I have designed the following Sallen-Key high pass filter (see image below). I would like to verify its stability using the phase margin of the open loop transfer function.

Sallen-Key HP filter

Inserting a voltage source at -IN does not break the feedback loop due to R1. So where do I have to insert a voltage source in order to get a good approximation of the open loop transfer function? Or is a more involved method such as the General Feedback Theorem needed? And if so, how to apply this here?


1 Answer 1


No - the general feedback theorem is not necessarily to be applied. The shown circuit has two feedback loops - one with 100% negative feedback (unity gain) and one frequency-dependent network for realizing the filter function. Due to the internal phase shifts, this second loop provides positive feedback for a certain frequency range (in the vicinity of the pole frequency).

As a consequence, it is this network which determines primarily the phase margin and you can treat the active device as an ideal unity-gain amplifier.

Therefore: Set Vin=0, open the loop at the upper node of R1 and inject the test signal Vt (AC 1V, for example) at this point. Then, the loop gain is LG=Vout/Vt.

(In your text you mention the phase margin and the "open-loop transfer function". Please note that for finding the phase margin we need the LOOP GAIN which is the product open-loop gain (of the opamp) times feedback function).

EDIT (remark): To be more detailed, one must realize that the shown circuit has three loops (three methods to open feedback networks) - with different loop gains and, hence, each with (at least theoreticallly) different stability margins. That means: The circuit itself does not have a certain stabiliy margin - instead, each loop is equipped with a margin.

  • loop 1: Open only the RC-feedback path. Result: Gain margin GM=6 dB, no phase margin to be defined because the loop gain is always < 0dB.

  • loop 2: Open both feedback loops (RC path and short to the inv. input). Result: Gain margin again app. 6dB, phase margin PM=63 deg.

  • loop 3: Open only the short to the inv. input. As a result, both PM and GM are app. as for loop 2.

Comment: What is the meaning of a stability margin? Answer: Both margins give us an information which additional gain resp. phase shift might be introduced into the respective loop (that means: is allowed) until the circuit becomes unstable. Having this in mind, it makes no sense to open loop 3 (and to find margins) because a short is a short and cannot have any remarkable deviations. However, such unwanted influences (parts tolerances) may happen for the RC-network and for the opamp. For this reason, the first two simulations (loop 1 and loop 2) give results, which may be interesting for the designer. (The mentioned GM and PM values are simulation results obtained using the LT1001 opamp)

  • \$\begingroup\$ You mean like !this (image)? By now I've also applied the GFT once and this yields a very different result GFT. The ruling plots are plot cut loop and plot GFT. T() is the open loop gain for GFT. \$\endgroup\$
    – opieters
    Mar 5, 2017 at 15:34
  • \$\begingroup\$ It is a bit more complex - see my EDIT (remark). \$\endgroup\$
    – LvW
    Mar 5, 2017 at 16:46
  • \$\begingroup\$ The resulting gain margin of GM=6dB is no surprise because the design formula for the pole Q shows that Qp will aproach infinity for a gain of +2. \$\endgroup\$
    – LvW
    Mar 6, 2017 at 14:25

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