The objective of this screencast is to show the basic entities, the nuts and bolts,
of block diagrams in a brief discussion about a few
mathematical properties that dictate block diagrams.
What is a block diagram?
A block diagram is a mathematical representation of a process.
When is comes to analyzing processes we have PNIDs and PFDs available,
but often times these diagrams don't really help us understand
from mathematical standpoint how everything relates to one another,
so block diagrams do a very nice job of showing this.
And at the core of any block diagram are four major components:
The first component is the arrow.
The arrow is relatively straight forward in that it represents how information or how streams
are moving from place to place, so it shows sequence.
The second is the block.
The block is the representation of a mathematical model of how an input affects an output,
so a block shows the relationship between the input and the output.
So, in our diagram here, this is the block, and G1 shows how the input E impacts the output M.
The third major aspect of a block diagram is the branching point.
And the branching point is shown on the right hand side of the diagram,
The branch point represents where information gets sent to multiple places.
The best corollary for this is the idea of composition in mass balances.
So if you think about it from the mass balance standpoint, when you have a stream,
Lets say of 80 kg per minute of a 50/50 solution of water and salt,
If you break the stream in half, assuming a well-mixed system, both streams will have the same composition,
and that is the idea of the branch point.
So if I look at the diagram here, we have M of s before the branch point, and what that means is
every point coming in and out of this is the same thing, so in other words
we have M of s representing the arrow coming out of the system
and we have the M of s representing the feedback which is going to the circular element.
This circular element represents the fourth and final important component in a block diagram,
and that represents the summing point.
And as the name indicates, it sums multiple pieces of information.
Here is our summing point and what you will notice at our summing point is that we have
signs which are at the outside of the circle. Some conventions put those signs inside the circle
some put it outside the circle, it doesn't really matter, they mean the same thing.
And all that is showing is whether or not that component should be added or subtracted
So if we were to look at our block diagram here, what we would see is the fact that
from this summing point is that E of s,
whats going out equals R of s with a positive, because there is a positive sign there,
minus M of s. M of s is the signal entering this summing point and it is a subtraction.
In order for the block diagram to be useful as a mathematical representation of the process,
you have to be aware of the rules that dictate block diagrams algebra.
We've already discussed a few of them here, with the branch point and the summing point,
but lets talk about two more rules that are of interest in use of block diagrams.
The first rule to note is the fact that block diagrams, like numbers, obey the commutative property.
So what this means is that there is no mathematical difference between the following block diagrams.
So here what the first diagram shows is the fact that x is going to equal y, the variable that comes in.
Every time you go through a block it represents a multiplication.
So therefore you will have y times G2, and then moving out onto the next block
you are going to multiply yG2, which is what is technically the stream represented here,
and multiply that by G1.
And therefore x will then equal yG2G1.
Similarly for the second diagram, x will equal y, the input variable,
multiplied by G1G2.
So therefore, these are the two answers for the respective block diagrams we have here,
but an important point to note here is that these are both equal to each other.
The second one to note here is the fact that block diagrams, like numbers, obey the distributive property.
So an example to show the distributive property could be done by developing the block diagram shown below.
If we start from the left to the right, a general procedural method to handle these, you'll hit the summing point here
which shows the fact that we have a positive x1 minus x2.
With the distributive property, the G1 will be distributed to both x1 and x2,
which means that this will become G1x1-G1x2.
And since this outlet of G1 represents the output of y,
We know the fact that these two relationships, y and G1x1-G2x2 are equivalent.