AC circuits - voltage: The current I and the impedance Z for...

AC circuits - voltage: The current $I$ and the impedance $Z$ for certain $A C$ circuits are given. Write $I$ and $Z$ in trigonometric form and find the voltage in each circuit. Recall $V=I Z$
$$I=3-2 j A \text { and } Z=2+3.75 j \Omega$$

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Key Concepts

Complex Numbers in AC Circuit Analysis

Complex numbers are used to represent sinusoidally varying signals in AC circuits. They enable the encapsulation of both magnitude and phase within a single entity, allowing for efficient mathematical operations that mirror the physical behavior of alternating currents.

Trigonometric (Polar) Form

Converting complex numbers to trigonometric form involves finding the magnitude and phase angle, which is particularly useful in AC circuits for simplifying calculations. In polar form, multiplication and division of complex quantities translate into operations on their magnitudes and phase angles.

Ohm's Law for AC Circuits

Ohm's Law in AC circuits is expressed as V = IZ, where V is the voltage, I is the current, and Z is the impedance. When represented as complex numbers, this relationship inherently accounts for the phase differences between the voltage and current.

Impedance in AC Circuits

Impedance extends the concept of resistance to include reactive elements, such as inductors and capacitors, in AC circuits. It is a complex number that incorporates both the resistive and reactive effects, influencing the amplitude and phase of current and voltage.

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