GATE Electrical Engineering (EE) Test: Signal Flow Graphs - 1 Free Online

Explanation: By definition signal flow graphs are the graphical representation of the relationships between the variables of set linear algebraic equations.

Explanation: Nodes are the point by which the branches are outgoing or ingoing and this can be input or output node and input node is the node having only outgoing branches.

Explanation: Using mason’s gain formula transfer function from signal flow graph can be calculated which relates the forward path gain to the various paths and loops.

Forward path: G₁ G₂, G₁ G₃

Loops: -G₂, -G₁G₂H, -G₃

Finding the transfer function using Mason's gain formula:

Explanation: Signal flow graphs are used to find the transfer function of control system by converting the block diagrams into signal flow graphs or directly but cannot be used for nonlinear systems.

Explanation: As one set technique and formula is used here but in block diagram technique various methods are involved which increases complexity.

Explanation: Loop is the part of the network in which the branch starts from the node and comes back to the same node and non touching loop must not have any node in common.

Explanation: The relationship between input and output variable of a signal flow graph is the overall gain of the system.

Explanation: Use mason’s gain formula to solve the signal flow graph and by using mason’s gain formula transfer function from signal flow graph can be calculated which relates the forward path gain to the various paths and loops.

Explanation: Using mason’s gain formula transfer function from signal flow graph can be calculated which relates the forward path gain to the various paths and loops.

Concept:

Mason’s Gain Formula

• It is a technique used for finding the transfer function of a control system. 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.

Suppose there are ‘N’ forward paths in a signal flow graph. The gain between the input and the output nodes of a signal flow graph is nothing but the transfer function of the system. It can be calculated by using Mason’s gain formula.

Mason’s gain formula is

Where,

C(s) is the output node

R(s) is the input node

T is the transfer function or gain between R(s) and C(s)

P_i is the i^th forward path gain

Δ = 1−(sum of all individual loop gains) + (sum of gain products of all possible two non-touching loops) − (sum of gain products of all possible three non-touching loops) + ........

Δi is obtained from Δ by removing the loops which are touching the ith forward path.

Calculations:

The forward paths are as follows:

P₁ = 5

P₂ = 2 × 3 × 4 = 24

The loops are as follows:

L₁ = -2, L₂ = -3, L₃ = -4, L₄ = -5

The two non-touching loops are:

L₁L₃ = 8

There is no three non-touching loops

By Mason’s gain formula:-

Concept:

Signal flow graph

• It is a graphical representation of a set of linear algebraic equations between input and output.
• The set of linear algebraic equations represents the systems.
• The signal flow graphs are developed to avoid mathematical calculation.

Maon gain formula is used to find the ratio of any two nodes or transfer function.

Where P_k = k^th forward path gain

Δ = 1- ∑ individual loop gain + ∑ two non-touching loops gain - ∑ the gain product of three non-touching loops + ∑ gain of four non-touching loops

Shotcut: while writing Δ take the opposite sign for the odd number of non-touching loops snd the same sign for the even the number of non-touching loops.

Δ_K is obtained from Δ by removing the loops touching the K^th forward path.

Calculation:

For the given SFG two forward paths

Since all loops are touching the paths P_K1 and P_K2 so Δ_K1 = Δ_K2 = 1
We have Δ = 1- ∑ individual loops + ∑ non-touching loops gain
Loops are

As all the loops are touching each other we have
Δ = 1 – ( L₁ + L₂ + L₃ + L₄)
Δ = 1 – ( - 4 – 4s^-1 – 2s^-2 -2s^-1 )
Δ = 5 + 6s^-1 + 2s^-2 

Nodes are the point by which the branches are outgoing or ingoing and this can be input or output node and input node is the node having only outgoing branches.

Concept:

• A signal flow graph is a graphical representation of a set of linear algebraic or differential equations. It is a diagram that represents a set of simultaneous linear equations using nodes and directed branches. In control system engineering, signal flow graphs are used to quickly solve the equations related to systems.
• Each node represents a system variable, and each directed branch represents a gain or a multiplication factor between two variables. The direction of the arrow represents the direction of the flow of the signal. The summing and branching points are used to represent system equations in a graphical way.
• Although signal flow graphs are used in the analysis of control systems (option 4), it's not a special type of graph solely for modern control systems. They can be used for a variety of applications involving sets of linear equations, not just modern control systems.

Node:

• A node that has only outgoing branches called input mode
• Which has only incoming branches, known as an output node
• Which has both incoming & outgoing branches, mixed node.             

Branch:

• It is an alone segment that joins two nodes.
• It has both gain & direction

Concept:

According to Mason’s gain formula, the transfer function is given by

Where, n = no of forward paths

M_k = k^th forward path gain

Δ_k = the value of Δ which is not touching the k^th forward path

Δ = 1 – (sum of the loop gains) + (sum of the gain product of two non-touching loops) – (sum of the gain product of three non-touching loops)

Application:

In the given signal flow graph,

Forward paths: P₁ = ad

Loops: L₁ = cd, L₂ = ade

Δ = 1 – (cd + ade)

Δ₁ = 1

Transfer function = 
Now, let us check the options.

Option 1:

Forward paths: P₁ = a(d + c)

Loops: L₁ = ae(d + c)

Δ = 1 – ae(d + c)

Δ₁ = 1

Transfer function = 

Option 2:

Forward paths: P₁ = d(a + c)

Loops: L₁ = de(a + c)

Δ = 1 – de(a + c)

Δ₁ = 1

Transfer function = 

Option 3:

Option 4:

Hence the signal graph in option (C) is the equivalent representation of the signal flow graph given in the question.