--- title: "Signal flow graph and Mason's formula: G3 feedforward" subject: "Control Systems" chapter: 3 key: "semester_3/control_systems:sfg-g1g2g3-h1h2-2076a" source: https://www.bctengineer.com/q/semester_3/control_systems/sfg-g1g2g3-h1h2-2076a site: BCT Engineering --- # Signal flow graph and Mason's formula: G3 feedforward Control Systems · Chapter 3: System Transfer Function and Responses - Asked in: 2076 Asoj, 2+6 marks; 2067 Asar (old course), 8 marks (asked 2 times) - 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-h1h2-2076a - Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-3 ## Question Develop signal flow graph for the block diagram model below and find transfer function using Mason's gain formula. [Figure: R → summing point S1 (+, −) → G1 → node A → summing point S2 (+, −) → G2 → node B → summing point S3 (+, +) → C. G3 from node A to S3 (+). H1 from node B fed back to S1 (−). H2 from C fed back to S2 (−).] ## Short answer C(s)/R(s) = (G1G2 + G1G3) / (1 + G1G2H1 + G2H2 − G1G2G3H1H2). ## Answer #### Signal flow graph Nodes: $R$, $e_1$ (after $S_1$), $A$ (after $G_1$), $e_2$ (after $S_2$), $B$ (after $G_2$), $C$. A dummy output node $C$ is drawn with unity gain. ```text 1 G1 1 G2 1 R ───►e1 ───►A ────►e2 ────► B ─────►C ──1──► C ▲ │ ▲ │ │ │ └───────┼──G3───┼───────►│ │ │ │ │ │ └──────(−H2)─────┘ └────────(−H1)─────────┘ ``` Branches: $R\to e_1$ (1), $e_1\to A$ ($G_1$), $A\to e_2$ (1), $e_2\to B$ ($G_2$), $B\to C$ (1), $A\to C$ ($G_3$), $B\to e_1$ ($-H_1$), $C\to e_2$ ($-H_2$). #### Mason's gain formula $$ T = \frac{1}{\Delta}\sum_k P_k\Delta_k $$ **Forward paths** - $P_1 = G_1G_2$ ($R\to e_1\to A\to e_2\to B\to C$) - $P_2 = G_1G_3$ ($R\to e_1\to A\to C$) **Individual loops** - $L_1 = -G_1G_2H_1$ ($e_1\to A\to e_2\to B\to e_1$) - $L_2 = -G_2H_2$ ($e_2\to B\to C\to e_2$) - $L_3 = +G_1G_3G_2H_1H_2$ ($e_1\to A\to C\to e_2\to B\to e_1$; two negative branches give a positive gain) **Non-touching loops:** none (all share node $e_2$ or $B$). **Determinant** $$ \Delta = 1 - (L_1 + L_2 + L_3) = 1 + G_1G_2H_1 + G_2H_2 - G_1G_2G_3H_1H_2 $$ **Cofactors:** both forward paths touch all loops, so $\Delta_1 = \Delta_2 = 1$. #### Transfer function $$ \frac{C(s)}{R(s)} = \frac{G_1G_2 + G_1G_3}{1 + G_1G_2H_1 + G_2H_2 - G_1G_2G_3H_1H_2} $$ This agrees with block diagram reduction of the same figure. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What are the forward paths from R to C? A. G1G2 and G1G3 B. G1G2G3 only C. G2 and G3 D. G1G2H1 and G1G3H2 Answer: A. One path goes through G1 and G2, the other through G1 and the G3 branch from A to C. (Key point: Forward paths) 2. Which set gives the three individual loops? A. -G1G2H1, -G2H2, +G1G2G3H1H2 B. -G1G2H1, -G2H2, -G1G2G3H1H2 C. +G1G2H1, +G2H2, +G1G3 D. -G1H1, -G3H2, -G2 Answer: A. The third loop passes through two negative branches (H1 and H2), so its gain is positive. (Key point: Loop gains) 3. What are Δ1 and Δ2 and why? A. Both 1, because every loop touches both forward paths B. Both 0, no loops C. Both equal to Δ D. 1 - L1 for both Answer: A. All loops share node e2 or B with both paths, so no loops remain after removing a path. (Key point: Cofactors) 4. What is Δ? A. 1 + G1G2H1 + G2H2 - G1G2G3H1H2 B. 1 - G1G2H1 - G2H2 + G1G2G3H1H2 C. 1 + G1G2H1 + G2H2 + G1G2G3H1H2 D. 1 + G1G2 + G1G3 Answer: A. Δ = 1 - (L1+L2+L3) with L1 and L2 negative and L3 positive. (Key point: Determinant) ## Related in Control Systems - [Signal flow graph and Mason's formula: G1–G4, H1–H3](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Asfg-mason-g1-g4-h1-h3) — Control Systems, chapter 3, asked 2× - [Signal flow graph of G1–G3, H1–H3 block diagram](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Asfg-g1g2g3-h1h2h3) — Control Systems, chapter 3 - [Block diagram reduction with G1, G2, H1, H2, feedforward](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Abdr-g1g2-h1h2-ff-unity) — Control Systems, chapter 3, asked 2× - [Mason's gain formula: G1–G4 with H1, H2, unity loop](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Amason-g1-g4-h1h2-2075a) — Control Systems, chapter 3, asked 2× - [Block diagram reduction: G3 feedforward, H1 and H2 loops](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Abdr-g1g2g3-h1h2-b) — Control Systems, chapter 3, asked 4× - [Block diagram reduction of Fig. 2(a): G1–G3, H1, H2](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Abdr-g1g2g3-h1h2-2066) — Control Systems, chapter 3, asked 2×