RMS values and phasor diagrams

Question

In RL circuits (AC source) we use vectors to calculate the total voltage, I understand that we cannot add them algebraically because they do not exist at the same time as we are using rms values for both voltages which represent the peak of each wave (multiplied by $0.707$); I understand the phase shift between the two.

My question is:

Why do we apply vector rules to calculate the peak(or rms value) of the total voltage wave whereas we use different peaks of another two waves?

I mean how could the phasor diagram represent the shift angle and, at the same time, calculate the value of the total voltages ?

Answer 1

The equation for the voltages in a driven RLC circuit can be put into the form:

   I Z cos(wt+ϕ) = I Xc cos(wt – π/2) + I R cos(wt) + I XL cos(wt + π/2)

Where I is the maximum current, Z the impedence, Xc the capacitive reactance (1/wC), XL the inductive reactance (wL), and w the angular frequency. (One or more of these terms may be missing in a simpler circuit.) (The I's can be divided out.) Either way, this equation resembles the equation for adding the x components of three vectors to get the x component of the resultant vector. (As function of time, all four (imaginary) vector are rotating around the origin of an xy coordinate system. As they rotate, the x components represent the instantaneous voltage on the corresponding circuit element and their angular separation represents the phase difference.) To simplify the calculation of Z and ϕ, the vectors are generally sketched at t = 0. The rotating vector are just a way of representing trig functions which are out of phase. In more advanced courses you may see the vectors in the complex plane represented by imaginary exponential functions. These may be used to represent wave functions as well as circuit voltages.