(2) For the circuit below, find and vc(t) and i(t) for all t. 4 – 3 u(t)( www 1 Vc 10 + 10 i 7 10 -H 3+ //F Vc

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**Problem Statement:**

For the circuit below, find \( v_c(t) \) and \( i(t) \) for all \( t \).

**Circuit Description:**

- The circuit consists of a voltage source, resistors, a controlled voltage source, an inductor, and a capacitor.
- Voltage source: \( 4 - 3u(t) \) volts, where \( u(t) \) is the unit step function.
- Resistor: \( 1 \, \Omega \)
- Controlled voltage source: with a voltage gain of \( \frac{V_c}{10} \)
- Controlled current source: \( 10i \), where \( i \) is the current through the circuit.
- Inductor: \( \frac{7}{10} \, \text{H (henries)} \)
- Capacitor: \( \frac{1}{7} \, \text{F (farads)} \)
- \( V_c \) is the voltage across the capacitor.

**Objective:**

Determine the expressions for \( v_c(t) \) and \( i(t) \) for the circuit.

**Note:** The circuit includes elements that have dependencies on the voltage across the capacitor and current through certain parts, which may require solving differential equations based on Kirchhoff’s laws and the relationships for inductors and capacitors.
Transcribed Image Text:**Problem Statement:** For the circuit below, find \( v_c(t) \) and \( i(t) \) for all \( t \). **Circuit Description:** - The circuit consists of a voltage source, resistors, a controlled voltage source, an inductor, and a capacitor. - Voltage source: \( 4 - 3u(t) \) volts, where \( u(t) \) is the unit step function. - Resistor: \( 1 \, \Omega \) - Controlled voltage source: with a voltage gain of \( \frac{V_c}{10} \) - Controlled current source: \( 10i \), where \( i \) is the current through the circuit. - Inductor: \( \frac{7}{10} \, \text{H (henries)} \) - Capacitor: \( \frac{1}{7} \, \text{F (farads)} \) - \( V_c \) is the voltage across the capacitor. **Objective:** Determine the expressions for \( v_c(t) \) and \( i(t) \) for the circuit. **Note:** The circuit includes elements that have dependencies on the voltage across the capacitor and current through certain parts, which may require solving differential equations based on Kirchhoff’s laws and the relationships for inductors and capacitors.
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