What is the relation between reactance, resistance and impedance?

Question

Examoogle · Accepted Answer

$Z = R + jX$

Explanation: In AC circuits, impedance ($Z$) is the total opposition to current flow, represented as a complex quantity comprising both real and imaginary parts. The resistance ($R$) represents the real part (dissipative component), while the reactance ($X$) represents the imaginary part (storage component), combined using the complex operator $j$.

Working Principle: The impedance is represented in the complex plane where the real axis corresponds to resistance $R$ and the imaginary axis corresponds to reactance $X$. For an inductive circuit, the reactance is positive, resulting in $Z = R + jX$, whereas for a capacitive circuit, the reactance is negative, resulting in $Z = R - jX$. The magnitude of impedance is given by $|Z| = \sqrt{R^2 + X^2}$.

Key Formulas: $Z = R + jX$ — Complex form of impedance; $|Z| = \sqrt{R^2 + X^2}$ — Magnitude of impedance; $\phi = 	an^{-1}(\frac{X}{R})$ — Phase angle of impedance

Key Points: Resistance ($R$) is independent of frequency and causes real power dissipation.; Reactance ($X$) is frequency-dependent and represents energy storage in magnetic or electric fields.; The operator $j$ indicates a $90^{\circ}$ phase shift between the resistive and reactive components.; Impedance is measured in Ohms ($\Omega$).

Advantages: Allows for easy calculation of voltage and current using Ohm's law in phasor form.; Enables phase angle analysis in AC circuits.

Disadvantages: Requires knowledge of complex algebra for circuit analysis.; Only applicable to linear circuits with sinusoidal inputs.

Applications: Analysis of RLC networks.; Power systems transmission modeling.; Filter design in telecommunications.

Additional Info: In a purely resistive circuit, $X = 0$, so $Z = R$.. Option D represents a capacitive impedance where the reactance is negative.. The format $Z = R + jX$ is the standard rectangular form used in electrical engineering.

Answer: A is correct — The impedance $Z$ in an AC circuit is defined as the complex sum of the resistance $R$ and the reactance $X$ represented as $Z = R + jX$.

Exam Tip: Always remember that in the complex plane, inductive reactance ($X_L = \omega L$) is positive, while capacitive reactance ($X_C = \frac{1}{\omega C}$) is negative; this determines the sign of the imaginary component in $Z$.

Related Questions

Discussion (0)