Introduction

In Chapter Modeling and specification of amplifiers,\ we have showed that all amplifiers can be characterized by their input impedance, their output impedance and their transfer characteristic. We determined nine different unilateral amplifier types and modeled their ideal behavior with the transmission-1 two port parameters. We found five different versions for each of those types if we also consider whether the ports are floating or if they have one of their terminals connected to the ground.

Design tasks

In this chapter, we will discuss means to design the port impedances and the source-to-load transfer. We will do this by designing the transmission-1 two-port parameters of the amplifier.

We have seen that the port impedances should provide optimum sense and drive conditions for the source and the load, respectively. Sensing the open circuit voltage of the signal source, as well as driving the load with a current, requires an infinite port impedance. In order to sense the short-circuit current of the signal source, as well as drive the load with a voltage, the port impedance should be zero. For accurate termination of transmission lines, the port impedance should equal the characteristic impedance of the transmission line.

Let us, for example, consider a voltage amplifier that is driven from a voltage source with an open-circuit voltage \(V_{s}\) and a source impedance \(Z_{s}\). The amplifier should provide a voltage \(V_{\ell}\) across the load impedance \(Z_{\ell}\). This voltage should be an accurately amplified copy of the source voltage. The input impedance of the amplifier is \(Z_{i}\) and the output impedance of the amplifier equals \(Z_{o}\), as shown in Fig. 212.

The source-to-load voltage transfer for this configuration can be found as:

\[\frac{V_{\ell}}{V_{s}}=\frac{Z_{i}}{Z_{i}+Z_{s}}A_{v}\frac{Z_{\ell}}{Z_{\ell }+Z_{o}},\]

where \(A_{v}\) is the voltage amplification factor of the amplifier: \(A_{v}=\frac{V_{o}}{V_{i}}\).

Fig. 212 Simple model of a voltage amplifier with source and load.

Accurate information transfer from source-to-load is only possible if:

1. All impedances are accurately known

2. In cases in which \(Z_{s}\) and/or \(Z_{i}\) are not accurately known, the input impedance of the amplifier should be much larger than the source impedance.

3. In cases in which \(Z_{\ell}\) and/or \(Z_{o}\) are inaccurately known, the output impedance of the amplifier should be much smaller than the load impedance.

Brute force port impedance design

Fig. 213 Voltage amplifier with brute force correction of its input and output impedance.

If the optimum drive or termination conditions are not met, accurate and linear impedances can be inserted into the signal path to improve the source sense and the load drive conditions. In the case of the voltage amplifier from Fig. 212, this can be done by inserting a linear and accurately known impedance \(Z_{se}\) in series with \(Z_{s}\) and a linear and accurately known impedance \(Z_{p}\) in parallel with \(Z_{\ell}\), as shown in Fig. 213.

Techniques by which accurate and linear port impedances are realized through their simple insertion in series or in parallel with the signal path, are called brute force techniques. In general, these techniques should be avoided, because they result in a deterioration of the signal-to-noise ratio and the power efficiency of the amplifier.

Example

Let us consider an amplifier with equivalent input noise sources \(V_{eq}\) and \(I_{eq}\). The amplifier is driven from a voltage source with an open source voltage \(V_{s}\) and an impedance \(Z_{s}\). The noise voltage associated with the source impedance is \(V_{ns}\). The spectral density of this voltage equals \(4kT\operatorname{Re}(Z_{s})\) [V\(^{\text{2}}\)/Hz]. An impedance \(Z_{se}\) has been placed in series with the signal path and an admittance \(Y_{p}\) has been placed in parallel with the signal path. The spectral density of the voltage noise \(V_{nY}\) associated with \(Z_{se}\) equals \(4kT\operatorname{Re}(Z_{se})\) [V\(^{\text{2}}\)/Hz]. The spectral density of the current noise \(I_{nY}\) associated with \(Y_{p}\) equals \(4kT\operatorname{Re}(Y_{p})\) [A\(^{\text{2}} \)/Hz]. The complete configuration is shown in Fig. 214.

Fig. 214 Impedances in parallel or in series with the signal path generally degrade the signal-to-noise ratio.

For the evaluation of the influence of the noise sources on the signal-to-noise ratio, the noise sources need to be transformed into one total equivalent noise voltage \(V_{ntot}\) in series with the signal voltage source. The spectral density \(S_{V_{ntot}}\) of this total equivalent input noise voltage for the circuit from Fig. 214, can be obtained as:

\[\begin{split}S_{V_{ntot}} & =4kT\left( \operatorname{Re}\left( Z_{s}\right) +\operatorname{Re}\left( Z_{se}\right) \left\vert 1+Z_{s}Y_{p}\right\vert ^{2}+\operatorname{Re}\left( Y_{p}\right) \left\vert Z_{s}\right\vert ^{2}\right) \nonumber\\ & +S_{V_{eq}}\left\vert 1+Z_{s}Y_{p}\right\vert ^{2}\\ & +S_{I_{eq}}\left\vert Z_{se}+Z_{s}\left( 1+Z_{se}Y_{p}\right) \right\vert ^{2}\nonumber\end{split}\]

This expression shows an enlarged contribution of both equivalent noise sources to the total equivalent input noise if \(Z_{se}\) and/or \(Y_{p}\) differ from zero. Only in narrow-band applications can the noise be improved by inserting impedances in series or in parallel with the signal path. This occurs if \(Z_{se}=-\operatorname{Im}(Z_{s})\). In cases in which current is the information-carrying quantity, this condition would be:\ \(Y_{p} =-\operatorname{Im}(Y_{s}).\) Such improvement of the signal-to-noise ratio, however, only occurs in the vicinity of some (resonance) frequency at which these conditions are approximately met.

From the above analysis, we may draw the following conclusion:

Definition

Impedances in series or in parallel with the signal source will deteriorate the signal-to-noise ratio of the amplifier. This deterioration has two causes:

1. In general, the contribution of the amplifier’s equivalent input noise sources to the total input noise will increase due to the insertion of these impedances as has been shown in the above example.

2. If the real part of these impedances differs from zero, the thermal noise associated with it, decreases the signal-to-noise ratio.

Impedances in series or in parallel with the load will deteriorate the power efficiency of the amplifier. This deterioration also has two causes:

1. Impedances in series or in parallel with the load require the amplifier’s output current or voltage excursions to be larger for the same amount of load power. This results in an increase of the power losses in the amplifier. Only in narrow-band applications, can improvement be achieved by tuning out unwanted effects from reactive parts of impedances in series or in parallel with the load, with impedances that have opposite reactive parts.

2. If the impedances in series or in parallel with the load have a real part, extra power has to be delivered by the amplifier.

Negative feedback amplifiers

The first negative feedback amplifier was built in 1927 by H. Black.[15] He applied negative feedback to obtain linear and stable-gain repeater amplifiers for long-distance telephone systems. Black’s patent was awarded in 1937.[42]

Negative feedback is a powerful error reduction technique that trades gain for quality improvement. With negative feedback, the source-to-load transfer of an amplifier, as well as the port impedances, can primarily be fixed with passive devices, while biased amplifying devices are used to provide power gain. Such an approach is useful because the characteristics of real world passive devices closely match those of their corresponding network abstraction. In other words, real world resistors, capacitors and inductors show approximate resistive, capacitive and inductive behavior over a wide operating range, respectively. This cannot be said of the biased amplifying devices that provide the power gain. Their characteristics are generally nonlinear, suffer from speed limitation, show relatively large fabrication tolerances and are rather sensitive to temperature variations.

We will show that the transmission-1 parameters of amplifiers can accurately be fixed with feedback elements around a high-gain amplifier or controller. Each nonzero parameter can be fixed to an accurate value with one feedback loop. In this way, we are able to give the port impedances and the source-to-load transfer their desired values, without significant degradation of other performance aspects, such as the noise performance, the accuracy, the linearity, the power efficiency and the dynamic response of the amplifier. As a matter of fact, with negative feedback, we are able to design those performance aspects almost independently.

This is usualy called {\em orthogonal design}.

This chapter

In section Design of feedback configurations, we will present a design procedure for fixing the transmission-1 parameters through application of negative feedback. We will introduce two different feedback techniques: direct feedback and indirect or model-based feedback.

In section Implementation of negative feedback, we will discuss various implementation techniques. The best performance can be obtained with direct nonenergic feedback. Nonenergic feedback amplifiers use nonenergic feedback elements.

A circuit element is nonenergic if the instantaneous power delivered to it is always zero. Hence, nonenergic network elements do not store energy and do not dissipate power. Nonenergic network elements are:

• Ideal transformers* Ideal gyrators* Short circuit* Open circuit.

The practical use of nonenergic feedback is restricted due to the limited availability of those elements. Other techniques that will be discussed are: passive feedback, active feedback and balanced feedback.

The following sections will be devoted to a more elaborate treatment of all the techniques introduced. Section Nonenergic feedback discusses the design and the performance aspects of nonenergic negative feedback amplifiers. Conceptually, the best performance can be achieved with this type of amplifiers. Section Passive feedback is devoted to passive feedback. With passive feedback amplifiers, the feedback elements adversely affect the signal-to-noise ratio and the power efficiency. These effects, however, can be kept small, when compared to brute-force techniques. In amplifiers with passive feedback networks, the sign of the source-to-load transfer as well as the source-load isolation, is indissolubly connected to the type of transfer. Active feedback, balanced feedback and indirect feedback can be used to design the sign of the transfer, as well as the source-to-load isolation independent from the type of transfer. These techniques will be discussed in sections Active feedback, Design of balanced amplifiers and Indirect feedback, respectively.