# second order, passive RC filters

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#### xXGiuseppeXx

##### Newbie level 2
Hello all, I have always referred to this forum for any electrical questions but this is my first time posting; never thought I would have to!

Anyway my problem is this: I am playing around with passive, analog filters. I have been reading some theory here:

https://www.electronics-tutorials.ws/filter/filter_2.html

It all seems to make sense and I have made excel sheets and LTSpice simulations to solidify the theory.
However, the one thing that bugs me is the 2 pole RC filter. Towards the bottom I quote:

"In practice, cascading passive filters together to produce larger-order filters is difficult to implement accurately as the dynamic impedance of each filter order affects its neighboring network. However, to reduce the loading effect we can make the impedance of each following stage 10x the previous stage, so R2 = 10 x R1 and C2 = 1/10th C1."

Q1:
Why a factor of 10 and how do they correspond to each other? I know it is to prevent loading from the filter stages but I am unsure why. In my LTSpice simulations, a factor of 100 is much better and is close to the -3dB point.

Q2:
In the next section, 2 pole, high pass, RC filters are discussed but they do not give a rule on which components to proportion to what. I have tried the rule and something similar to that in my Q1 but it simply does not work.

I figure if I can understand how they came to the factor in Q1, Q2 should be easier to understand and perhaps I can synthesize the factors (that is, if proportioning the filter's components is possible to prevent loading of each other).

#### FvM

##### Super Moderator
Staff member
The cascaded RC filter shown in the tutorial isn't a good second order filter. Some points said about second order filters in the text don't apply to this filter. All commonly used filter types (Bessel, Butterworth, Chebychev) require complex poles for second order filters, which can be only implemented as LC or active RC filter. The said factor 10 impedance is a resonable rule of thumb to get almost decoupled first order filters.

#### Tunelabguy

##### Full Member level 5
Sure, 100x impedance at each stage is better than 10x, but at what cost? Like most things in electronic design this choice is a compromise. How close to 2nd order response do you need? How much impedance-raising are you willing to tolerate? If you did use 100x then after two stages of filtering you would have an output impedance that is 10,000 times higher than the input impedance. So if your signal is driving from a 500-Ohm source, the filtered version of that signal can only drive a 5,000,000-Ohm load. This may be OK if you are driving a FET-input amplifier, but most other loads are going to present a lower impedance to your filtered signal. Then depending on the frequency range you are thinking about, the capacitors necessary to implement this filter might be difficult-to-achieve sub picofarad capacitors, likely to be swamped by stray circuit capacitance. Overall you give yourself lots of headaches by forcing an exceptionally high output impedance.

#### xXGiuseppeXx

##### Newbie level 2
@ FvM
I see. So is there a site/book where I can a nice introduction to the subject? I kinda want to see where the equation come from. I have used this book for my introduction to electric filters:

https://www.amazon.com/Electric-Cir...&keywords=electric+circuit+nilson+8th+edition

However, they really did not explain any second order filters and that knowledge I picked up through online searching. I was looking into this book but I am not sure if this will point me in the right direction:

https://www.amazon.com/gp/offer-listing/0070704341/ref=dp_olp_used?ie=UTF8&condition=used

I have also look at LC filter calculators but there is no explanation where it all comes from:

https://www.the12volt.com/caraudio/cross12db.asp

@Tunelaguy
Yes the trade off was obvious to me as well. I didn't think about the stray capacitance though. I was more concerned on the availability of the components. It would be nice to get to 5% - 10% within the cutoff frequency with the pass band "hugging" the 0dB line (I know 0dB is impossible to achieve but one can get close no?). And yes, the output impedance is poor as well unless it is going to into a FET or a high impedance input of a meter or oscilloscope. I guess an LC filter fixes the output impedance but doesn't it make the input impedance so low that it would "stress" the source which I guess would be bad if it can't deliver too much current? I know active filters can get around all of this but I would like to deal with passive ones right now.

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#### FvM

##### Super Moderator
Staff member
I'm sure that the Electronic Filter Design Handbook or other profound text books can answer all your questions. Depending on how deeply you want to get involved with electronic filter design, it might be too extensive.

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