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Signal Flow Graph Mason Gain Rule Feedback Loop Solver

· Last updated：2026-09-01

Solve control-system signal flow graphs with Mason's gain formula: forward paths, loops, non-touching combinations, Δ/Δk determinants and the exact transfer function.

Details

What this tool helps you do

Enter edges as "from -> to = gain" (one per line). Gains may be numbers (0.5, -2), symbols (G1, H2) or products (2G1, b1z^-1); arithmetic stays exact via fractions. The solver enumerates forward paths and simple cycles, finds pairwise non-touching loop combinations, expands Δ = 1 − ΣL + ΣLL − …, computes each Δk, and returns T = Σ Pk·Δk / Δ as a symbolic ratio plus a numeric value when possible.

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Written by the Elysia Tools team

Tool usage guide

Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Category
Science & Education
Input types
textarea, text
Output type
html
Sample coverage
4
API ready
Yes

Overview

The Signal Flow Graph Mason Gain Rule Feedback Loop Solver computes exact transfer functions for control-system graphs using Mason's gain formula. By entering directed edges with symbolic or numeric gains, the tool identifies forward paths, individual loops, and non-touching loop combinations to evaluate system determinants and overall gain step by step.

When to use

  • Simplifying complex block diagrams converted into signal flow graphs with multiple nested feedback loops.
  • Evaluating the symbolic transfer function T = Σ(Pk·Δk)/Δ for linear control systems coursework or design analysis.
  • Validating exact rational transfer functions with numeric gain values to prevent calculation errors in manual determinant expansion.

How it works

  1. 1Input your graph edges line by line using the format 'from -> to = gain', supporting numerical constants, symbols like G1, or expressions like b1*z^-1.
  2. 2Specify the input (source) node and output (sink) node to define the start and end boundaries of the transmission path.
  3. 3The solver enumerates forward paths (Pk), finds all simple feedback cycles (L), and groups mutually non-touching loops.
  4. 4It expands the system determinant Δ and path cofactors Δk, outputting the complete symbolic transfer function and exact rational numeric value.

Use cases

  • Analyzing multi-loop analog feedback amplifier networks and active filter signal flow graphs.
  • Deriving transfer functions for digital filter topologies defined by delay blocks and discrete coefficient gains.
  • Solving control engineering homework problems and verifying manual expansions of Mason's gain formula.

Examples

1. Dual Forward Paths with Non-Touching Self-Loops

Control Engineering Student

Background

Analyzing a textbook signal flow graph with parallel paths and isolated feedback nodes.

Problem

Determine the symbolic transfer function from source node R to sink node out.

How to use

Enter the directed edges with symbols G1 through G4 and feedback gains H1 and H2, then set R as input and out as output.

Edges:
R -> n1 = 1
n1 -> n2 = G1
n2 -> n3 = G2
n3 -> out = 1
n1 -> n4 = G3
n4 -> n3 = G4
n2 -> n2 = H1
n4 -> n4 = H2
Input Node: R
Output Node: out

Outcome

Finds 2 forward paths and 1 non-touching loop pair, outputting T = (G1·G2·(1−H2) + G3·G4·(1−H1)) / (1 − H1 − H2 + H1·H2).

2. Exact Numeric Closed-Loop Gain Calculation

Systems Engineer

Background

Testing a multi-branch signal graph with calibrated numeric gains.

Problem

Compute the exact rational transfer function without floating-point rounding errors.

How to use

Input numeric edge gains for all branches, setting input to R and output to out.

FAQ

What gain formats are supported in the edge list?

You can use numeric values (such as 2, -0.5), symbolic variables (such as G1, H2), and products (such as 2*G1 or b1*z^-1).

How are non-touching loops evaluated in the determinant Δ?

The solver automatically identifies pairs and higher-order combinations of loops that share no common nodes and includes their products in the Δ expansion.

Can this tool handle self-loops on a single node?

Yes, self-loops entered in the format 'node -> node = gain' are recognized and factored into loop gains and non-touching calculations.

Does the solver provide numeric values as fractions or decimals?

For numeric inputs, calculations maintain exact fractional arithmetic and display both exact rational forms and decimal approximations.

What happens if there are no non-touching loop pairs?

The system determinant Δ simplifies directly to 1 minus the sum of all individual loop gains (1 − ΣL).

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Edges:
R -> n1 = 1
n1 -> n2 = 2
n2 -> n3 = 3
n3 -> out = 1
n1 -> n4 = 4
n4 -> n3 = 5
n2 -> n2 = 0.1
n4 -> n4 = 0.2
Input Node: R
Output Node: out

Outcome

Calculates Δ = 18/25, numerator = 114/5, and exact transfer function T = 95/3 (≈ 31.6667).