The output voltage of an AC generator is given by Δv= (170 V) sin(60 πt) . The generator is conn

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The output voltage of an AC generator is given by Δv= (170...

The output voltage of an AC generator is given by $\Delta v=$ $(170 \mathrm{V}) \sin (60 \pi t) .$ The generator is connected across a $20.0=\Omega$ By inspection, what are the (a) maximum voltage and (b) frequency? Find the (c) rms voltage across the resistor, (d) rms current in the resistor, (e) maximum current in the resistor, (f) power delivered to the resistor, and (g) current when t 5 0.005 0 s. (h) Should the argument of the sine function be in degrees or radians?

Key Concepts

Sinusoidal Functions

AC signals are often modeled as sinusoidal functions, which describe periodic oscillations in terms of amplitude, frequency, and phase. This representation allows for the analysis of time-varying voltages and currents, facilitating the determination of key characteristics such as maximum (peak) values and phase relations.

Angular Frequency and Frequency

Angular frequency (?) is a measure of how rapidly a sine wave oscillates and is related to the regular frequency (f) by the formula ? = 2?f. Understanding this relationship is crucial for converting between the mathematical representation of a waveform and its physical oscillation rate in hertz (Hz).

Root Mean Square (RMS) Values

The RMS value of an AC voltage or current is a statistical measure that represents the equivalent DC value which would deliver the same power to a resistive load. For a sinusoidal waveform, the RMS value is obtained by dividing the amplitude by the square root of two, making it a fundamental concept in AC circuit analysis.

Ohm’s Law

Ohm’s Law establishes a linear relationship between voltage, current, and resistance in electrical circuits (V = IR). This principle is fundamental when analyzing circuits with resistive loads, allowing for the determination of current from known voltage and resistance values, and vice versa.

Instantaneous versus Maximum Values

In AC circuits, the instantaneous value refers to the voltage or current at any given moment in time, whereas the maximum (or peak) value represents the highest magnitude reached by the waveform. Differentiating between these values is essential for correctly analyzing circuit behavior over time.

Electrical Power in Resistive Circuits

The power delivered to a resistor in an AC circuit can be calculated using either the instantaneous values or more commonly, the RMS values. This is important for designing and analyzing circuits to ensure components operate within safe power limits.

Radians versus Degrees

When dealing with trigonometric functions in physics and engineering, the arguments are typically expressed in radians due to their natural relationship with the properties of the circle and their prevalence in calculus-based derivations. Recognizing the unit of angular measure is essential for correctly interpreting sinusoidal functions.

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