Block Diagram

Block diagrams consist of four basic symbols:

• Arrow – Used to represent a variable and the direction of the cause-and-effect relation
• Circle – Represents addition as well as subtraction, depending on the sign associated with the variable’s arrow
• Block – Used to represent the input-output relation of a transfer function
• Take-off point – used to obtain the value of a variable from its arrow, for use in another part of the diagram

• (a) $X(s)=F(s)K$
• (b) $X(s)=\frac{F(s)}{s}$

Equivalent Block Diagrams

Let’s say we have a system described by:

$$\dot{x}(t)+ax(t)=f(t)$$

In Laplace domain:

$$\begin{array}{rl}sX(s)+aX(s)& =F(s)\\ (s+a)X(s)& =F(s)\end{array}$$

Then, our transfer function is:

$$\frac{X(s)}{F(s)}=\frac{1}{s+a}$$

or

$$\begin{array}{rl}(s+a)X(s)& =F(s)\\ sX(s)& =F(s)-aX(s)\\ X(s)& =\frac{1}{s}[F(s)-aX(s)]\end{array}$$

In block diagram form:

Examples

• Example 1: Mass-spring system with Dampener
• Example 2: Simple feedback loop system
• Example 3: Numerically defined system
• Example 4: Complex multi-DOF mass-spring system