Problem 50 - Precalculus: Functions and Graphs

Question

Lissajous figures are used in the study of electrical circuits to determine the phase difference \(\phi\) between a known voltage \(V_{1}(t)=A \sin (\omega t)\) and an unknown voltage \(V_{2}(\vec{t})=B \sin (\omega t+\phi)\) having the same frequency. The voltages are graphed parametrically as \(x=V_{1}(t)\) and \(y=V_{2}(t)\) If \(\phi\) is acute, then $$\phi=\sin ^{-1} \frac{y_{\mathrm{int}}}{y_{\max }}$$ where \(y_{\text {int }}\) is the nonnegative \(y\) -intercept and \(y_{\max }\) is the maximum \(y\) -value on the curve. (a) Graph the parametric curve \(x=V_{1}(t)\) and \(y=V_{2}(t)\) for the specified range of \(t\) (b) Use the graph to approximate \(\phi\) in degrees. $$\begin{aligned}&V_{1}(t)=163 \sin (120 \pi t), \quad V_{2}(t)=163 \sin (120 \pi t+\pi / 4)&-0 \leq t \leq 0.02\end{aligned}$$

Step-by-Step Solution

Answer

The phase angle \(\phi\) is approximately 45 degrees.

Step 1: Understand Parametric Equations

Identify that the given voltages \(V_1(t)\) and \(V_2(t)\) are \(x(t) = 163 \sin(120\pi t)\) and \(y(t) = 163 \sin(120\pi t + \frac{\pi}{4})\) respectively.

Step 2: Determine Maximum Values

The maximum value of \(y\) (i.e., \(y_{max}\)) is 163, as it is the amplitude of \(V_2(t)\).

Step 3: Find y-intercept

Set \(t = 0\) in the equation for \(y(t)\). This results in \(y(0) = 163\sin(\frac{\pi}{4}) = 163 \cdot \frac{\sqrt{2}}{2} \approx 115.45\). Hence, \(y_{int} \approx 115.45\).

Step 4: Calculate \(\phi\) using the formula

Apply the formula \(\phi = \sin^{-1}\left(\frac{y_{int}}{y_{max}}\right)\) to calculate the phase angle. Substitute \(y_{int} = 115.45\) and \(y_{max} = 163\). Calculate \(\phi \approx \sin^{-1}\left(\frac{115.45}{163}\right)\approx \sin^{-1}(0.7071)\approx \frac{\pi}{4}\) radians.

Step 5: Convert Phase Angle to Degrees

Convert the phase angle from radians to degrees. As \(\phi = \frac{\pi}{4}\) rad is equivalent to 45 degrees, the angle is approximately 45 degrees.

Key Concepts

Parametric Equations

Parametric equations are a powerful mathematical tool used to describe curves by defining both the x and y coordinates as separate functions of a parameter, often denoted by \( t \). This approach allows us to easily model complex motion and shapes. In the context of Lissajous figures, we use parametric equations to simultaneously represent two signals or voltages.

For instance, consider the equations \( x(t) = 163 \sin(120\pi t) \) and \( y(t) = 163 \sin(120\pi t + \frac{\pi}{4}) \). These describe the x and y coordinates as functions of time \( t \), capturing how each voltage changes over time. By plotting these parametric curves, we obtain not just the relationship between the two voltages, but also visualize the effect of any phase difference between them.

This graphical representation is particularly handy in electrical engineering, where it aids in visualizing differences in timing between signals, resulting from various effects such as delays or shifts.

Phase Difference

Phase difference, commonly referred to as \( \phi \), is a measure of how "out of step" two periodic signals are with respect to each other. It's crucial when analyzing waveforms in contexts such as electrical circuits or signal processing, where synchronization and timing are vital.

For the given problem, the phase difference between the voltages \( V_1(t) \) and \( V_2(t) \) is illustrated using Lissajous figures plotted from parametric equations. The phase difference can be calculated once we know the y-intercept and the maximum y-value of the plot. Here, \( \phi \) was found using the formula \( \phi = \sin^{-1}\left(\frac{y_{int}}{y_{max}}\right) \), where \( y_{int} \) is the initial y-value at \( t = 0 \), and \( y_{max} \) is the peak value.

This angle tells us how much one signal is leading or lagging behind the other, often expressed in degrees, and is essential for accurately characterizing and analyzing circuit behaviors.

Sine Function

The sine function is a fundamental trigonometric function that describes smooth, repetitive oscillations. Defined as the vertical position of a point on a unit circle as it rotates around the origin, it's essential for modeling periodic behaviors like waves.

In our equations \( V_1(t) = 163 \sin(120\pi t) \) and \( V_2(t) = 163 \sin(120\pi t + \frac{\pi}{4}) \), the sine function represents the oscillating nature of the voltages, typical in AC circuits. The amplitude of 163 indicates the peak voltage value, while the angular frequency \( 120\pi \) dictates how quickly the waveform cycles through its oscillations. The phase shift \( \frac{\pi}{4} \) in \( V_2(t) \) shifts its waveform forward, a crucial aspect that impacts phase difference interpretation.

Sine functions enable precise descriptions of waveforms in mathematical models, making them indispensable in both theoretical and practical applications, especially in describing signals in electrical circuits.

Electrical Circuits

Electrical circuits consist of components like resistors, capacitors, and inductors, which create paths for the flow and transformation of electric current. In analyzing such circuits, understanding the behavior of waveforms—like the voltages we study—becomes essential.

Electrical circuits often involve alternating current (AC) where voltages and currents are sinusoidal. Determining the phase difference between signals within these circuits is fundamental, particularly in complex configurations where signals might interfere or need to be synchronized. Using Lissajous figures plotted from parametric equations offers a visual method to assess the phase relationship between different voltages.

This visualization assists in assessing circuit performance, enabling engineers to design and troubleshoot efficiently by clarifying how components' behaviors are affected by timing differences. Therefore, mastering concepts like phase difference and sine function within the circuit helps in creating more efficient and reliable systems.