VIPSolutionsGive step-by-step solution with explanation and final answer:Consider the block diagram shown in Fig. 2. Find the equivalent transfer function < Hs)
Question:
Give step-by-step solution with explanation and final answer:
Consider the block diagram shown in Fig. 2. Find the equivalent transfer function <
Hs)
This is AI generated Response.
Asked by: Test User 100199
Created at: 2025-05-17 05:02:14
Asked by: Test User 100199
Created at: 2025-05-17 05:02:14
Final Answer:
Rendered
Raw Text
Did you like this answer?
Let's solve the block diagram step by step to find the equivalent transfer function \( \frac{C(s)}{R(s)} \):
### **Step 1: Label all signals**
Let's denote signals at different points:
- Output of summing point 1: \( X(s) \)
- Output of \( G_1(s) \): \( Y(s) \)
- Output of summing point 2: \( Z(s) \)
- Output of \( G_2(s) \): \( W(s) \)
- Output of summing point 3 (final output): \( C(s) \)
### **Step 2: Write equations for each summing point**
#### **Summing Point 1:**
\[
X(s) = R(s) - H(s)C(s)
\]
#### **After \( G_1(s) \):**
\[
Y(s) = G_1(s)X(s)
\]
#### **Summing Point 2:**
\[
Z(s) = Y(s) - G_4(s)C(s)
\]
#### **After \( G_2(s) \):**
\[
W(s) = G_2(s)Z(s)
\]
#### **Summing Point 3:**
\[
C(s) = W(s) + G_3(s)Y(s)
\]
---
### **Step 3: Substitute back and simplify**
#### Substitute \( W(s) \) and \( Y(s) \):
\[
C(s) = G_2(s)Z(s) + G_3(s)Y(s)
\]
But \( Z(s) = Y(s) - G_4(s)C(s) \), so:
\[
C(s) = G_2(s)[Y(s) - G_4(s)C(s)] + G_3(s)Y(s)
\]
\[
C(s) = G_2(s)Y(s) - G_2(s)G_4(s)C(s) + G_3(s)Y(s)
\]
\[
C(s) + G_2(s)G_4(s)C(s) = [G_2(s) + G_3(s)]Y(s)
\]
\[
C(s)[1 + G_2(s)G_4(s)] = [G_2(s) + G_3(s)]Y(s)
\]
\[
C(s) = \frac{[G_2(s) + G_3(s)]Y(s)}{1 + G_2(s)G_4(s)}
\]
#### Now, recall \( Y(s) = G_1(s)X(s) \), and \( X(s) = R(s) - H(s)C(s) \):
\[
Y(s) = G_1(s)[R(s) - H(s)C(s)]
\]
Substitute for \( Y(s) \) in the equation for \( C(s) \):
\[
C(s) = \frac{[G_2(s) + G_3(s)]G_1(s)[R(s) - H(s)C(s)]}{1 + G_2(s)G_4(s)}
\]
\[
C(s)[1 + G_2(s)G_4(s)] = [G_2(s) + G_3(s)]G_1(s)[R(s) - H(s)C(s)]
\]
\[
C(s)[1 + G_2(s)G_4(s)] = [G_2(s) + G_3(s)]G_1(s)R(s) - [G_2(s) + G_3(s)]G_1(s)H(s)C(s)
\]
Bring all \( C(s) \) terms to one side:
\[
C(s)[1 + G_2(s)G_4(s) + [G_2(s) + G_3(s)]G_1(s)H(s)] = [G_2(s) + G_3(s)]G_1(s)R(s)
\]
\[
C(s) = \frac{[G_2(s) + G_3(s)]G_1(s)}{1 + G_2(s)G_4(s) + [G_2(s) + G_3(s)]G_1(s)H(s)}R(s)
\]
---
### **Step 4: Final Transfer Function**
\[
\boxed{
\frac{C(s)}{R(s)} = \frac{[G_2(s) + G_3(s)]G_1(s)}{1 + G_2(s)G_4(s) + [G_2(s) + G_3(s)]G_1(s)H(s)}
}
\]
---
#### **Summary of Method:**
- Write node equations at each summing point.
- Substitute equations backward from the output.
- Collect all terms with \( C(s) \) and \( R(s) \).
- Solve for \( \frac{C(s)}{R(s)} \).
If you need further simplification or a specific case, let me know!
📢 Instructions:
- Ask anything relatd to above question only.
- For new Question goto :