Signal Flow Graph and Block Diagram MCQ Quiz - Objective Question with Answer for Signal Flow Graph and Block Diagram - Download Free PDF

Explanation:

Signal Flow Graph

Definition: A Signal Flow Graph (SFG) is a topological representation of a set of linear algebraic equations or differential equations. It is a graphical method used to represent the relationships between variables in a system. The graph consists of nodes (representing system variables) and directed branches (representing causal relationships and transfer functions) that connect the nodes. SFG is widely used in control systems to analyze and simplify the representation of complex systems.

Working Principle:

In an SFG, each variable in the system is represented as a node, and the relationships between these variables are represented as directed branches. The direction of the branch indicates the cause-and-effect relationship, while the weight of the branch represents the transfer function or gain between the variables. By using graphical techniques such as Mason's Gain Formula, the overall transfer function of the system can be determined efficiently.

Key Features of Signal Flow Graphs:

• Nodes represent system variables (e.g., input, output, and intermediate variables).
• Directed branches represent causal relationships between variables, with weights indicating the transfer function or gain.
• It is applicable only to linear time-invariant systems.
• SFGs are particularly useful for analyzing feedback systems and determining overall system behavior.

Applications:

• Analyzing control systems, particularly feedback systems.
• Simplifying and solving complex systems of linear equations.
• Deriving the transfer function of a system.
• Modeling and analyzing electrical circuits, mechanical systems, and other dynamic systems.

Correct Option Analysis:

The correct option is:

Option 3: Topological representation of set of differential equations.

This option correctly describes the Signal Flow Graph. It is a topological representation of a set of differential equations or linear algebraic equations, showcasing the relationships between system variables in a graphical form. The SFG provides an intuitive and systematic way to analyze and compute the transfer function of a system, making it a valuable tool in control system analysis.

Important Information

To further understand the analysis, let’s evaluate the other options:

Option 1: Polar plot

A Polar Plot is a graphical representation of a complex function, typically used in control systems and signal processing to represent the frequency response of a system. It plots the magnitude and phase of a system's transfer function in polar coordinates. While polar plots are useful in frequency domain analysis, they are not related to the Signal Flow Graph, which is a topological representation of equations in the time domain.

Option 2: Bode plot

A Bode Plot is another frequency domain analysis tool that represents the magnitude and phase of a system's transfer function as separate plots on a logarithmic scale. It is widely used for stability analysis and design in control systems. However, like the polar plot, it is not related to the Signal Flow Graph, which focuses on the relationships between variables in a system using a topological approach.

Option 4: Truth table

A Truth Table is a tabular representation of the logical relationships between inputs and outputs in a digital system. It is commonly used in digital electronics and Boolean algebra to analyze and design logic circuits. While truth tables are essential in digital system design, they have no connection to Signal Flow Graphs, which are used for analyzing linear systems.

Option 5: Not provided

No specific information is given for this option. However, it is clear that none of the options other than Option 3 correctly describe the Signal Flow Graph.

Conclusion:

The Signal Flow Graph is a powerful tool for representing and analyzing linear systems. It uses a topological approach to model the relationships between system variables and provides an efficient way to compute the transfer function. By understanding its principles and applications, one can effectively analyze complex systems and gain insights into their behavior. The correct answer, Option 3, accurately reflects the nature and purpose of the Signal Flow Graph, while the other options pertain to unrelated concepts in control systems, signal processing, or digital electronics.

Explanation:

Control System: Movement of Summing Point in Block Diagrams

Definition: In control systems, a summing point is a node where different signals are algebraically summed. These summing points are crucial for defining the relationships between input and output signals. The movement of a summing point in a block diagram impacts the mathematical representation of the control system and its feedback path.

Working Principle: When a summing point is moved across a block in a control system, it alters the feedback path and the overall system behavior. The movement of the summing point follows specific rules to ensure that the system's mathematical representation remains consistent. These rules are based on the nature of the block and the feedback configuration.

Correct Option Analysis:

The correct option is:

Option 1: Multiplication of the G(s) in the feedback path.

Explanation: When a summing point is moved to the right side of a block (with transfer function G(s)) in a block diagram, the feedback path gets multiplied by the transfer function G(s). This happens because the signal passing through the summing point now encounters the block G(s) before reaching its destination. Mathematically, if the original feedback path was H(s), it becomes H(s) × G(s) after the movement of the summing point.

Example: Suppose we have a system with the following configuration:

• The summing point is initially on the left of the block G(s).
• The feedback path is represented by H(s).

After moving the summing point to the right side of the block G(s), the feedback path changes to H(s) × G(s). This rule ensures that the system's mathematical representation remains consistent while modifying the block diagram.

Importance: This concept is important in control system analysis and design because it helps engineers simplify or rearrange block diagrams without altering the system's behavior. Understanding these transformations is crucial for tasks such as deriving transfer functions and analyzing system stability.

Additional Information:

To further understand the analysis, let’s evaluate the other options:

Option 2: Multiplication of the 1/G(s) in the feedback path.

This option is incorrect. Moving the summing point to the right side of the block G(s) does not lead to multiplication by 1/G(s). Instead, the feedback path is multiplied by G(s), as explained in the correct option analysis. Multiplication by 1/G(s) would occur under a different transformation, such as moving a takeoff point across a block.

Option 3: Addition of gain block.

This option is incorrect. Moving a summing point does not involve the addition of a gain block to the system. Instead, it modifies the feedback path by altering the mathematical relationship between the signals, as described in the correct option analysis.

Option 4: Subtraction of gain block.

This option is also incorrect. Moving a summing point does not involve the subtraction of a gain block. The movement affects the feedback path by introducing a multiplication factor, not by adding or subtracting blocks.

Conclusion:

Understanding the rules for moving summing points and blocks in control system diagrams is essential for accurate analysis and design. In this case, moving a summing point to the right side of a block results in the multiplication of the feedback path by the block's transfer function (G(s)). This transformation preserves the system's behavior while allowing for simplifications or modifications in the block diagram representation.

Concept

The transfer function of the given system is given by:

When you are calculating  then put u=0 and while calculating  then put x₁ = 0.

Mason gain formula:

Calculation

Considering x₁ as input and u=0.

P_k = G₁G₂

Δ_k = 1 because if we remove P_k = G₁G₂, no loop exists. In such conditions, Δk becomes 1.

Individual Loop = -G₁G₂H

Considering u as input and x1=0.

Pk = 1 and Δk = 1

It is given that G1 G2 H ≫ 1

Consider, the first term =  =  .......(i)

second term = 

if G1 G2 H ≫ 1, then 1+G1 G2 H >>1

second term =  ...........(ii)

• When blocks are connected in parallel, the overall transfer function is obtained by adding the transfer functions of the individual blocks.
• In a block diagram, when blocks are connected in parallel, it means that they all receive the same input and provide outputs that contribute to the overall output.
• If you have two or more blocks connected in parallel with transfer functions then overall transfer function with G(s) for the parallel configuration is given by:
  
• The transfer function of the whole system will be the addition of the transfer function of each block.

Signal flow graph:

Forward paths,

P₁ = 3, Δ₁ = 1

P2 = , Δ₂ = 1

Loops : L₁ = 

Using Masson's graph formula,

Summing point:

It is the point where two signals are added or subtracted.

Take-off point

Take off point is the point from where the signal is taken and feedback or forward to a summing point in the system.

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.

T F =  

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

The disturbance response at the output y(t) is zero mean.

At ω = 0, Y(j0) = 0

⇒ K = -2

Concept:

Mason's Gain Formula is used to evaluate an overall transmittance (gain), which can be expressed as,

Where

Pk = forward path transmittance of kth path

Δ = graph determinant comprising closed-loop transmittances & mutual interactions between non-touching loops.

ΔK = path factor consisting of all isolated closed loops from the forward path in the graph.

Analysis:

Forward path: G₁ G₂, G₁ G₃

Loops: -G₂, -G₁G₂H, -G₃

Finding the transfer function using Mason's gain formula:

Concept:

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

Where, n = no of forward paths

Mk = kth forward path gain

Δk = the value of Δ which is not touching the kth 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:

Forward paths: 

Loops: 

Δ₁ = 1

Transfer function 

Option 4:

Forward paths: 

Loops: 

Δ₁ = 1

Transfer function 

Hence the signal graph in option (3) is the equivalent representation of the signal flow graph given in the question.

Concept:

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

Where, n = No of forward paths

Mk = kth forward path gain

Δk = the value of Δ which is not touching the kth 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:

Given signal flow graph is

Forward paths

M₁ = s × s × 1/s = s

M₂ = 1/s × 1/s = 1/s²

Loops:

L₁ = s × s (-1) = - s²

L₂ = -1/s 

L₃ =  s × s × 1/s × (-s) = -s²

L₄= 1/s × 1/s × (-s) = - 1/s

Δ = 1 - ( - s2  - s2  - 1/s  - 1/s) = 1 + 2s² + 2/s

Concept:

If the open-loop transfer function G(s) is connected in positive feedback with a feedback gain of H(s), then the transfer function of the closed-loop system is: 

If the open-loop transfer function G(s) is connected in negative feedback with a feedback gain of H(s), then the transfer function of the closed-loop system is: 

When two systems are connected in parallel, then the overall gain of the system will be the sum of their individual gains.

When two systems are connected in a cascade connection, then the overall gain of the system will be the product of their individual gains.

Calculation:

We have,

Here,

G(s) = G

H(s) = 2

Feedback is positive,

from above concept,

TF =  = 

Explanation:

Signal Flow Graph:

• A graphical method of representing the control system using the linear algebraic equations is known as the signal flow graph.
• It is abbreviated as SFG
• The equation representing the system holds multiple variables that perform a crucial role in forming the graph.
• The signal from a node to another flows through the branch in the direction of the arrowhead.
• The graphical method is valid only for linear time-invariant systems.
• A block diagram can be converted to SFG as shown.

Self Loop:

• A loop consisting of a single branch and a single node.
• The paths in such loops are never defined by any forward path or feedback loop as these never trace any other node of the graph.
• For the given SFG it is formed at node 4 by branch d.

Non-Touching Loops:  

• When two or more loops have not shared a common node then that type of loop is called non-touching loops.
• For the given SFG it is formed by nodes 2-6-2 and 3-4-3
   

Mixed Node:

• It is also known as a chain node and consists of branches having both entering as well as leaving signals.
• For the given SFG node 2 to node 7 are mixed nodes.
• For this SFG, node 2 to node 7 are mixed nodes.

Touching Loops: 

• When two or more loops have shared a common node then that type of loop is called touching loops.
• For the given SFG it is formed by nodes 2-6-2 and 3-4-3

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)

Pi is the ith 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:

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)

Pi is the ith 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.

Calculation:

Given block diagram is,

There are two forward paths,

Δ₁P₁ = G₁, Δ₂P₂ = G₂

There are two loops,

- G₁H, - G₂H

Δ = 1 - (- G1H - G2H) = 1 + H (G₁ + G₂)

From mason's gain formula