Design filter with specific phase response

I am trying to build and eventually implement digitally a Filter with specific phase delay at a single frequency.

So for example at 60Hz the signal should be delayed 45 degrees, etc.

How would one approach this on paper, circuit and say Matlab?

You can use a simple RC filter. Adjust R with a potentiometer (within limits).

There are formulas which relate time constant/ frequency/ attenuation/ phase shift/ etc.

A simulation is invaluable for conceptualizing performance.
Phase delay is different in the three filters.



You could use an inductor to create the phase delay too.

A simple 1st order lag filter consisting of a resistor and capacitor, as suggested by BradtheRad can be used. Choose a capacitor value, then R = Xc*tan(Theta).
. R in Ohms, Xc = 1/(2*pi*f*C), f in Hertz, C in Farads.
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The analog transfer function is f(s) = 1/(Tau*s + 1), where Tau = RC.
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This filter can be transformed to a digital filter using the impulse-invariant transform. Just make sure that the sample frequency is considerably higher than the Nyquist frequency of 2f. Otherwise aliasing will affect the phase shift.
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Incidentally, the phrase "phase delay" is improperly used in this context. You are after a specified "phase shift". Phase delay is defined as phase shift divided by frequency, and has the unit of time. Group delay is defined as the derivative of phase with respect to frequency, and also has the unit of time.