Question

a. A series LCR circuit is connected to an ac source. Using the phasor diagram, derive the expression for the impedance of the circuit.

b. Plot a graph to show the variation of current with frequency of the ac source , explaining the nature of its variation for two different resistances R1 and R2 (R1< R2) at resonance.

Answer 1

a.

E = E₀ sinωt is applied to a series LCR circuit. Since all three of them are connected in series the current through them is same. But the voltage across each element has a different phase relation with current. The potential difference V_L, V_C and V_R across L, C and R at any instant is given by V_L = IX_L, V_C = IX_C and V_R = IR, where I is the current at that instant.

V_R is in phase with I. VL leads I by 90° and VC lags behind I by 90° so the phasor diagram will be as shown Assuming V_L > V_C, the applied emf E which is equal to resultant of potential drop across R, L & C is given as

E² = I² [ R² + ( X_L - X_C)² ]

Or
\(\frac{E}{\sqrt{[R^2+(X_L\ -\ X_C)^2]}}=\frac{E}{Z}\) ,where Z is impedance.

Emf leads current by a phase angle φ as tan φ = \(\frac{V_L-V_C}{R}=\frac{X_L-X_C}{R}\)

b. The curve (i) is for R₁ and the curve (ii) is for R₂