Determine the initial and final conditions on iL from the given figure. Assume R3 = 19.5 Ω and R4 = 22 Ω. What are the initial and final conditions of the current through the induc... Determine the initial and final conditions on iL from the given figure. Assume R3 = 19.5 Ω and R4 = 22 Ω. What are the initial and final conditions of the current through the inductor?
Understand the Problem
The question asks us to determine the initial (t=0) and final (t=infinity) current through the inductor (iL) in a given circuit. The circuit involves resistors, an inductor, a voltage source, and a switch that changes position at t=0. We need to calculate the current iL at these two time instances, given the resistor values and the voltage source.
Answer
The initial and final conditions of the current through the inductor is $3125$ mA and $0.0981$ A.
Answer for screen readers
The initial and final conditions of the current through the inductor is $3125$ mA and $0.0981$ A.
Steps to Solve
- Determine the circuit before t=0
When $t < 0$, the switch is connected to the $10 \Omega$ resistor. The inductor acts as a short circuit in steady state. The current flows through the $10 \Omega$ resistor and the $R_4$ resistor. With the inductor as a short, $R_3$ is effectively shorted, so no current flows through it.
- Calculate the current through the inductor at t=0-
The current $i_L(0^-)$ can be calculated using Ohm's Law: $$ i_L(0^-) = \frac{100}{10 + R_4} = \frac{100}{10 + 22} = \frac{100}{32} = 3.125 \text{ A} $$
- Determine the initial condition iL(0+)
Since the current through an inductor cannot change instantaneously, the current at $t=0^+$ is the same as the current at $t=0^-$. Therefore, $i_L(0^+) = i_L(0^-) = 3.125 \text{ A} = 3125 \text{ mA}$.
- Determine the circuit after t=0
When $t > 0$, the switch is connected to the $1000 \Omega$ resistor. After a long time ($t = \infty$), the inductor acts as a short circuit. The current now flows through the $1000 \Omega$ resistor and $R_3$. Resistor $R_4$ is effectively shorted, so no current flows through it.
- Calculate the final current through the inductor iL(∞)
The final current $i_L(\infty)$ can be calculated using Ohm's Law: $$ i_L(\infty) = \frac{100}{1000 + R_3} = \frac{100}{1000 + 19.5} = \frac{100}{1019.5} \approx 0.0981 \text{ A} $$
The initial and final conditions of the current through the inductor is $3125$ mA and $0.0981$ A.
More Information
The initial condition is determined by considering the circuit's state before the switch changes position, assuming it has reached a steady state. The inductor acts as a short circuit in DC steady-state. The final condition is found by considering the circuit's state after the switch has been in its new position for a long time, again assuming a steady state is reached.
Tips
- Failing to recognize the inductor's behavior as a short circuit in steady-state DC analysis.
- Incorrectly calculating the total resistance in the circuit for both initial and final conditions.
- Forgetting that current through an inductor cannot change instantaneously, thus $i_L(0^+) = i_L(0^-)$.
- Incorrectly calculating the current using Ohm's law.
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