The power in AC circuit is given by:

P = Erms Irms cosφ

The value of power factor cosφ in series LCR circuit at resonance is-

Concept:

Resonance: In series LCR circuits, Resonance is a condition in which the inductive reactance and capacitive reactance are equal and lie opposite in phase, so thery cancel out each other and only resistance is left as impedance.

The formula for power factor is calculated as the ratio of resistance to total impedance.

\(⇒ ϕ =\frac{Resistance~of~circuit}{Total~impedance}\)

Fig: a Series LCR circuit

Figure Phasor diagram for series RLC circuit

Explanation:

For a series LCR circuit, the total potential difference of the circuit is given by:

⇒ \(V = \sqrt {{V_R^2} + {{\left( {{V_L} - {V_C}} \right)}^2}} \)

Where VR = potential difference across R, VL =  potential difference across L, and VC =  potential difference across C

For a series LCR circuit, Impedance (Z) of the circuit is given by:

⇒ \(Z = \sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} \)

Where R = resistance, X_L =induvtive reactance, and X_C = capacitive reactive

The resonant frequency of a series LCR circuit is given by

⇒ \(f = \frac{1}{{2π }}\sqrt {\frac{1}{{LC}}}\)

At Resonance X_L = X_C

​∴ The impedance of the circuit (Z) = \(\sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} \)

∴ Z = \(\sqrt{R^2+(X_L-X_L)^2}\)

∴ Z = \(\sqrt{R^2+(0)}\)

∴ Z = R    ............... (1)

Power factor: It is the ratio of resistance to impedance in an LCR given by

⇒ \({\cos Φ = \frac{R}{Z}}\)

Where R = resistance and Z = impedance.

⇒ cos ϕ = \(\frac{R}{R}\) = 1           (∵ Z = R from eq 1)

The power factor of a series LCR circuit at resonance will be 1 due to the presence of only resistance as an effective impedance.