Question

Answer the following question.
In a series LCR circuit connected across an ac source of variable frequency, obtain the expression for its impedance and draw a plot showing its variation with frequency of the ac source. 

Answer 1

Consider the LCR circuit 

An AC source E with voltage `ν = ν_m` sin ωt is applied across LCR circuit

As the inductor, capacitor, and the resistor are connected in series so the current through all of them is the same (same amplitude and same phase)
Let the current be 1= I_m sin wt
The voltage across each component has a different phase relation with the current.  

1. Let the maximum voltage across the resistor be V_R = I_mR that is in the same phase of the current hence it is represented by OA in the phasor diagram.
2. Let the maximum voltage across the inductor be V_L = I_mX_L and that leads the current by `pi/2` it is represented by OD in the phasor diagram.
3. Let the maximum voltage across the capacitor be V_c = I_mX_c and that lags behind the current by `pi/2`, it is represented by OC in the phasor diagram.

Resultant voltage can be found by using the vector sum of the phasors. The resultant voltage is represented by OF.

It can be written as:

`V_m = sqrt(V_R^2 + (V_L - V_c)^2)`

`V_m = sqrt((I_mR)^2 + (I_mX_L - I_mX_c)^2)`

`V_m = I_m sqrt(R^2 + (X_L - X_c)^2)`

`Z = V_m/I_m = sqrt(R^2 + (X_L - X_c)^2)`

or, `Z = V_m/I_m = sqrt(R^2 + (ωL - 1/(ωC))^2)`

Variation of impedance Z with frequency f: