Calculate the current in the resistor in Fig. P 8.5 if the...

Calculate the current in the resistor in Fig. $\mathrm{P} 8.5$ if the voltage input is
(a) $v_{1}(t)=10 \cos \left(377 t+180^{\circ}\right) \mathrm{V}$
(b) $v_{2}(t)=12 \sin \left(377 t+45^{\circ}\right) \mathrm{V}$
Give the answers in both the time and frequency domains.

Key Concepts

Phasor Representation

Phasor representation is the method of expressing sinusoidal functions as complex numbers, which capture both the magnitude and the phase angle of the waveform. This simplifies the analysis of AC circuits by converting time-varying signals into constant complex quantities, enabling the use of algebraic techniques instead of solving differential equations.

Frequency Domain Analysis

This concept involves representing and analyzing signals and circuits in the frequency domain instead of the time domain. By expressing sinusoids as phasors (complex numbers that encode amplitude and phase information), differential equations governing circuit behavior become simple algebraic equations. Frequency domain analysis simplifies the study of linear circuits subject to sinusoidal inputs, allowing for easier manipulation and understanding of steady-state behavior.

Signal Conversion

Signal conversion is the process of rewriting sinusoidal signals into equivalent forms that facilitate analysis. Typically, sinusoids expressed as sine or cosine functions with phase shifts are converted into a consistent format (often cosine form) to easily derive their phasor equivalents. This step is important for ensuring a uniform approach when adding or comparing different signals in the frequency domain.

Ohm's Law in AC Circuits

Ohm's Law in AC circuits relates the phasor voltage across a resistor to the phasor current through it using the resistance (which, for a resistor, is a real number and does not vary with frequency). This law is expressed as V = I·R in the frequency domain, simplifying calculations of current when the voltage and resistance are known, and allows transitioning back to the time domain after analysis.

Time Domain Representation

Time domain representation involves expressing circuit quantities, like voltage and current, as time-varying functions. After completing analysis in the frequency domain, results can be converted back into the time domain to obtain the actual sinusoidal waveforms. This representation is essential for understanding how the circuit behaves over time in real-world applications.

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