Block Diagram Math and Manipulations

Electronic systems and circuits are often represented by block diagrams, and block diagrams have a unique algebra and set of transformations[1]. Block diagrams are used because they are a shorthand pictorial representation of the cause-and-effect relationship between the input and output in a real system. They are a convenient method for characterizing the functional relationships between components. It is not necessary to understand the functional details of a block to manipulate a block diagram.

The input impedance of each block is assumed to be infinite to preclude loading. Also, the output impedance of each block is assumed to be zero to enable high fan-out. The systems designer sets the actual impedance levels, but the fan-out assumption is valid because the block designers adhere to the system designer’s specifications. All blocks multiply the input times the block quantity (see Figure 5–1) unless otherwise specified within the block. The quantity within the block can be a constant as shown in Figure 5–1(c), or it can be a complex math function involving Laplace transforms. The blocks can perform time-based operations such as differentiation and integration.

Figure 5–1. Definition of Blocks

Adding and subtracting are done in special blocks called summing points. Figure 5–2 gives several examples of summing points. Summing points can have unlimited inputs, can add or subtract, and can have mixed signs yielding addition and subtraction within a single summing point. Figure 5–3 defines the terms in a typical control system, and Figure 5–4 defines the terms in a typical electronic feedback system. Multiloop feedback systems (Figure 5–5) are intimidating, but they can be reduced to a single loop feedback system, as shown in the figure, by writing equations and solving for V OUT /V IN . An easier method for reducing multiloop feedback systems to single loop feedback systems is to follow the rules and use the transforms given in Figure 5–6.

 

Figure 5–2. Summary Points

Figure 5–3. Definition of Control System Terms

Figure 5–4. Definition of an Electronic Feedback Circuit

Figure 5–5. Multiloop Feedback System

Block diagram reduction rules:

1. Combine cascae blocks.
2. Combine parallel blocks.
3. Eliminate interior feeback loops.
4. Shift summing points to the left.
5. Shift takeoff points to the right.
6. Repeat until canonical form is obtaine.
   
   

Figure 5–6 gives the block diagram transforms. The idea is to reduce the diagram to its canonical form because the canonical feedback loop is the simplest form of a feedback loop, and its analysis is well documented. All feedback systems can be reduced to the canonical form, so all feedback systems can be analyzed with the same math. A canonical loop exists for each input to a feedback system; although the stability dynamics are independent of the input, the output results are input dependent. The response of each input of a multiple input feedback system can be analyzed separately and added through super position.

Figure 5–6. Block Diagram Transforms