Cm Lm v v v v Substituting the values of v Rm , v Cm , and v Lm from Eq. ( . ) into the above equation, we have ( ( L v i R i X i X ( L i X X ⎡ ⎤ ⎣ ⎦ or, ( L v i X X [ . (a)] By analogy to the resistance in a circuit, we introduce the impedance Z in an ac circuit: v i Z [ .
(b)] where ( L Z X X ( . ) FIGURE . (a) Relation between the phasors V L , V R , V C , and I , (b) Relation between the phasors V L , V R , and ( V L + V C ) for the circuit in Fig. .
. Since phasor I is always parallel to phasor V R , the phase angle φ is the angle between V R and V and can be determined from Fig. . : tan Cm Lm Rm v v v φ Using Eq.
( . ), we have tan L X X φ ( . ) Equations ( . ) and ( .
) are graphically shown in Fig. ( . ). This is called Impedance diagram which is a right-triangle with Z as its hypotenuse.
Equation . (a) gives the amplitude of the current and Eq. ( . ) gives the phase angle.
With these, Eq. ( . ) is completely specified. If X C > X L , φ is positive and the circuit is predominantly capacitive.
Consequently, the current in the circuit leads the source voltage. If X C < X L , φ is negative and the circuit is predominantly inductive. Consequently, the current in the circuit lags the source voltage. Figure .
shows the phasor diagram and variation of v and i with ω t for the case X C > X L . Thus, we have obtained the amplitude and phase of current for an LCR series circuit using the technique of phasors. But this method of analysing ac circuits suffers from certain disadvantages. First, the phasor diagram say nothing about the initial condition.
One can take any arbitrary value of t (say, t