Signal-flow graph

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A signal-flow graph (SFG) is a special type of block diagram,[1] constrained by rigid mathematical rules, that is a graphical means of showing the relations among the variables of a set of linear algebraic relations. Nodes represent variables, and are joined by branches that have assigned directions (indicated by arrows) and gains. A signal can transmit only in the direction of the arrow.

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[edit] Example 1 Simple amplifier

Figure 1: Simple SFG
Figure 1: Simple SFG
Amplification of a signal x1 to become a larger output y2 by an amplifier with gain a11 is described mathematically by:
y_1 = a_{11}x_1 \,
which becomes the signal flow graph of Figure 1.

Although this equation is represented by the SFG of Figure 1, the algebraically equivalent relation

x_1 = {y_1} / {a_{11}} \

is not considered to be implied by Figure 1. That is, the SFG is unilateral, sometimes emphasized by calling x1 a "cause" and y1 an "effect", or by calling x1 an "input" and y1 an "output".

[edit] Example 2 Two-port network

Figure 2: Two-port SFG
Figure 2: Two-port SFG
The two coupled equations below can represent the current-voltage relations in a two-port network:
y_1 = a_{11}\ x_1 + a_{12}\ x_2
y_2 = a_{21}\ x_1 + a_{22}\ x_2 \ ,
which equations become the signal flow graph of Figure 2.

[edit] Example 3 Asymptotic gain formula

Figure 3: A possible signal-flow graph for the asymptotic gain model
Figure 3: A possible signal-flow graph for the asymptotic gain model
Figure 4: A different signal-flow graph for the asymptotic gain model
Figure 4: A different signal-flow graph for the asymptotic gain model
A possible SFG for the asymptotic gain model for a negative feedback amplifier is shown in Figure 3, and leads to the equation for the gain of this amplifier as
G = \frac {y_2}{x_1} = G_{\infty} \left( \frac{T}{T + 1} \right) + G_0 \left( \frac{1}{T + 1} \right) \ .
The interpretation of the parameters is as follows: T = return ratio, G∞ = direct amplifier gain, G0 = feedforward (indicating the possible bilateral nature of the feedback, possibly deliberate as in the case of feedforward compensation). Figure 3 has the interesting aspect that it resembles Figure 2 for the two-port network with the addition of the extra feedback relation x2 = T y1.
From this gain expression an interpretation of the parameters G0 and G∞ is evident, namely:
G_{\infty} = \lim_{T \to \infty }G\ ; \ G_{0} = \lim_{T \to 0 }G \ .
There are many possible SFG's associated with any particular gain relation. Figure 4 shows another SFG for the asymptotic gain model that can be easier to interpret in terms of a circuit. In this graph, parameter β is interpreted as a feedback factor and A as a "control parameter", possibly related to a dependent source in the circuit. Using this graph, the gain is
G = \frac {y_2}{x_1} = G_{0} + \frac {A} {1 - \beta A} \ .
To connect to the asymptotic gain model, parameters A and β cannot be arbitrary circuit parameters, but must relate to the return ratio T by:
T = - \beta A \ ,
and to the asymptotic gain as:
G_{\infty} = \lim_{T \to \infty }G = G_0 - \frac {1} {\beta} \ .
Substituting these results into the gain expression,
G = G_{0} + \frac {1} {\beta} \frac {-T} {1 +T}
= G_0 + (G_0 - G_{\infty} ) \frac {-T} {1 +T}
= G_{\infty} \frac {T} {1 +T} + G_0 \frac {1}{1+T} \ ,
which is the formula of the asymptotic gain model.

Signal flow graphs are used in many different subject areas besides control and network theory, for example, stochastic signal processing.[2]

[edit] References

  1. ^ DiStephano, J. J., Stubberud, A. R., & Williams, I. J. (1995). Schaum's outline of theory and problems of feedback and control systems, Second Edition, New York: McGraw-Hill Professional, §8.8 pp. 187-189. ISBN 0070170525. 
  2. ^ Barry, J. R., Lee, E. A., Messerschmitt, D. G., & Lee, E. A. (2004). Digital communication, Third Edition, New York: Springer, p. 86. ISBN 0792375483. 

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