Supposing I have a network filter of 3 RC filters connected in cascade configuration producing the filter H(w).
The single RC filter has a frequency response H_{RC}(w)
Is it correct the following?
phase(H(w)) = 3*phase(H_{RC}(w))
My doubt comes from the fact that the first and second filters when connected to form the cascade, exhibit a dependency from the network following them (the rest of the cascade).
Thanks in advance R
It's approximately true if the Rs go up rapidly as you go towards the output, but not otherwise.
For instance, say your circuit goes
In 0---------R1R1---*---R2R2---*---R3R3---*--0 Out | | | C1C C2C C3C C1C C2C C3C | | | GND GND GND
with R1 = 1k, R2 = 10k, and R3 = 100k, with the caps scaled to the same time constant, e.g. C1 = 1uF, C2 = 0.1 uF, C3 = 0.01 uF.
In this case, the phase shift is three times that of a single section, to reasonable accuracy (a few degrees).
If you swap all the caps, it works even better, though there are flatter spots between the RCs rolloffs of the individual sections.
However, if you make all the Rs the same, the approximation falls apart completely. You can see this by looking at the limits.
At very low frequency, well below the lowest RC corner frequency, you get the equivalent of
phase((R1+R2+R3)*C3) + phase((R1+R2)*C2) + phase(R1*C1). (1)
That's because the capacitors are hardly doing anything, so you can ignore the interactions between them.
At higher frequency, the source impedance seen by later sections drops, so their phase shift is less than you'd expect.
At very high frequency, the caps are very low impedance, so the phase becomes
phase(R1*C1) + phase(R2*C2) + phase(R3*C3). (2)
If R3 >> R2 >> R1, expressions (1) and (2) are nearly equivalent, so the additive approximation works fine for many purposes. You can see that that isn't the case if the resistors are of comparable size.
You can also put an op amp buffer between sections, which will make (2) correct up to the start of the buffer rolloff.
Cheers
Phil Hobbs
-- Dr Philip C D Hobbs Principal Consultant ElectroOptical Innovations LLC Optics, Electro-optics, Photonics, Analog Electronics 160 North State Road #203 Briarcliff Manor NY 10510 hobbs at electrooptical dot net http://electrooptical.net
Il giorno sabato 14 giugno 2014 13:09:58 UTC+2, riccardo manfrin ha scritto :
onfiguration producing the filter H(w).
cted to form the cascade, exhibit a dependency from the network following t hem (the rest of the cascade).
Gotcha. basically the 10 factor used in each successive RC stage for the re sistance has the task of "disconnecting" the following circuit from the pre vious RC filter (R_N looks like an open circuit to the R_{N-1}C_{N-1} filte r), hence making the RC filters independent and therefore allowing their ph ase to be additive with good aproximation.
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