculating the component values. See details below.

calculations

I have written a simple C program to do the above calculations. The source code is here: filter.c
It is written in ANSI C, so it should be portable to just about any system where a C compiler is available. I have compiled and tested it with GCC on Linux and TCC on Linux and Windows. This simple tool was used to verify the design above and to design filters for all of the other amateur bands from 1.8MHz to 52MHz. T50-2 toroid cores were used for frequencies below 10MHz. T50-6 cores were used for frequencies above 10MHz. The 17m (18MHz) filter is shown below.

17m BPF
Testing the 17m filter


John Charlton G3VRF has produced a very useful spreadsheet which can be used for calculating the filter component values. Fourth Order Butterworth.xls


References:
[1] Handbook of FILTER SYNTHESIS. Anatol I. Zverev. Wiley 1967.

[2] http://w7zoi.net/tech.html

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