Topic 10
Block Diagrams
10.1
Motivation
Block diagrams are an easy way to represent feedback loops, which will be used for the block diagrams
remainder of the course. Figure 10.1 shows several example block diagrams which I will feedback
use in these notes to describe strategies to analyze and reduce block diagrams.
U(s) +
U(s) +
_
Y(s)
G1
Σ
_
G1
Σ
G2
Y(s)
G3
(a)
U(s)
(b)
K
+
G2
G1
Σ
+
Y(s)
G3
(c)
Figure 10.1: Example Block Diagrams
10.2
Reducing Simple Blocks
There are a set of tricks to reduce simple feedback loops. Figure 10.1a represents the
simplest negative feedback loop possible, and it can be reduced to the following simple
transfer function:
Y (s)
G1
=
.
U (s)
1 + G1
The transfer function is defined in two parts, the numerator and the denominator. The
numerator is the product of everything in the direct line from input to output. The
denominator is one minus everything in the entire feedback loop (the minus sign on the
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TOPIC 10. BLOCK DIAGRAMS
summation block changes that to positive). So, using the same logic, Figure 10.1b can
be reduced to:
Y (s)
G1 G2
=
,
U (s)
1 + G1 G2 G3
and Figure 10.1c reduces to:
Y (s)
KG1 G2
.
=
U (s)
1 G1 G2 G3
10.3
Reducing Complex Blocks
It is easiest to use the aforementioned method when you are only dealing with single
feedback loops. However, when you are dealing with a block diagram, like the one given
in Figure 10.2, the simple method can become tedious to apply. As such, it is often
G3
U(s)
G1
+
G2
Σ
_
+
+
Σ
_
+
G4
+
Σ
Y(s)
G5
G6
Figure 10.2: A Complex Block Diagram
easier to apply an algebraic approach described presently. We will start by naming
several of the points within the block diagram, which I have done in Figure 10.3 (A and
B). Then we will define a series of algebraic equations that define each of the named
points (and the output) in terms of the input, output, or other named points; I have
defined these as Equations 10.1–10.3.
A = U (s)G1
B = A + AG2
Y (s)G6
Y (s)G5 = A (1 + G2 )
(10.1)
Y (s)G5
Y (s) = AG3 + BG4
(10.2)
(10.3)
Substituting Equation 10.2 into Equation 10.3 and simplifying should yield Equation 10.4.
Y (s) = A(G2 G4 + G3 + G4 )
Y (s)G4 G5
(10.4)
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10.3. REDUCING COMPLEX BLOCKS
G3
U(s)
G1
+
Σ
_ A
G2
+
+B
Σ
_
G4
+
+
Σ
Y(s)
G5
G6
Figure 10.3: An Annotated Complex Block Diagram
Substituting Equation 10.1 into this will give us Equation 10.5.
Y (s) = U (s)(G1 G2 G4 +G1 G3 +G1 G4 ) Y (s)(G2 G4 G6 +G3 G6 +G4 G5 +G4 G6 ) (10.5)
Finally, you can solve this equation down to the standard form for transfer functions
(Equation 10.6).
Y (s)
G1 G2 G4 + G1 G3 + G1 G4
=
U (s)
1 + G2 G4 G6 + G3 G6 + G4 G5 + G4 G6
(10.6)
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TOPIC 10. BLOCK DIAGRAMS