Find the transfer function of the signal flow graph shown below.

To find the transfer function (x_8)/(x_1) of the signal flow graph, we will use Mason's Gain Formula: T = (1)/() _k=1^N P_k _k where P_k is the gain of the k-th forward path, is the system determinant, and _k is the determinant of the graph when all loops touching the k-th forward path are removed. Step 1: Identify all forward paths (P_k) and their gains. A forward path is a path from the input node (x_1) to the output node (x_8) that does not traverse any node more than once. Path 1 (P_1): x_1 x_2 x_3 x_4 x_5 x_6 x_7 x_8 The gain is the product of the branch gains along this path: P_1 = (1)(G_1)(G_2)(G_3)(G_4)(G_5)(1) = G_1 G_2 G_3 G_4 G_5 Path 2 (P_2): x_1 x_2 x_3 x_4 x_5 x_7 x_8 (via G_6) The gain is the product of the branch gains along this path: P_2 = (1)(G_1)(G_2)(G_3)(G_6)(1) = G_1 G_2 G_3 G_6 Step 2: Identify all individual loops (L_i) and their gains. A loop is a path that starts and ends at the same node, traversing no other node more than once. Loop 1 (L_1): x_2 x_3 x_2 L_1 = G_1 · (-H_1) = -G_1 H_1 Loop 2 (L_2): x_3 x_4 x_3 L_2 = G_2 · (-H_2) = -G_2 H_2 Loop 3 (L_3): x_4 x_5 x_4 L_3 = G_3 · (-H_3) = -G_3 H_3 Loop 4 (L_4): x_2 x_3 x_4 x_5 x_6 x_2 L_4 = G_1 · G_2 · G_3 · G_4 · (-H_4) = -G_1 G_2 G_3 G_4 H_4 Step 3: Identify all non-touching loops. Non-touching loops are loops that do not share any common nodes. Pairs of non-touching loops: L_1 (nodes x_2, x_3) and L_3 (nodes x_4, x_5) are non-touching. L_1 L_3 = (-G_1 H_1)(-G_3 H_3) = G_1 G_3 H_1 H_3 All other pairs of loops (L_1 L_2, L_1 L_4, L_2 L_3, L_2 L_4, L_3 L_4) share common nodes and are therefore touching. Triplets or higher order non-touching loops: Since there is only one pair of non-touching loops, there are no triplets or higher order sets of non-touching loops. Step 4: Calculate the system determinant . = 1 - L_i + L_i L_j - L_i L_j L_k + = 1 - (L_1 + L_2 + L_3 + L_4) + (L_1 L_3) = 1 - (-G_1 H_1 - G_2 H_2 - G_3 H_3 - G_1 G_2 G_3 G_4 H_4) + (G_1 G_3 H_1 H_3) = 1 + G_1 H_1 + G_2 H_2 + G_3 H_3 + G_1 G_2 G_3 G_4 H_4 + G_1 G_3 H_1 H_3 Step 5: Calculate _k for each forward path. _k is the determinant of the graph when all loops touching the k-th forward path are removed. For P_1 = G_1 G_2 G_3 G_4 G_5: P_1 passes through nodes x_2, x_3, x_4, x_5, x_6, x_7. All loops (L_1, L_2, L_3, L_4) touch P_1. Therefore, _1 = 1. For P_2 = G_1 G_2 G_3 G_6: P_2 passes through nodes x_2, x_3, x_4, x_5, x_7. All loops (L_1, L_2, L_3, L_4) touch P_2. Therefore, _2 = 1. Step 6: Apply Mason's Gain Formula. T = (P_1 _1 + P_2 _2)/() T = ((G_1 G_2 G_3 G_4 G_5)(1) + (G_1 G_2 G_3 G_6)(1))/(1 + G_1 H_1 + G_2 H_2 + G_3 H_3 + G_1 G_2 G_3 G_4 H_4 + G_1 G_3 H_1 H_3) T = (G_1 G_2 G_3 G_4 G_5 + G_1 G_2 G_3 G_6)/(1 + G_1 H_1 + G_2 H_2 + G_3 H_3 + G_1 G_2 G_3 G_4 H_4 + G_1 G_3 H_1 H_3) We can factor out G_1 G_2 G_3 from the numerator: T = (G_1 G_2 G_3 (G_4 G_5 + G_6))/(1 + G_1 H_1 + G_2 H_2 + G_3 H_3 + G_1 G_2 G_3 G_4 H_4 + G_1 G_3 H_1 H_3) Send me the next one 📸