Signal Flow Graph Construction

A signal flow graph (SFG) is a graphical representation of the algebraic relationships between variables in a system. It was developed as an alternative to block diagrams and forms the foundation for Mason's gain formula, which gives the system transfer function directly without step-by-step reduction.

Core Concept Explanation

A signal flow graph consists of nodes and branches. Each node represents a signal (or variable), and each directed branch represents the dependency of one signal on another. The gain of a branch is the coefficient relating the two nodes it connects. If a branch from node xi to node xj has gain a_ij, then it contributes a_ij × xi to the value of xj.

A source node has only outgoing branches and no incoming branches. It represents an independent input. A sink node has only incoming branches and represents the output. Nodes with both incoming and outgoing branches are called mixed nodes. The variable at any mixed node is the sum of all products of incoming branch gains and their respective node values.

A forward path is any continuous path from the input (source) node to the output (sink) node, without passing through the same node twice. The forward path gain is the product of all branch gains along that path. A loop is a closed path starting and ending at the same node, without repeating any node. The loop gain is the product of all branch gains around the loop.

Mathematical Expression

The relationship at any node j in an SFG is expressed as: xj = sum over all i of (a_ij × xi), where a_ij is the branch gain from node i to node j. For example, in a simple three-node SFG with nodes x1, x2, x3, forward gains a12 and a23, and a feedback branch a32 (from x3 back to x2): x2 = a12 × x1 + a32 × x3 and x3 = a23 × x2.

When constructing an SFG from a block diagram, each summing point and take-off point becomes a node, and each block with its gain becomes a branch. The negative sign of feedback in a block diagram is absorbed into the branch gain of the feedback path in the SFG (making it -H(s) instead of +H(s)). This is an important point that is often asked in GATE.

Practical Understanding

To convert a block diagram to an SFG, the following mapping is used. Each signal (input, output, error signal, intermediate signal) becomes a node. Each block G(s) between two nodes becomes a branch with gain G(s). Each summing point that sums signals is replaced by multiple branches feeding into the same node, with gains of +1 or -1 depending on their sign. Take-off points simply allow the same node to have multiple outgoing branches.

For a standard closed-loop system with forward gain G(s) and feedback gain H(s), the SFG has four nodes: R(s), E(s), C(s) at intermediate point, and the final C(s). The forward branch from E to C has gain G(s), and the feedback branch from C back to E has gain -H(s) (negative because of the negative feedback sign absorbed from the summing point).

Given:
Block diagram with G1(s) = 4, G2(s) = 3/(s+2), H(s) = 0.5 (feedback)
Convert to SFG and identify forward paths and loops.

Why this formula applies:
SFG construction directly maps block diagram elements to nodes and branches.

Nodes:
x1 = R(s) [source node]
x2 = E(s) [after summing point]
x3 = output of G1 [intermediate]
x4 = C(s) [sink node]

Branches:
x1 → x2 : gain = +1 (input to error node)
x2 → x3 : gain = G1 = 4
x3 → x4 : gain = G2 = 3/(s+2)
x4 → x2 : gain = -H = -0.5 (feedback, negative sign included)

Forward path:
P1 = 1 × 4 × 3/(s+2) = 12/(s+2)

Loop:
L1 = 4 × [3/(s+2)] × (-0.5) = -6/(s+2)

Final Answer:
Forward path gain P1 = 12/(s+2), Loop gain L1 = -6/(s+2)

Exam Tip: The key difference between SFG and block diagram is that the negative sign of feedback is explicitly included in the branch gain in SFG (as -H(s)), whereas in block diagram the negative sign appears at the summing point. GATE problems often test whether you correctly include this sign in loop gain calculations.

Quick Revision

• SFG nodes represent signals; branches represent gains between signals.
• Source node: only outgoing branches (input). Sink node: only incoming branches (output).
• Forward path: continuous path from source to sink, no node repeated.
• Loop: closed path starting and ending at same node, no node repeated.
• In SFG, the negative feedback sign is absorbed into the feedback branch gain as -H(s).
• Converting block diagram: each signal is a node, each block is a branch, summing point signs become branch gains.
• GATE trap: In block diagram the minus sign is at summing point; in SFG it is part of the branch gain. Not including it makes loop gain positive and gives wrong CLTF.