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Find Vo (t) for t20 0. 25 H auct-2) E 0-25 H

Find Vo (t) for t20 0. 25 H auct-2) E 0-25 H

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Find V(t) for t>=0

### 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
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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