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Chapter 23: Problem 32

What trigonometric function describes how voltage and current vary with time in an AC circuit?

Short Answer

Expert verified
Voltage and current in an AC circuit are described by sine or cosine functions.

Step by step solution

01

Identify the Nature of AC Signals

In an AC (Alternating Current) circuit, the voltage and current vary periodically with time. The periodic nature of these signals can be described using trigonometric functions.
02

Analyze the Periodic Function

Trigonometric functions such as sine (\(\sin\)) and cosine (\(\cos\)) are used to model periodic phenomena because of their repetitive oscillations over a specified interval. These functions are well-suited for representing how electrical quantities like voltage and current change over time in an AC circuit.
03

Choose the Appropriate Function

In standard AC circuits, the voltage (\(V(t)\)) and the current (\(I(t)\)) are typically represented by the sine function or cosine function. The sine function is often used:\[V(t) = V_m \sin(\omega t + \phi)\]and\[I(t) = I_m \sin(\omega t + \phi)\]where:- \(V_m\) and \(I_m\) are the maximum (peak) voltage and current,- \(\omega\) is the angular frequency,- \(t\) is time,- \(\phi\) is the phase angle.
04

Conclusion

Both voltage and current in an AC circuit can be modeled using sine or cosine functions. These functions effectively capture the periodic nature of AC signals, where the voltage and current oscillate smoothly over time.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometric Functions
Trigonometric functions are fundamental in mathematics, especially when dealing with phenomena that repeat periodically. These functions include sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)), among others, but the first two are most relevant in AC circuits. Trigonometric functions are essential because they:
  • Exhibit repetitive oscillations over a specified interval, which is ideal for modeling cycles and waves.
  • Make it possible to represent angles and circular motion in a wave format.
In AC (Alternating Current) circuits, the values of voltage and current oscillate following these functions, allowing engineers and students to model how these values change over time. Understanding these functions helps maintain the efficiency and safety of electrical systems.
Sine Function
The sine function (\(\sin\)) is one of the most widely used trigonometric functions in AC circuits. It helps model how electrical quantities, like voltage and current, vary in a smooth, wave-like pattern. The sine wave is characterized by:
  • A repetitive pattern that cycles through positive and negative values.
  • A period, which is the time it takes for the wave to complete one full cycle.
  • An amplitude, representing the maximum value reached during the cycle.
In the context of AC circuits, the sine function is often used as follows:\[V(t) = V_m \sin(\omega t + \phi)\]and\[I(t) = I_m \sin(\omega t + \phi)\]where:
  • \(V_m\) and\(I_m\) are the peak voltage and current.
  • \(\omega\) is the angular frequency, giving the rate of oscillation.
  • \(t\) represents time.
  • \(\phi\) is the phase angle, indicating the offset of the wave.
Cosine Function
The cosine function (\(\cos\)) is another critical trigonometric function used in AC circuits. While similar to the sine function, it has a different phase shift, starting at its maximum value rather than zero. Key aspects of the cosine function include:
  • A similar periodic behavior to sine, with cycles repeating at regular intervals.
  • A maximum amplitude reflective of the peak value, just like the sine wave.
  • Often used interchangeably with sine for mathematical flexibility.
In many AC circuit analyses, the cosine function is employed to describe alternating quantities in equations like:\[V(t) = V_m \cos(\omega t + \phi)\]and\[I(t) = I_m \cos(\omega t + \phi)\]This choice often depends on the initial conditions of the circuit or the specific application involved. Understanding both sine and cosine functions allows for a more comprehensive analysis of all AC waveforms.

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Most popular questions from this chapter

A rectangular loop \(3.2 \mathrm{~cm}\) wide and \(5.1 \mathrm{~cm}\) long is placed in a magnetic field. The angle between the normal to the loop and the magnetic field is \(32^{\circ}\), and the magnetic flux through the loop is \(2.2 \times 10^{-3} \mathrm{~T} \cdot \mathrm{m}^{2}\). What is the magnitude of the magnetic field?

A transformer has twice the number of loops on its secondary coil as on its primary coil. (a) What is the ratio of the secondary voltage to the primary voltage? (b) What is the ratio of the secondary current to the primary current?

A Boeing KC-135A airplane has a wingspan of \(39.9 \mathrm{~m}\) and flies at a constant altitude near the North Pole with a speed of \(850 \mathrm{~km} / \mathrm{h}\). If Earth's magnetic field is \(5.0 \times 10^{-6} \mathrm{~T}\) at that location, what is the induced emf between the wing tips of the airplane?

A transformer has 50 loops in the primary coil and 125 loops in the secondary coil. The voltage in the primary circuit is \(25 \mathrm{~V}\). (a) Is the voltage in the secondary circuit greater than, less than, or equal to \(25 \mathrm{~V}\) ? (b) What is the voltage in the secondary circuit?

Consider four transformers (A, B, C, and D) for which the voltage in the primary coil is \(V_{\mathrm{p}}\), the number of loops in the primary coil is \(N_{\mathrm{p}}\), and the number of loops in the secondary coil is \(N_{\mathrm{s}}\). Rank the transformers in order of increasing voltage in the secondary coil. Indicate ties where appropriate. $$ \begin{array}{|c|c|c|c|} \hline \text { Transformer } & \boldsymbol{V}_{p}(\mathbf{V}) & \boldsymbol{N}_{p} & \boldsymbol{N}_{s} \\ \hline \text { A } & 100 & 20 & 100 \\ \hline \text { B } & 100 & 100 & 20 \\ \hline \text { C } & 20 & 50 & 50 \\ \hline \text { D } & 50 & 400 & 800 \\ \hline \end{array} $$
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