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Transfer Function C/RC/R from a Signal Flow Graph (SFG) Using Mason’s Gain Formula

Transfer Function C/RC/R from a Signal Flow Graph (SFG) Using Mason’s Gain Formula

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Nov 4, 2026

When a signal flow graph (SFG) is given, the overall transfer function C/RC/R can be found systematically using Mason’s gain formula.2

Core idea: enumerate all forward paths from input RR to output CC, compute the gain of each path, enumerate all individual loops, and compute the determinant terms that account for non-touching loops.

In your problem, the SFG in the provided figure is used to compute the transfer function C/RC/R by applying the above enumeration and determinant rules.

Footnotes

  1. (Mason’s gain formula overview) https://en.wikipedia.org/wiki/Signal_flow_graph - Defines signal flow graphs and Mason’s gain formula with C/R=sumPkDeltak/DeltaC/R=\\sum P_k\\Delta_k/\\Delta. ↩ ↩2 ↩3

  2. (Control systems / SFG transfer function method) https://www.controlanywhere.com/control-systems-signal-flow-graph-mason-gain-formula - Stepwise explanation of loops, non-touching loops, and determinant terms. ↩

Mason’s Gain Formula (Signal Flow Graph) - Transfer Function

Key definitions (used directly in the formula)

  • Forward path: any route from RR to CC.
  • Loop: any feedback cycle.
  • Non-touching loops: used to form higher-order terms in Δ\Delta.
  • Path gain: multiply gains along that path.

Mason’s gain formula is:

CR=∑k(PkΔkΔ)\frac{C}{R}=\sum_k \left(P_k \frac{\Delta_k}{\Delta}\right)

where PkP_k are forward path gains, Δ\Delta is the determinant for the whole graph, and Δk\Delta_k is the determinant for the graph after removing loops that touch the kk-th forward path.2

Footnotes

  1. (Mason’s gain formula overview) https://en.wikipedia.org/wiki/Signal_flow_graph - Defines signal flow graphs and Mason’s gain formula with C/R=sumPkDeltak/DeltaC/R=\\sum P_k\\Delta_k/\\Delta. ↩

  2. (Control systems / SFG transfer function method) https://www.controlanywhere.com/control-systems-signal-flow-graph-mason-gain-formula - Stepwise explanation of loops, non-touching loops, and determinant terms. ↩

Compute $C/R$ from the SFG (Mason’s Gain Formula Workflow)

  1. 1
    Step 1

    List every directed path from input node to output node. For each path, compute its gain PkP_k as the product of branch gains along that path. keyword

  2. 2
    Step 2

    List every directed closed loop. For each loop LiL_i, compute loop gain as the product of branch gains around the loop. Sum these into the term ∑Li\sum L_i.

  3. 3
    Step 3

    Δ=1−∑Li+∑(LiLj)−∑(LiLjLm)+⋯\Delta = 1-\sum L_i + \sum (L_i L_j) - \sum(L_i L_j L_m)+\cdots where the sums beyond the first are formed from non-touching loop combinations (pairs, triples, etc.).

  4. 4
    Step 4

    For a given forward path kk, exclude from Δ\Delta all loops that touch (share nodes with) that forward path, then recompute the determinant to get Δk\Delta_k.

  5. 5
    Step 5

    CR=∑k(PkΔkΔ)\frac{C}{R} = \sum_k\left(P_k\frac{\Delta_k}{\Delta}\right).

  6. 6
    Step 6

    Simplify the resulting rational expression. Commonly you end with a denominator that matches Δ\Delta and numerators formed from contributions of each forward path weighted by Δk\Delta_k.

What you must extract from the figure

To produce the final closed-form C/RC/R, you need the following from the specific SFG diagram you provided:

  1. Each branch gain value (e.g., G1,G2,k,sT+1G_1, G_2, k, sT+1, etc.).
  2. Every forward path from RR to CC.
  3. Every loop gain (closed cycles).
  4. Which loops are non-touching with each other, and which loops touch a given forward path.

Mason’s gain formula guarantees that if you compute PkP_k, Δ\Delta, and Δk\Delta_k correctly from that exact topology, the resulting C/RC/R is correct.

Footnotes

  1. (Mason’s gain formula overview) https://en.wikipedia.org/wiki/Signal_flow_graph - Defines signal flow graphs and Mason’s gain formula with C/R=sumPkDeltak/DeltaC/R=\\sum P_k\\Delta_k/\\Delta. ↩

Pro Tip

When listing loops, mark nodes used by each loop on paper. This makes it fast to decide which loops are non-touching (for Δ\Delta) and which must be removed (for each Δk\Delta_k).

Common Pitfall

A loop is defined by a directed closed path. Accidentally treating an undirected cycle or a path that doesn’t follow arrow directions will produce an incorrect LiL_i, which then corrupts Δ\Delta and the final C/RC/R.

From Diagram to Transfer Function

Label directed paths

Step A

Find all R→CR\to C forward paths and compute PkP_k."

Find closed loops

Step B

Enumerate all loops and compute loop gains LiL_i."

Build determinant terms

Step C

Compute 0˘394\u0394 using non-touching loop products."

Compute per-path determinants

Step D

Compute 0˘394k\u0394_k after removing loops touching path kk."

Final assembly

Step E

Use C/R=∑kPk(Δk/Δ)C/R=\sum_k P_k(\Delta_k/\Delta) and simplify."

Compact reference (Mason’s formula)

Given forward paths {Pk}\{P_k\} and all loops {Li}\{L_i\}:

Delta=1−sumiLi+sumi<j,textnon−touchingLiLj−sumi<j<k,textpairwisenon−touchingLiLjLk+cdots\\Delta = 1-\\sum_i L_i + \\sum_{i<j,\\text{ non-touching}} L_iL_j - \\sum_{i<j<k,\\text{ pairwise non-touching}} L_iL_jL_k + \\cdots

and

Deltak=1−sumiintextloopsnottouchingpathkLi+sumtextnon−touchingpairs(w.r.t.pathk)LiLj−cdots\\Delta_k = 1-\\sum_{i\\in \\text{loops not touching path }k} L_i + \\sum_{\\text{non-touching pairs (w.r.t. path }k)} L_iL_j-\\cdots

Then:

fracCR=sumkleft(PkfracDeltakDeltaright).\\frac{C}{R}=\\sum_k \\left(P_k\\frac{\\Delta_k}{\\Delta}\\right).

This is the standard Mason gain approach used to compute transfer functions from SFGs.2

Footnotes

  1. (Mason’s gain formula overview) https://en.wikipedia.org/wiki/Signal_flow_graph - Defines signal flow graphs and Mason’s gain formula with C/R=sumPkDeltak/DeltaC/R=\\sum P_k\\Delta_k/\\Delta. ↩

  2. (Control systems / SFG transfer function method) https://www.controlanywhere.com/control-systems-signal-flow-graph-mason-gain-formula - Stepwise explanation of loops, non-touching loops, and determinant terms. ↩

Knowledge Check

Question 1 of 4
Q1Single choice

In Mason’s gain formula, what does Δ\Delta represent?

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