At this page I stumbled upon the following section and highlighted the part in question:

enter image description here

How come the total cascaded RC filters' response be the multiplication of each RC filter's transfer function? Shouldn't there be an ideal non-inverting buffer in between them to claim this?

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    \$\begingroup\$ FYI for those interested, the three stage passive network has been approached here (but not placed into a more standard form -- I see I may need to fix that when I get a moment.) \$\endgroup\$ – jonk Nov 30 '20 at 5:52
  • \$\begingroup\$ If the impedance scale as you go down the filter, then you can approximate what he's asserting, but it rapidly becomes impractical as the number of stages increases. So you could have two stages as perhaps 1k/1uF followed by 100k/10nF, but you run out of useful resistor range very quickly with more stages. \$\endgroup\$ – Neil_UK Nov 30 '20 at 8:19

You're absolutely correct. Just cascading RC stages as in the schematic causes each stage to load the others. While the result is a lowpass filter with an order equal to the number of stages, the poles end up distributed along the real number line. They pretty much can't lie one atop the other (i.e., they can't follow the transfer function given on that page) without the buffer amplifiers you mention.

It just shows that you can't trust self-styled experts on the web!

  • \$\begingroup\$ Thanks I was doubting myself since they have a very good format and seems they spent so much time on its style with all those plots. \$\endgroup\$ – pnatk Nov 30 '20 at 3:03
  • \$\begingroup\$ Is the reason the buffer has to be there is to ensure the stage 2 doesn’t draw current from stage 1? \$\endgroup\$ – Aaron Nov 30 '20 at 4:30
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    \$\begingroup\$ Yes -- that's pretty much what I meant when I said one stage would load the other. \$\endgroup\$ – TimWescott Nov 30 '20 at 5:47
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    \$\begingroup\$ @Aaron Also see this. \$\endgroup\$ – jonk Nov 30 '20 at 5:51

It's not actually a lie but it's definitely misleading.

The full version is that the transfer function of each block is not fully defined without considering the source and load impedances.

The simple form of the transfer function for a single section

H(w) = 1/(1+jwRC)

is only true when fed from 0 ohms source and driving infinite load impedance.

Thus, as you correctly state, raising this to the n'th power is only correct with a buffer (Zin = inf, Zout = 0) between stages.

Allowing for the first stage as the source impedance of the second stage (and the third stage as its load impedance), the statement that the overall response is the product of each section becomes true again.

But the maths quickly becomes much more complex, hence Spice simulators...

however for some purposes, you can approximate this to some level of accuracy by decreeing R/10 to be approximately 0, and 10R to be approximately infinity, and cascading three stages with the same RC product, as R/10 * 10C, R*C, and 10R * C/10.

By minimising the loading of each stage on its predecessors, and by minimising the source impedance of each following stage, this can get close to teh desired N'th order response.

I would simulate this to find its limits, and it can't realistically be pushed beyond 2 or 3 stages.

In any case it's massively overdamped; once you introduce a buffer you are into the realms of much more optimal (e.g. Sallen and Key) filters where the second-order sections give you better control over frequency response and damping.


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