Mason’s Gain Formula

It is a technique used for finding the transfer function of a control system. Basically, a formula that determines the transfer function of a linear system by making use of the signal flow graph is known as Mason’s Gain Formula.

It shows its significance in determining the relationship between input and output.

Introduction

In the previous article, we have seen how a signal flow graph is constructed. We have already seen that a signal flow is a graph is generally formed by using the algebraic equations that define the system, in which the variables of the equations play a crucial role.

Previously we have discussed that S J Mason introduced the idea of signal flow graph to make the analysis of the system easy. Also, the formula for finding the transfer function of the SFG was stated by Mason. Thus, it is named so.

In the block diagram reduction technique, we have seen that in order to simplify a system for the purpose of analysis, we must apply some reduction rules in a proper manner. And after the simplification of the system, its overall transfer function C(s)/R(s) is determined.

This means that to simplify the system, after applying each rule, a reduced block diagram must be formed after every step. This makes the simplification time-consuming.

So, to deal with the problems regarding block diagram reduction, signal flow graph is adopted and used.

In SFG, once the graph is obtained then the transfer function can be easily determined by mason’s gain formula.

Mason’s Gain Formula

To determine the transfer function of the system using signal flow graph, the formula given below is used.

Here every term has its own significance, as these values vary with signal flow graph.

K denotes the number of forward paths,

T_K shows the gain of the k^th forward path,

Δ is the determinant of the system calculated as:

Δ = 1 – (sum of loop gains of total individual loops in the SFG) + (sum of the product of gains of all pairs of two non-touching loop in SFG) + (sum of the product of all pairs of three non-touching loops in the SFG) + ——

Δ_K denotes the Δ for the region of SFG which do not touches the K^th forward path.

We will see the application of mason’s gain formula on the signal flow graph using some examples but first, let us understand the different terms associated with the mason’s gain formula.

So, basically by noticing the above formula, we get the idea that firstly the total number of forward paths in the graph must be calculated as this will play a crucial role in determining the overall gain between input and output of the system.

Further, all the number of loops in the SFG must be noted down with their respective gain as their summation is required. Also, the gains of each loop in the SFG must be summed but only those with two non-touching loops.

In a similar way, the combination of gains with three non-touching loops must be summed. This continues till the SFG contains even higher number of non-touching loops.

Once this all is determined, then by substituting the respective gains in the above-described formula, the overall gain of the signal flow graph is determined and so the system.

Let us now see some examples.

Examples of Mason’s Gain Formula

In the previous article, we have already described the way to construct a signal flow graph.

So, now consider that we have a signal flow graph shown below and we have to determine its respective gain.

• Example 1:

Here we will apply mason’s gain formula but first, determine each component of the formula step by step.

Here the total number of forward paths are two.

T₁ = G₁G₂G₃G₄G₅

T₂ = G₁G₂G₆

Also, the above signal flow graph contains, 2 individual feedback loops

L₁ = – G₁H₁

L₂ = – G₄H₂

These two loops of the SFG are also the two non-touching loops.

Therefore, substituting the values in the formula to calculate Δ, we will get,

Δ = 1 – (L₁ + L₂) + (L₁L₂)

Δ = 1 – (- G₁H₁ – G₄H₂) + [(- G₁H₁) (- G₄H₂)]

So,

Δ = 1 + G₁H₁ + G₄H₂ + [(G₁G₄ H₁H₂)]

Further, we will now calculate ΔK

So,

Δ₁ = 1 – (loops that are not touching first forward path)

And here there is no such loop which is not touching the first forward path, hence,

Δ₁ = 1 – (0)

Δ₁ = 1

Now,

Δ₂ = 1 – (loops not touching the second forward path)

And here L₂ is not touching the second forward path, therefore,

Δ₂ = 1 – (L₂)

Δ₂ = 1 – (- G₄H₂)

On simplifying

Δ₂ = 1 + G₄H₂

So, now substituting the values in the mason’s gain formula:

This is the gain of the system with the above given SFG.

• Example 2:

Consider the SFG shown below:

Here we have only a single forward path.

Thus, K = 1

So,

T₁ = G₁G₂G₃G₄G₅

The 4 individual feedback loops of the above shown signal flow graph are:

L₁ = – G₁H₁

L₂ = – G₂H₂

L₃ = – G₄H₃

L₄ = – G₄G₅H₄

The various combinations of 2 non-touching loop of the SFG are:

L₁L₃ = (- G₁H₁)(- G₄H₃) = G₁H₁G₄H₃

L₁L₄ = (-G₁H₁)(-G₄G₅H₄) = G₁H₁G₄G₅H₄

L₂L₃= (-G₂H₂)(-G₄H₃) = G₂H₂G₄H₃

L₂L₄= (-G₂H₂)(-G₄G₅H₄) = G₂H₂G₄G₅H₄

In this SFG, there are no 3 non-touching loops, thus we will stop right here.

So, Δ will be given as:

Δ = 1 – (-G₁H₁ – G₂H₂ – G₄H₃– G₄G₅H₄) + [(G₁H₁G₄H₃) + (G₁H₁G₄G₅H₄) + (G₂H₂G₄H₃) + (G₂H₂G₄G₅H₄)]

So, we will have,

Δ = 1 + G₁H₁ + G₂H₂ + G₄H₃ + G₄G₅H₄ + G₁H₁G₄H₃ + G₁H₁G₄G₅H₄ + G₂H₂G₄H₃ + G₂H₂G₄G₅H₄

Now, as we have a single forward path,

Therefore, Δ₁ = 1 – (0)

This is so because we have no such loops that are not touching the first forward path.

So, Δ₁ = 1

Hence, the transfer function of the above signal flow graph will be:

On substituting the values, we will get,

This is the transfer function of the system with the above-given signal flow graph.

Hence, in this way, the input relationship of a complex system using signal flow graph can be determined.