Question

The current in a circuit, \(i(t)\), is given by $$ i(t)=25 \mathrm{e}^{-0.2 t} \quad t \geq 0 $$ (a) State the current when \(t=0\). (b) Calculate the value of the current when \(t=2\) (c) Calculate the time when the value of the current is \(12.5\).

Short answer

Answer: The current is 25 A when t = 0, approximately 18.34 A when t = 2, and the current is 12.5 A at approximately 3.47 seconds.

Step-by-step solution

(a) Current at t = 0

To find the current when t = 0, we need to plug t = 0 into the given current function: $$ i(t) = 25 e^{-0.2 t} $$ When \(t = 0\): $$ i(0) = 25 e^{-0.2 \times 0} = 25 e^0 = 25(1) = 25 $$ So the current when \(t = 0\) is 25 A.

(b) Current at t = 2

To find the current when t = 2, we need to plug t = 2 into the given current function: $$ i(t) = 25 e^{-0.2 t} $$ When \(t = 2\): $$ i(2) = 25 e^{-0.2 \times 2} = 25 e^{-0.4} \approx 18.34 $$ So the current when \(t = 2\) is approximately 18.34 A.

(c) Time when current is 12.5

To find the time when the current is 12.5, we need to set the current function equal to 12.5 and solve for t: $$ 12.5 = 25 e^{-0.2 t} $$ First, divide both sides by 25: $$ 0.5 = e^{-0.2 t} $$ Now take the natural logarithm of both sides: $$ \ln(0.5) = -0.2 t $$ Finally, divide by -0.2 to get t: $$ t = \frac{\ln(0.5)}{-0.2} \approx 3.47 $$ So, the time when the current is 12.5 A is approximately 3.47 seconds.

Key concepts

Current Calculations

Understanding current calculations in electrical circuits involves applying exponential functions to determine the behavior of current over time. In this specific scenario, the current equation is given by \(i(t) = 25\, \mathrm{e}^{-0.2 t}\). This formula represents exponential decay, where the initial current, 25 A, decreases over time as \(t\) increases.
To find the current at any specific point in time, we substitute the value of \(t\) into this equation. For example, when \(t = 0\), the equation simplifies to \(i(0) = 25\, \mathrm{e}^{0} = 25\), since \(\mathrm{e}^{0} = 1\). At \(t = 0\), the current is at its maximum, 25 A.
Another example is when \(t = 2\), the equation becomes \(i(2) = 25 \cdot \mathrm{e}^{-0.4}\). To solve this, you calculate the exponent \(-0.4\), determine \(\mathrm{e}^{-0.4}\), and finally, multiply this by 25 to find the current at \(t = 2\), which is approximately 18.34 A.
To recap:

• The initial current value is found by setting \(t = 0\).
• For any given time \(t\), substitute \(t\) into the current formula \(i(t)\).
• Simplify the function using basic exponential rules to find the current value.

Natural Logarithm Application

The natural logarithm is a key mathematical tool used to solve exponential equations, like those in circuits involving exponential decay.
In many cases, you are asked to find a specific time \(t\) when the current reaches a particular value. This involves using the natural logarithm to "remove" the exponential term. For instance, if you want the current to be 12.5 A, you start with the equation \(12.5 = 25\, \mathrm{e}^{-0.2 t}\). To isolate \(t\), divide both sides by 25, which gives you \(0.5 = \mathrm{e}^{-0.2 t}\).
The following steps are:

• Utilize the natural logarithm: \(\ln(0.5) = -0.2 t\).
• Solve for \(t\) by dividing the logarithmic result by the coefficient of \(t\), which is \(-0.2\).
• The calculation \(t = \frac{\ln(0.5)}{-0.2}\) gives the required time.

With these steps, you find that \(t \approx 3.47\) seconds when the current is 12.5 A. Using natural logarithms makes handling the exponential decay process straightforward.

Electrical Engineering Mathematics

Electrical engineering mathematics involves a wide range of concepts that help engineers understand and design circuits. Exponential decay is a crucial concept that explains how quantities decrease over time within circuits.

The function \(i(t) = 25\, \mathrm{e}^{-0.2 t}\) models processes where the current is strong initially but diminishes as time passes. This happens in many real-world scenarios, like charging or discharging a capacitor.
Understanding this involves:

• Exponential Functions: They describe rapid changes, reducing complexities into simpler calculations.
• Natural Logarithms: Essential for undoing exponential effects, allowing for time determination when knowing a target current value.
• Physics and Electronics: The core principles, such as Ohm's law and circuit analysis, often intertwine with these mathematical techniques for thorough comprehension.

Applying these mathematical techniques within electrical engineering ensures reliable circuit design and analysis. The mathematical modeling of current behaviors allows engineers to predict and control circuit responses effectively.