https://r2.erweima.ai/mathbot/image/17251502812800r7hiecv.jpg

Obtenha a função de transferência para o seguinte circuito F(s)=Vs(s)/Ve(s):

To find the transfer function \( F(s) = \frac{V_s(s)}{V_e(s)} \) for the given circuit, we need to analyze the circuit's components and apply circuit analysis techniques in the Laplace domain.

### Circuit Components:
- \( R_1 \): Resistor
- \( C_1 \): Capacitor
- \( R_2 \): Resistor
- \( C_2 \): Capacitor

### Step 1: Analyze the Circuit

1. **Input Voltage \( V_e(s) \):** This is the voltage applied to the circuit.
2. **Output Voltage \( V_s(s) \):** This is the voltage across the capacitor \( C_2 \) and resistor \( R_2 \).

### Step 2: Impedances in the Laplace Domain
- The impedance of a resistor \( R \) remains \( R \).
- The impedance of a capacitor \( C \) is \( \frac{1}{sC} \), where \( s \) is the Laplace variable.

### Step 3: Circuit Analysis
Let's write down the equations based on the topology:

1. **Voltage Division for \( V_x(s) \):**
   \[
   V_x(s) = V_e(s) \times \frac{R_1}{R_1 + \frac{1}{sC_1}}
   \]

2. **Output Voltage \( V_s(s) \):**
   This voltage is the result of a voltage division between \( R_2 \) and \( C_2 \):
   \[
   V_s(s) = V_x(s) \times \frac{\frac{1}{sC_2}}{R_2 + \frac{1}{sC_2}}
   \]

### Step 4: Simplify and Obtain the Transfer Function
Let's substitute \( V_x(s) \) from the first equation into the second equation:

1. Simplify \( V_x(s) \):
   \[
   V_x(s) = V_e(s) \times \frac{R_1 \cdot sC_1}{R_1sC_1 + 1}
   \]

2. Now, substitute into \( V_s(s) \):
   \[
   V_s(s) = V_e(s) \times \frac{R_1 \cdot sC_1}{R_1sC_1 + 1} \times \frac{\frac{1}{sC_2}}{R_2 + \frac{1}{sC_2}}
   \]

3. Simplify the expression:
   \[
   V_s(s) = V_e(s) \times \frac{R_1 \cdot sC_1}{R_1sC_1 + 1} \times \frac{1}{sC_2(R_2sC_2 + 1)}
   \]

4. Finally, the transfer function \( F(s) = \frac{V_s(s)}{V_e(s)} \):
   \[
   F(s) = \frac{R_1 \cdot C_1}{(R_1sC_1 + 1)(R_2sC_2 + 1)}
   \]

### Conclusion:
The transfer function of the given circuit is:
\[
F(s) = \frac{R_1 \cdot C_1}{(R_1sC_1 + 1)(R_2sC_2 + 1)}
\]

Would you like a detailed explanation of any part of this solution, or do you have any additional questions?

Here are some related questions you might be interested in:
1. How does the time constant of the circuit affect the transfer function?
2. What is the physical meaning of the poles in the transfer function?
3. How would the transfer function change if \( C_2 \) were replaced by an inductor?
4. How do you find the frequency response from the transfer function?
5. What are the conditions for the circuit to be critically damped?

**Tip:** Always check the dimensional consistency of your transfer function to ensure that the units make sense.

Learn how to find the transfer function F(s) = Vs(s) / Ve(s) for a circuit using circuit analysis techniques in the Laplace domain. Includes step-by-step solutions, formulas for impedance, and voltage division. Suitable for advanced learners and engineers.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation