Question

Alternating Current. Given an alternating voltage \(v=4.27 \sin \left(463 t+27^{\circ}\right),\) find the maximum voltage, the period, frequency, and phase angle, and the instantaneous voltage at \(t=0.12 \mathrm{s}.\)

Short answer

The maximum voltage is \(4.27\) V, the period \(T = \frac{2\pi}{463} \approx 0.01357\) s, the frequency \(f = \frac{1}{T} \approx 73.69\) Hz, the phase angle is \(27^\circ\), and the instantaneous voltage at \(t = 0.12\) s is approximately \(4.27 \sin(463 \times 0.12 + 27^\circ)\) V.

Step-by-step solution

Identifying the Maximum Voltage

The maximum voltage (also called amplitude) in an AC circuit is the coefficient of the sine function in the voltage equation. In the given voltage equation, the maximum voltage is simply the coefficient before the sine function.

Finding the Period

The period of an alternating current is the time it takes to complete one full cycle. It can be calculated from the angular frequency, which is the coefficient of \(t\) in the sine function. The angular frequency \(\omega\) is related to the period \(T\) by the equation \(\omega = \frac{2\pi}{T}\).

Calculating the Frequency

Frequency \(f\) is the reciprocal of the period \(T\). Once the period is found, calculate the frequency using the relationship \(f = \frac{1}{T}\).

Determining the Phase Angle

The phase angle is given directly in the sine function as \(27^\circ\). It signifies the shift of the wave from its standard position.

Computing the Instantaneous Voltage at \(t = 0.12 \text{s}\)

To find the instantaneous voltage at a specific time, substitute the value of \(t\) into the voltage equation and perform the calculation. Ensure that the phase angle is in radians if the calculator is in radian mode.

Key concepts

Maximum Voltage

When discussing alternating current (AC), we often refer to the maximum voltage, which is also known as the amplitude of the voltage waveform. This value represents the peak or highest voltage level achieved by the AC signal over its cycle. In the given exercise, the maximum voltage is determined directly from the coefficient preceding the sine function in the voltage equation, which is 4.27 volts. This value is crucial as it helps in assessing the power capabilities and insulation requirements of electrical systems.
Understanding the maximum voltage also allows us to evaluate the potential stress on various electrical components, ensuring their safe operation within rated voltage levels.

Period of AC Circuit

The period of an AC circuit is the duration of time taken to complete one full cycle of the waveform. This period is derived from the angular frequency, denoted as \( \omega \). The angular frequency is the constant that multiplies the variable \( t \) in the sine function and is related to the period \( T \) by the equation \( \omega = \frac{2\pi}{T} \). For students to find the period, they would use the given angular frequency to solve for \( T \) in this equation. The period is a fundamental characteristic of an AC signal as it defines the signal's repetition rate, which is vital for the timing and synchronization in electrical circuits and electronic devices.

Angular Frequency

In the realm of alternating current, angular frequency is a measure of how quickly the AC waveform oscillates. It is closely tied to both the period and the frequency of the waveform. The angular frequency is represented by \( \omega \) and is used in the equation \( \omega = 2\pi f \) where \( f \) is the conventional frequency. This relationship implies that as angular frequency increases, the waveform cycles more rapidly. Students often encounter angular frequency expressed in radians per second, which differs from the standard frequency represented in Hertz (Hz). The understanding of angular frequency is integral for analyzing the behavior of AC circuits and for engineering applications that involve signal processing or resonance phenomena.

Phase Angle

A key concept in AC circuits is the phase angle, which determines the initial angle of the sinusoidal function representing the AC signal. In the context of the given exercise, the phase angle is \( 27^\circ \) as indicated in the equation. This angle signifies the amount by which the voltage waveform is shifted horizontally along the time axis from a standard reference point. Phase angle is crucial when comparing or combining two AC signals, as it affects how the peaks and troughs of the waveforms align with one another. For students to make sense of phase angles, they need to consider the impact on power delivery and the constructive or destructive interference that can occur when waveforms with different phases are overlaid.

Instantaneous Voltage

The concept of instantaneous voltage pertains to the voltage level of the AC signal at any given moment in time. Unlike direct current where the voltage remains constant, the voltage in an AC circuit varies with time based on a sinusoidal pattern. To calculate instantaneous voltage, one would substitute a specific time point into the voltage equation. In our exercise, the goal was to find the voltage at \( t = 0.12s \), which required inputting this time value into the equation and computing the result. Instantaneous voltage is practical for understanding what happens in an AC circuit at precise moments, and it is vital for myriad applications, such as determining the output of power supplies and real-time analysis of electrical waveforms.