Help with graphical technique of adding 2 sinusoids

[p75213]

Homework Statement

I have a problem understanding how equation 9.11 (see attached) works with Fig. 9.4 (a). Any help would be appreciated.

Homework Equations

The Attempt at a Solution

 

[Simon Bridge]

The two equations (9.11) are being represented as a 2D vector (fig 9.4 a).

The authors have used the phasor representation of the sinusoids and the diagrams represent the vector sums of the phasors. I'd have explained it a bit differently.

Basically, cosine and sine functions have a relative phase of π/2 - so you can use them as x and y axes. The amount each contributes to the result is it's amplitude, which is what is being plotted on the axis. This does not work if the sinusoids have other than nπ/2: n=0,1,2,...

In general - use the phasor representation for arbitrary sinusoids and add the phasor arrows head-to-tail just like vectors.

 

[JHamm]

Try expanding the right hand side, you'll see that it works out :)

Edit: Oops beaten, cutthroat business this physics stuff :P

 

[p75213]

Thanks for that. Makes sense now that I realize they are vectors. Makes you wonder why they didn't make it clearer. Phasors is the subject of the next chapter.

 

[Simon Bridge]

Phasors is the subject of the next chapter.

Ah well, they are trying to be kind to you by building up to them.
I always just launch right into it. By now you've had vectors drilled into you so you see them in your sleep so you can handle rotating vectors no problem.

Never mind - skip ahead to phasors and you'll see it makes better sense.

 

[NascentOxygen]

I have a problem understanding how equation 9.11 (see attached) works with Fig. 9.4 (a).

Remember, wherever you see A.cos(wt) you can replace it by A.sin(wt+Pi/2) because this is an exact equivalence.

By similar reasoning, -cos(wt) = cos(wt+Pi) = cos(wt-Pi)