# Why do four cascaded all-pass filters cause a 180° phase shift?

This op-amp all-pass filter is reported to introduce a 90° phase shift at the cutoff frequency of the RC network placed at the + input.

(The FET is used as a controlled variable resistor to move the phase shifted frequency "up and down.")

In phasers, this signal is then added to the original one to have a cancellation (notch) at that particular frequency.

The famous MXR PHASE 90 phaser, instead, creates two notches to enhance the effect: it does so cascading four elementary phase shifting units like this:

How does that work?

I would expect the series of four identical units to create a 360° phase shift at the RC cutoff frequency, not two 180° shifts at two different frequencies.

It is also reported that in the phase 90 the phase shift is distributed like this (among the elementary units)

How can you have a 45° phase shift per unit using an all-pass filter that introduces a 90° shift?

It all seems very confusing to me.

I would greatly appreciate your help.

• What is a "phaser"? Commented Mar 24, 2020 at 17:00
• Of course, you can have a phase shift of 45 deg per unit - however, not at the frequency w=1/tau (with tau=RC)
– LvW
Commented Mar 24, 2020 at 17:05
• @Hearth "Phaser" is the fancy name for Phase Shifter, an audio effect.
– Dag
Commented Mar 24, 2020 at 17:10
• @LvW but how? And why do the cascading of four identical all-pass filters create two different notches instead of just creating a 360 phase shift at just one particular frequency?
– Dag
Commented Mar 24, 2020 at 17:12
• Notches ??? An allpass section has constant magnitude over the frequency and shifts the phase only.
– LvW
Commented Mar 24, 2020 at 17:17

I'm here to help since I got the same question some time ago, now I got it.

What happens is: a first order all pass filter shifts the cut-off frequency to 90º and shifts the f=infinity to 180º.

If you get 4 filters (with the same value for componentes) cascated, you will get the cut-off frequency to shift not 90º, but 90º x 4 = 360º (at the output). In the end makes no difference in that frequency.

However, the first filter shifts one specific frequency to 45º (as well as one to 50º, one to 55º ). Four filters in a row will shift that frequency to 45º x 4, which happens to be 180º at the output.

Now that is the frequency which will have the first notch. Other way to get 180º of shift is adding 360º (discovering a total shift of 180º + 360º=540º) because it will have the same phase carachteristics. In that way, we do the reverse engineering, and knowing that it will be 540º at the output, we can discover how much does it shifts at each filter (divide by 4 filters, equaling 135º if i'm not mistaken).

In that way, you can use the equation of the phase of first order all pass filter to discover what frequencies are notched, dependding on the values of your components, substituting the phase on the equation, with 45º and 135º!

Hope you understood, as I am also doing something along the lines of the MXR-90 for a college work.

Any doubts you can ask me!

• thank you very much! it does make more sense, now. All the explanations of the phase-90 were kinda vague, and kept taking for granted that one already had this concept clear in mind.
– Dag
Commented Jun 4, 2020 at 10:13
• Exactly! I had the same feeling, even more in relation to the power source, which gives 9 to 0 volt, given that the JFET must receive a negative voltage. But that must be a falut of mine. Cheers Commented Jun 18, 2020 at 15:55