Precisely phase shifting very low frequency signal with RC filter

Hi,

I have a 0.3 V amplitude signal in an extremely low frequency range (1-2 Hz) that I'm putting phase shifted into 4 amplifiers. I need one signal at the original phase, one shifted by (pi/2) radians with respect to the original, one shifted by (pi) radians with respect to the original, and lastly one shifted by (3pi/2) radians with respect to the original. I think there's a way to do this with RC filters, possibly even just by sticking a capacitor in series, but can anyone give me and idea of how I should go about it? I would appreciate it greatly.

The output impedance of my signal generator is 0.2 ohms if that is important.


Best regards,
Michael

What you need is an all pass filter. One issue will be the RC values, they will be quite large, and therefore some thought has to go into the design and choosing appropriate op amps with respect to bias and offset current.
Additionally, temperature coefficients need to be low enough to meet any drift requirements you establish, for example, what sort of tolerence is there associtated with the phase shift, particularly if you cascade filter sections. There are active filter programs that will allow you to design and simulate your circuits.
Cheers,
Richard

In amateur-radio literature from 1950s and 1960s, look up phasing-type single-sideband SSB exciters. The audio phase-shifting network does 90-degree phase-shift over audio frequencies, so would do what you want if you multiply the capacitor values by 1000. As Richard points out, the capacitor values will be quite large.
How tight tolerances do you need? You might do better doing this digitally using an all-pass FIR filter approximating the Hilbert Transform.
Charles Proteus Steinmetz taught that what you want to do with four phases 90-degrees apart can also be done with three phases 120-degrees apart. This is why large power lines have three current-carrying conductors.

Thanks for your response. I think you're right that an active all-pass filter with an op amp would be best. I think I could actually do it with just the one filter to phase shift by pi/2, then invert that with an op amp to get the 3pi/2 phase shift, and invert the original to get the pi phase shift.

With regards to choice of resistor and capacitor. Using the equation for the frequency corresponding to a pi/2 phase shift in an all pass filter, f=1/(2*pi*R*C), if I wanted to use a 10 uF capacitor for example, I would have a resistance range of 8-16 kohms for my 1-2 Hz signals, which seems pretty possible to attain with a potentiometer.

Is there a reason to prefer smaller capacitors and larger resistors here or vice versa with regards to the transfer function of the filter?

I would like my output to be as close as possible in amplitude to my input, so should I set the op amp up as larger than unity gain to compensate for a little loss?

Best,
Michael

Hi, Michael. Generally, in an RC filter, you keep the resistance(s) low to avoid susceptibility to noise but not too low to avoid dissipating a lot of power or loading the signal source down too much. So it all depends on your situation. With a 0.2 ohm signal generator you don't need to be loading it down much except that you need to not exceed its current sourcing capability. The values you mentioned seem quite reasonable to me. Capacitors can take up a lot of space so using higher resistance will allow you to use smaller caps. I know of no other reason to prefer larger or smaller caps transfer function wise.

You should not need to increase gain to compensate for losses, unless the filter is attenuating your signals too much at a frequency of interest. But it would be better to adjust your cut-off frequency instead.

Be aware that many capacitors have wide tolerances on capacitance value. More precise capacitors are more expensive which is another reason to try and keep them smaller.

Best, -JD

Since you need to use film capacitors, it make sense to keep them below a microfarad or so. Using a potentiometer can be an issue, usually they have large temperature coefficients (unless you spend up)and you need to consider what happens if the wiper goes open(this dictates using a pot with a small resistace compared to the total required resistance) I can't judge if this will be a problem since you have not said what sort of stability you need. Usually fixed resistors are more useful, put provision in for resistor pairs, either in series or parallel is one method. I use an RCL meter to sort my capacitors, measure their values, then buy E96 resistors to suit. As I said, there are simulation packages that for this sort of problem can be useful, since a Monte Carlo analysis is possible in many of them, you are probably aware of LTSpice from Analog Devices, for example.
Cheers,
Richard

Thank you very much guys. You are all quite helpful. I built a prototype all-pass filter to see if the phase shift looked like what I wanted. My signals are a little unusual in that they're superpositions of a sine wave and cosine wave at twice the frequency, (Sin(omega*t) + Cos(2*omega*t)). I'd like to attach the image from my scope, where you can see the two equal height humps in the original signal have been skewed so that the first is shorter and the second taller. Unfortunately there doesn't seem to be a way to attach anymore, and the comment is filtering out my external link. https://ibb.co/dbhMr7H

I assume the skew is because the filter is phase shifting the sine wave which has frequency = 1/(2*pi*R*C) by pi/2, but shifting the cosine wave moving at twice that frequency by -2*Arctan(2*omega*R*C), which is different, leading to the skewed appearance. Plotting the sum of the 2 differently phase shifted waves does seem to confirm this.

I guess the only way around this that I can think of is to generate the sine and cosine waves independently, phase shift them separately with the appropriate RC filters, then sum them with a summing amplifier.

Additionally, I realized inverting the signals to make the pi rotated, and 3*pi/2 rotated signals is probably not a good idea. The reason I'm contemplating using these filters in the first place instead of independently generating 4 signals is that there is a small DC offset (10 mV) between the analog outputs that I'm unable to calibrate out of my DAC. So inverting a signal with a +10mV offset will give me one with a -10mV offset leading to exactly the problem I'm trying to avoid. I figure if I use precision op amps to rotate the original signal, I could get a smaller DC offset between the signals and improve my results.

Thanks again for the help. This forum is quite a fun resource.

Best,
Michael

That being said, with the appropriate choice of components, do you guys think it would be possible to phase-shift the 3 signals and amplify them to 30 V amplitude with less than 1 mV DC offset accumulated between them?

Any drift in the phase shift is preferable to a DC offset.

Yes, but you have to pay close attention to a number of issues. Firstly, what temperature do you wish to operate over. Have you established that the input voltage has no offset. What is the maximum size of resistor you are likely to use. From that you can establish the temperature coefficients needed for your resistors, the capacitors don't matter if the phase is not an issue.
You then have to chosse an op amp that meets the specs for offset voltage and current with temperature using the given resistors. The offset voltage can be compensated for normally in a number of ways if needs be, and if the resitors are small enough, input offset current may not be an issue. Whilst most modern op amps are low voltage, there are some very good ones that will operate at 30V, look at TI and Analog Devices for starters. Long term drift of op amp specs may mean that you will need to make adjustments in the future.
Cheers,
Richard

Since you are dealing only with AC signals, you can eliminate DC offset by using AC coupling. In AC coupling, stages are coupled through one-pole hipass filter built from series capacitor and shunt resistor.
What loads are you intending to drive with your 30-volt signals? You might be able to conveniently use capacitors in series with your loads. This has been standard practice in audio and radio worlds for decades.
Why are you running your DAC output at only 0.3 volts? Running close to the DAC fullscale would reduce the impact of DAC DC offset.