Problem 48 - 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. $$\begin{aligned}&V_{1}(t)=6 \sin (120 \pi t), \quad V_{2}(t)=5 \cos (120 \pi t)&0 \leq t \leq 0.02\end{aligned}$$

Step-by-Step Solution

Answer

Phase difference \( \phi \) is approximately \( 90^\circ \).

Step 1: Understanding the Problem

We need to graph the parametric equations where \( x = V_1(t) = 6 \sin(120 \pi t) \) and \( y = V_2(t) = 5 \cos(120 \pi t) \) over the interval \( 0 \leq t \leq 0.02 \). From this graph, we need to approximate the phase difference \( \phi \) using the formula \( \phi = \sin^{-1} \left( \frac{y_{\text{int}}}{y_{\max}} \right) \).

Step 2: Determine Parametric Equations with t

The given voltages are expressed as sine and cosine functions of the same angular frequency. Therefore, \( x = 6 \sin(120 \pi t) \) and \( y = 5 \cos(120 \pi t) \) describe the Lissajous figure where both functions are based on \( 120 \pi t \).

Step 3: Plot the Parametric Equations

To plot \( x \) and \( y \), we need to evaluate these functions over the range \( t = 0 \) to \( t = 0.02 \). We can use graphing calculators or computer software to draw these parametric equations. This graph will look like an ellipse, as both functions have a phase shift and are orthogonal to each other.

Step 4: Identify Key Points on the Graph

From the graph, determine the maximum \( y \) value, \( y_{\text{max}} \), and the \( y \) intercept, \( y_{\text{int}} \). Here, \( y_{\max} = 5 \) since this is the amplitude of \( V_2(t) \), and the graph intersects the \( y \)-axis at \( y = 5 \).

Step 5: Calculate \( \phi \) Using the Formula

Now that we have \( y_{\text{int}} = 5 \) and \( y_{\max} = 5 \), substitute these values into the formula: \( \phi = \sin^{-1} \left( \frac{5}{5} \right) = \sin^{-1}(1) \). This implies \( \phi = 90^\circ \).

Key Concepts

Parametric Equations

Parametric equations are a way to represent geometric shapes, like curves, using a common variable, often denoted as \( t \). In the context of Lissajous figures, these equations help in plotting the x and y coordinates of a graph as a function of time.
This means rather than plotting a direct relationship between x and y, each is independently expressed in terms of \( t \). For our problem, \( x = V_1(t) = 6 \sin(120 \pi t) \) and \( y = V_2(t) = 5 \cos(120 \pi t) \).
This allows each coordinate to vary based on the parameter \( t \), encompassing a specific interval \( 0 \leq t \leq 0.02 \). This interval dictates the section of the curve you can visualize.

• The sine and cosine functions ensure a smooth and periodic curve, in this case, forming a pattern called the Lissajous figure.
• As \( t \) progresses, both equations change, creating a dynamic illustration of their combined behavior, captured over the time domain.

Phase Difference

Phase difference expresses how out of sync two cyclical processes are, and it plays a critical role in comparing voltages in electrical circuits. It's the angular difference between two waves having the same frequency but differing in their starting points. In our exercise, the phase difference \( \phi \) is measured between the two waveforms represented by \( V_1(t) = 6 \sin(120 \pi t) \) and \( V_2(t) = 5 \cos(120 \pi t) \).
Understanding phase difference is essential in identifying the offset between the starting points of signals in circuits.

• A zero phase difference would mean the two waves peak at the same time.
• A phase difference of \( 90^{\circ} \) indicates a quarter of a cycle lag of one wave behind the other.
• In practice, engineers use this concept to synchronize or deliberately de-synchronize waveforms for varied applications, such as in filters and signal processing.

Graphing Electrical Circuits

Graphing electrical circuits visually represents voltage variation over time, which helps in diagnosing and designing circuits. By graphing parametric equations derived from voltages, one can visualize the into-reactive nature of the waves. In our context, plotting \( x = V_1(t) = 6 \sin(120 \pi t) \) and \( y = V_2(t) = 5 \cos(120 \pi t) \) over an interval highlights the relationship between two voltages.
This graph forms what's known as a Lissajous curve, often elliptical, showcasing how distinct phase shifts appear in practice.

• Lissajous figures are practical tools for comparing signal characteristics in real-time, hence crucial in signal analysis.
• Engaging these methods can reveal critical points like amplitude and phase shifts, offering insights into better circuit designs.
• Ultimately, by using oscilloscopes or specialized software, engineers can display and interrogate these complex wave interactions effectively.