Problem 68 - Precalculus: Functions and Graphs

Question

Exer. \(67-68\) : For certaln applications in electrical englneering, the sum of several voltage signals or radio waves of the same frequency is expressed in the compact form \(y=A \cos (B t-C) .\) Express the given signal in this form. $$y=10 \sin \left(120 \pi t-\frac{\pi}{2}\right)+5 \sin 120 \pi t$$

Step-by-Step Solution

Answer

Combine the two cosine representations using phasor addition to get: \(y = A \cos(120\pi t - C)\).

Step 1: Recognize the Sine to Cosine Relationship

The formula for a sine function can be converted to a cosine function using the identity \(\sin(x) = \cos(x - \frac{\pi}{2}))\). In this problem, the first sine term already matches the form: \(10 \sin \left(120\pi t - \frac{\pi}{2}\right)\). This can be directly rewritten as \(10 \cos(120\pi t - \pi)\).

Step 2: Express the Second Term as Cosine

For the second term \(5 \sin(120\pi t)\), apply the sine to cosine identity: \(\sin(x) = \cos(x - \frac{\pi}{2})\), which gives:\(\5 \sin(120\pi t) = 5 \cos(120\pi t - \frac{\pi}{2})\).

Step 3: Re-write in a Common Form

Both terms are in the cosine form, so they can be combined if applicable. Rewrite each term:- \(10\cos(120\pi t - \pi)\)- \(5\cos(120\pi t - \frac{\pi}{2})\)These are two distinct cosine terms, expressed in the desired form: \(A \cos(Bt - C)\).

Step 4: Combine Using Phasor Addition

Since we want a single expression \(y = A \cos(Bt - C)\), we apply phasor addition to the terms:1. Convert each cosine term into phasor (imaginary exponentials).2. Add the phasors.3. Convert the sum of phasors back to the cosine form.This step might involve solving simultaneous equations; however, notice the first term already has a shift equivalent to half rotation \(\pi\), which might be depicted directly.

Key Concepts

Voltage Signals

Voltage signals are fundamental elements in electrical engineering. They represent the variation of voltage over time and are frequently used to model how electrical currents behave.
Voltage signals are often visualized as waveforms, which are crucial for both analysis and design in systems such as radios, computers, and communication devices.

• Voltage signals can be either direct current (DC) or alternating current (AC).
• AC signals often form wave-like patterns, like sine waves, essential for describing oscillations.
• The amplitude of a voltage signal represents its maximum strength.

Understanding voltage signals involves breaking them down into their essential components like amplitude, frequency, and phase shift. This analysis helps engineers to manipulate these signals for desired applications.

Sine and Cosine Conversion

The conversion between sine and cosine functions is a common technique in waveform analysis. It is pivotal when consolidating multiple waveforms into a single expression.
One of the identities used for conversion is \( \sin(x) = \cos(x - \frac{\pi}{2}) \).

• This means that any sine function can be expressed as a cosine function with a phase shift of \( \frac{\pi}{2} \).
• Such conversions simplify the mathematical manipulation of waveforms, allowing for easier algebraic handling.

The sine and cosine relationship is vital when optimizing circuits and systems, as it allows for efficient signal processing and provides a unified framework for handling oscillations and vibrations.

Electrical Engineering Applications

In electrical engineering applications, the analysis of voltage signals through waveform analysis plays a crucial role. Such analysis helps in understanding and designing communication systems, power systems, and electronic devices.

• Applications include designing filters to remove unwanted frequencies.
• Signal conversion mechanisms use these patterns to transform and transmit data effectively.

Using a compact form such as \( y = A \cos(Bt - C) \), engineers can simplify complex waveforms.
This simplification aids in creating reliable and efficient electrical circuits, which is key in everything from radio transmissions to high-speed computing systems.

Waveform Analysis

Waveform analysis involves scrutinizing the detailed shape and form of voltage signals. This process helps in decoding the information carried by these waves, such as when and how they oscillate.

• Fourier analysis is a technique that breaks down waveforms into simpler sinusoidal components.
• Understanding the phase, amplitude, and frequency of waveforms is important for modulation and signal processing.

Through waveform analysis, we gain insights into how different signals interact.
This interaction is crucial for tasks like interference reduction, where unwanted signals are minimized to ensure the clarity and reliability of data transmission.
Thus, analyzing waveforms effectively leads to the optimization of various electronic systems.