0Block Diagram Reduction Method
Step 1: Initial Setup
In the original block diagram:
- You have blocks G1 and G2 in series in the forward path.
- These feed into a summing point with a feedback loop containing H1.
- There’s also another block G3 in a parallel path.
- Finally, the result is fed into a second feedback loop with H2.
Step 2: Replace G1 and G2 with G12
Identify the Series Combination: Since G1 and G2 are in series, we can replace them with a single block, G12, where:
G12 = G1 * G2
Draw the Simplified Diagram:
Replace G1 and G2 with G12 in the block diagram.
You should now have:
- A single block G12 in the forward path, leading to the summing point where the feedback loop with H1 is attached.
This simplified block diagram at this step would look like this:
R(s) → [ G12 ] → ⊕ → [ G3 ] → C(s)
↑
[ H1 ]
Step 3: Apply Feedback Reduction with G12 and H1
Calculate the Feedback Reduction: Now, apply the feedback formula for G12 and H1:
G_feedback1 = G12 / (1 + G12 * H1) = (G1 * G2) / (1 + G1 * G2 * H1)
Draw the Updated Diagram: Replace the feedback loop with this single block G_feedback1.
Your new diagram should look like this:
R(s) → [ G_feedback1 ] → [ G3 ] → C(s)
Step 4: Combine G_feedback1 and G3 in Parallel
Parallel Combination: Since G_feedback1 and G3 are in parallel, combine them:
G_parallel = G_feedback1 + G3 = (G1 * G2) / (1 + G1 * G2 * H1) + G3
Draw the Diagram After Parallel Combination: Replace G_feedback1 and G3 with the combined G_parallel block.
The diagram now looks like this:
R(s) → [ G_parallel ] → ⊕ → C(s)
↑
[ H2 ]
Step 5: Final Feedback Loop Reduction with G_parallel and H2
Apply the Feedback Reduction: Finally, reduce the feedback loop with G_parallel and H2 using the feedback formula:
G_total = G_parallel / (1 + G_parallel * H2)
This results in the final transfer function C(s)/R(s):
C(s)/R(s) = ((G1 * G2) / (1 + G1 * G2 * H1) + G3) / (1 + ((G1 * G2) / (1 + G1 * G2 * H1) + G3) * H2)
