Question

Electrical Current Suppose that at any time \(t(\mathrm{sec})\) the current \(i(\mathrm{amp})\) in an alternating current circuit is \(i=2 \cos t+2 \sin t .\) What is the peak (largest magnitude) current for this circuit?

Short answer

The peak current for this circuit is \(2.83\) amp.

Step-by-step solution

Apply Pythagorean Identity

Transform the current function into an equivalent form by using the Pythagorean identity. \(\cos(t)^2 + \sin(t)^2 = 1\). So the equivalent form can be written using a trigonometric identity as \(i = 2 \sqrt{2} \sin(t + \pi/4)\). This simplifies the function making it easier to solve for the maximum.

Determine sinusoidal peak current

For a sinusoidal current, peak current is obtained when the sinusoidal function achieves its maximum value which is 1. As such, substitute \(\sin(t + \pi/4)\) with its maximum value (1) to solve for the peak current, \(i_{peak} = 2 \sqrt{2} \times 1\).

Compute the peak current

Simplify \(i_{peak} = 2 \sqrt{2}\) to find that the peak current for this circuit is \(i_{peak} = 2.83\) amp.