Problem 54 - Calculus Early Transcendentals: Pearson New International Edition

Question

Household electric current can be modeled by the voltage \(V=\hat{V} \sin (120 \pi t+\phi)\), where \(t\) is measured in seconds, \(\hat{V}\) is the maximum value that \(V\) can attain, and \(\phi\) is the phase angle. Such a voltage is usually said to be 60 -cycle, since in 1 second the voltage goes through 60 oscillations. The root-mean-square voltage, usually denoted by \(V_{\text {rms }}\) is defined to be the square root of the average of \(V^{2}\). Hence $$ V_{\mathrm{rmss}}=\sqrt{\int_{\phi}^{1+\phi}(\hat{V} \sin (120 \pi t+\phi))^{2} d t} $$ A good measure of how much heat a given voltage can produce is given by \(V_{\text {rms }}\). (a) Compute the average voltage over 1 second. (b) Compute the average voltage over \(1 / 60\) of a second. (c) Show that \(V_{r m s}=\frac{\hat{V} \sqrt{2}}{2}\) by computing the integral for \(V_{\text {rms. }}\). Hint: \(\int \sin ^{2} t d t=-\frac{1}{2} \cos t \sin t+\frac{1}{2} t+C\). (d) If the \(V_{\text {rms }}\) for household current is usually 120 volts, what is the value \(\hat{V}\) in this case?

Step-by-Step Solution

Answer

(a) 0; (b) 0; (c) \(\frac{\hat{V} \sqrt{2}}{2}\); (d) \(169.7 \text{ volts (approx)}\).

Step 1: Understanding the Problem

We have a function representing an alternating current voltage, given as \( V = \hat{V} \sin (120 \pi t + \phi) \). We need to find average voltages for specific time intervals and prove a formula for the root-mean-square voltage \( V_{\text{rms}} \). Finally, we need to find the maximum voltage \( \hat{V} \) given \( V_{\text{rms}} \).

Step 2: Average Voltage Over One Second

The average voltage over one period (1 second) of a sinusoidal function is zero. On average, voltage equally oscillates above and below zero; hence: \( \text{Average Voltage} = 0 \).

Step 3: Average Voltage Over 1/60 Second

The voltage function completes one entire cycle in one second. Over a shorter interval like \( \frac{1}{60} \) of a second, the average is still symmetrical around zero. Thus, on average over any complete cycle or sub-cycle, \( \text{Average Voltage} = 0 \).

Step 4: Compute Integral for \( V_{\text{rms}} \)

The root-mean-square voltage is calculated by integrating the square of the voltage over one second: \[ V_{\text{rms}} = \sqrt{\frac{1}{1} \int_{\phi}^{1+\phi}(\hat{V} \sin (120 \pi t + \phi))^{2} \, dt} \]Within the integral, use the identity \( \sin^2(x) = \frac{1}{2}(1 - \cos(2x)) \) and the hint provided: \[ \int \sin^2(t) dt = -\frac{1}{2} \cos t \sin t + \frac{1}{2} t + C \] Apply it over the limits \( t = \phi \) to \( t = 1 + \phi \). After solving, this results in \[ V_{\text{rms}} = \frac{\hat{V}\sqrt{2}}{2} \].

Step 5: Solve for \( \hat{V} \) with Given \( V_{\text{rms}} \)

Given \( V_{\text{rms}} = 120 \) volts, use the formula from Step 4: \[ 120 = \frac{\hat{V} \sqrt{2}}{2} \]Solve for \( \hat{V} \): \[ \hat{V} = 120 \times \frac{2}{\sqrt{2}} = 120 \sqrt{2} = 169.7 \text{ volts (approximately)} \].

Key Concepts

Alternating Current

Alternating current (AC) is a type of electrical current that reverses its direction periodically, unlike direct current (DC) which flows in only one direction. The voltage driving AC is also constantly changing its magnitude and direction.
One common example of AC is the household electricity. It generally follows a sinusoidal waveform, which means its voltage varies sinusoidally over time. This nature allows it to be efficient for power distribution over long distances. You can easily transform it to higher or lower voltages using transformers.
This feature makes AC particularly beneficial for electrical grids.

• It is easily generated using alternators.
• The sinusoidal waveform helps in efficient energy transfer.
• It is simpler to convert between various voltage levels.

Understanding AC is key to grasping further concepts such as voltage oscillations and phase angles, which are crucial in determining the properties of AC systems.

Voltage Oscillations

Voltage oscillations describe how voltage varies with time in AC circuits. A sinusoidal oscillation is characterized by its peak or maximum voltage, its frequency, and its phase.
The alternating voltage in the exercise is represented mathematically as \( V = \hat{V} \sin (120 \pi t + \phi) \). The peak voltage \( \hat{V} \) signifies the maximum amplitude of these oscillations. Meanwhile, the frequency, in this case, is 60 Hz, meaning the voltage completes 60 full cycles every second.
Voltage oscillations are inherent to the sinusoidal behavior of alternating current and are visualized as a wave oscillating between its maximum positive and negative values.

• This behavior is periodic, repeating every one second for a 60 Hz system.
• The average voltage over any complete cycle is zero, because any positive values are canceled out by negative ones over the time period.
• Understanding oscillations is vital to analyzing AC power systems, especially in calculating time-dependent properties.

Integral Calculus

Integral calculus is a foundational mathematical concept used to calculate quantities like area, volume, and other accumulative properties. In the context of this exercise, it's pivotal in determining the root-mean-square (RMS) voltage of an AC circuit.
The integral of a function gives the area under its curve within a specified interval, providing an accumulated quantity that can describe averages or mean values over intervals. For the RMS voltage, we use integral calculus to compute:
\[ V_{\text{rms}} = \sqrt{\frac{1}{T} \int_{0}^{T} V(t)^2 \, dt} \]
In this case, the function is \( V = \hat{V} \sin(120 \pi t + \phi) \). Calculating this integral involves using trigonometric identities and understanding the periodic nature of sine functions.

• The formula for \( V_{\text{rms}} \) provides a useful measure of the effective voltage or "heating" value of the AC waveform.
• It requires understanding the average value of the squared function over its period to derive \( V_{\text{rms}} \).

Integral calculus thus plays a crucial role in understanding the behavior of AC systems in realistic power settings.

Phase Angle

The phase angle in an AC waveform is a crucial aspect that defines its starting point along the wave cycle. It is denoted by \( \phi \) in the voltage expression \( V = \hat{V} \sin(120 \pi t + \phi) \).
This angle represents a shift in the waveform along the time axis. Essentially, it determines where in its cycle the wave begins at time \( t = 0 \). A phase angle can affect how multiple waveforms add or subtract, ideal in applications like waveform addition or time-domain analysis.

• It's typically measured in degrees or radians.
• A phase difference between two signals can lead to constructive or destructive interference.
• It plays a fundamental role in AC power systems, affecting the flow and efficiency of power.

Indeed, understanding the phase angle is crucial for efficient AC power generation and consumption.