Introduction#
In Chapter Design of feedback amplifier configurations, we showed that low-noise and power-efficient amplifiers can be realized with the aid of negative feedback. When applying negative feedback, the input impedance, the output impedance and the source-to-load transfer of an amplifier obtain their required value with the aid of feedback networks around high-gain loop amplifiers or controllers. If we use nullors as ideal controllers, the transfer characteristics of the amplifier are completely determined by the feedback networks. In fact, when designing this so-called feedback configuration, we perform the first design step in amplifier design. The result of this step is a conceptual design in which primary performance aspects such as the ideal gain, the port impedances and the port isolation configuration of the amplifier have been designed.
In the second design step, we would like to design the controllers without changing anything that has been designed during the first step.
Two-step design approach#
If the design of a controller can be done without changing the amplifier concept, we have a straightforward two-step design approach with no iterative loops. Such a design approach requires decomposition of the allowed information processing errors in two independent contributions that can be designed during the two subsequent steps:
Error contributions due to imperfections of the feedback networks
These error contributions have been discussed in Chapter Design of feedback amplifier configurations. Since the feedback network determines the ideal gain of the negative feedback amplifier, tolerances of devices in the feedback network result in tolerances of the ideal gain. We have also seen that application of passive devices in the feedback network may result in an increase of noise and a decrease in power efficiency. It will also be clear that nonlinear and/or dynamic behavior of feedback elements results in a nonlinear and/or dynamic ideal transfer of the negative feedback amplifier. Sometimes, such effects are intended. In logarithmic amplifiers, for example, the output quantity intentionally changes logarithmically with the input quantity. Similarly, active filters exhibit an intended dynamic behavior.
Error contributions due to imperfect implementation of the controller(s)
The transfer of a negative feedback amplifier only equals its ideal gain if all of the controllers are nullors. In practice, this will never be the case. As a matter of fact, it doesn’t have to be the case. In general, small deviations from the ideal transfer of the negative feedback amplifier can be allowed. Hence, a part of the total error budget can be reserved for error contributions resulting from imperfect behavior of the controller. In order to assign such error budgets, or to judge whether given error budgets are realistic, we need to know the way in which and to what extent the performance aspects of the controller affect those of the negative feedback amplifier. We have already studied the influence of controller noise and are able to assign budgets for the equivalent-input noise voltage and noise current source of the controller. We have not yet studied the influence of gain and bandwidth limitations of the controller, nor do we have a clear understanding of the way in which the nonlinearity that occurs in the controller manifests itself in the transfer of the negative feedback amplifier.
At this stage, we would like to have a modeling technique at our disposal that clearly relates all kinds of behavioral aspects of the controller to relevant behavioral aspects of the negative feedback amplifier. However, not every model that provides this insight is useful. We need to have a model that supports our two-step design approach. Hence, it should split error contributions into two parts: changes in the ideal gain due to imperfections of the feedback network and deviations from this (changed) ideal gain due to controller imperfections.
This chapter#
In this chapter, we will discuss various techniques for modeling of feedback systems and circuits. Some techniques will only briefly be discussed, while a model that supports the two-step design method will be discussed in more detail.
A model that is widely used for analysis of negative feedback systems is the feedback model by Black.
Black’s feedback model is very useful during system-level design with building blocks that have unilateral transfer and that show no interaction. These conditions are usually not satisfied in electronic circuits. As a matter of fact, satisfaction of these conditions would put impractical constraints on the design of application-specific negative feedback amplifiers: they would introduce requirements that adversely affects the feasibility of the design. Black’s feedback model will be discussed in section Black’s feedback model.
One feedback model that supports the two-step design approach is the asymptotic gain model. It will be presented in section Asymptotic gain model. This model is based upon the superposition model, which models the behavior of circuits that exploit negative feedback as accurately as network theory.
A feedback model that is very suited for measurements and simulations of complete feedback circuits has been described by Middlebrook’s generalized feedback theorem. It is suited for analysis rather than for synthesis. It will not be discussed in this book.
In this book we will use the asymptotic gain model to relate the performance aspects of the controller to those of the negative-feedback amplifier. This will be done in Chapter Amplifier performance and controller requirements.