### Construction of SFG from simultaneous equations

Construction of SFG from simultaneous equations

Rules For Construction of a Signal Flow Graph:

1. The actual signal only travels through branch and towards the direction specified through an arrow indicator.

2. The Output from a branch is the product of the gain and input of that branch.

3. Input to a specific node is the sum of all the signals that enter the node.

Let us consider the following set of simultaneous equations:
$\small x_1$=$\small g_1x_0+h_1x_2$
$\small x_2=g_2x_1$
$\small x_3$=$\small g_3x_2+h_2x_4$
$\small x_4$=$\small g_4x_3$
$\small x_5=g_5x_4$
There are total 6 variables and 7 coefficients, i.e there will be 6 nodes and 7 branches.
Signal Flow Graph for equation $\small x_1$=$\small g_1x_0+h_1x_2$ :

Signal Flow Graph for equation $\small x_2=g_2x_1$ is given by:

Signal Flow Graph for equation $\small x_3$=$\small g_3x_2+h_2x_4$ is given by:

Signal Flow Graph for $\small x_4$=$\small g_4x_3$:

Signal Flow Graph for $\small x_5=g_5x_4$

Combining All the Signal Flow Graph we get:

Signal Flow Graph From Block Diagram

We can convert a Block Diagram Representation into a Signal Flow Graph Representation using the following steps :

1. Convert all the variables, summing points and takeoff points to nodes.

2. Represent all the blocks as branches and represent the functions in the block as transmittance of the branch.

3. Connection of nodes and branches has to remain the same and if there is any connection without block, represent it  a branch with unity gain.