Block Diagrams

Abstract

A block diagram of a system is a graphical representation of the interconnections between the components of the system and the flow of signals. The diagram is composed of functional blocks which contain the mathematical operations of the components.

Ex:

Consider this system: The equations of motion of this system are defined by:

m \overset{x}{¨} + c \overset{x}{˙} + k x Laplace Transform: (m s^{2} + cs + k) X (s) \frac{X ( s )}{Y ( s )} = c \overset{y}{˙} + k y = (cs + k) Y (s) = \frac{cs + k}{m s ^{2} + cs + k}

Basic Elements

Rules for Reduction

Rule 1: Cascaded Blocks

Rule 2: Summing Two Signals

Rule 3: Moving a summing point back

Rule 4: Moving a summing point forward

Rule 5: Moving a branch forward

Rule 6: Moving a branch behind

Rule 7: Eliminating Feedback Loop

$E (s)$ is known as the Error Signal and $B (s)$ as the Feedback Signal

Worked Examples of Block Diagram Reduction

Ex 1:

The second loop can be restated as $\frac{G _{2} ( s )}{1 + G _{2} ( s ) H _{2} ( s )}$ , then following that we get: $G_{1} (s) + \frac{G _{2} ( s )}{1 + G _{2} ( s ) H _{2} ( s )}$ ( $\frac{G _{1} ( s ) + G _{1} ( s ) G _{2} ( s ) H _{2} ( s ) + G _{2} ( s )}{1 + G _{2} ( s ) H _{2} ( s )}$ ) in a feedback loop with $H_{1} (s)$

This can be restated as a single block as:

\frac{G _{1} ( s ) + \frac{G _{2} ( s )}{1 + G _{2} ( s ) H _{2} ( s )}}{1 + G _{1} ( s ) H _{1} ( s ) + \frac{G _{2} ( s ) H _{1} ( s )}{1 + G _{2} ( s ) H _{2} ( s )}}

\frac{\frac{G _{1} ( s ) + G _{1} ( s ) G _{2} ( s ) H _{2} ( s ) + G _{2} ( s )}{1 + G _{2} ( s ) H _{2} ( s )}}{1 + \frac{H _{1} ( s ) G _{1} ( s ) + G _{1} ( s ) G _{2} ( s ) H _{1} ( s ) H _{2} ( s ) + H _{1} ( s ) G _{2} ( s )}{1 + G _{2} ( s ) H _{2} ( s )}}

Block_Diagram_Worked_Examples.excalidraw

Quartz 4

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