Understanding Power Triangle and Box Conversions: Explained and Simplified

[influx]

I understand that S = P + jQ however I am confused how they got from that to the yellow box. Also, how did they get from the yellow box to the pink box and from the pink box to the blue box?

Thanks

 

[Simon Bridge]

I understand that S = P + jQ however I am confused how they got from that to the yellow box.

Definition of RMS values.

Also, how did they get from the yellow box to the pink box and from the pink box to the blue box?

Following the definitions. eg. by definition: $\phi=\theta_V-\theta_I$ so you get the pink box.
The blue box does not follow directly from the pink box - you get it from the definition of S and some trigonometry. S is the hypotenuse of a triangle with opposite side length Q and adjacent side P. Use SOH CAH TOA.

 

[FOIWATER]

It is not obvious to first time learners why the current angle possesses a negative sign in the yellow box.

It is simply because it is defined that inductors consume positive reactive power.

Here's what I mean, You might know that inductive reactance has a positive 'j' associated with it? inductive reactance is jXL. j is a place-holder that rotates a vector by 90 degrees. This means that the impedance associated with an inductor is at 90 degrees phase shift. This causes the current to be at a negative 90 degrees based on ohms law (V@0/X@90 = I@-90). If we want to define reactive power consumption as negative for a capacitor and positive for an inductor, we need to take the conjugate of the current angle. Hence the reason for the negative current angle in the yellow box.

Now, the transition from the yellow to the pink box is just defining the difference in the voltage and current angles as an angle. So now you know that the angle associated with apparent power is the phase difference between the voltage and current.

The transition from the pink to the blue box is an application of Euler's formula.

http://en.wikipedia.org/wiki/Euler's_formula

Have a look at the first paragraph.

I sincerely hope this helps and if it doesn't I welcome any further questions or corrections from others