Application, modeling and characterization of amplifiers#

In Chapter Modeling and specification of amplifiers, we will give applications of amplifiers, define the amplification function and discuss the ideal behavior of amplifiers. We will then discuss the nature of information processing errors of practical amplifiers and introduce related performance aspects.

In this chapter, we will also discuss some physical appearances of amplifiers. The construction of amplifiers results from trade-offs between performance and costs made during the design process. Cost factors of amplifiers can be numerous, and some examples of cost factors will be given. For design decisions, it is convenient to have a figure of merit at one’s disposal, and some suggestions will be given.

We will then discuss the modeling and the characterization of the ideal behavior of amplifiers.

The fundamental physical limitations of information processing and the technological limitations cause the behavior of practical amplifiers to deviate from their desired behavior. How the observer experiences these deviations depends on how the information is present in the signal. This will be elucidated with some examples.

Principle of amplification#

In Chapter Amplification Mechanism, we will study how the principle of amplification is materialized in electronic amplifiers. We will show that signal amplification can be obtained through the proper interconnection of electrical power sources and passive, electronic devices. Examples of such devices are MOS transistors, bipolar transistors, and vacuum tubes. It will be shown that these devices, when used in combination with power sources, can provide an available power gain larger than unity; which is a distinguishing property of amplifiers. For this reason, these devices are often called active devices. Combining power sources with such devices is usually called biasing. This chapter concludes with a conceptual approach to biasing. We will show that the quiescent operating conditions of a correctly biased amplifier stage do not depend on the stage’s drive and termination resistances.

Modeling of semiconductor devices#

Knowledge of the operation and modeling of modern semiconductor devices is indispensable when designing analog electronic circuits. However, in-depth treatment of semiconductor physics and modeling techniques is beyond the scope of this book. Chapter Active Devices briefly describes the construction, operation, and modeling of BJTs, JFETs, and MOS transistors. The main goal is to provide a basic understanding of the construction and operation of the devices as a basis for modeling the device’s performance during various stages of the design. The Gummel-Poon model for BJTs \cite[-2.5cm]{GummelPoon1970} and the Shichman and Hodges model for JFETs \cite[-1cm]{ShichmanHodges1968} will be presented and simple models for hand calculations will be derived from them. For MOS devices, the basic Shichman and Hodges model, the Meyer capacitance model \cite[-0.25cm]{Meyer1971} and the Ward and Dutton capacitance model [11] will be discussed. The latter model is used in the BSIM3 small-signal model [12], often used in SPICE.

With the EKV model [13], the small-signal parameters can be written expressed in the device geometry, the drain current, and the drain-source voltage. These expressions are valid from weak to strong inversion, including short-channel effects. In this way, it is possible to design the small-signal dynamic transfer and the noise performance independent from the biasing circuitry. SLiCAP has built-in small-signal models for CMOS18 devices whose parameters depend on the channel width and length, the operation current and voltage, and only a few EKV model parameters. This way of working facilitates the design of CMOS circuits using the inversion coefficient or the transconductance efficiency, as described by Binkley[14].

Basic amplification with CS stage#

In Chapter Basic amplification: CS stage, we will study the performance limitations and the design considerations for the common-source (CS) stage, which can be considered as the basic MOS transistor amplifier stage. At a later stage, we will show that other MOS amplifier stages can be derived from the CS stage through the application of error reduction techniques such as compensation or negative feedback.

The common-emitter (CE) stage can be regarded as the basic BJT amplifier stage. Performance limitations and design considerations for this stage will be added in a future version of this book.

For these basic amplifier stages, we will discuss the way in which their performance can be altered by design. We will see that the designer does not have many degrees of freedom to optimize the performance of such elementary amplifier stages. The operating conditions, the fabrication technology, and the geometry or the device type are the only design variables at the disposal of the designer to optimize their performance-to-costs ratio. Moreover, the various performance aspects of single-transistor amplifiers cannot be designed independently

and compromises between performance aspects often need to be made.

Application of balancing techniques: differential pair and push-pull stages#

If the desired performance-cost ratio of an amplifier cannot be achieved with basic amplifier stages, error-reduction techniques can be applied for its improvement. In Chapter Balancing techniques, we will study the application of balancing techniques and their impact on the performance-cost ratio of an amplifier. Two particular applications of balancing techniques will be discussed in more detail: anti-series connection of equal devices, and parallel connection of complementary devices.

Anti-series connection of basic amplifier stages provides a four-terminal stage with an odd transfer characteristic and improved isolation between the input port and the output port. The behavioral modifications that are a result of series, complementary-series, and anti-series connection, will be investigated. It will be shown that the properties of the MOS and the BJT differential pair can easily be related to those of the CS and the CE stage, only by considering such behavioral modifications. We will see that, when applied in a truly balanced environment, the small-signal transfer and the noise performance of the differential pair can equal those of the basic CE or CS stage at the costs of four times the area and four times the operating current. As a result of the anti-series connection, offset voltages are canceled and the bias sources change from differential-mode to common-mode. The large-signal transfer of these anti-series stages has an odd characteristic with current saturation.

The complementary-parallel connection of amplifier stages split an input signal into a push and a pull current. These currents can be much larger than the quiescent operating current of the stage and can be used as high-efficiency amplifier stages. The CMOS inverter is a complementary parallel stage.

The behavioral modifications resulting from parallel, anti-parallel, and complementary-parallel connection of amplifiers or amplifier stages, will be investigated. The properties of the MOS and the BJT push-pull stage will be related to those of their constituting CS or CE stage.

Not all performance aspects of an amplifier or amplifier stage can be improved with balancing. The (gain) accuracy, the dynamic transfer, and their temperature dependencies of balanced stages equal those of their unbalanced version. Improvement of those aspects requires the application of techniques with better error-reduction capabilities.

Design of negative feedback amplifier configurations#

Negative feedback is very a powerful error-reduction technique. The characteristics of negative feedback amplifiers are primarily fixed with reference networks or feedback networks. These feedback networks create an accurate copy of the source signal from the load signal. They can be passive or nonenergic elements of which the electrical behavior is accurately defined. A high-gain controller or error amplifier minimizes the difference between the source signal and this copy. In this way, the properties of the feedback amplifier are predominantly determined by its feedback networks; the error amplifier provides the available power gain but does not define the source-to-load transfer.

The design of application-specific negative feedback amplifiers is discussed in Chapter Design of feedback amplifier configurations. In this chapter, we will discuss the design of negative feedback amplifier configurations for specific sources and loads. We will see that the amplifier types, introduced in Chapter Modeling and specification of amplifiers can be synthesized by combining voltage and or current sensing at the load with voltage and or current comparison at the signal source. During the conceptual design of negative feedback amplifiers, nullors are used as ideal controllers. The nullor can be considered a network element with an infinite available power gain and no speed limitation. At a later stage of the design, these nullors need to be replaced with error amplifiers, implemented with amplifier stages, or with operational amplifiers.

In this chapter, we will show that the amplifier types from Chapter Modeling and specification of amplifiers can all be designed using nonenergic feedback elements. Nonenergic feedback elements are lossless and have no energy storage. In addition, two of them, the ideal transformer and gyrator behave like natural two ports. Unfortunately, transformers are not ideal and suffer from energy storage and losses, while physical operating mechanisms for gyrators have not (yet) been discovered. The use of more practical passive feedback elements limits the number of negative feedback configurations. If this is the case, the application of active feedback, balancing, or indirect feedback may be considered.

The operational amplifier as controller in feedback amplifiers#

Operational amplifiers are intended as controllers for negative feedback amplifiers. With their high voltage gain, high common-mode and differential-mode input impedance, high common-mode rejection ratio and low output impedance early types were versatile building blocks for negative feedback voltage amplifiers. Nowadays, the behavior of current-feedback and rail-to-rail output operational amplifiers strongly deviates from this behavior, which complicates their application.

Chapter Application and specification of operational amplifiers deals with the modeling of operational amplifiers. Aside from modeling all behavioral aspects with so-called macro models, attention will be paid to the modeling of individual performance aspects, which is considered to be more useful for deriving budgets for different performance limitations and for taking early-stage design decisions.

Another aspect that is limiting the application of operational amplifiers as universal controller is the fact that their output port, which is usually a high-efficiency push-pull output stage, has a split return path connected to both supply terminals. This imposes difficulties to the implementation of grounded current sensing techniques and limits the number of amplifier configurations that can be realized using solely operational amplifiers as controllers. This limitation as well as various ways to deal with it will also be discussed in this chapter.

Introduction to biasing#

An introduction to the biasing of negative feedback amplifiers will be presented in Chapter Introduction to amplifier biasing. In this book, we will advocate a strict separation between the design of the signal transfer and the design of the biasing of amplifiers and amplifier stages. At an earlier stage we have already shown that the principle of amplification requires the application of properly interconnected power sources and passive nonlinear electronic devices. Biasing refers to the derivation of all these power sources from the power supply source(s).

Biasing of (cascaded) amplifier stages will only be presented after we have discussed the design of the signal processing properties of an amplifier. The reason for this is that biasing of stages and of interconnected stages only needs to be done if the signal processing by the conceptually biased stages is adequate. Biasing of a configuration of which the signal processing is not according to the requirements is meaningless and regarded as a lost of valuable design time. In this introductory chapter, we will only discuss the consequences of errors that are a result of imperfect biasing of controllers. Such errors occur due to device tolerances and temperature deviations. These errors are usually modeled with the aid of equivalent-input offset and bias currents and voltages. Statistical description methods will be given and error reduction techniques to minimize their effects, will be discussed. Examples will be given for negative feedback amplifiers equipped with operational amplifiers, but the theory is not limited to these cases. Compensation, AC coupling and negative feedback biasing will be introduced as methods for the reduction of biasing errors. The latter two can only be applied if frequency components of the signal differ from those of temperature changes. These techniques establish a high-pass character of the signal transfer and design criteria for the high-pass cut-off frequency will be given. Proper high-pass filter characteristic can be established using frequency compensation techniques.

Modeling of negative feedback circuits#

After we are able to design all kinds of application-specific amplifier configurations with nullors as controllers, we need to find specifications for practical controllers. To this end, we need a way of feedback modeling that facilitates a two-step design:

1. Design of the ideal transfer which is fixed by the feedback network

2. Design of an acceptable deviation from this ideal behavior caused by the nonideal controller.

The widely used feedback model introduced by Black [15] provides accurate performance analysis of negative feedback amplifiers only under limited conditions. It does not account for the so-called direct transfer from the source to the load, and it assumes unilateral transfer and ideal sensing at the load and comparison at the source. As a result of these limitations, it is suited for the analysis of negative-feedback systems rather than for the two-step design of negative-feedback amplifier circuits. The feedback theory introduced by Bode \cite[-1.5cm]{Bode1945} in 1945 and described by Chen [16] gives a method to analyze the stability of a feedback loop. Middlebrook [17] introduced the double injection theory to measure the loop gain. However, all these models focus on stability analysis, rather than on a two-step design of a well-defined dynamic behavior of the feedback amplifier.

The only feedback model that facilitates the two-step design is the asymptotic gain model as described by Rosenstark \cite[-0.5cm]%{Rosenstark1974}. This model shows that the design of a negative feedback amplifier can be performed in the two subsequent and independent steps that have been mentioned above. Moreover, the source-to-load transfer obtained from this model equals the one found from network analysis. The asymptotic gain model will be discussed in Chapter Modeling of negative feedback circuits.

Setting up controller performance specifications#

With the aid of the asymptotic gain model, we are able to relate performance aspects of the controller to those of the negative feedback amplifier. This enables use to derive budgets for the performance aspects of the controller, which is a minimum requirement for the two-step design approach described above. In Chapter Amplifier performance and controller requirements this will be done for the static accuracy, the nonlinearity and the bandwidth of the amplifier. We will find that:

1. The static error of a feedback amplifier sets a requirement for the controller’s contribution to the static or DC loop gain.

2. The low-pass cut-off frequency of a feedback amplifier sets a requirement for the contribution of the controller to the gain-poles product of the dominant poles of the loop gain.

3. The static differential-gain error of the negative-feedback amplifier sets a requirement for the contribution of the controller to the static differential error to gain ratio of the DC loop gain.

Design conclusions for other performance aspects such as the high-pass cut-off frequency will also be derived. The derivation of budgets for noise and power losses of the controller has already been dealt with in Chapter Design of feedback amplifier configurations.

In this chapter, we will also introduce techniques for the evaluation of the stability of negative feedback amplifiers. Techniques known from control theory, such as the Nyquist stability criterion, the Routh array analysis method and the root-locus technique will be summarized and elucidated with examples. Frequency compensation techniques for establishing the desired filter characteristics will be discussed at a later stage.

Frequency compensation#

Frequency compensation techniques for establishing proper high-pass or low-pass filter characteristics will be extensively discussed in Chapter Frequency compensation. Concepts and strategies for frequency compensation will be introduced and implementation examples will be given. Special attention will be paid to the impact that frequency compensation may have on other performance aspects such as bandwidth, linearity, overdrive recovery and noise. It will be shown that frequency compensation with the aid of phantom zeros is the most powerful method because it has the lowest interaction with other performance aspects. Implementation of both active and passive phantom zeros will be discussed and illustrated with examples. Other techniques such as pole-splitting techniques, resistive broadbanding and nested control will be discussed as well.

Design of local feedback stages#

Amplifier stages that use a single, unbalanced or balanced CE or CS stage as controller are called local feedback stages. Local feedback stages can be used as single-stage amplifiers, or as stages in negative feedback amplifiers. In Chapter Local feedback stages, we will discuss the design of local feedback stages. We will show that the well-known CD stage or source follower as well as its bipolar version, known as the emitter follower or CC stage, can be considered as unity-gain voltage amplifiers that exploit nonenergic feedback and that have the CS or CE stage as controller, respectively. Similarly, the common-gate (CG) or the common-base (CB) stage can be considered as nonenergic negative feedback current followers. The advantage of such an description method is evident. If those stages are feedback versions of the basic CS or CE stages, then the loop gain, which can be regarded as a measure for the amount of negative feedback, indicates the extent to which their behavior deviates from that of the CE or CS stage. It will then become clear that commonly known properties, such as the low output impedance of the CC stage, are only true if the stage is driven from an impedance that establishes a relatively large loop gain. If such a stage is driven from a current source, accurate input voltage comparison cannot be performed, the loop gain will be low and the output impedance does not differ from that of a CS stage. In addition, since the CS and CE stages are nonenergic negative feedback amplifiers, they inherit the properties of nonenergic feedback amplifiers. Without further analysis it then becomes clear that the equivalent input noise sources of a CD or the CC stage equal those of the CS or the CE stage of which they are constituted, respectively. Similar things can be said about their power efficiency.

Aside from the design of CD, CC, CG and CB stages, the design of other basic local feedback stages, such as the series, the shunt stage and some dual-loop local feedback stages, as well as the application of balancing techniques will be discussed as well. A separate section will be devoted to the so-called cascode stage. This stage consists of a cascade connection of the CS and CG stage (MOS version) or a CE and a CB stage (BJT version). Its interesting properties makes it an ideal inverting amplifier stage in multiple-stage negative feedback amplifiers that ensures low interaction between stages. It will be shown that the CD-CG cascode and its bipolar version, the CC-CB cascode, can similarly be regarded as basic non-inverting amplifier stages.

Design of multiple-stage negative feedback amplifiers#

High-gain amplifiers may be constructed from a cascade connection of amplifier stages. High-performance negative feedback amplifiers, however, require controllers that comprise multiple amplifier stages. The CS or the CE stage, the cascode stages, the local feedback stages as well as the balanced versions of all these stages may be candidates for amplifier stages in such a multiple-stage controller. In Chapter Multi-stage Feedback Amplifiers we will discuss the design of multiple-stage controllers. We will show that by selecting a CS or CE (cascode) stage, or their balanced version, the controller will have the best possible noise performance if the noise performance of this stage is optimized for the given source impedance and feedback network(s). Similarly, by selecting a CE or CS (cascode) stage or its balanced version the contribution to the differential error to gain ratio of the loop gain will be as low as possible.

Design criteria for the number of stages and the preferred type(s) for intermediate stages will also be given. It is important to have a rough estimate for the number of stages at an early stage of the design. If the number of stages is more that two or three, frequency compensation may become difficult and one may consider to construct the amplifier from a number of multiple-stage feedback amplifiers. Nested feedback techniques will also be discussed and illustrated with examples.

The motivation of the type of stages inside the controller may also be driven from practical limitations such as the power supply voltage and the complexity of the biasing. In modern analog CMOS design, the low power supply voltage may put a serious constraint to the architecture of the controller. Since biasing considerations may seriously influence the design of amplifiers, they need to be accounted for during all stages of the design process. However, this does not change the design approach for amplifiers. If the signal processing performance of an optimally designed signal path does not leave room for any degradation possibly resulting from biasing, the detailed design of the bias sources is of no use. Hence, the design of the noise performance, the bandwidth, the linearity and the frequency compensation should always be done before implementing the bias sources.

Biasing#

The biasing and design of the biasing elements is performed in four steps:

1. Simplification of the biasing scheme.

   During the design of the signal path of the amplifier we use conceptually biased amplifier stages as introduced in Chapter Amplification Mechanism. Such stages use four bias sources. During this design step, this biasing scheme will be simplified and the remaining bias sources will be replaced with the power supply and nonlinear resistive elements that exhibit a voltage or current source character. This step will be elucidated in Chapter Amplifier Biasing.

2. Setting up specifications for the resulting bias sources.

   After a biasing scheme has been developed, the performance requirements for the bias sources need to be derived from error budgets for noise, bandwidth and nonlinearity. This step will also be elucidated in Chapter Amplifier Biasing.

3. Design of the bias sources.

   The design of bias sources will be added to a future edition of this book.

4. Application of error reduction techniques for minimization of biasing errors resulting from device tolerances and temperature changes.

This has been discussed in Chapter Introduction to amplifier biasing.