Understanding Phasor Analysis in RC Circuits

[Physics_5]

I understand that the voltage phasor for a resistor is in phase with the current phasor in a simple ac circuit consisting of only an AC voltage source and the resistor.

I understand that the voltage phasor for a capacitor is pi/2 radians behind the current phasor for an AC circuit consisting of only an AC voltage source and a capacitor.

When an RC circuit consisting of a resistor and capacitor in series is analyzed, I don't understand how the above still holds. Both proofs were based on the fact that the voltage across each circuit element varies by E = E0*cos(wt), but now the sum of the voltages across each element vary by this NOT each individual voltage.

I feel like this is a pure assumption. Here are the links to the book proof. I have a problem with the second picture of the phasor diagrams.
http://imgur.com/wmjzJ6C
http://imgur.com/aeh1ESb

 

[Baluncore]

We know that the same current, i, flows in series through the resistor and capacitor. We know the voltage across the resistor will be Vr = i * R. We know the voltage across the capacitor will be Vc = i * Xc, with phase lagging Vr by 90°. At this point we do not know the value of i.
When we realize the vector sum of Vr and Vc is the applied voltage, Eo = Vr + Vc, we can solve for i, which identifies the scale of the diagram.

If we know the value of R and Xc, then we know the complex impedance of the series combination is (R + jXc ).
The applied voltage is Eo, so i = Eo / (R + jX ).

 

[tech99]

Further to Baluncore's explanation, if you are analysing a series combination of impedances then assume a fixed current (maybe 1 amp) through both, and if analysing a parallel combination assume a fixed voltage (maybe 1 volt) across both.