SFG

A signal-flow graph is a directed diagram used to represent the mathematical relationships and signal pathways within a linear system, most commonly in control theory and electrical engineering. Each node represents a variable or signal, and each directed edge (arrow) between nodes represents a gain or transfer function that acts upon the signal. Unlike block diagrams, which show functional blocks and their connections, an SFG is more abstract and algebraic, making it especially useful for analyzing feedback systems, cascade amplifiers, and electrical networks where multiple signals interact.

The construction of an SFG follows strict rules. A source node has no incoming edges and represents an input signal. A sink node has no outgoing edges and represents an output or variable of interest. All other nodes are intermediate and receive inputs along their incoming edges, sum those signals, and transmit the result along outgoing edges. Each edge is labeled with a gain constant or transfer function that multiplies the signal passing along it. A feedback path, shown as an edge returning to an earlier node, allows the system to use its own output as input, which is central to analyzing closed-loop control and stability.

Using SFGs to solve systems

The power of an SFG lies in Mason's gain formula, which provides a deterministic way to calculate the overall transfer function of a system without needing to solve a set of simultaneous equations. This formula uses the concept of forward paths and loops: a forward path is any route from source to sink that does not traverse any node twice, and a loop is any closed path. By identifying all such paths and loops, and applying Mason's rule algebraically, an engineer can extract the system's input-output relationship directly from the graph. This is far faster than conventional manipulation of block diagrams or state-space equations, especially for systems with complex feedback.

SFGs are drawn using standard symbols: nodes as small circles or dots, edges as arrows labeled with gains, and gain values placed near the arrows. The graphs are often arranged left to right, with the input source on the left and output sink on the right, though this is a convention rather than a requirement. When a system contains multiple feedback loops or cross-coupling between branches, the visual clarity of an SFG can reveal the structure more readily than a textual equation would.

In practice, SFGs appear most often in design of analog control systems, active filter networks, and stability analysis of amplifier feedback circuits. They are less common in modern digital and software systems, where simulation and numerical methods dominate, but they remain a core analytical tool in analog electronics and classical control engineering curricula. The ability to read and construct an SFG correctly is expected of anyone working with operational amplifier feedback networks or designing compensators for servo systems.