Phasor Addition Techniques: Adding Frequencies, Amplitudes & Phases

[omertech]

Hello,

I was wondering about the adition of phasors with different amplitude, frequency and phase.

Wikipedia supplied the technique of adding phasors with the same frequency but different amplitude and phase (http://en.wikipedia.org/wiki/Phasor#Addition).

When it comes to adding phasors with different frequencies, I found an explenation involving beats regarding phasors with the same amplitude and phase but different frequency(http://www.animations.physics.unsw.edu.au/jw/beats.htm).

I want to know if there is a way to add phasors with different frequency, amplitude and phase to come up with a single, analytically solveable function. If not, how is it possible then to predict such system?

Thanks in advance

 

[sophiecentaur]

Phasors are, essentially, only used when you have the same frequencies because they are supposed to be a stationary 'snapshot' of the phase and amplitude vectors. If two signals are not at the same frequency, the relative phase is constantly increasing / decreasing so a snapshot means nothing - you need to describe the situation in the 'full' form.

You sometimes see diagrams with one long phasor with a smaller circle around its tip representing a second signal with its constantly rotating phasor - or, sometimes, just a fuzzy ball at the end, representing noise or an interfering signal which is randomly changing in amplitude and phase. OK for illustrative purposes but no so useful for serious 'analysis'.

 

[omertech]

Ok, so if we would treat the signals as waves, is it possible to describe the resultant interference with an equation proper for analysis? For example the intereference of the signals:
$$x_1=A_1\cos{(\omega_1t+\theta_1)}$$
$$x_2=A_2\cos{(\omega_2t+\theta_2)}$$
is:
$$x_3=x_1+x_2=A_1\cos{(\omega_1t+\theta_1)}+A_2\cos{(\omega_2t+\theta_2)}$$
How would could you transform x₃ to something proper for analysis? (that is you could get analytical solutions to certain values of x₃, its derviatives, integrals etc.)

Thanks again

 

[sophiecentaur]

Those are not waves. They are just time varying signals, remember.

You can differentiate that expression (the sum of two signals) or treat it in any way you want. What other analysis would you want to do? Are you looking for something more than there is, perhaps?

 

[omertech]

Yes but they are essentially oscillations, this means that they behave as waves (superpose and interfere for example) doesn't it?

 

[sophiecentaur]

Oscillations are not waves. Interference is a phenomenon involving space as well. These two signals just add up and the phase is not dependent on any 'position'.

 

[omertech]

Well, as far as I can tell thery are both described mathemtically the same. The equations above could describe oscillations as well as one dimensional waves, and the expression for x₃ could describe interference as well as just any other linear combination of oscillations.

Regardless of the nature of the phasor, if we would treat the equations above as oscillations, could we transform x₃ to become proper for analysis?

 

[sophiecentaur]

No. Look at the equation for a one dimensional wave. It has the variable x in it. There is a subtle difference.

 

[omertech]

Yes I see what you meant. I am still curious about the oscillating equations, do you have any idea about that?

 

[sophiecentaur]

They just describe signals with sinusoidal time variation. In a linear medium, they will superpose.

 

[omertech]

Yes, so mathemtically speaking, x₃ describes the linear combination of x₁ and x₂. Is there a way to transform it into a form proper for anaylsis? (that is you could get analytical solutions to certain values of x₃, its derviatives, integrals etc.)

 

[sophiecentaur]

Its time derivatives? Go ahead and follow the rules. It works, as ever.