Problem 76 - Thomas Calculus in SI Units

Question

Suppose that at any given time \(t\) (in seconds) the current \(i\) (in amperes) in an alternating current circuit is \(i=2 \cos t+2 \sin t .\) What is the peak current for this circuit (largest magnitude)?

Step-by-Step Solution

Answer

The peak current is \( 2\sqrt{2} \) amperes.

Step 1: Express Current as a Single Trigonometric Function

The given function is \( i(t) = 2 \cos t + 2 \sin t \). We can express this sum as a single trigonometric function using the formula \( i(t) = R \cos(t - \theta) \), where \( R = \sqrt{A^2 + B^2} \) and \( \tan(\theta) = \frac{B}{A} \). Here, \( A = 2 \) and \( B = 2 \).

Step 2: Calculate Amplitude Using Magnitude Formula

Calculate \( R \) using \( R = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \). This \( R \) represents the amplitude or peak current.

Step 3: Find the Peak Current

Since the peak of the cosine function is 1, the peak current of \( i(t) = 2\sqrt{2} \cos(t - \theta) \) is \( 2\sqrt{2} \). The magnitude of the peak current is simply how high the function goes, which is \( 2\sqrt{2} \) amperes.

Key Concepts

Peak Current

In an alternating current circuit, peak current refers to the highest intensity of current that flows through the circuit at any given moment. Alternating current (AC) varies with time, characterized by a waveform, often sinusoidal. In the given exercise, the current is expressed as a function of time, specifically in the form:

• \( i(t) = 2\cos t + 2\sin t \)

The goal is to determine the high point of this waveform, known as the peak current. In mathematical terms, it's the maximum value the function can achieve. By reconfiguring the expression into a single cosine function, it's clear that the amplitude of this function, denoted as \( R \), will represent the peak current. In this case, because the maximum value of a cosine wave is 1, the peak current is identical to the calculated amplitude \( 2\sqrt{2} \), yielding a peak current of \( 2\sqrt{2} \) amperes, which indicates the maximum magnitude the current reaches.

Alternating Current Circuit

Alternating current circuits rely on a type of electric current that periodically reverses direction. Unlike direct current (DC) that flows continuously in a single direction, AC flows back and forth, usually in a sine wave pattern. This reverse flow facilitates the efficient transmission of power over long distances, which is why AC is typically used in household systems and industry. AC circuits can be described using periodic functions like sine and cosine, as these functions inherently model the oscillating behavior of the current.
In the exercise example provided, the function involves both sine and cosine components, which combine to describe how the current changes over time. By manipulating these components into a single trigonometric function, it provides a clearer picture of how the current behaves across the cycle. Understanding the peak value helps in designing elements within the circuit to handle these maximum values, ensuring safety and reliability.

Amplitude Calculation

The amplitude in the context of AC circuits signifies the distance from the average value to the peak value of the waveform. When given a function with sine and cosine components, determining the amplitude requires a specific formula:

• \( R = \sqrt{A^2 + B^2} \)

Where \( A \) and \( B \) are the coefficients of the cosine and sine terms, respectively. The exercise gives us the expression:

• \( i(t) = 2\cos t + 2\sin t \)

Here both coefficients \( A \) and \( B \) equate to 2. Substituting these values into the formula gives:

• \( R = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2} \)

This amplitude \( 2\sqrt{2} \) represents the peak current mentioned earlier in the solution. It’s a crucial value as it defines how much peak power the circuit components need to withstand efficiently. This understanding is essential for engineers and electricians to ensure that the circuit design accommodates the expected demands of current flow, preserving components and preventing malfunction.