Answer :
Final Answer
To find the complete solution for i(t) for t ≥ 0, we first need to identify the given parameters:v₁(t) = 14e(-4t) V, r₁ = 8ω, and l₁ = 11 H. Using these values, the expression for i(t) can be derived. The final answer is:
i(t) = (14 * 8ω) / (11 * r₁) * (e(-4t) - e(4t)) * sin(8ωt) A
Explanation
1. Understanding the Problem:
The problem asks us to find the complete solution for the current i(t) in an RL circuit with given voltage source v₁(t) and known parameters r₁ and l₁. The equation for the current in an RL circuit is:
i(t) = (v(t) / r₁) * (1 - e(-r₁ / (2 * l₁) * t)) * sin(ωt)
where v(t) is the voltage across the inductor, r₁ is the resistance, l₁ is the inductance, and ω is the angular frequency.
2. Analyzing the Given Parameters:
We are provided with the voltage source v₁(t) = 14e(-4t) V, r₁ = 8ω, and l₁ = 11 H. We need to find the angular frequency ω and then substitute the given values into the equation for i(t).
3. Finding the Angular Frequency (ω):
Since r₁ = 8ω, we can express ω as:
ω = r₁ / 8 = (8ω) / 8
Now, we can rewrite the equation for i(t) using the given v₁(t) and the calculated ω:
i(t) = (14 / 11 * r₁) * (1 - e(-r₁ / (2 * l₁) * t)) * sin((r₁ / 8) * t)
Substituting r₁ = 8ω:
i(t) = (14 * 8ω) / (11 * r₁) * (1 - e(-8ω / (2 * 11) * t)) * sin((8ω) / 8 * t)
Now, we can simplify the expression:
i(t) = (14 * 8ω) / (11 * r₁) * (1 - e(-8ω / 22 * t)) * sin(ωt) A
In conclusion, using the given voltage source v₁(t) = 14e^(-4t) V, resistance r₁ = 8ω, and inductance l₁ = 11 H, we derived the complete solution for the current i(t) in the RL circuit. The final expression for i(t) is:
i(t) = (14 * 8ω) / (11 * r₁) * (e(-4t) - e(4t)) * sin(8ωt) A
