1. Calculating Impedance in a Parallel Combination of Resistors and Inductors

The total impedance $Z_{eq}$ of a parallel combination of a resistor $R$ and an inductor with inductive reactance $X_L$ is found using the formula:

$\frac{1}{Z_{eq}} = \sqrt{\left(\frac{1}{R}\right)^2 + \left(\frac{1}{X_L}\right)^2}$

Alternatively, using complex notation:

$Z_{eq} = \frac{R jX_L}{R + jX_L}$

where:

• $j = \sqrt{-1}$ (the imaginary unit),
• $X_L = \omega L = 2\pi f L$ is the inductive reactance,
• $f$ is the frequency of the AC supply,
• $L$ is the inductance.

Expanding the denominator:

$Z_{eq} = \frac{R jX_L (R - jX_L)}{(R + jX_L)(R - jX_L)}$

$= \frac{R jX_L - j^2 X_L^2}{R^2 + X_L^2}$

Since $j^2 = -1$:

$= \frac{R jX_L + X_L^2}{R^2 + X_L^2}$

$= \frac{X_L^2 + jR X_L}{R^2 + X_L^2}$

This gives the equivalent impedance as a complex number, which can also be converted into polar form.

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2. Relationship Between Power Factor and Phase Angle in AC Circuits

Power factor ($\text{PF}$) is the cosine of the phase angle $\theta$ between voltage and current:

$\text{Power Factor} = \cos \theta$

where:

• $\theta$ is the phase angle, which depends on the impedance $Z$ of the circuit.

For different circuit elements:

• Pure resistor: $\theta = 0^\circ$, so $\cos 0^\circ = 1$ (Power Factor = 1, purely real power).
• Pure inductor: $\theta = 90^\circ$, so $\cos 90^\circ = 0$ (Power Factor = 0, purely reactive power).
• RL Circuit: The power factor is between 0 and 1, depending on the ratio of resistance and reactance.

A low power factor indicates a high reactive power component, reducing efficiency in power transmission.

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Do you need more details or examples?

Related Questions:

1. How does impedance affect current flow in an AC circuit?
2. What happens to the total impedance when adding more parallel branches with resistors and inductors?
3. How can power factor be improved in an inductive load?
4. What are the advantages of using complex impedance notation in AC circuit analysis?
5. How do capacitors influence impedance in parallel AC circuits?

Tip:

A higher power factor (closer to 1) improves energy efficiency and reduces losses in electrical systems. Using power factor correction capacitors helps balance inductive loads! 🚀