Signal Flow Graphs (SFGs) are graphical representations of linear time-invariant systems used to analyze and understand the behavior of complex systems. SFGs are composed of nodes representing variables and edges representing the flow of signals between variables.

By using SFGs, we can easily analyze and solve complex systems in a step-by-step manner. In this article, we will go through some solved examples of SFGs to illustrate their usage.

**Example 1:**

Consider a linear time-invariant system with the transfer function H(s) = (s+1)/(s+2). Draw the SFG for this system and find the overall transfer function.

To draw the SFG, we first identify the input and output variables. In this case, the input variable is u and the output variable is y. We then represent these variables as nodes in the SFG. Next, we represent the transfer function H(s) as a block with an arrow pointing from the input node to the output node, as shown below:

u —-> (s+1)/(s+2) —-> y

We then convert the transfer function block into a set of two nodes connected by an edge with a gain of (s+1)/(s+2), as shown below:

u ——-[1]——> [s+1]/[s+2] ——[1]——-> y

The overall transfer function is obtained by applying Mason’s gain formula, which states that the transfer function is given by the sum of all the individual paths from the input to the output, each path multiplied by its respective gain. The formula can be written as:

H(s) = (sum of all individual paths from u to y, each path multiplied by its gain) / (1 – sum of all individual loops multiplied by their gains)

Using this formula, we can calculate the overall transfer function as:

H(s) = [1/(s+2)] * [(s+1)/(s+2)]

————

1 – [1/(s+2)] * [1]

H(s) = (s+1)/(s+3)

**Example 2:**

Consider a linear time-invariant system with the transfer function H(s) = (s+1)/(s^2 + 2s + 5). Draw the Signal Flow Graph for this system and find the overall transfer function.

To draw the SFG, we first identify the input and output variables. In this case, the input variable is u and the output variable is y. We then represent these variables as nodes in the SFG. Next, we represent the transfer function H(s) as a block with an arrow pointing from the input node to the output node, as shown below:

u —-> (s+1)/(s^2 + 2s + 5) —-> y

We then convert the transfer function block into a set of three nodes connected by edges, as shown below:

u —-[1]—-> [s+1] —-[1]—-> [1/s^2+2s+5] —-[1]—-> y

We can now apply Mason’s gain formula to obtain the overall transfer function. The individual paths and loops are shown below:

Paths:

u -> [s+1] -> [1/s^2+2s+5] -> y (gain = (s+1)/(s^2+2s+5))

u -> [1/s^2+2s+5] -> y (gain = 1/(s^2+2s+5))

Loops:

[1/s^2+2s+5] (gain = 1)

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