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Find V(t) for t>=0
Transcribed Image Text:### Educational Explanation: Analyzing an RL Circuit with a Step Function Voltage Source
#### Problem Statement:
The objective is to find \( V_o(t) \) for \( t \geq 0 \).
#### Circuit Description:
The given circuit features the following components and configuration:
- A voltage source of 9 times the unit step function, \( 9U(t-2) \), is connected in series.
- A 3-ohm resistor (\( 3 \Omega \)) is connected in series with the voltage source.
- Following the resistor is an inductor with an inductance of 0.25 Henry (\( 0.25 H \)).
- This series combination is then connected to a parallel combination that includes:
- A 2-ohm resistor (\( 2 \Omega \))
- Another inductor with the same inductance of 0.25 Henry (\( 0.25 H \))
The circuit forms a series-parallel RL configuration.
#### Step-by-Step Analysis:
1. **Identify the Key Variables**:
- **Current through the circuit**.
- **Voltage across the inductors**.
2. **Analyze the Impact of the Step Input**:
- The voltage source \( 9U(t-2) \) implies a step function that turns on at \( t = 2 \) with a magnitude of 9V.
- For \( t < 2 \), the voltage source is 0V.
- For \( t \geq 2 \), the voltage source is 9V.
3. **Write the Differential Equations**:
- The impedance of the inductors needs to be considered.
- Initially assume steady-state solutions to find initial conditions and then solve using Laplace transforms or classical methods.
4. **Calculation and Simplification**:
- Translate the circuit into the Laplace domain to simplify the differential equations.
- Solve for \( V_o(s) \) and then perform the inverse Laplace transform to find \( V_o(t) \).
5. **Initial and Final Conditions**:
- At \( t = 0 \), everything behaves as if the circuit has been at rest for a long period.
- Evaluate the current and voltage for both resistors and inductors at \( t = 2 \), just before and after the input steps up.
#### Diagram Explanation:
- **Voltage Source**: The circle with a
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