Any design starts with the clear conception of the system goals and the knowledge of the component base (process) at hand. The system description includes the list of input and output variables, the functional connections between them, input and output limitations, disturbances (noise, mismatch, etc.), and acceptable static and dynamic errors. It can be presented in the forms of algebraic and differential equations, text descriptions, graphs, tables, and macromodels. In the OpAmp design this is often formulated by a similarity criterion (make an amplifier similar to another but with certain improvements). Some requirements may not be present in the initial description, and the familiarity with the applications is essential for the designer. There are always a number of trade-offs and the possibility of additional functional features to add to product. Familiarity with the application helps to acquire an understanding of what is important for customers.
The next step is theoretical or, better yet, experimental confirmation that the design goals are achievable and do not contradict each other.
Examples of the trade-offs are the power to noise ratio or the output power to the power consumption ratio (always < 1). The discovery of the theoretical limits can be a very creative process. A challenging and feasible functional description may determine an innovative solution.
The graphical presentation of the functional description is easiest for the mind to grasp. There are two equivalent methods of the graphical presentation: block diagrams and signal flow graphs. Block diagrams are used very often. However, the signal flow graph may be preferable, as it is faster to draw, and the formal rules for equivalent graph transformations have been developed [25]. Long time ago the signal flow graphs used to be a common tool for electric circuits analysis and, sometimes, for synthesis as
well [93, 94]. Fortunately, these applications of graphs get more attention again [23, 24, 28].
The difference between a structure (or a graph) and a circuit reflects the difference in the level of abstraction. The components of a structure are macromodels, transfer functions, links and nodes of the signal flow graph or the block diagram. The components of a circuit are real transistors, resistors, capacitors, etc. or their symbols in the circuit diagram. The transition from the structure to the circuit is not unique, and there is always a set of circuits implementing the structure.
Fig. 2-1 gives a simple OpAmp circuit diagram (fig. 2-1a) and its signal graphs with different degrees of detailing.
Figure 2-1. OpAmp functional presentation with different signal flow graphs
Fig. 2-1b represents the functional definition of the OpAmp as a unit with the transfer function A(s). Any OpAmp can be described this way.
Fig. 2-1c describes this OpAmp as a two-stage circuit with feedback path via Miller compensation capacitor CM. This description neglects the signal feed forward via this compensation capacitor.
The effects of the signal feedforward through CM are included in the graph of Fig. 2-1d. This feedforward causes the faster phase degradation at the high frequencies in comparison with other compensation methods.
Canceling of this effect can be done, as it is follows from the graph, by breaking the feed forward link, for example, by adding a follower in series with Miller capacitor. Such break of an undesirable link is one of the steps in the structural design.
This last graph (fig. 2-1e) represents in more detail the operation of input differential stage and the current mirror. The left part of this graph is what is called in the structural design “the general structure with common-mode feedback”. This graph may represent a rather general multidimensional system as discussed in the Appendix.
Using the graph of fig. 2-1c, one can see that the amplifier open-loop gain is equal toA=gm01Reqvg4gm4ZL. The transconductances gm01 of the input differential pair and gm4 of the output device M4 are limited by the corresponding values of current. There is no control over the load, and the only viable way left for designer to increase the open-loop gain is to increase the equivalent resistance Reqvg4 at the node g4 (fig. 2-1a). This resistance is defined by the internal feedbacks in the transistors M1 and M2. The graph detailing these feedbacks and the ways to break these links and to boost the gain are discussed in chapter 4.
It is important to realize that the application of traditional graph theory rules to the transformations of system graphs is limited by the linear systems. The real systems with limitations can be presented by the signal flow graphs with limitation links. The inverted L-shape line is chosen here to denote these links. An example representing this limiting link for gm4 is shown in fig. 2-1b. A word of caution should be said about these systems.
Smooth nonlinearities may be represented by the links with transfer functions that can vary in some range. Using the piece-wise approximation and choosing different combinations of the link transfer values one can represent a nonlinear system by a set of graphs. One can then apply the traditional graph theory to each graph of this set. Yet, the graph transformations of these partial graphs will not necessary lead to valid realization of the system or its blocks (units). Hence, the system that includes nonlinear links needs more attention, as the rules of traditional theory being applicable to each graph of the set may be not applicable to the whole system.
Every connection of two components or units may create a new feedback loop. There are two practical rules for electronic circuits that should be obeyed to eliminate unnecessary interactions:
- do not load a current source (high output impedance) on the high- impedance load;
- do not load a voltage source (low output impedance) on the low- impedance load.
The system design starts using ideal current and voltage sources. Every time when the ideal component is replaced with a real one the effects of the
new “natural” feedback loop should be anticipated. If the exclusion of nonideality is essential for the system operation then these components should be “idealized” in the structural design way, namely, by adding the efficient feedback loop consisting of the sensor, amplifier and actuator.
The description of connections of passive R, C and active components as parts of feedback loops in a signal flow graph allows the designer abstracting from the nature of energy in these components. This means that one can use the design solutions found on the structural level, regardless of the physical nature of the component base. This similarity of the models describing different kinds of energy has been noticed long ago [33]. It can be shown that this similarity is a consequence of the energy preservation law. On the structural level there is no difference between design of the bipolar or CMOS OpAmp, of the cruise missile control or of the corporation management structure.