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Mason's gain formula (MGF) is a method for finding the
transfer function In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, models the system's output for each possible ...
of a linear
signal-flow graph A signal-flow graph or signal-flowgraph (SFG), invented by Claude Shannon, but often called a Mason graph after Samuel Jefferson Mason who coined the term, is a specialized Flow graph (mathematics), flow graph, a directed graph in which nodes rep ...
(SFG). The formula was derived by Samuel Jefferson Mason, for whom it is named. 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. MGF comes up often in the context of
control system A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large industrial ...
s,
microwave Microwave is a form of electromagnetic radiation with wavelengths shorter than other radio waves but longer than infrared waves. Its wavelength ranges from about one meter to one millimeter, corresponding to frequency, frequencies between 300&n ...
circuits and digital filters because these are often represented by SFGs.


Formula

The gain formula is as follows: : G = \frac = \frac : \Delta = 1 - \sum L_i + \sum L_iL_j- \sum L_iL_jL_k + \cdots + (-1)^m \sum \cdots +\cdots 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'' = path gain of the ''k''th forward path between ''y''in and ''y''out *''L''''i'' = loop gain of each closed loop in the system *''L''''i''''L''''j'' = product of the loop gains of any two non-touching loops (no common nodes) *''L''''i''''L''''j''''L''''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.


Definitions

Source: *Path: a continuous set of branches traversed in the direction that they indicate. *Forward path: A path from an input node to an output node in which no node is touched more than once. *Loop: A path that originates and ends on the same node in which no node is touched more than once. *Path gain: the product of the gains of all the branches in the path. *Loop gain: the product of the gains of all the branches in the loop.


Procedure to find the solution

# Make a list of all forward paths, and their gains, and label these ''G''''k''. # 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''''i''''L''''j''). Make a list of all pairwise non-touching loops taken three at a time (''L''''i''''L''''j''''L''''k''), then four at a time, and so forth, until there are no more. # Compute the determinant Δ and cofactors Δ''k''. # Apply the formula.


Examples


Circuit containing two-port

The transfer function from Vin to V2 is desired. There is only one forward path: :* Vin to V1 to I2 to V2 with gain G_1 = -y_ R_L \, There are three loops: :* V1 to I1 to V1 with gain L_1 = -R_\text y_ \, :* V2 to I2 to V2 with gain L_2 = -R_L y_ \, :* V1 to I2 to V2 to I1 to V1 with gain L_3 = y_ R_L y_ R_\text \, : \Delta = 1 - ( L_1 + L_2 + L_3 ) + ( L_1 L_2 ) \, note: L1 and L2 do not touch each other whereas L3 touches both of the other loops. : \Delta_1 = 1 \, note: the forward path touches all the loops so all that is left is 1. : G = \frac = \frac \,


Digital IIR biquad filter

Digital filter In signal processing, a digital filter is a system that performs mathematical operations on a Sampling (signal processing), sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other ma ...
s are often diagramed as signal flow graphs. : There are two loops :* L_1 = -a_1 Z^ \, :* L_2 = -a_2 Z^ \, : \Delta = 1 - ( L_1 + L_2 ) \, Note, the two loops touch so there is no term for their product. :There are three forward paths :* G_0 = b_0 \, :* G_1 = b_1 Z^ \, :* G_2 = b_2 Z^ \, : All the forward paths touch all the loops so \Delta_0 = \Delta_1 = \Delta_2 = 1 \, : G = \frac \, : G = \frac \,


Servo

The signal flow graph has six loops. They are: :* L_0 = - \frac \, :* L_1 = \frac \, :* L_2 = \, \frac :* L_3 = \frac \, :* L_4 = \frac \, :* L_5 = \frac \, : \Delta = 1 - (L_0+L_1+L_2+L_3+L_4+L_5) + (L_0 L_1 + L_0 L_3 )\, There is one forward path: :* g_0 = \frac \, The forward path touches all the loops therefore the co-factor \Delta_0 = 1 And the gain from input to output is \frac = \frac \,


Equivalent matrix form

Mason's rule can be stated in a simple matrix form. Assume \mathbf is the transient matrix of the graph where t_ = \left mathbf\right 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 u_ = \left mathbf\right, where : \mathbf = \left ( \mathbf - \mathbf \right ) ^ , and \mathbf 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.


Complexity and computational applications

Mason's Rule can grow factorially, because the enumeration of paths in a directed graph grows dramatically. To see this consider the complete directed graph on n vertices, having an edge between every pair of vertices. There is a path form y_\text to y_\text for each of the (n-2)! permutations of the intermediate vertices. Thus
Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of row-wise operations performed on the corresponding matrix of coefficients. This method can a ...
is more efficient in the general case. Yet Mason's rule characterizes the transfer functions of interconnected systems in a way which is simultaneously algebraic and combinatorial, allowing for general statements and other computations in algebraic systems theory. While numerous inverses occur during Gaussian elimination, Mason's rule naturally collects these into a single quasi-inverse. General form is ::\frac, Where as described above, q is a sum of cycle products, each of which typically falls into an ideal (for example, the strictly causal operators). Fractions of this form make a
subring In mathematics, a subring of a ring is a subset of that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and that shares the same multiplicative identity as .In general, not all s ...
R(1+\langle L_i\rangle)^ of the rational function field. This observation carries over to the noncommutative case, even though Mason's rule itself must then be replaced by Riegle's rule.


See also

*
Signal-flow graph A signal-flow graph or signal-flowgraph (SFG), invented by Claude Shannon, but often called a Mason graph after Samuel Jefferson Mason who coined the term, is a specialized Flow graph (mathematics), flow graph, a directed graph in which nodes rep ...
* Riegle's rule


Notes


References

* * {{DEFAULTSORT:Mason's Rule Control theory