Signal flow graph with G3 subtracted at output

Signal flow graph

Nodes: $R$, $A$ (output of $S_1$), $B$ (output of $G_1$), $C$.

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

              -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:

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