Obtain the relation of phase between instantaneous current and voltage with the help of phase diagram for series LCR circuit.

Explanation

In circuit L-C-R are in series. Therefore, the act current in each element is the same at any time having the same amplitude and phase.

Let it be `I = I_(m ) sin ( omega t + phi )` …(1)
where `phi` is the phase difference between the voltage across the source and the current in the circuit.
Let `vec(I )` be the phasor representing the current in the circuit and `vec( V_(L)) , vec( V_(C )) , vec( V_(R )) ` and `vec( V )` represent the voltage across the inductor, resistor, capacitor and the source respectively.
`vec( V _(R ))` is parallel to `vec( I )`
`vec( V_(C ))` is `( pi )/( 2)` behind ` vec( I)` and
`vec( V_(L))` is `( pi )/(2)` ahead of `vec(I)`.
`vec( V_(L)), vec( V_(C )), vec( V_(R ))` and `vec( I)` are shown in figure with appropriate phase relation.

The amplitude of phasor are as follow.
`V_(Rm) = I_(m) R, V_(CM ) = I_(m) X_(c ), V_(Lm) = I_(m) X_(L)`
The voltage equation for the circuit can be written as,
`L(dI)/( dt) + IR + ( q)/( C ) = V ` can be written as below
`vec(V_(L)) + vec( V_(R )) + vec( V_(C )) = vec( V )` where `V_(L) = L(dI)/( dt) , V_(R ) = IR ` and `V_( C ) = ( q )/( C )`
`:. ` The phasor relation `vec( V_(L)) + vec( V_(R )) + vec( V_(C )) = vec( V )`
This relation is represented in below figure.

Since `vec( V_( C ))` and `vec( V_(L))` are in opposite directions, so the resultant value of phasor.
`vec( V _(C )) - vec( V_(1)) = V_(Cm ) - V_(Lm)`
Since `vec( V )` is represented as the hypotenuse of a right triangle whose sides are `vec( R )` and `vec( V_(C )) + vec( V _(L))` the Pythagorean theorem gives.
`V_(m)^(2)= V_(Rm)^(2) + ( V_(Cm ) - V_(Lm))^(2)`
`:. V_(m)^(2) = ( I_(m)R)^(2) + [(I_(m) X_(C ))-(I_(m)X_(L))]^(2)`
`:. V_(m)^(2)= I_(m)^(2) [ R ^(2) + ( X_(C ) - X_(L))^(2) ]`
`:. I_(m)^(2) = ( V_(m)^(2))/(R^(2) + ( X_(C )-X_(L))^(2))`
`:. I_(m)= ( V_(m))/([R^(2) + ( X_(C )-X_(L))^(2) ]^((1)/(2)))` but `sqrt(R^(2) + ( X_(C )-X_(L))^(2)) =Z` where Z is called impedance.
`:. I_(m) = ( V_(m))/(Z)` is the amplitude of current.
Since phasor `vec(I)` is always parallel to phaosr `vec( V_( R ))` the phase angle `phi` is the angle between `vec( V_(R ))` and `vec( V )` can be determined from figure,

`tan phi = ( V_(Cm ) - V_(Lm))/(V_(Rm))`
`= ( I_(m) X_(C ) - I_(m) X_(L))/(I_(m)R )`
`:. tan phi = ( X_(C ) - X_(L))/(R )` gives phase angle.
Impedance Z of circuit can be determined by figure. This is called impedance diagram which is a right triangle with Z as its hypotenuse.
Impedance from impedance diagram, `Z = sqrt( R^(2) + (X_(C ) - X_(L))^(2))`