VIPSolutionsGive step-by-step solution with explanation and final answer:Q1/ Using block diagram reduction find the closed loop transfer function of the system whose block diagram is given below R + + bh oo C(s) ’ Ta i il
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Give step-by-step solution with explanation and final answer:
Q1/ Using block diagram reduction find the closed loop transfer
function of the system whose block diagram is given below
R + + bh oo C(s)
’ Ta i il
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Asked by: hhhh
Created at: 2025-05-27 17:50:10
Asked by: hhhh
Created at: 2025-05-27 17:50:10
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Let's solve the block diagram step-by-step using block diagram reduction techniques.
---
### **Step 1: Redraw and Identify Paths and Loops**
Let’s assign the outputs at each summing point for clarity:
- Let the output after \(G_1\) be \(X\).
- After \(G_2\) is \(Y\).
- After \(G_3\) is \(Z\).
- After \(G_4\) is \(W\).
---
### **Step 2: Start from the Innermost Feedback Loop**
#### **Innermost Loop (with \(H_2\))**
The forward path from \(G_2\) to \(G_3\) has a feedback via \(H_2\):
\[
\text{Equivalent Transfer Function:} \quad G_2 \to G_3 \text{ with feedback } H_2
\]
This is a standard negative feedback:
\[
\text{Equivalent block} = \frac{G_2 G_3}{1 + G_3 H_2}
\]
---
### **Step 3: Add \(G_4\) in Parallel**
At the summing point before \(C(s)\), \(G_4\) is in parallel to the reduced forward path (from above):
\[
\text{Combined Forward Path:} \quad \frac{G_2 G_3}{1 + G_3 H_2} + G_4
\]
---
### **Step 4: Combine Forward Path with \(G_1\)**
Before the above segment, there is \(G_1\). So the forward path from \(R(s)\) is:
\[
G_1 \left( \frac{G_2 G_3}{1 + G_3 H_2} + G_4 \right)
\]
---
### **Step 5: Feedback Loops (\(H_1\))**
Now, there is a feedback \(H_1\) from the output to the summing point before \(G_1\):
- The entire forward transfer function from \(R(s)\) to \(C(s)\) is what we derived above.
- The feedback path is \(H_1\).
This is a standard negative feedback system:
\[
T(s) = \frac{\text{Forward Path}}{1 + (\text{Forward Path}) \cdot (\text{Feedback Path})}
\]
So,
\[
T(s) = \frac{G_1 \left( \frac{G_2 G_3}{1 + G_3 H_2} + G_4 \right)}{1 + G_1 \left( \frac{G_2 G_3}{1 + G_3 H_2} + G_4 \right) H_1}
\]
---
### **Step 6: Final Answer (Closed Loop Transfer Function)**
\[
\boxed{
\frac{C(s)}{R(s)} = \frac{G_1 \left( \frac{G_2 G_3}{1 + G_3 H_2} + G_4 \right)}{1 + G_1 \left( \frac{G_2 G_3}{1 + G_3 H_2} + G_4 \right) H_1}
}
\]
---
#### **Summary of Steps:**
1. Reduce the inner feedback loop (\(G_3\) with \(H_2\)).
2. Add \(G_4\) in parallel.
3. Multiply by \(G_1\).
4. Apply the main feedback (\(H_1\)).
5. Write the final closed-loop transfer function.
If you need the steps broken down further, let me know!
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