Introduction

In the previous chapter, we discussed the design of the bandwidth of a negative feedback amplifier. We have studied effects that contribute to high-pass cut-off and effects that contribute to low-pass cut-off. We have seen that speed limitations cause low-pass cut-off. Low-pass cut-off should occur above the frequency range of interest, and we found design criteria to achieve this. We have also seen that DC blocking elements in the loop or, more generally, the use of AC coupling techniques may result in high-pass cut-off. High-pass cut-off should occur below the frequency range of interest, and we have also found ways to achieve this.

However, until now, the actual character of the roll-off behavior has not yet been studied. We have found ways to determine the stability of the amplifier, but we have not yet defined the means to achieve stable behavior, nor have we studied ways to design a specific roll-off character. This will be the subject of this chapter. In order to deal with this topic in a structured way, we will assume that the high-frequency cut-off and the low-frequency cut-off can be designed independently. In other words, the pole-zero pattern that determines the high-pass behavior can be designed independently from the pole-zero pattern that determines the low-pass behavior, and vice versa. Since in this book, we confine ourselves to the design of wide-band amplifiers, this will be the case.

In practice, this means that capacitors and inductors that play a role in the low-frequency cut-off can be considered as short circuits and open circuits at high-frequency cut-off, respectively. Similarly, capacitors and inductors that determine the high-frequency roll-off can be considered as open circuits and short circuits during the low-frequency roll-off, respectively.

The design of the desired cut-off characteristic is usually referred to as frequency compensation. The use of this term implies that after we have designed the bandwidth, the pole zero pattern of the transfer is usually not as desired and should somehow be corrected. This is usually the case, and frequency compensation is a collective term for application of such corrections:

Definition

Frequency compensation comprises a collection of techniques that can be used to correct pole-zero patterns in such a way as to obtain a desired frequency response or time response.

Frequency compensation techniques can be applied to obtain different types of responses. Examples of such responses are:

• Maximally flat magnitude (MFM) response

• No overshoot in step response

• Maximally flat group delay

In this chapter, we will focus on establishing an MFM response, but the techniques discussed can also be applied for obtaining other types of responses. Only the extent to which such techniques will be or can be applied, as well as the related design equations, will differ for other types of responses.

We will first study frequency compensation techniques that do not affect the designed bandwidth. This ensures that the design of the frequency response of a negative feedback amplifier can be performed in two subsequent steps:

1. The design of an adequate bandwidth

2. The design of a desired pole-zero patterns

Only in cases in which the amplifier’s bandwidth exceeds the required bandwidth may frequency compensation result in bandwidth reduction. However, in order to obtain the largest possible performance-to-cost ratio, we will show that a limitation of the bandwidth of the ideal gain and maximization of the bandwidth of the servo function is the best strategy in such cases.

Frequency compensation should also not result in an unacceptable degradation of other performance aspects, such as, among others

• Signal-to-noise ratio

• Distortion and overdrive recovery

• Accuracy

Filter design approach

Before we will discuss frequency compensation techniques, we will formulate the goal of frequency compensation in a mathematical way. This facilitates the development of strategies and procedures for frequency compensation.

In the previous chapters, we have seen that the source-load transfer \(A_{f}(s)\) of a negative feedback amplifier can be approximated by the product of its ideal transfer \(A_{i}(s)\) and the servo function \(\frac{-L(s)}{1-L(s)} \):

(154)\[A_{f}(s)=A_{i}(s)\frac{-L(s)}{1-L(s)}. \label{eq-gainFBamp}\]

This is the case if the loop gain reference variable has been selected such that the controller becomes a nullor for \(\left\vert L\right\vert \rightarrow\infty\) and the influence of the direct transfer \(\rho\) is negligible.

In that case, the asymptotic gain equals the ideal gain and the loop gain is a measure for the difference between the ideal gain and the gain.

From now on, we will assume this to be the case.

Low-pass cut-off

We will assume that the desired filter characteristic of the amplifier has been designed in the ideal transfer \(A_{i}(s)\).

See example 7.example-IntegratorHighPass: the design of a current integrator.

The bandwidth over which the source-load transfer \(A_{f}(s)\) approaches the ideal transfer is determined by the bandwidth of the servo function. During the design of the low-pass cut-off we will assume that there exists a midband frequency range where the loop gain equals \(L_{MB},\) with \(\left\vert L_{MB}\right\vert \gg1\), at higher frequencies, but below the low-pass cut-off frequency \(\omega_{h},\) the loop gain has \(n\) dominant poles and no zeros.

At a later stage, we will see that poles with a frequency above \(\omega_h\) can also be dominant. A proper definition of a dominant pole is “a pole that contributes to the bandwidth”.

For studying the low-pass cut-off the loop gain may then be written as an \(n-th\) order all-pole low-pass filter :

\[L(s)=\frac{L_{MB}}{%{\displaystyle\prod\limits_{i=1}^{n}} \left( 1-\frac{s}{p_{i}}\right) }.\]

Fig. 376 shows two asymptotes of the magnitude characteristic of the loop gain, as well as the asymptotes of the magnitude characteristic of the servo function for this situation. The two asymptotes of the loop gain are:

1. The asymptote at midband frequencies

2. The asymptote at the low-pass cut-off frequency \(\omega_{h}\)

Fig. 376 Asymptotes of the magnitude characteristics of the loop gain and the servo function for a loop gain with a midband value of \(L_{MB} \) and \(n\) dominant poles above midband frequencies.

With the aid of ((140)), \(A_{f}(s)\) can be written as:

\[A_{f}(s)=A_{i}(s)\frac{-L_{MB}}{1-L_{MB}}\frac{1}{1+a_{1}s+a_{2}s^{2}+\cdots a_{n}s^{n}},\]

where

\[a_{n}=\frac{1}{\left( 1-L_{MB}\right) \prod_{i=1}^{n}p_{i}}.\]

In case of an MFM characteristic , the low-pass cut-off frequency \(\omega_{h}\) can be obtained as:

\[\omega_{h}=\left\vert a_{n}\right\vert ^{-\frac{1}{n}}.\]

The goal of high-frequency compensation is to give the coefficients \(a_{1}\cdots a_{n-1}\) the values that correspond to those of an \(n-th\) order low-pass filter with the desired characteristic and with a cut-off frequency close to \(\omega_{h}\).

High-pass cut-off

Fig. 377 Asymptotes of the magnitude characteristics of the loop gain and the servo function for a loop gain with a midband value of \(L_{MB} \) and \(n\) dominant poles below midband frequencies.

Fig. 377 illustrates a high-pass cut-off. For this type of transfer, we will assume that a midband frequency range can be defined where the value of the loop gain equals \(L_{MB},\) with \(\left\vert L_{MB}\right\vert \gg1\) , at lower frequencies, but above the high-pass cut-off frequency \(\omega_{\ell},\) the loop gain has \(k\) dominant poles \(p_{j}\), where \(k\) equals the number of zeros below \(\omega_{\ell}\).

At a later stage, we will see that poles with a frequency below \(\omega _{\ell}\) can also be dominant. Again, a proper definition of a dominant pole is “a pole that contributes to the bandwidth”.

For studying high-pass cut-off the loop gain may then be approximated by a \(k-th\) order high-pass filter and may be written as

\[L(s)=L_{MB}\frac{%{\displaystyle\prod\limits_{j=1}^{k}} \left( -\frac{s}{p_{j}}\right) }{%{\displaystyle\prod\limits_{j=1}^{k}} \left( 1-\frac{s}{p_{j}}\right) }.\]

The source-load transfer \(A_{f}(s)\) can then be written in the form

\[A_{f}(s)=A_{i}(s)\frac{-L_{MB}}{1-L_{MB}}\frac{b_{k}s^{k}}{1+b_{1}s+b_{2} s^{2}+\cdots b_{k}s^{k}},\]

where

\[b_{k}=\frac{1-L_{MB}}{\prod_{j=1}^{k}p_{j}}.\]

In case of an MFM characteristic , the high-pass cut-off frequency \(\omega_{\ell}\) can be obtained as

\[\omega_{\ell}=\left\vert b_{k}\right\vert ^{-\frac{1}{k}}.\]

The goal of low-frequency compensation is to give the coefficients \(b_{1}\cdots b_{k-1}\) the values that correspond to those of an \(k-th\) order high-pass filter with the desired characteristic and with a cut-off frequency close to \(\omega_{\ell}\).

Design conclusion

We may draw the following conclusions:

• If, below certain frequencies, the loop gain drops below unity, the servo function will obtain a high-pass character.

• If, above certain frequencies, the loop gain drops below unity, the servo function will obtain a low-pass character.

• If both situations occur, the servo function will have a band-pass character. If the low-pass cut-off and high-pass cut-off are well separated, the source-load transfer can be written as a product of four terms:

  \[A_{f}(s)=A_{i}(s)\ \frac{-L_{MB}}{1-L_{MB}}\ \frac{b_{k}s^{k}}{1+b_{1} s+b_{2}s^{2}+\cdots b_{k}s^{k}}\ \frac{1}{1+a_{1}s+a_{2}s^{2}+\cdots a_{n}s^{n}},\]

  in which:

• \(A_{i}(s)\) is the ideal transfer, which has been designed using nullors as controllers.

• The term \(\frac{-L_{MB}}{1-L_{MB}}\) is a measure for the accuracy at midband frequencies.

• The term \(\frac{b_{k}s^{n}}{1+b_{1}s+b_{2}s^{2}+\cdots b_{k}s^{n}}\) describes the high-pass roll-off with respect to the ideal transfer. The MFM high-pass cut-off frequency \(\omega_{\ell}\) is found as \(\left\vert b_{k}\right\vert ^{-\frac{1}{k}}\), in which \(b_{k}=\frac{1-L_{MB}}{\prod _{j=1}^{k}p_{j}}\) with \(p_{j}\) being a pole that contributes to high-pass cut-off , and \(L_{MB}\) the midband frequency loop gain.

• The term \(\frac{1}{1+a_{1}s+a_{2}s^{2}+\cdots a_{n}s^{n}}\) describes the low-pass roll-off with respect to the ideal transfer. The MFM low-pass cut-off frequency \(\omega_{h}\) is found as \(\left\vert a_{n}\right\vert ^{-\frac{1}{n}}\), in which \(a_{n}=\frac{1}{\left( 1-L_{MB}\right) \prod_{i=1}^{n}p_{j} }\) with \(p_{i}\) being a pole that contributes to the low-pass cut-off and \(L_{MB}\) the midband frequency loop gain.

Compensation techniques

There are different ways to adjust the coefficients \(a_{i},\) \(\left( i<n\right) \)and \(b_{j}\), \(\left( j<k\right) \) to their desired values. Separate sections will be devoted to each of the techniques listed below.

1. The most powerful frequency compensation technique is the insertion of so-called phantom zeros.

2. Another technique is to change the positions of two (dominant) poles of the loop gain in such a way that their product remains the same, while their sum changes. In this way, the bandwidth of the servo function is preserved, but lower order coefficients of \(s\) in the denominator of the servo function can be modified. This technique is often referred to as pole-splitting.

3. Instead of splitting two poles by changing their interaction, poles can also be split with the aid of pole-zero canceling techniques.

4. An alternative to changing the sum of two poles, is to trade midband loop gain with the frequency of a dominant pole such that the product of the midband loop gain and the dominant poles is not affected. This technique is known as resistive broadbanding.

Compensation strategies

We have seen that the product of the midband loop gain and the dominant poles determine the bandwidth of the servo function. We have also seen that a large value of the loop gain is also beneficial to a high accuracy and a high linearity of the source-load transfer. Hence, there may exist situations in which severe requirements for the linearity or the accuracy result in a servo bandwidth that is far more than that required. In such cases, we may reduce the bandwidth while performing frequency compensation.

In general, there are three different strategies for frequency compensation:

1. Maintain the designed bandwidth of the source-load transfer. This is a useful strategy if the requirement for the midband loop gain that follows from the bandwidth design prevails over the one that follows from linearity or accuracy design considerations and the bandwidth has been designed to its desired value.

2. Exchange the bandwidth of the ideal transfer with the bandwidth of the servo function. This is a useful strategy if the requirement for the midband loop gain that follows from linearity and/or accuracy requirements dominates over the one obtained from bandwidth design considerations and if the bandwidth obtained in this way is more than required. This is the most powerful method of bandwidth reduction. However, we will see that a complicating side effect is that non-dominant poles of the loop gain before compensation may have become dominant after compensation.

3. Reduce the bandwidth of the servo function through reduction of the loop gain poles product. This may be a useful strategy if the bandwidth is larger than required while there exist too many non-dominant poles in the loop gain such that the previous method could not be implemented.

This chapter

Phantom zero compensation will be discussed in section Phantom zero compensation. We will introduce the concept of phantom zeros, calculate their values for second and third order systems and present implementation methods. We will also present a technique for bandwidth reduction with the aid of phantom zeros and discuss the influence of non-dominant poles.

Pole-splitting by means of capacitive negative feedback across a gain stage will be discussed in section Pole-splitting. This technique is often referred to as Miller compensation [41].

Pole-splitting by means of pole-zero canceling will be presented in section Pole-zero canceling.

Resistive broadbanding is a brute-force technique for exchanging a pole frequency with the midband loop gain, while maintaining the product of the midband loop gain and the dominant poles. It will be discussed in section Resistive broadbanding.

Phase margin and amplitude margin are properties of the loop gain that are often used as a measure for the stability of a negative feedback amplifier. Although the method is not advocated here, frequency compensation driven by phase margin improvement will be illustrated in section Phase margin design.

Reduction of the servo bandwidth can be useful for reducing the number of dominant poles. It can be achieved either by reducing the midband loop gain, lowering the frequency of one or more poles or adding one dominant pole. It will be discussed in section Reduction of the servo bandwidth.

In section Nested control, we will discuss the design of the controller using cascaded feedback amplifiers. In control theory this is known as nested loop control. This technique should be applied if the controller should have a well-defined dynamic behavior, such as in analog PID controllers.

In section Feedback biasing frequency compensation, we will discuss the frequency compensation of circuits that use negative feedback biasing. This is often referred to as low-frequency compensation.

Large variations in the drive and termination impedance of an amplifier may require special measures to ensure stability. Such variations may be part of the character of the signal source or the load, but they may also be a result of a shorted or a disconnected amplifier port. In section Compensation for open and shorted ports we will discuss techniques for dealing with these impedance variations.

Non-dominant poles are poles that do not contribute to the bandwidth of the servo function. The frequency of these poles exceeds the unity-gain frequency of the loop gain. Although they cannot be included in the design of the bandwidth of the servo function, they may cause severe deviations from the MFM characteristic designed considering dominant poles alone. Their influence and measures for its reduction will be discussed in section Influence of non dominant poles.