---
title: "Signal flow graph with G3 subtracted at output"
subject: "Control Systems"
chapter: 3
key: "semester_3/control_systems:sfg-g1g2g3-h-2069"
source: https://www.bctengineer.com/q/semester_3/control_systems/sfg-g1g2g3-h-2069
site: BCT Engineering
---

# Signal flow graph with G3 subtracted at output

Control Systems · Chapter 3: System Transfer Function and Responses

- Asked in: 2069 Chaitra, 6 marks
- Video: [Problem 2 on Converting Block Diagram to Signal Flow Graph](https://www.youtube.com/watch?v=uCm4n897cOY) — TutorialsPoint
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/sfg-g1g2g3-h-2069
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-3

## Question

Convert the given block diagram to signal flow graph and determine the overall transfer function using Mason's Gain Formula. [Figure: R(s) → summing point S1 (+, −) → node A → G1(s) → node B → G2(s) → summing point S2 (+, −) → C(s). G3(s) from node A to S2 (−). H(s) from node B fed back to S1 (−).]

## Short answer

Forward paths: P1 = G1G2, P2 = −G3; loop L1 = −G1H gives Δ = 1+G1H, and all Δ_k = 1. Using Mason's rule, C(s)/R(s) = (G1G2 − G3)/(1+G1H).

## Answer

#### Signal flow graph
Nodes: $R$, $A$ (output of $S_1$), $B$ (output of $G_1$), $C$.

From the diagram: $A = R - HB$, $B = G_1A$, $C = G_2B - G_3A$.

```text
              -G3
        +----------------+
        |                v
R --1--> A --G1--> B --G2--> C
         ^         |
         +---(-H)--+
```
Branches: $R\to A$ (1), $A\to B$ ($G_1$), $B\to C$ ($G_2$), $A\to C$ ($-G_3$), $B\to A$ ($-H$).

#### Mason's gain formula
$$
T = \frac{\sum P_k\Delta_k}{\Delta}
$$

**Forward paths**
- $P_1 = G_1G_2$ (R, A, B, C)
- $P_2 = -G_3$ (R, A, C)

**Loops**
- $L_1 = -G_1H$ (A, B, A)

There is only one loop, so no non-touching loops.
$$
\Delta = 1 - L_1 = 1 + G_1H
$$

**Cofactors:** both paths pass through node $A$, which is on $L_1$, so $\Delta_1 = \Delta_2 = 1$.

#### Transfer function
$$
T = \frac{P_1\Delta_1 + P_2\Delta_2}{\Delta}
$$

**Answer:**
$$
\frac{C(s)}{R(s)} = \frac{G_1G_2 - G_3}{1 + G_1H}
$$

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. From the block diagram, C equals
   A. G2*B + G3*A
   B. G2*B - G3*A
   C. G1*G2*A
   D. G2*B - H*A
   Answer: B. S2 is a (+,-) summing point with G2 positive and G3 negative, so C = G2B - G3A. (Key point: Node equations)

2. The forward paths from R to C are
   A. G1G2 only
   B. G1G2 and G3
   C. G1G2 and -G3
   D. G1G2H
   Answer: C. Path R-A-B-C = G1G2 and path R-A-C = -G3. (Key point: Forward paths)

3. The single loop is -G1H, so Delta is
   A. 1 - G1H
   B. 1 + G1H
   C. 1 + G1G2H
   D. 1 - G3
   Answer: B. Delta = 1 - L1 = 1 + G1H. (Key point: Loop and delta)

4. C/R is
   A. (G1G2 + G3)/(1 + G1H)
   B. (G1G2 - G3)/(1 - G1H)
   C. (G1G2 - G3)/(1 + G1H)
   D. G1G2/(1 + G1H)
   Answer: C. Both cofactors are 1 because both paths touch the loop at node A. (Key point: Overall TF)

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