Mason's gain formula is a method for finding the transfer function

Transfer function

A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...

 of a linear signal-flow graph

Signal-flow graph

A signal-flow graph is a special type of block diagram—and directed graph—consisting of nodes and branches. Its nodes are the variables of a set of linear algebraic relations. An SFG can only represent multiplications and additions. Multiplications are represented by the weights of the branches;...

 (SFG). The formula was derived by Samuel Jefferson Mason

Samuel Jefferson Mason

Samuel Jefferson Mason was an American electronics engineer. Mason's invariant and Mason's rule are named after him.He was born in New York City, but he grew up in a small town in New Jersey. It was so small, in fact, that it only had a population of 26. He received a B.S...

 and is named after its discoverer. MGF is an alternate method to finding the transfer function algebraically by labeling each signal, writing down the equation for how that signal depends on other signals, and then solving the multiple equations for the output signal in terms of the input signal. MGF provides a step by step method to obtain the transfer function from a SFG. Often, MGF can be determined by inspection of the SFG. The method can easily handle SFGs with many variables and loops including loops with inner loops. The method, being simple and mechanical and can often require less effort and is less error prone than the direct algebraic method. MGF comes up often in the context of control systems and digital filters because control systems and digital filters are often represented by SFGs.

Formula

The gain formula is as follows:

where:

• Δ = the determinant of the graph.
• y_in = input-node variable
• y_out = output-node variable
• G = complete gain between y_in and y_out
• N = total number of forward paths between y_in and y_out
• G_k = gain of the kth forward path between y_in and y_out
• L_i = loop gain of each closed loop in the system
• L_iL_j = product of the loop gains of any two non-touching loops (no common nodes)
• L_iL_jL_k = product of the loop gains of any three pairwise nontouching loops
• Δ_k = the cofactor value of Δ for the k^th forward path, with the loops touching the k^th forward path removed. I.e. Remove those parts of the graph which form the loop, while retaining the parts needed for the forward path.

Procedure

To use this technique,

1. Make a list of all forward paths, and their gains, and label these G_k.
2. Make a list of all the loops and their gains, and label these L_i (for i loops). Make a list of all pairs of non-touching loops, and the products of their gains (L_iL_j). Make a list of all pairwise non-touching loops taken three at a time (L_iL_jL_k), then four at a time, and so forth, until there are no more.
3. Compute the determinant Δ and cofactors Δ_k.
4. Apply the formula.

Circuit containing two-port

The transfer function from V_in to V₂ is desired.

There is only one forward path:

• V_in to V₁ to I₂ to V₂ with gain

There are three loops:

• V₁ to I₁ to V₁ with gain
• V₂ to I₂ to V₂ with gain
• V₁ to I₂ to V₂ to I₁ to V₁ with gain

note: L₁ and L₂ do not touch each other whereas L₃ touches both of the other loops.
note: the forward path touches all the loops so all that is left is 1.

Digital IIR biquad filter

Digital filter

Digital filter

In electronics, computer science and mathematics, a digital filter is a system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other major type of electronic filter, the analog filter, which is...

s are often diagramed as signal flow graphs.

There are two loops

• 
• 

Note, the two loops touch so there is no term for their product.

There are three forward paths

• 
• 
• 

All the forward paths touch all the loops so

Servo

The signal flow graph has six loops. They are:

• 

• 

• 

• 

• 

• 

There is one forward path:

• 

The forward path touches all the loops therefore the co-factor
And the gain from input to output is

Equivalent matrix form

Mason's rule can be stated in a simple matrix form. Assume is the transient matrix of the graph where is the sum transmittance of branches from node m toward node n. Then, the gain from node m to node n of the graph is equal to , where

,

and is the identity matrix.

Mason's Rule is also particularly useful for deriving the z-domain transfer function of discrete networks that have inner feedback loops embedded within outer feedback loops (nested loops). If the discrete network can be drawn as a signal flow graph, then the application of Mason's Rule will give that network's z-domain H(z) transfer function.