Problem 70 - Thomas Calculus

Question

Peak alternating current 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: Rewriting the Current Function

The given function is \( i(t) = 2\cos t + 2\sin t \). We can express this as \( i(t) = 2(\cos t + \sin t) \) to make further calculations easier.

Step 2: Using Trigonometric Identity

To find the peak current, we can use the identity \( \cos t + \sin t = \sqrt{2}\cos(t - \frac{\pi}{4}) \). Therefore, \( i(t) = 2\sqrt{2}\cos(t - \frac{\pi}{4}) \).

Step 3: Finding the Maximum Value

The function \( i(t) = 2\sqrt{2}\cos(t - \frac{\pi}{4}) \) reaches its maximum value when the cosine term is equal to 1 (as cosine ranges between -1 and 1).

Step 4: Calculating the Peak Current

Thus, the peak current is given by \( 2\sqrt{2}\times 1 = 2\sqrt{2} \).

Step 5: Conclusion

The largest magnitude of the current, or the peak current in this circuit, is \( 2\sqrt{2} \) amperes.

Key Concepts

Peak Current

In alternating current (AC) circuits, the peak current is the maximum absolute value of the current that occurs in the circuit. It is essential to understand because it can help in designing circuits that are safe and efficient. In our exercise, the current in the AC circuit is described by the equation \[ i(t) = 2\cos t + 2\sin t \]. To find the peak current, we look for the maximum value that this function can reach.
This is done by transforming the function into a more manageable form utilizing trigonometric identities, which we will explore further in the following sections. Once the function is expressed in its new form, identifying the peak current becomes straightforward by determining when it reaches its maximum value.

Trigonometric Identity

A trigonometric identity is an equation that holds true for all values of the variable. In the context of our problem, we use the identity \[ \cos t + \sin t = \sqrt{2}\cos(t - \frac{\pi}{4}) \] to simplify the expression for the current.

• This identity allows us to consolidate the two trigonometric functions \(\cos t\)and \(\sin t\)into a single cosine function, which is easier to analyze.
• The factor \(\sqrt{2}\)scales the cosine function, preparing it for evaluation at its peak.
• By shifting the angle \(t\)by \(\frac{\pi}{4}\), we adjust the phase of the cosine function.

Trigonometric identities are very powerful tools in simplifying complex trigonometric expressions, which ultimately aid in finding peak values of functions.

Maximum Value of a Function

To find the peak current, we must determine the maximum value of the transformed current function \[ i(t) = 2\sqrt{2}\cos(t - \frac{\pi}{4}) \]. The concept is straightforward: find when the cosine function reaches its maximum.
The cosine function \(\cos x\)varies between -1 and 1. Therefore, the maximum value is 1. Multiplying this maximum by whatever multiplier the cosine function might have gives us the peak value.

• In this problem, since \(2\sqrt{2}\)scales the cosine expression, the peak is \(2\sqrt{2}\times 1 = 2\sqrt{2} \).
• This peak represents the maximum positive deviation of the current's magnitude.
• Finding such maxima is crucial in circuit design for ensuring that the components can handle extreme conditions without failure.

Understanding this allows effective analysis and design of AC circuits, focusing on their performance and reliability.

Cosine Function

The cosine function is one of the fundamental trigonometric functions and is widely used in physics and engineering to model periodic phenomena like alternating current. Its basic properties include:

• The cosine function oscillates between -1 and 1 over its domain, which means it has predictable rhythmic peaks and valleys.
• Cosine is a continuous function, essential for modeling smooth and seamless curves like those found in waveforms.
• In our problem, the cosine function is used to represent the alternating current waveform, which is naturally periodic.

The strategic use of the cosine function in solving peak current equations lies in its regularity and symmetry. By maximizing the cosine term to 1, we directly find the peak of the original alternating waveform, providing us with insightful and clear results. Understanding the properties of the cosine function is crucial for analyzing and solving problems in electrical circuits.