Given the following circuit: a) Derive the transfer function for v_capacitor(s)/Vin (s) b) Find the damping ratio of the system given the component values in the figure c) Find the resonance frequency of the circuit

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Given the following circuit:

a) Derive the transfer function for v_capacitor(s)/Vin (s)

b) Find the damping ratio of the system given the component values in the figure

c) Find the resonance frequency of the circuit

The image shows an electrical circuit diagram. Here is a detailed transcription and description:

**Circuit Components:**

1. **Voltage Source:** 
   - Represented on the left side of the circuit.
   - Labeled as "Vin".
   - It is a cosine source, denoted as \(10 \cos t \, \text{V}\).

2. **Resistors:**
   - Two resistors, each with a resistance of \(1 \, \Omega\), are arranged in series and parallel.
   - The first resistor is directly connected in series with the voltage source.
   - The second resistor is connected in parallel with an inductor and capacitor.

3. **Inductor:**
   - Labeled with an inductance of \(1 \, \text{H}\).
   - Placed in parallel with the second \(1 \, \Omega\) resistor and the capacitor.

4. **Capacitor:**
   - Labeled with a capacitance of \(1 \, \text{F}\).
   - Also connected in parallel with the second resistor and inductor.
   - The voltage across the capacitor is labeled as \(v(t)\).

**Circuit Function:**
- The circuit is an RLC (Resistor, Inductor, Capacitor) circuit driven by an AC voltage source. 
- The objective might be to analyze the behavior of the circuit in response to the AC input, particularly looking at the voltage across the capacitor, represented as \(V_{\text{capacitor}}\).

This type of circuit is often studied in electrical engineering to understand resonance, impedance, and the phase relationship between voltage and current in AC circuits.
Transcribed Image Text:The image shows an electrical circuit diagram. Here is a detailed transcription and description: **Circuit Components:** 1. **Voltage Source:** - Represented on the left side of the circuit. - Labeled as "Vin". - It is a cosine source, denoted as \(10 \cos t \, \text{V}\). 2. **Resistors:** - Two resistors, each with a resistance of \(1 \, \Omega\), are arranged in series and parallel. - The first resistor is directly connected in series with the voltage source. - The second resistor is connected in parallel with an inductor and capacitor. 3. **Inductor:** - Labeled with an inductance of \(1 \, \text{H}\). - Placed in parallel with the second \(1 \, \Omega\) resistor and the capacitor. 4. **Capacitor:** - Labeled with a capacitance of \(1 \, \text{F}\). - Also connected in parallel with the second resistor and inductor. - The voltage across the capacitor is labeled as \(v(t)\). **Circuit Function:** - The circuit is an RLC (Resistor, Inductor, Capacitor) circuit driven by an AC voltage source. - The objective might be to analyze the behavior of the circuit in response to the AC input, particularly looking at the voltage across the capacitor, represented as \(V_{\text{capacitor}}\). This type of circuit is often studied in electrical engineering to understand resonance, impedance, and the phase relationship between voltage and current in AC circuits.
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