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Transfer Function from a Signal Flow Graph (SFG): Finding (\frac{C}{R})

Transfer Function from a Signal Flow Graph (SFG): Finding (\frac{C}{R})

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Nov 4, 2026

A signal flow graph (SFG) is a graphical representation of a set of linear relationships, where directed branches carry gains and nodes represent variables. To find the overall transfer function (\frac{C}{R}), the standard approach is Mason’s Gain Formula. In this section we’ll use that framework to compute (\frac{C}{R}) from an arbitrary SFG by (i) enumerating forward paths, (ii) enumerating loops, and (iii) applying Mason’s formula determinants. 2

Key terms you’ll use: signal flow graph , forward path , loop , Mason’s gain formula.

Footnotes

  1. Mason's gain formula (Signal flow graph) - Defines Mason’s gain formula and the determinant terms. ↩

  2. Signal flow graph (general) - Background on SFGs and their relationship to transfer functions. ↩

Mason's Gain Formula (Signal Flow Graph) - Derivation & Examples

Mason’s Gain Formula (the computation template)

For an SFG with:

  • Forward paths (P_k) from (R) to (C),
  • Loop gains (L_i) for all individual loops,
  • Non-touching loop groups,

Mason’s formula states:

CR=∑kPk ΔkΔ\frac{C}{R}=\frac{\sum_k P_k\,\Delta_k}{\Delta}

where:

Δ=1−∑iLi+∑i<jnon-touchingLiLj−∑i<j<kpairwise non-touchingLiLjLk+⋯\Delta = 1 - \sum_i L_i + \sum_{i<j \atop \text{non-touching}} L_iL_j - \sum_{i<j<k \atop \text{pairwise non-touching}} L_iL_jL_k+\cdots

and (\Delta_k) is the same determinant as (\Delta), but only including loops that do not touch forward path (P_k). 3

Important: If the figure you provided contains specific gains (e.g., blocks like (G(s)), (H(s)), constants, etc.), you plug those branch gains into (P_k), (L_i), and (\Delta), then simplify algebraically.

Footnotes

  1. Mason's gain formula (Signal flow graph) - Defines Mason’s gain formula and the determinant terms. ↩

  2. Signal flow graph (general) - Background on SFGs and their relationship to transfer functions. ↩

  3. Control system block diagram reduction & loop/path concepts (general references) - Useful for interpreting feedback loops and their influence on transfer functions. ↩

Step-by-step: Compute \(\frac{C}{R}\) from the given SFG

  1. 1
    Step 1

    Label the source node as (R) and the sink node as (C). If the diagram uses intermediate summing nodes, treat them as graph nodes with incoming/outgoing edges.

  2. 2
    Step 2

    List every distinct directed path from (R) to (C) that does not pass through the same node more than once.

  3. 3
    Step 3

    Multiply the gains of all branches along path (P_k): (P_k=\prod (\text{branch gains on that path})).

  4. 4
    Step 4

    List every directed cycle that starts and ends at the same node (no repeated intermediate nodes in the cycle definition).

  5. 5
    Step 5

    Multiply branch gains around each loop. Sign is handled automatically if branch gains are negative at summing junctions.

  6. 6
    Step 6

    Use $$ \Delta = 1 - \sum_i L_i + \sum_{\text{non-touching } i,j} L_iL_j - \cdots

  7. 7
    Step 7

    For each (P_k), remove (exclude) every loop that touches (shares any node with) (P_k), then compute the determinant of the remaining non-touching loops using the same alternating-sign rule.

  8. 8
    Step 8

    Compute CR=∑kPk ΔkΔ\frac{C}{R}=\frac{\sum_k P_k\,\Delta_k}{\Delta}, then algebraically simplify.

Visual structure (how paths and loops appear)

A forward path shares nodes with (R \to C). Loops are “detours” that start/end at a node and wrap around back. Non-touching loops are loops that do not share nodes. Mason’s determinant bookkeeping depends entirely on which loops touch which forward paths. 2

Footnotes

  1. Mason's gain formula (Signal flow graph) - Defines Mason’s gain formula and the determinant terms. ↩

  2. Signal flow graph (general) - Background on SFGs and their relationship to transfer functions. ↩

Mason’s formula components checklist

You compute each quantity from the SFG, then combine them.

Pro Tip: Determining loop “touching”

A loop touches a forward path (P_k) if they share at least one node. In practice, mark the nodes belonging to (P_k), then discard any loops that pass through any marked node when forming (\Delta_k).

Common pitfall: Missing a non-touching loop group

In (\Delta), you must include products of all non-touching loop combinations (pairs, triples, etc.). Missing even one combination changes (\Delta) and therefore (\frac{C}{R}) significantly.

Worked-algebra checklist for the provided diagram

Roadmap to get (\frac{C}{R}) from any SFG

Identify nodes and gains

1. Parse graph

Recognize (R), (C), and every branch gain."

Enumerate all \(P_k\)

2. Build forward paths

List all (R \to C) node-disjoint-by-path paths."

Enumerate all \(L_i\)

3. Build loops

Find cycles and compute loop products."

Compute \(\Delta\) and each \(\Delta_k\)

4. Determinants

Use non-touching loop combinations only."

Apply Mason’s formula

5. Combine

Compute CR=∑kPkΔkΔ\frac{C}{R}=\frac{\sum_k P_k\Delta_k}{\Delta}."

SFG / Mason’s Gain Formula Quick Checks

1 / 5
Question · Term

What is a forward path \(P_k\)?

Click to reveal
Answer · Definition

A directed path from (R) to (C) that doesn’t pass through any node more than once.

Knowledge Check

Question 1 of 3
Q1Single choice

In Mason’s gain formula, (\Delta_k) is computed by excluding which loops?

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