# constant phase shifter (simplified)

Thread Starter

#### PeteHL

Joined Dec 17, 2014
358
The circuit that I'm seeking should give me a constant relative phase shift of 90 degrees perhaps over a decade frequency range. I'm after an analog solution with op-amps. For now at least it doesn't have to be very accurate, something like within +/- 20 % would be fine.

Searching on the web, I have come across procedures to do this. However they are all very high-level math intensive, all of which I don't understand. So what I am seeking is a relatively simple procedure for designing the shifter from a few op-amps. My preference would be to basically understand how it works, but that isn't necessary. Just a bunch of equations that I could use without knowing what I'm doing would be okay.

At this point I'm not certain that the phase shifter will do what I want in my scheme of audio signal processing, so I don't wish the design and testing to be very laborious, nor do I expect high tolerance over a wide audio bandwidth.

Thanks if anyone can help,
Pete

#### alfacliff

Joined Dec 13, 2013
2,458
look up "synchro control transformer" looks like a motor, is a two phase variable phase transformer. feed the signal into the rotor, take it out eithr of the two outputs. rotate the shaft for phase shift.

#### crutschow

Joined Mar 14, 2008
27,237
look up "synchro control transformer" looks like a motor, is a two phase variable phase transformer. feed the signal into the rotor, take it out eithr of the two outputs. rotate the shaft for phase shift.
Won't that still require the audio to be phase-shifted by 90° for the synchro two-phase input?

#### crutschow

Joined Mar 14, 2008
27,237
How about this?

• PeteHL
Thread Starter

#### PeteHL

Joined Dec 17, 2014
358
look up "synchro control transformer" looks like a motor, is a two phase variable phase transformer. feed the signal into the rotor, take it out eithr of the two outputs. rotate the shaft for phase shift.
Not having ever heard of a syncro control transformer before, and I've been interested in audio electronics for many years, I would bet that such a device is not widely available and also expensive. So I'm not inclined to seek it out, as I'm not certain that constant 90 degree phase shift is a solution for what I'm attempting. But thanks anyway for the reply.

What I'm suggesting should be possible, as I've seen a circuit composed of eight op-amps, and many resistors and capacitors that provides the 90 degree shift with high tolerance and over the entire audio frequency spectrum.

-Pete

Thread Starter

#### PeteHL

Joined Dec 17, 2014
358

#### Tesla23

Joined May 10, 2009
445
What I'm suggesting should be possible, as I've seen a circuit composed of eight op-amps, and many resistors and capacitors that provides the 90 degree shift with high tolerance and over the entire audio frequency spectrum.
Look up articles about generating SSB by the phasing method, this may help:
http://www.tonnesoftware.com/quad.html

#### crutschow

Joined Mar 14, 2008
27,237
That looks interesting, but the schematic is illegible for me.
Click on the schematic to enlarge it.

Thread Starter

#### PeteHL

Joined Dec 17, 2014
358
Click on the schematic to enlarge it.
Thanks, I didn't think to do that (embarrassment)! I figured that I just had blurry vision.

Thread Starter

#### PeteHL

Joined Dec 17, 2014
358

#### KL7AJ

Joined Nov 4, 2008
2,229
Yipper, skipper. This is topologically the same as the massive passive RC networks they used in the early days of Single Sideband.

#### Tesla23

Joined May 10, 2009
445

#### MrAl

Joined Jun 17, 2014
8,256
Hello there,

Welcome to the world of circuit analysis The 7 op amp circuit, despite it's apparent complexity, is actually very simple to analyze, given a little patience. This is partly because all the op amp stages are exactly the same so the analysis is the same for each stage, and partly because each stage is like an "all pass" where the gain is exactly 1 for each stage regardless of frequency. This means we only have to analyze one of the 6 stages and then compound the results.

To start, the gain for each stage is 1 because of the two 100k resistors in each stage, and that actually aides in eliminating those two resistors from each stage so they dont have to appear in the analysis. That leaves us with one resistor, one pot, and one cap for each of the 6 stages.
Second, the pot can be eliminated from the analysis by making that one resistor remaining part of the pot itself. The cap of course can not be eliminated, but now we are down to only TWO parts for each stage:
1. The resistor in series with the capacitor.
2. The capacitor itself.

Next we have to identify each of these resistors and capacitors for each stage because they are all different,so we number them from left to right, starting at the top left stage:
R1,R2,R3, and respectively C1,C2,C3

and at the bottom left from left to right:
R4,R5,R6, and respectively C4,C5,C6.

Those are the only components needed for the basic analysis.

Since the gain for each stage is 1, that means the gain for the top section is 1 and the gain for the bottom section is 1, so we dont really need to include a gain factor.

The phase shift for each stage is however different for each stage because all the R's and C's are different. But since the phases in the top or bottom section add, the output of either the top or bottom is just the sum of the three individual phase shifts. The output phase shift however is the difference between the top and bottom section phases because the output is not single ended. If we want a single ended output (which is probably the case) we have to add one more op amp section as a subtractor on the output, to subtract the bottom section output from the top section output, and that is not too difficult using a difference amplifier.

The output therefore comes out to be fairly simple and is:

TotalPhaseShift (in radians)=
atan2((2*w*C6*R6)/(w^2*C6^2*R6^2+1),-(w^2*C6^2*R6^2-1)/(w^2*C6^2*R6^2+1))
+atan2((2*w*C5*R5)/(w^2*C5^2*R5^2+1),-(w^2*C5^2*R5^2-1)/(w^2*C5^2*R5^2+1))
+atan2((2*w*C4*R4)/(w^2*C4^2*R4^2+1),-(w^2*C4^2*R4^2-1)/(w^2*C4^2*R4^2+1))
-atan2((2*w*C3*R3)/(w^2*C3^2*R3^2+1),-(w^2*C3^2*R3^2-1)/(w^2*C3^2*R3^2+1))
-atan2((2*w*C2*R2)/(w^2*C2^2*R2^2+1),-(w^2*C2^2*R2^2-1)/(w^2*C2^2*R2^2+1))
-atan2((2*w*C1*R1)/(w^2*C1^2*R1^2+1),-(w^2*C1^2*R1^2-1)/(w^2*C1^2*R1^2+1))

and as you can see this is just three phase shifts added and three subtracted as explained above.
The phase of any one stage can be computed by using just that one line "atan2(...)" for that particular stage if desired.
Multiply the above by 180/pi to get the result in degrees.

The output plot from 0 to 20kHz without any adjustment is shown in the attachment. It is fairly constant around 90 degrees, but not perfect.

If we wanted to compute the possible error due to one or more of the 100k resistors (and/or other components) being off a little we'd have to include the gain for each stage, but for the first basic analysis we can see that this circuit works at least somewhat like it is supposed to work.

Last edited:
Thread Starter

#### PeteHL

Joined Dec 17, 2014
358
Mr Al,
Thank you for your explanation. My assumption is that in your post #15 you are discussing the "WIDEBAND ACTIVE PHASE SHIFTER" that Crutschow provided a link to (Ham Radio, Jan. 1978). What I noticed about it is that the frequency above or below each phase shifter is the frequency at which that shifter produces a 90 degree phase shift (after fine tuning). Is there any simple explanation for how those particular frequencies are arrived at? My guess is that there isn't a simple explanation. For example, see
http://webpages.charter.net/wa1sov/technical/allpass/allpass.html

#### Veracohr

Joined Jan 3, 2011
752

#### MrAl

Joined Jun 17, 2014
8,256
Mr Al,
Thank you for your explanation. My assumption is that in your post #15 you are discussing the "WIDEBAND ACTIVE PHASE SHIFTER" that Crutschow provided a link to (Ham Radio, Jan. 1978). What I noticed about it is that the frequency above or below each phase shifter is the frequency at which that shifter produces a 90 degree phase shift (after fine tuning). Is there any simple explanation for how those particular frequencies are arrived at? My guess is that there isn't a simple explanation. For example, see
http://webpages.charter.net/wa1sov/technical/allpass/allpass.html
Hi again,

Yes that is the network i had looked at, and i included the schematic with the parts labeled to reduce confusion about which resistors where which, like R1, R2, and the capacitors are all labeled too.

If you just want to calculate the frequency you can look at the post just before this one, but i have a feeling you want to know how the frequencies are arrived at in the first place.

I did not look at that aspect of it yet, so perhaps you can provide more information on what you said you found and we can look at that first. Maybe explain a little better like were you talking about each stage or the whole upper or lower section, etc.

Another way to do it would be to assume a given stage design, which would not be unreasonable since they all use an All pass network, and then use the equations that describe the phase shift (as i posted myself previously, possibly with additional stages) in a fitting algorithm that uses several chosen points as reference. Using matrix math, the matrixes are then transformed and various correlations computed, and component values varied systematically until a close match between the predicted response and the desired response is obtained. This would be interesting to try really.

Some problems do require a digital computer to perform the various tasks, but we can take a look at what you found first if you like. Maybe we can find a simple explanation that at least approximates the behavior.

We might also look at a possible AGC method, by first designing a 90 degree phase shifter that shifts all frequencies by 90 degrees but allows a fairly large gain change, but then uses an AGC to correct the gain. That would probably be the simplest method if we can get the AGC to be fast enough. That's if this idea is practical.

Also, modern methods would probably include DSP most likely, so that would be a whole different design strategy.

Thread Starter

#### PeteHL

Joined Dec 17, 2014
358
Mr Al,
My point was that it seems that the designer of the Wideband Active Phase Shifter (the entire circuit) might have gone about designing it by first determining the frequency at which each section should shift the signal by 90 degrees, then solve for component values on that basis. If this is in fact how he proceeded, then the question is how he arrived at the specific 90 degree frequency for each of the 6 sections.

For example from the schematic where you assigned resistor/ capacitor numbers, the first section of the upper three,

1/2*pi*R1*C1 = 1/2*pi*38.46k Ohm*20nF = 206 Hz

where tan(90/2)= 1 (the numerator)

My thought was perhaps first selecting the six different frequencies might be intuitively understandable which the equations that you gave are not. For example, each phase shifting section produces phase lag, yet in your equations of post #15, phase shifting of the bottom three sections subtract from the shifting of the upper three.

Regards and thanks,
Pete