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Having trouble to tell what kind of filter (Low-Pass, High-Pass, Band-Pass, Band-Reject(Notch) are present given a topology in cascade.

In the following picture I see 3 stages, is this ok? (2 active stages, 1 passive stage) but cannot tell if LP, HP, BP, or Notch filter is which one and why.

/!\Note: the values of resistors and capacitors showed are of no importance right now and are the values by default in circuit lab.

Cascade filter

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  • \$\begingroup\$ If it is already in CircuitLab, why not run a simulation of the circuit to prove the theory in the answers? Brian mentioned 1.6kHz, I was thinking somewhere between 1 and 2kHz for cut off frequency. So I bet I should give a nice diagram. From the steepness of the edge you can determine the order of the filter. \$\endgroup\$ – jippie Mar 16 '13 at 17:16
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    \$\begingroup\$ @jippie where's the fun then? The value is being able to "read" the circuit, rather than rely on simulations :) \$\endgroup\$ – clabacchio Mar 16 '13 at 17:21
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    \$\begingroup\$ @jippie: Simulating and watching the bode plot solves the problem for now, but if this were asked in an interview (or worse yet, in an exam), clearly I won't ask them to let me simulate in in my laptop. \$\endgroup\$ – mongoose85 Mar 16 '13 at 17:43
  • \$\begingroup\$ @clabacchio I already solved the Bode plot in my mind, I want to know if I'm right. \$\endgroup\$ – jippie Mar 16 '13 at 17:47
  • \$\begingroup\$ @mongoose85 do you want to know how to analyze a cascaded filter or what that specific filter does? If it's the second, please edit the title accordingly... \$\endgroup\$ – clabacchio Mar 16 '13 at 17:51
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In a purely academic sense, a bandpass or notch filter by itself must have an even number of stages so the roll off rate is the same on each side of the filter. This isn't always the case in the real world, but that is beyond the scope of this question.

A low or high pass filter can have an odd or even number of stages. The easiest way to determine filter type is by looking at the reactive components. Typically, in active filters capacitors are used exclusively. They are easier and cheaper to manufacture, and with tighter tolerances than inductors. Since the impedance of a capacitor goes to zero as frequency goes to infinity, a capacitor to ground is a low pass filter. In the case of the Sallen-Key topology, a capacitor to ground and in the feedback loop will form a low pass filter. As the others have mentioned, your circuit is a 5th order low pass.

Sallen-Key Low Pass

A high pass filter is formed by switching the Rs and Cs.

Sallen-Key High Pass

There is also a Sallen-Key bandpass filter, but it's only first order roll off on each side.

Sallen-Key Bandpass

Another common topology that is pretty handy to know about is the state variable. It isn't as dense as the Sallen-Key, but one circuit contains a high pass, low pass, and bandpass with the same critical frequency.

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It looks to me like a 5th order low pass filter made from two sallen-key 2nd order filters with a simple RC low pass in between them. It's low pass because of the configuration of the R's and C's. The key thing to look for is the RC directly before the non-inverting input of each op-amp - if there's a C to ground then it's low pass. Ditto the RC between the two 2nd order op-amp stages. The easiest way to think of it is that C's will conduct more as the frequency rises therefore a series R and C to ground will pass DC and low frequencies but at some frequency point in the spectrum the C will start to dominate. This is determined by the frequency of 1/(2*PI*R*C) and for the circuit above it is approx 1.5kHz. I haven't got my calculator with me so maybe someone else will correct my brain math.

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  • \$\begingroup\$ That's about right. My not-even-on-an-envelope guess said 1.6kHz. The distinguishing feature of the Sallen-and-Key second order stage is the unity gain amplifer. The Q of the first one would be familiar to a real expert because the R-C time constants are equal ( 1.0? 1.4?) I have forgotten. The second active stage will have a lower Q because it's driven from the passive R-C, not a perfect voltage source. \$\endgroup\$ – Brian Drummond Mar 16 '13 at 17:04
  • \$\begingroup\$ @Andy aka: That's for R1 and C1. But what about R2 and C2? C2 is not connected to ground. \$\endgroup\$ – mongoose85 Mar 16 '13 at 17:22
  • \$\begingroup\$ @mongoose85 do you mean R3-C2. The drawing isn't very large and it does look like you mean R3. Anyway assuming you do... it's the nature of the beast - a car can usually be recognized by it having 4 wheels that touch the ground but it can't necessarily be recognized by counting the number of windows. Same sort of words for the cap that feeds back from the output to the junction of R3 and R1 - it's not easy to simply explains what it does - best to look at the math but when you look at R3&C2 and R1&C1 built around an op-amp then it's a 2nd order low pass sallen key \$\endgroup\$ – Andy aka Mar 16 '13 at 17:33
  • \$\begingroup\$ R1=R2, C1=C2 => Q=0.5, which makes it a Linkwitz-Riley filter. \$\endgroup\$ – user207421 Mar 18 '13 at 5:28
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It is low pass, because it has a DC path (low frequency) from input to both non-inverting amplifiers and an AC path to ground (high frequency). So DC is propagated to the output and AC is filtered to ground (or through the low output impedance of the amplifier).

This is a extremely simplified analysis of things. It doesn't account for order, nor for the cut off frequency.

If you want to know for sure, you can always simulate the circuit in a tool like spice or CircuitLab.com

A high pass filter would have capacitors and resistors swapped, and a band or notch filter would have mixed positions. To differentiate between band and notch filter, you probably have to do some actual calculations on the specified parts.

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  • \$\begingroup\$ I wouldn't be surprised if cut off frequency is in kHz range (from a back of envelope estimation). \$\endgroup\$ – jippie Mar 16 '13 at 16:51
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You can recognize the category of filter from whether the capacitors shunt the signal, or pass the signal. Both the filter stages are low-pass filters because in all the RC networks, the capacitors shunt the signal to a low impedance.

The tricky capacitors are the ones in the feedback path, but a mental shortcut to classifying them is to regard the amplifier output as a ground-like low impedance. What's happening there is that those capacitors are "bootstrapped" from the amplifier output instead of going to the ground. Do not mistake these capacitors as a bypass path for high frequencies around the amp. Rather, it is more helpful to pretend that the output of the filter is an alternative ground.

As a shortcut, mentally redraw the circuit diagram such that the amplifier is removed, and the positive feedback element (resistor or capacitor) terminates to ground instead. Then apply your understanding of RC networks to determine the combination of high and low pass filtering.

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Having trouble to tell what kind of filter (Low-Pass, High-Pass, Band-Pass, Band-Reject(Notch) are present given a topology in cascade.

For a start, find the gain as the frequency goes to zero (open the capacitors, short the inductors) and the gain as the frequency goes to infinity (short the capacitors, open the inductors).

Given just the four categories of filters you've listed, you have the following possibilities:

If the gain goes to zero at both extremes, you have a band-pass filter.

If the gain is the same at both extremes, you have a band-reject filter

If the gain only goes to zero at one extreme, you have either a low-pass or a high-pass filter.

Of course, there are other possible filters to consider such as, e.g., shelving filters and all-pass filters etc.

Now, looking at the circuit you've posted...

  • with the capacitors open, the gain is 1.

  • with the capacitors shorted, the gain is 0.

Thus, this circuit is a low-pass filter.

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