[sdiy] Passive filters and impedances

[Neil Johnson]

Tom,

Don is right in promoting the virtue of learning the process by hand, both
to get a feel for the problem at hand, and to know how to drive the tools
later on.

To give you a taster, here are the expressions for the transfer function
for the three cases:
(%i = the imaginary operator or "j" as used by electrical/electronics
folks, %pi = PI)

1/ Single-pole R/C/R case:

Rimp/((2*%i*%pi*f*Rimp*C1+1)*R1+Rimp)

2/ Two-pole case:

Rimp/(((4*%pi^2*f^2*Rimp*C1*C2-2*%i*%pi*f*C1)*R1-2*%i*%pi*f*Rimp*C2-1)*R2+(-2*%i*%pi*f*Rimp*C2-2*%i*%pi*f*Rimp*C1-1)*R1-Rimp)

3/ Three-pole case:

-(Rimp)/(
(((8*%i*%pi^3*f^3*Rimp*C1*C2*C3+4*%pi^2*f^2*C1*C2)*R1+4*%pi^2*f^2*Rimp*C2*C3-2*%i*%pi*f*C2)*R2+((4*%pi^2*f^2*Rimp*C2+4*%pi^2*f^2*Rimp*C1)*C3-2*%i*%pi*f*C2-2*%i*%pi*f*C1)*R1-2*%i*%pi*f*Rimp*C3-1)*R3+
((4*%pi^2*f^2*Rimp*C1*C3+4*%pi^2*f^2*Rimp*C1*C2-2*%i*%pi*f*C1)*R1-2*%i*%pi*f*Rimp*C3-2*%i*%pi*f*Rimp*C2-1)*R2+(-2*%i*%pi*f*Rimp*C3-2*%i*%pi*f*Rimp*C2-2*%i*%pi*f*Rimp*C1-1)*R1-Rimp)

>From these expressions you can then extract the amplitude and phase
responses.  To find out where the poles are just work out the roots of the
denominators.  Things are much simpler without Rimp.

I would certainly recommend having a play with the ABCD matrix method - it
is very easy to do by hand for the 1-pole and 2-pole cases; it explodes for
the 3-pole case though (see above).

Although for simple designs follow Richie's advice and make each following
stage's resistor 10x the previous one, e.g., R1=1k,
R2=10k,R3=100k,Rimp=1Meg and scale the caps according to taste.

Good luck!

Neil
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