Block diagrams can be represented as signal flow graphs. Signals flows from left to right, hence the name. Each signal in the system is represented by a node.
Here is an example of a SFG:
The arrow pointing to the right represents Forward Path, and the gain is the multiplication of the functions such as in order to get somewhere. (The forward path gain from , the forward path gain from )
Non-Touching Loops are set of loops that do not share any common node.
The algorithm to drawing a signal flow graph is as follows.
The following steps are option, tedious, but is needed later on to find the transfer function of the entire system.
Example: expressing a circuit as a SFG
First, we need to specify the relationship between all the signals (voltage and current):
So we have five signals nodes: with being the input and being the output.
We draw the SFG using the algorithm described above.
Note that the equations we wrote down is not unique, thus the SFG is also not unique. But the resulting transfer function is unique.
We identify 3 loops:
We identify 1 pair of non-touching loops:
We identify 1 forward path:
The mason’s gain formula is used to compute the transfer function from SFGs.
is the gain of path
is the determinant of the entire graph
is with out all the loops touching path (setting those loops to 0)
Consider 7 loops, and are non-touching. Then
Example: determinant practice