Introduction
We discussed passive filters a bit, and now we are going to make life a little easier for us. As you may have observed, the filters are designed with the same equations everytime, so that brings up an interesting idea: why not just design the filter ONCE for a corner frequency of 1 radian per second and a characteristic impedance of 1 ohm, then just scale the design for other frequencies and impedances? Good idea. Other people thought so too, so that's exactly what they did.
Lowpass Prototype Filters
Filters are usually designed from the lowpass prototypes suggested above, with a corner at 1 r/s and source and load at 1 ohm. Image parameter or constant-K lowpass prototypes filters are really easy to remember. In the half-section, the series inductor is 1 and the shunt capacitor is 1. That makes the filter values always 1-2-2-2-...-2-1. The m-derived end section is 0.6 for the series L, and 1.067 for the shunt L and 0.6 for the shunt C.
Scaling Frequency
This is pretty straightforward. Just reduce the prototype L's and C's by a factor of the radian frequency of the desired corner.
Scaling Impedance
For characteristic impedances other than 1 ohm, raise the series leg's impedance and the shunt leg's impedance by a factor of the new impedance value. That's all there is to it.
Example: 5-pole lowpass, image-parameter design, 40 MHz corner, 50 ohm source and load
Here we can write the values of the prototype lowpass half-section as 1-1, and the PLPF values as 1-2-2-2-1. If we scale to 40 MHz (250E6 r/s) and 50 ohms, the half-section L's become 0.2E-6 or 0.2 uH and the half-section C's become 8E-11 or 80 pF. So the values are 0.2 uH, 160 pF, 0.4 uH, 160 pF, and 0.2 uH (series-leg input). The same can be done for the m-derived end sections ... add 0.12 uH in series leg and 0.212 uH and 48 pF in the shunt leg. Neat!
Transformation to Highpass
We can transform the lowpass structure to a highpass structure easily by replacing all the inductive reactances by capacitive reactances and vice-versa. This would even apply to the m-derived sections.
Example: 5-pole highpass, 40 MHz corner, 300 ohms, balanced, shunt leg input.
The lowpass prototype half-section is L = 1 and C = 1. Transforming to highpass puts C = 1 in the series leg and L = 1 in the shunt leg. Scaling frequency and impedance gives a half-section of 13 pF series and 1.2 uH shunt. Balancing the structure (2x27 pF and 1.2 uH) and cascading half-sections gives us the same values we came up with before, 1.2 uH shunt, 2x13 pF series, 0.6 uH shunt, 2x13 pF series, 1.2 uH shunt.
Transformation to Bandpass
This gets a little more confusing, harder to keep things straight. I myself just get the values for the half-section, scaling for frequency width and impedance, resonate the legs at the center frequency, then cascade the transformed half-sections to come up with the final bandpass filter.
Example: 3-pole bandpass, 18 MHz center, 6 MHz wide, 50 ohms.
18 MHz: 113E6 r/s
6 MHz: 37.7E6 r/s
Scaled half-section (6 MHz, 50 ohms): L = 1.32 uH series, C = 530 pF shunt Resonated at 18 MHz: add C = 58 pF series, L = 0.147 uH shunt
Final values come out to ... 0.147 uH and 530 pF shunt, 2.64 uH and 29 pF series, 0.147 uH and 530 pF shunt ... just as before.
Conclusion to Part 3
The ideas of transforming prototype lowpass filters, frequency scaling, and impedance scaling can be applied to other filter designs such as Butterworth, Chebychev, or Cauer-elliptic. This is a very powerful tool. It is regularly used throughout filter design. Again, after the filter is synthesizd, check your work by doing a circuit analysis of the proposed filter design. It is S-O-O-O easy to drop a term. A computer spreadsheet might be a good investment here.
The next section will talk about the methods that can be used in narrow bandpass filter design and also consider using resonators not made out of the usual L's and C's.