Derive Power Factor Expression for a Series RL Choke Coil

A practical choke coil is modeled as a pure resistor (R) in series with a pure inductor (L). When an AC voltage (V) is applied, a current (I) flows.

1. Impedance Triangle: The relationship between resistance (R), inductive reactance (X_L), and total impedance (Z) can be visualized with a right-angled “impedance triangle”.
   • The base of the triangle is the resistance (R).
   • The perpendicular side is the inductive reactance (X_L = ωL = 2πfL).
   • The hypotenuse is the total impedance (Z).
2. Pythagorean Theorem: From the triangle, the magnitude of the impedance Z is found using the Pythagorean theorem:
   Z² = R² + X_L²
   Z = √(R² + X_L²)
3. Phase Angle (φ): In this series RL circuit, the current (I) lags behind the voltage (V) by a phase angle φ. This same angle φ is found in the impedance triangle, between the resistance (R) and the impedance (Z).
4. Definition of Power Factor: The power factor is defined as the cosine of this phase angle φ.
   cos φ
5. Derivation: From basic trigonometry in the impedance triangle (Cosine = Adjacent / Hypotenuse):
   cos φ = R / Z
6. Final Expression: By substituting the expression for Z from Step 2, we get the final derived expression for the power factor:
   cos φ = R / √(R² + X_L²)
   or
   cos φ = R / √(R² + (2π f L)²)