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Obtain the relation of phase between instantaneous current and voltage with the help of phase diagram for series LCR circuit. |
Answer» Solution :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)` aheadof `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 representedas 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 determinedby 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))` |
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