Problem 59 - Calculus with Concepts in Calculus

Question

Suppose the current \(I(t)\) flowing in an electrical circuit at time \(t\) is given by $$ I(t)=\frac{100}{1+t^{2}}+3 \sin \frac{30 t}{\pi} \text { for } t \geq 0 $$ Show that $$ \lim _{t \rightarrow \infty}\left(I(t)-3 \sin \frac{30 t}{\pi}\right)=0 $$ Thus for large values of \(t, I(t)\) is very nearly equal to \(3 \sin (30 t / \pi)\). The expression \(3 \sin (30 t / \pi)\) is called the steady-state current, and the expression \(100 /\left(1+t^{2}\right)\) is the transient current (since it is significant only for small values of \(t\) ).

Step-by-Step Solution

Answer

\( \lim_{t \to \infty} (I(t) - 3 \sin \frac{30 t}{\pi}) = 0 \), so \( I(t) \approx 3 \sin \frac{30 t}{\pi} \) for large \( t \).

Step 1: Identify Components of Current Function

Given the function for current is \( I(t) = \frac{100}{1+t^2} + 3 \sin \frac{30 t}{\pi} \). It consists of a transient component \( \frac{100}{1 + t^2} \) and a steady-state component \( 3 \sin \frac{30 t}{\pi} \).

Step 2: Understand the Limit

The problem asks us to show that \( \lim_{t \to \infty} \left( I(t) - 3 \sin \frac{30 t}{\pi} \right) = 0 \). This implies that as \( t \to \infty \), the contribution from the transient component becomes negligible.

Step 3: Simplify the Expression

Simplify the expression by isolating the transient component: \( I(t) - 3 \sin \frac{30 t}{\pi} = \frac{100}{1 + t^2} \). This expression needs to evaluate to zero as \( t \to \infty \).

Step 4: Calculate the Limit of Transient Component

Evaluate \( \lim_{t \to \infty} \frac{100}{1 + t^2} \). As \( t \to \infty \), the denominator \( 1 + t^2 \) grows infinitely large, causing the fraction to tend towards zero.

Step 5: Conclusion on Limit

Since \( \lim_{t \to \infty} \frac{100}{1 + t^2} = 0 \), we conclude that \( \lim_{t \to \infty} \left( I(t) - 3 \sin \frac{30 t}{\pi} \right) = 0 \). This confirms that for large values of \( t \), the current \( I(t) \) is approximately equal to the steady-state component \( 3 \sin \frac{30 t}{\pi} \).

Key Concepts

Transient Current

Transient current in an electrical circuit refers to the component of the current that is only significant for a short period after a change in the circuit occurs. In the given current function, the transient component is represented by \( \frac{100}{1+t^2} \). This part of the current decreases quickly over time, as its influence diminishes as time progresses.
As \( t \) increases, the denominator \( 1 + t^2 \) becomes very large, leading to the entire fraction approaching zero. Thus, the transient current has a fleeting nature, being notable only when \( t \) is small.
This characteristic is why transient currents are often the focus of early-time behavior analysis in circuits, where sudden changes, such as those from a switch opening or closing, temporarily affect the overall current. Over time, however, these effects fade away, emphasizing the steady-state behavior of the circuit.

Limit as t approaches infinity

Understanding the limit as \( t \to \infty \) is crucial for determining the long-term behavior of the circuit's current. In the given problem, evaluating \( \lim_{t \to \infty} \left( I(t) - 3 \sin \frac{30 t}{\pi} \right) = 0 \) helps us recognize which components of the current persist over time.
In this expression, we isolate the transient part \( \frac{100}{1+t^2} \) from the steady-state component \( 3 \sin \frac{30 t}{\pi} \).

• The calculation simplifies by focusing on \( \lim_{t \to \infty} \frac{100}{1+t^2} \), which equals zero.

This demonstrates that over time, the transient current's influence vanishes, and the current primarily reflects the steady-state component.
Seeing limits in action helps students grasp how circuits stabilize over time, highlighting which effects are temporary and which are enduring.

Electrical Circuit Analysis

Electrical circuit analysis involves understanding the different components of a circuit and how they influence its behavior over time. In this exercise, we have dissected a function representing the current \( I(t) \) in a circuit.
The main components at play are transient and steady-state currents, which require analyzing how they affect the circuit at different time scales.

• Transient currents reveal initial responses to changes, important for short-term analysis.
• Steady-state currents indicate long-term behaviors, providing insights into the sustained functioning of the circuit.

By applying circuit analysis, students can predict how real-world circuits will function under various conditions. This problem shows the importance of separating these components to study their individual as well as combined effects.
Such analytical skills are foundational for electrical engineers, enabling them to design and troubleshoot circuits effectively by understanding both immediate alterations and stable states.