Question

Using phasor diagram for a series LCR circuit, obtain an expression for impedance.

Answer 1

We assume the inductor, capacitor, and resistor are perfect. Because these are connected in series, they always have the same current (i = i₀ sin ωt). The voltage across the resistor (e_R = R_i) is in phase with the current. The voltage across the inductor (eL= X_Li) leads the current by π/2 rad, whereas the voltage across the capacitor (e_C = X_Ci) lags behind the current by π/2 rad. This is illustrated in the phasor diagram.

From this figure,

`e_0^2 = e_R^2 + (e_L − e_C)^2`

= `R^2 i_0^2 + (X_L i_0 − X_C i_0)^2` 

= `i_0^2 [R^2 + (X_L − X_C)^2]`

∴ `e_0 = i_0 sqrt(R^2 + (X_L − X_C)^2)`

= i₀Z

`Z = e_0/i_0`

= `sqrt(R^2 + (X_L - X_C)^2`

∴ The effective resistance of the circuit. It is called the impedance.