What is sinϕ from impedance triangle?

Question

Examoogle · Accepted Answer

$X_L/Z$

Explanation: In an R-L series AC circuit, the impedance triangle is a right-angled triangle where the horizontal side represents resistance (R), the vertical side represents inductive reactance (X_L), and the hypotenuse represents impedance (Z). The phase angle (\phi) between voltage and current is defined by the trigonometric relations within this triangle.

Working Principle: The angle \phi represents the phase displacement between the applied voltage and current. According to trigonometry in a right-angled triangle, the sine of the angle is defined as the ratio of the opposite side to the hypotenuse. In the impedance triangle, the opposite side to angle \phi is the inductive reactance X_L, and the hypotenuse is the impedance Z.

Key Formulas: $Z = \sqrt{R^2 + X_L^2}$ — Total impedance formula; $\sin\phi = \frac{X_L}{Z}$ — Ratio definition of sine from impedance triangle; $\cos\phi = \frac{R}{Z}$ — Power factor definition

Key Points: The impedance triangle is derived from the voltage triangle by dividing all sides by the current (I).; \phi is the angle by which the current lags behind the voltage in an inductive circuit.; The sine of the phase angle represents the reactive factor of the circuit.

Advantages: Provides a simple visual representation of circuit parameters.; Enables easy calculation of power factor and phase angle.

Applications: Analysis of AC series circuits.; Power factor correction calculations.

Additional Info: Option A ($X_L/R$) represents the tangent of the phase angle (tan \phi).. Option C ($R/Z$) represents the cosine of the phase angle (cos \phi), which is the power factor.. Option D ($Z/R$) represents the secant of the phase angle (sec \phi).

Answer: B is correct — By trigonometric definition in a right-angled impedance triangle, $\sin\phi$ is the ratio of the perpendicular side ($X_L$) to the hypotenuse ($Z$).

Exam Tip: Always remember that in the impedance triangle, the horizontal component is always Resistance (R) and the vertical is Reactance (X). Use SOH-CAH-TOA to recall the ratios: Sine is Opposite/Hypotenuse.

Related Questions

Discussion (0)