GENERALIZED FEEDBACK CIRCUIT ANALYSISmsn/nitiphat.pdf · GENERALIZED FEEDBACK CIRCUIT ANALYSIS Scott K. Burgess and John Choma, Jr. Department of Electrical Engineering–Electrophysics,

Chapter 6

GENERALIZED FEEDBACK CIRCUITANALYSIS

Scott K. Burgess and John Choma, Jr.Department of Electrical Engineering–Electrophysics, University of Southern California

6.1. Introduction

Feedback, whether intentionally incorporated or parasitically incurred, per-vades all electronic circuits and systems. A circuit is a feedback network ifit incorporates at least one subcircuit that allows a circuit branch current orbranch voltage to modify an input signal variable in such a way as to achieve anetwork response that can differ dramatically from the input/output (I/O) rela-tionship observed in the absence of the subcircuit. In general, the subcircuitthat produces feedback in the network, as well as the network without the feed-back subcircuit, can be nonlinear and/or time variant. Moreover, the subcircuitand the network in which it is embedded can process their input currents orvoltages either digitally or in an analog manner. In the discussion that follows,however, only linear, time-invariant analog networks and feedback subcircuitsare addressed.

There are two fundamental types of feedback circuits and systems. In a pos-itive, or regenerative feedback network, the amplitude and phase of the fedback signal, which is effectively the output response of the feedback subcir-cuit, combine to produce an overall system response that may not be boundedeven when the input excitation to the overall system is constrained. Althoughregenerative feedback may produce unbounded, and hence unstable, responsesfor bounded input currents or voltages, regeneration is not synonymous withinstability. For example, regenerative amplifiers have been designed to deliverreproducible I/O voltage gains that are much larger than the gains achievablein the absence of positive feedback [1,2]. In another application of regener-ation, high-frequency compensation has been incorporated to broadband thefrequency response of bipolar differential amplifiers that are otherwise bandlimited [3]. The most useful utility of positive feedback is electronic oscillators[4], while the most troubling ramification of positive feedback derives from theparasitic capacitances and inductances indigenous to high-performance analogintegrated circuits. These elements interact with on chip active elements toproduce severely underdamped or outright unstable circuit responses [5].

169C. Toumazou et al. (eds), Trade-Offs in Analog Circuit Design: The Designer’s Companion, 169–206.© 2002 Kluwer Academic Publishers. Printed in the Netherlands.

170 Chapter 6

The companion to positive feedback is negative or degenerative feedback,which is the most common form of intentionally invoked feedback architecturein linear signal processing applications. Among the most important of theseapplications are amplifiers [6] for which degeneration serves at least four pur-poses. First, negative feedback desensitizes the gain of an open loop amplifier(an amplifier implemented without feedback) with respect to uncertainties inthe model parameters of passive elements and active devices. This desensiti-zation property is crucial in view of open loop parametric uncertainties causedby modeling approximations, temperature variations, biasing perturbations,and non-zero fabrication and manufacturing tolerances. Second, and princi-pally because of the foregoing desensitization property, degenerative feedbackreduces the dependence of circuit response on the parameters of inherently non-linear active devices, thereby improving the linearity otherwise attainable inopen loops. Third, negative feedback displaying non-zero feedback at zero sig-nal frequencies broadbands the dominant pole of an open loop amplifier, whichconduces at least the possibility of a closed loop network with improved high-frequency response. Finally, by modifying the driving point input and outputimpedances of the open loop circuit, negative feedback provides a convenientvehicle for implementing voltage buffers, current buffers [7] and circuits thateffect impedance transformation [8]. Other applications of negative feedbackinclude active RC filters [9,10], phase-locked loops [11], and a host of compen-sation circuits that offset common mode biasing difficulties [12], circumventthe deleterious effects of dense poles in open loop amplifiers [13,14], and allowfor low power biasing of submicron CMOS devices used in low voltage circuitand system applications [15].

Despite Bode’s pathfinding disclosures [16], which framed a mathematicallyelegant and rigorous strategy for investigating generalized feedback architec-tures, most analog circuit designers still perceive feedback circuit analysis anddesign as daunting tasks. Their perceptions doubtlessly derive from traditionalliterature which often simplifies feedback issues through such approximationsas dominant pole open loops, global feedback (feedback applied only betweenthe output and input ports of a considered structure), frequency-invariantfeedback, and feedback subcircuits that presumably conduct signals only uni-laterally, in a direction opposite to the flow of signals through the open loop.While these and other commonly invoked assumptions are generally accept-able in relatively low-frequency signal processors, their validity is dubiousin broadband and/or low-voltage/low-power analog circuits. In such applica-tions, second-order effects – non-dominant open loop poles and zeros, energystorage associated with on-chip interconnects and packaging, complex modelsnecessitated by device scaling requirements, etc. – often surface as significantphenomenology whose tacit neglect precludes an insightful understanding offeedback dynamics.

Generalized Feedback Circuit Analysis 171

The objective of this chapter is to formulate an easily understandable math-ematical strategy for the meaningful analysis of electronic feedback circuitsrealized in any device technology. The procedure developed herewith is under-standable because it exploits only such conventional tools of linear circuitanalysis as the Kirchhoff laws, superposition principles, network branch sub-stitution theory and the elementary features of two port network theories [17].As is the case with most design-oriented analytical techniques, the intentof the procedures disclosed on the following pages is to illuminate networkresponse characteristics whose understood attributes and limitations breedthe engineering insights that necessarily underpin prudent engineering circuitdesign.

6.2. Fundamental Properties of Feedback Loops

The transfer function and driving point impedance characteristics of themajority of electronic feedback systems respectively subscribe to the samemathematical forms. It is, therefore, instructive to precede the circuit level dis-closure of feedback principles with a generalized system level study of feedbackdiagrams, parameters and performance metrics. In this section of material, theparameters governing the electrical signatures of open loop gain and feedbackfactor are reviewed, as are the interrelationships among the parameters of theclosed loop gain, open loop gain and feedback factor. Included among theseparameters are the frequencies, the damping factor and the undamped naturalfrequency of oscillation of the open and closed loops. These parameters areexploited to delineate the closed loop sensitivity to open loop gain, the relativestability of the feedback loop, and the phase margin as a function of open loopcritical frequencies. The small signal step response of a second order closedloop is then examined to forge the open loop design guidelines commensuratewith acceptable settling times.

6.2.1. Open Loop System Architecture and Parameters

If the feedback undergoing study is global in the sense that the feedbacksubcircuit routes a portion of the output port signal to the input port, the I/Odynamics of the subject system can be modeled as the block diagram abstractedin Figure 6.1. In this diagram, is the frequency domain transfer functionof the open loop amplifier, while f(s), the feedback factor, represents thefrequency domain transfer function of the feedback subcircuit. If signals flowonly in the direction indicated by the arrows in the diagram, the closed looptransfer function, is easily verified to be

172 Chapter 6

The foregoing expression shows that if f(s) = 0, which effectively opensthe loop formed by the signal processing blocks whose transfer functions are

and f(s), the resultant closed loop gain, is the open loop gain,In lowpass electronics, the open loop invariably contains gain stages,

buffers, broadband compensation networks and other topologies that renderlarge over stipulated frequency passbands. Although elementary treat-

ments of feedback systems commonly represent this open loop gain as a singlepole transfer function, a more realistic representation is

where is the frequency of the lower frequency or more dominant pole, isthe frequency of the less dominant pole, and is the frequency of the transferfunction zero. Open loop stability mandates that both and have positivereal parts. Open loop physical realizability requires that the single zero be areal number and if and are complex numbers, they must be complexconjugate pairs. For the zero lies in the right half complex frequencyplane; implies a left plane zero. Finally, is the zero, or low,frequency gain of the open loop.

An alternative expression for the open loop gain is

In this relationship, is the damping factor. In concert with (6.2), it is givenby

On the other hand,


symbolizes the undamped natural frequency of oscillation of the open loopnetwork.

A practical implication of the undamped frequency parameter is thatis a measure of the open loop 3-dB bandwidth. Indeed, if andif the frequency of the open loop zero is infinitely large, is precisely theopen loop 3-dB bandwidth. On the other hand, is a measure of open loopstability. This contention is supported by Figure 6.2, which depicts the openloop unit step response, normalized to its steady-state value, as a functionof the normalized time, for the special case of a right half plane zero,

lying at infinitely large frequency. Observe that for which impliesidentical real poles, a well-behaved step response – albeit one having a relativelylarge rise time – is produced. In contrast, damping factors smaller than one,which correspond to complex conjugate poles, deliver responses displayingprogressively more pronounced ringing. The extreme case of zero dampingresults in a sinusoidal oscillation. Because an open loop having a second-ordertransfer function is invariably a simplified approximation of a third or higherorder system, inferring potential instability from unacceptably small dampingfactors in a second-order model comprises prudent engineering interpretation.

6.2.2. Closed Loop System Parameters

An expression for the closed loop transfer function of the feedback systemdepicted in Figure 6.1 derives from substituting either (6.3) or (6.2) into (6.1).To this end, consider the simplifying case of a frequency-invariant feedbackfactor; that is Then

174 Chapter 6

where

is the loop gain, and the zero frequency closed loop gain is

A necessary condition for degenerative, or negative, feedback is that the feed-back factor, and the zero frequency open loop gain, have the samealgebraic sign. For this negative feedback constraint, the closed loop undampednatural frequency is meaningfully expressed as

Finally, the closed loop damping factor is

The foregoing five relationships highlight both advantages and disadvan-tages of feedback purposefully applied around an open loop circuit. Perhapsthe most obvious attribute of feedback is that it desensitizes the closed looptransfer function with respect to perturbations in open loop gain. For example,if the magnitude of the loop gain, is large over a specified frequencypassband, (6.6) and (6.7) show that the closed loop gain over this passbandreduces to that is

In most integrated circuit amplifiers, the open loop gain depends on poorlycontrolled or ill-defined processing and active and passive device parameters.Equation (6.11) suggests that a tightly controlled feedback ratio can renderpredictable and reproducible closed loop performance that is nominally unaf-fected by the parametric vagaries of the open loop. However, a closed loopgain magnitude of at least unity mandates a feedback factor whose magnitudeis at most unity. It follows that the maximum practical value of zero frequencyloop gain, T(0), is the magnitude of the open loop gain,

A second advantage of negative feedback is the potential broadbanding thatit affords. Recalling that the undamped natural frequency is a measure of,but certainly not identically equal to, the 3-dB bandwidth, (6.9) alludes tobandwidth improvement by a factor of nominally the square root of one plus thezero frequency loop gain. Since this loop gain is necessarily large for acceptably


small closed loop sensitivity to open loop gain, the bandwidth enhancementafforded by negative feedback is potentially significant.

Unfortunately, the factor by which the undamped frequency is increasedis roughly the same as the factor by which closed loop stability is degraded.The special case of a right half plane zero lying at infinitely large frequencyconfirms this contention. From (6.10), observe that the resultant closed loopdamping factor is smaller than its open loop counterpart by a factor of thesquare root of one plus the zero frequency loop gain. As an example, con-sider a feedback structure at low signal frequencies which has an open loopgain of 24 or 26.6 dB, and an open loop damping factor of 2. The latter stip-ulation assuredly suggests a dominant pole open loop, since (6.4) yields anopen loop pole ratio, of 13.9 for But (6.10) gives a closed loopdamping factor of only 0.4, which implies complex conjugate closed loop poleswith corresponding significant ringing and overshoot in the closed loop stepresponse.

The preceding numerical example casts a shadow on commonly invokedpole splitting stability compensation measures [18,19]. Pole splitting aims toachieve a large open loop pole ratio, so that the resultant closed loopdamping factor, is suitably large. Assume that the desired closed loop dampingfactor satisfies where delivers, in the absence ofa finite frequency zero, a maximally flat magnitude second-order frequencyresponse. Then with T(0) = 24, (6.10) confirms that the requisite open loopdamping factor must be at least 3.54 whence by (6.4), the non-dominant-to-dominant pole frequency ratio must be at least 48. Since the bandwidth ofa dominant pole amplifier is essentially prescribed by the frequency of thedominant pole, a pole separation ratio of 48 may be plausible for amplifiersthat need deliver only relatively restricted open loop 3-dB bandwidths. But foramplifier applications that mandate large 3-dB bandwidths, pole splitting aloneis likely an inadequate or impractical stability compensation measure.

Note that worse case damping factor degradation derives from unity gainclosed loop designs for which the loop gain lies at its practical maximum value.This observation explains why general purpose circuits are routinely compen-sated to ensure stability under unity gain closed loop operating circumstances.Although compensation to ensure unity gain stability is both prudent and desir-able in general applications, it usually proves to be overly constraining in manyRF amplification and other special purpose integrated circuits.

The damping factor degradation for is exacerbated by a finite fre-quency right half plane zero since the term involving on the right handside of (6.10) subtracts from the term proportional to the open loop dampingfactor. On the other hand, a left half plane zero, say is seen toimprove the stability situation in the sense of increasing the closed loop damp-ing attributed to the first term on the left hand side of (6.10). Prudent feedback

176 Chapter 6

compensation scenarios, particularly in high-frequency signal processing appli-cations, therefore, combine procedures aimed toward realizing appropriate lefthalf plane zeros in the loop gain with traditional pole splitting methodologies[14,20].

6.2.3. Phase Margin

Although the preceding damping factor arguments convey a qualitative pic-ture of the stability of a closed feedback loop, they fail to offer a design-orientedguideline that quantifies the degree to which a feedback circuit realization isstable. If the loop gain magnitude response is a well-behaved, monotonicallydecreasing function of signal frequency, either the phase margin or the gainmargin proves to be an expedient stability metric. Of these two metrics, thephase margin is more easily evaluated mathematically.

For steady-state sinusoidal operating conditions, the closed loop gain in (6.6)becomes

where, from (6.7) and (6.2), the loop gain is

For frequency-invariant feedback, this loop gain displays a frequency responsethat mirrors that of the open loop transfer function. Moreover, the loop gainequals the open loop transfer function for the special case of a closed loopdesigned for unity gain.

Let denote the radial frequency at which the magnitude of the loop gainis unity. Then

where is the phase angle of the loop gain at the frequency wherethe loop gain magnitude is one. The phase angle, is the phase margin ofthe closed loop. Its significance can be appreciated by noting that if

whence (6.12) predicts sinusoidal closed loop oscillations. Itfollows that closed loop stability requires that the phase angle of the loop gainat the loop gain unity gain frequency be sufficiently less negative than –180°;that is, a sufficiently large and positive phase margin, is required.

The phase margin can be quantified if a few simplifying approximationsappropriate to pragmatic design objectives are invoked. In particular, assumethat the open loop amplifier, and hence, the loop gain, possesses a dominantpole frequency response. This approximation implies and gives riseto a greater than unity open loop damping factor, which has been noted as


conducive to an acceptably large closed loop damping factor. Assume furtherthat the frequency, of the right half plane zero, like the frequency of thenon-dominant pole, is also very large. This requirement also reflects designpracticality since small results in an uncompromisingly small closed loopdamping factor. As a result, the 3-dB bandwidth of the loop gain approximates

and the gain bandwidth product of the loop gain is simply If bothand are larger than the loop gain unity gain frequency, it follows that

It is convenient to normalize the frequencies, to the approximateloop gain unity gain frequency defined by (6.15). In particular, let

and

Equations (6.15)–(6.17) allow the closed loop damping factor in (6.10) and theclosed loop undamped natural frequency in (6.9) to be expressed respectively as

and

The approximations in these last two relationships are premised on the assump-tion of very large zero frequency loop gain, a condition observed earlier as onethat encourages closed loop response desensitization to open loop parameters.

Returning to the phase margin problem, (6.13)–(6.17) deliver

Upon introducing the constant, k, such that

the application of the appropriate trigonometric identities to (6.20) provides

178 Chapter 6

For large zero frequency loop gain, (6.22) collapses to the simple result,

It should be remembered that because of the presumption that the unitygain frequency of the loop gain in (6.13) closely approximates the product,

(6.22) and (6.23) provide realistic estimates of phase margin onlywhen the frequencies of the non-dominant pole and zero are each larger thanthe estimated unity gain frequency. This is to say that (6.22) and (6.23) arevalid insofar as

For the deleterious circumstance of a right half plane zero and thus,positive), Figure 6.3 graphically displays the dependence of phase margin onparameter k for various values of the zero frequency loop gain, T(0).

Example 6.1. A second-order negative feedback amplifier is designed tohave a loop gain at zero frequency of 25 (28 dB). The loop gain displays aright half plane zero at a frequency that is four times larger than the loop gainunity gain frequency. What phase margin is required if, ignoring the right halfplane zero, the closed loop amplifier is to establish a maximally flat magnitudefrequency response?

Solution 6.1.

A maximally flat lowpass amplifier implies (ignoring the effects of anyzeros) a closed loop damping factor, of Since the right halfplane zero is four times the unity gain frequency of the loop gain,The approximate form of (6.18) therefore, suggests This is to

1


2

3

say that the non-dominant pole of the open loop amplifier must be morethan 3.5 times larger than the unity gain frequency of the loop gain!

With and in (6.21) is 1.75.

Given k = 1.75 and T(0) = 25, (6.22) implies a phase margin of

Comment. To protect against oscillations incurred by parasitic energy storageand related interconnect phenomena, practical analog integrated circuits designedto be stable under unity closed loop gain conditions must generally have phasemargins in the range of 60–70 degrees. This constraint typically translates intothe requirement that the non-dominant amplifier pole be at least 3–4 times largerthan the amplifier unity gain frequency. Since such an operating prerequisitecomprises a formidable design task for amplifiers that must operate at RF signalfrequencies, the stability condition is often relaxed to ensure adequate phasemargin for only closed loop gains in the neighborhood of the specified closedloop gain performance.

6.2.4. Settling Time

The preceding section of material demonstrates that the phase margin, whicheffectively is the degree to which a closed feedback loop is stable, is stronglyinfluenced by the frequencies of both the non-dominant pole and the zeroimplicit to the loop gain. These critical frequencies have an equally strongeffect on the closed loop damping factor, which, in turn, determines the timedomain nature of the closed loop transient response. For the often encounteredcase of a closed loop damping factor that is smaller than one, it follows that thephase margin influences the time required by the step response to converge towithin a suitably small percentage of the desired steady-state output value. Thistime domain performance metric is commonly referred to as the settling time.

An investigation of the settling time of a closed feedback loop commenceswith designating the input to the system abstracted in Figure 6.1 as a unit step,for which the Laplace transform is X(s) = 1 /s. The resultant transformedoutput is, from (6.6)

where now represents the steady-state value of the unit step response.This step response, say y(t), is obviously the inverse Laplace transform of theright hand side of (6.25). If y(t) is normalized to its steady-state value,

180 Chapter 6

signifies an error between the normalized steady-state response and theactual normalized step response. An error, of zero corresponds to aninstantaneously settling output; that is, zero settling time.

Introducing the constants, M and such that

and

and letting

denote a normalized time variable, it can be shown that the error functiondefined by (6.26) is given by

This result presumes an underdamped closed loop and a zero lying in the righthalf plane.

Figure 6.4 pictures the time domain nature of the error function in (6.30).The presence of a right half plane zero causes the error to be positive in theneighborhood of the origin. Equivalently, the step response displays undershootshortly after time t = 0. Thereafter, the step response error is a damped sinusoid


for which maxima are manifested with a period of As expected, the rateat which the error converges toward its idealized value of zero increases forprogressively larger damping factors, thereby suggesting that small dampingfactors imply long settling times.

The strategy for determining the closed loop settling time entails determin-ing the time domain slope, of the error function. From the precedingdiscussion, this slope is periodically zero. The smallest value of normalizedtime x corresponding to zero slope of error defines the maximum error asso-ciated with initial undershoot. The second value of x, say correspondingto zero slope of error defines the maximum magnitude of error, say If thismaximum error magnitude at most equals the specified design objective forallowable error in the steady-state response, defines the normalized settlingtime for the closed loop.

Upon adoption of the foregoing analytical strategy, the settling time, isimplicitly found as

which conforms to an error maximum of

Very small closed loop damping factors are obviously undesirable. Thus, forreasonable values of damping and/or large M (large righthalf plane zero frequency), (6.31) and (6.32) respectively reduce to

and

Given large M, (6.18), (6.21), and (6.23) allow expressing the preceding tworelationships in the more useful forms,

and

182 Chapter 6

Example 6.2. A second order feedback amplifier is to be designed so thatits response to a step input settles to within 2% of steady-state value within750 pSEC. The low frequency loop gain is very large and to first order, thefrequencies of any right half plane circuit zeros can also be taken as large.Determine the requisite unity gain frequency of the loop gain, the frequency ofthe non-dominant loop gain pole and the phase margin.

Solution 6.2.

1 From (6.36),

implies Thus, the non-dominant pole of the loop gain functionmust be more than 2.4 times larger than the unity gain frequency of saidloop gain.

2

3

4

With and in view of the 750 pSEC settling time specification,(6.35) delivers Recalling (6.19), this resultmeans that the requisite unity gain frequency must be at least as large as

(682.8MHz).

Since symbolizes the ratio of the frequency of the non-dominant poleto the unity gain frequency, the preceding two computational steps yield

(1.66 GHz).

When the frequency of the right half plane zero is very large, k in (6.21)and (6.23) is very nearly The latter of these two relationships deliversa required phase margin of

Comment. Since the impact of the right half plane zero is tacitly ignoredin this calculation, a prudent design procedure calls for increasing the computedphase margin by a few degrees. Although the resultant phase margin and otherdesign requirements indigenous to this example are hardly trivial, they are achiev-able with appropriate device technologies and creative circuit design measures.The latter are likely to entail open loop pole splitting and/or the incorporation ofa compensating zero within the feedback factor.

6.3. Circuit Partitioning

From a purely computational perspective, the preceding section of materialis useful for determining the steady-state performance, transient time-domainperformance, sensitivity, and stability of practical feedback networks. Butfrom the viewpoint of circuit design, the practicality of the subject materialmight logically be viewed as dubious, for it promulgates results that dependon unambiguous definitions of the open loop gain and feedback factor. Stated


more directly, the results of Section (6.2) are useful only insofar as the trans-fer function of interest for a given circuit can be framed in the block diagramarchitecture of Figure 6.1. Unfortunately, casting a circuit transfer functioninto the form of Figure 6.1 is a non-trivial task for at least three reasons. First,neither the open loop amplifier nor the feedback function conducts signals uni-laterally. This is to say that amplifiers, and especially amplifiers operated athigh signal frequencies, invariably have intrinsic feedback. Moreover, sincethe feedback subcircuit is generally a passive network, it is clearly capableof conducting signals from circuit input to circuit output ports, as well asfrom output to input ports. Second, the open loop amplifier function is notcompletely independent of the parameters of the feedback subcircuit, whichinvariably imposes impedance loads on the amplifier input and output ports.Third, Figure 6.1 pertains only to global feedback structures. But practical feed-back circuits may exploit local feedback; that is, feedback imposed betweenany two amplifier ports that are not necessarily the output and input ports of theconsidered system. Local feedback is often invoked purposefully in broadbandanalog signal processing applications. On the other hand, parasitic local feed-back is commonly encountered in high-frequency systems because of energystorage parasitics associated with proximate on-chip signal lines, bond wireinterconnects and packaging.

Fortunately, theoretical techniques advanced originally by Kron [21,22] existto address this engineering dilemma. As is illustrated below, these techniques,which are now embodied into modern circuit partitioning theory [23], havebeen shown to be especially utilitarian in feedback circuit applications [24].

6.3.1. Generalized Circuit Transfer Function

Consider the arbitrary linear circuit abstracted in Figure 6.5(a). A voltagesignal having Thévenin voltage and Thévenin impedance is presumedto excite the input port of the circuit, while a load impedance, terminatesthe output port. If the subject linear circuit can be characterized by a lumpedequivalent model, the voltage gain, the input impedance,seen by the applied signal source, and the output impedance, facingthe load termination can be evaluated straightforwardly. Although a voltageamplifier is tacitly presumed in the senses of representing both the input andoutput signals as voltages, the same statement regarding gain and the drivingpoint input and output impedances applies to transimpedance, transadmittanceand current amplifiers.

Let the network under consideration be modified by applying feedback fromits k-th to c-th ports, where port k can be, but is not necessarily, the output portof the circuit, and port c can be, but is not necessarily, the input port. An ele-mentary representation of this feedback is a voltage controlled current source,

184 Chapter 6

as diagrammed in Figure 6.5(b). The implication of this controlled sourceis that a feedback subcircuit is connected from the k-th to c-th ports of the orig-inal linear network, as suggested in Figure 6.6(a). In the interest of analyticalsimplicity, this subcircuit is presumed to behave as the ideal voltage controlledcurrent source delineated in Figure 6.6(b). In particular, the input voltage tothe feedback subcircuit is the controlling voltage, of the dependent currentsource, which emulates a simplified Norton equivalent circuit ofthe output port of the feedback subcircuit.

Superposition theory applied with respect to the independent signal source,and the dependent, or controlled, current, in Figure 6.5(b) yields

and

In these relationships, and are frequency dependent constantsof proportionality that link the variables, and to the observable circuitvoltages, and In (6.37), it should be noted that is the voltage gain,

under the condition of This observation corroborates with thecircuit in Figure 6.5(a), for which the voltage gain in the absence of feedback,which implies P, and hence is zero, has been stipulated as

Recalling that (6.38) implies

whence


Assuming is non-zero and bounded, the insertion of the last result into(6.37) establishes the desired voltage transfer function relationship,

Equation (6.41) properly defines the closed loop gain in that through non-zeroP, an analytical accounting of the effects of feedback between any two networkports has been made.

In the denominator on the right hand side of (6.41), respectively define

and

as the normalized return ratio with respect to feedback parameter P and thereturn ratio with respect to P. Although not explicitly delineated, bothand are functions of frequency because in general, the parameters, Pand as well as the source and load impedances, and are frequency

186 Chapter 6

dependent. Analogously, introduce

as the normalized null return ratio with respect to P and

as the null return ratio with respect to P. Like and andare functions of frequency. Equation (6.41) is now expressible as

Either form of the preceding relationship is a general expression for the voltagegain of feedback structures whose electrical characteristics subscribe to thoseimplied by Figure 6.6. Equation (6.46) is actually a general gain expression forall feedback architectures, regardless of either the electrical nature of parameterP or the electrical model that emulates the terminal characteristics of the feed-back subcircuit. Because of this generality contention, it may be illuminatingto observe that (6.46) gives rise to the block diagram representation offeredin Figure 6.7. This architecture portrays the null return ratio,as a feedforward transfer function from the source signal node to the node atwhich the output signal produced by the feedback subcircuit is summed. Thetransfer function of the feedback subcircuit is clearly dependent on the returnratio, so that no feedback prevails when the normalized returnratio is zero. Both the null return ratio and the return ratio are directly propor-tional to the parameter, P, which causes feedback to be incurred between two


network ports. It might, therefore, be stated that the return ratio is a measureof the feedback caused by the feedback subcircuit, while the null return ratiomeasures feedforward phenomena through the feedback subcircuit.

It is also interesting to speculate that the general feedback system ofFigure 6.6 can be viewed, at least insofar as the I/O transfer function isconcerned, as an equivalent global feedback network. To this end, a com-parison of (6.46) with (6.6) and the abstraction in Figure 6.1 suggests definingan equivalent open loop gain as

while the equivalent loop gain, T(s), follows as

Since the feedback factor, f (s), in (6.6) is the loop gain divided by the openloop gain, (6.47) and (6.48) imply

As conjectured earlier, the open loop gain and feedback factor are difficult toseparate in practical feedback structures. In particular, (6.47) shows that theopen loop gain is dependent on the feedback parameter, P, and (6.48) depictsa feedback factor that is not independent of the open loop gain function.

Equation (6.46) underscores the fact that the voltage gain of the architecturedepicted in Figure 6.6 relies on only three metrics. These metrics are the gain,

for parameter P = 0, the return ratio, and the null return ratioSince a straightforward nodal or loop analysis of a feedback network such asthat shown in Figure 6.6(a) is likely to be so mathematically involved as toobscure an insightful understanding of network volt–ampere dynamics, it maybe productive to investigate the propriety of alternatively evaluating the forego-ing three metrics. In other words, it may be wise to partition the single problemof gain evaluation into three, presumably simpler, analytical endeavors.

Recalling (6.42)–(6.45), the voltage gain, in (6.46) is the gain for thespecial case of P = 0. Since P = 0 corresponds to zero feedback from k-th toc-th ports, and since the configuration in Figure 6.5(b) is the model of the feed-back network in Figure 6.6(a), can be evaluated by analyzing the reducednetwork depicted symbolically in Figure 6.8(a). Observe that the calculationof is likely to be simpler than that of because the subcircuit causingvoltage controlled current feedback from k-th to c-th ports in Figure 6.6(a) iseffectively removed.

From (6.38),

188 Chapter 6

Because of (6.42),

This result suggests a return ratio evaluation that entails (1) setting the indepen-dent signal source to zero, (2) replacing the dependent current generator at theoutput port of the feedback subcircuit by an independent current source, and(3) calculating the negative ratio of to In short, is parameter P mul-tiplied by the negative ratio of controlling variable to controlledvariable under the condition of nulled input signal. The computationalscenario at hand is diagrammed in Figure 6.8(b), where the original polarity ofvoltage is reversed, and hence denoted as while the original directionof current is preserved.

Return to (6.37) and (6.38) but now, constrain the output voltage, tozero. With the generator replaced by an independent current source,


the signal source voltage, necessarily assumes the value,

If this source voltage is inserted into (6.38), it follows that

whence by (6.44)

As suggested in Figure 6.8(c), the null return ratio computation entails(1) nulling the output response, (2) replacing the dependent current genera-tor at the output port of the feedback subcircuit by an independent currentsource, and (3) calculating the negative ratio of to In short,is parameter P multiplied by the ratio of phase inverted controlling variable

to controlled variable under the condition of a nulled outputresponse.

6.3.2. Generalized Driving Point I/O ImpedancesThe driving point input impedance, seen by the signal source applied to

the feedback network in Figure 6.5(b) derives from replacing the source circuitby an independent current source, say and computing the ratio, where

is the disassociated reference polarity voltage developed across the source.This computational scenario is illustrated in Figure 6.9(a). Since is atransfer function, (6.46) prescribes the form of this transfer relationship as

In (6.55) is the P = 0 value of the input impedance. This null parame-ter input impedance is the ratio evidenced when parameter P is set tozero, as diagrammed in Figure 6.9(b). The functions, and respec-tively represent the normalized return ratio and the normalized null return ratioassociated with the network input impedance.

Figure 6.9(c) is appropriate to the computation of wherein (1) theindependent signal source, applied to the circuit in Figure 6.9(a) is set to zero,(2) the dependent current generator at the output port of the feedback subcircuitis supplanted by an independent current source, and (3) the negative ratio

190 Chapter 6

of to is evaluated. Observe, however, that Figure 6.9(c)is similar to Figure 6.8(b), which is exploited to determine the normalizedreturn ratio pertinent to the voltage transfer function of the considered network.Indeed, if were infinitely large in Figure 6.8(b), both circuits would beidentical since nulling in Figure 6.9(c) is tantamount to open circuiting thesource circuit. Accordingly,

The normalized null return ratio, is evaluated from a circuit analysisconducted on the system shown in Figure 6.9(d). In this diagram, (1) the outputresponse in the circuit of Figure 6.9(a), which is is set to zero, (2) thedependent current generator at the output port of the feedback subcircuit isreplaced by the current source, and (3) the negative ratio ofto is evaluated. But Figure 6.9(d) is also similar to Figure 6.8(b). Bothstructures are topologically identical if in Figure 6.8(b) is zero since nulling

in Figure 6.9(d) amounts to grounding the source input port. Thus,


and it follows that the input impedance in (6.55) is expressible as

The last equation suggests that since is already known from workleading to the determination of the circuit transfer function, the null parameterimpedance, is the only function that need be determined to evaluate thedriving point input impedance. It is noteworthy that the evaluation of islikely to be straightforward since, like the evaluation of the null parametergain, it derives from an analysis of a circuit for which the dependentsource emulating feedback from k-th to c-th ports is nulled.

Figure 6.10 is the applicable circuit for determining the driving point outputimpedance, facing the load impedance, Observe that the Théveninsource voltage, is nulled and that the load is replaced by an independentcurrent generator, The analytical disclosures leading to the input impedancerelationship of (6.58) can be adapted to Figure 6.10 to show that

where is the output impedance under the condition of a nulled feedbackparameter; that is, when P = 0.

6.3.3. Special Controlling/Controlled Port Cases

Equations (6.46), (6.55), and (6.59) are respectively general gain, inputimpedance, and output impedance expressions for any linear network in whichfeedback is evidenced between any two network ports. In addition to theirvalidity, these relationships are quite useful in modern electronics and canbe confidently applied as long as the null metrics, and arenon-zero and finite. Despite their engineering utility, several special cases

192 Chapter 6

commonly arise to justify particularizing the subject relationships for theapplications at hand

Controlling feedback variable is the circuit output variable. In

.

Figure 6.5(b) consider the case in which the variable, which controls theamount of current fed back to the controlled or c-th port of the network, is theoutput response, voltage in this case. The present situation is delineated inFigure 6.11 (a). The pertinent input and output impedance expressions remaingiven by (6.55) and (6.59), respectively, where and are impedancesevaluated under the condition of P = 0. Similarly, in (6.46) is the P = 0value of the input-to-output voltage gain.


Figure 6.11 (b) shows the circuit pertinent to evaluating the normalized returnratio, In accordance with the procedures set forth above, the Théveninsignal voltage, is set to zero, the dependent current generator isreplaced by an independent current source, and the polarity of the controllingvariable, of the dependent source is reversed and noted as From (6.51),the normalized return ratio is

The normalized null return ratio, derives from an analysis of theconfiguration depicted in Figure 6.11(c). In this circumstance, the Théveninsignal voltage is not nulled, but the output voltage, is. Moreover, the

dependent current source is replaced by an independent current,and the polarity of the controlling variable, of the dependent source is

reversed and noted as But since (6.54) delivers

Since the normalized null return ratio is zero, the resultant gain equation in(6.46) simplifies to

Global feedback. In the global feedback system abstracted inFigure 6.12(a), the controlling feedback variable, is the output variable,and in addition, the controlled, or c-th, port is the network input port. Since thecontrolling and output variables are the same, the normalized null return ratiois zero, as in the preceding special case. Although the null parameter voltagegain, and the normalized return ratio, can be computed in the usualfashion, for global feedback circumstances it is expedient to model the signalsource as the same type of energy source used to emulate the fed back signal. Inthis case, the fed back signal happens to be a current source. Thus, as shown inFigure 6.12(b), the signal source is converted to an independent current source,

where is obviously Note that the fed back current and the signalcurrent flow in opposite directions and hence, the fed back current subtractsfrom the source current at the input node of the linear circuit. This situationreflects the negative feedback inferred by the block diagram in Figure 6.1. Thesource conversion renders the null gain a transimpedance, say where in

194 Chapter 6

concert with the definition of a null parameter gain and Figure 6.12(c),

As is suggested by Figure 6.12(d), the normalized return ratio derives from(1) nulling, or open circuiting, the applied independent signal current source,(2) replacing the dependent source by an independent generator, and(3) computing the ratio of the resultant phase inverted output voltage,to But since is applied across the same input port to which is applied,

reflects a polarity opposite to that of and is a phase inverted version ofis identical to the previously determined transimpedance, This

is to say that

Resultantly, the closed loop transimpedance, is

whose mathematical form is precisely the same as the gain expression for thesystem abstraction of global feedback in Figure 6.1. Since the


corresponding closed loop voltage gain is

Controlling feedback variable is the branch variable of thecontrolled port. Consider Figure 6.13(a) in which there is no obviousfeedback from k-th to c-th ports but instead, a branch admittance, say isincident with the c-th port. As illustrated in Figure 6.13(b), this branch topologyis equivalent to a voltage controlled current source, where the control-ling voltage, is the voltage established across the c-th port. By comparisonwith Figure 6.5(b), the latter figure shows that the feedback parameter, P, iseffectively the branch admittance, while the controlling voltage, is thevoltage, developed across the controlled port.

The gain, input impedance, and output impedance of the network inFigure 6.13(a) subscribe to (6.46), (6.58), and (6.59), respectively. The zeroparameter gain, input impedance, and output impedance, areevaluated by open circuiting the branch admittance, as per Figure 6.14(a).Note that open circuiting in Figure 6.13(a) is equivalent to nulling thecontrolled generator, in Figure 6.13(b).

The evaluation of the normalized return ratio, mirrors relevantprevious computational procedures. In particular, and as is delineated inFigure 6.14(b), (1) the independent signal source is set to zero, (2) the depen-dent current generator, across the c-th network port is replaced by anindependent current source, and (3) the ratio of the negative of indi-cated as to is calculated. However, this ratio is identically the Thévenin

196 Chapter 6

impedance, “seen” by admittance Accordingly,

Similarly, and as highlighted in Figure 6.14(c), the normalized null returnratio, is the null Thévenin impedance, seen by that is,the normalized null return ratio is the Thévenin impedance facing under thecondition of an output response constrained to zero. It follows from (6.46),(6.58) and (6.59) that the closed loop voltage gain, the driving pointinput impedance, and the driving point output impedance, are givenrespectively by


and

Two special circumstances can be extrapolated from the case just considered.The first entails a short circuit across the controlled c-th port, as is depicted inFigure 6.15. Recalling Figure 6.13(a), this situation corresponds towhence (6.68)–(6.70) become

and

These three expressions imply that for the case of a short circuit critical parame-ter, the gain, input impedance and output impedance are simply scaled versionsof their respective null (meaning open circuited c-th port branch) values. Thescale factors are related to the ratio of the null Thévenin impedance to theThévenin impedance facing the short circuited branch of interest.

The second of the aforementioned two special circumstances involves acapacitive branch admittance connected to the c-th port of a memoryless net-work driven by a source whose internal impedance is resistive and terminated ina load resistance, as abstracted in Figure 6.16. With reference to Figure 6.13(a),the condition at hand yields which can be substituted directly into(6.68)–(6.70). But in addition, the memoryless nature of the network to whichcapacitance C is,connected gives Thévenin and null Thévenin impedances that

198 Chapter 6

are actually Thévenin resistances, and “seen” by the subject branchcapacitance. It follows that

and

The pole incurred in the voltage transfer function by the branch capacitancelies at while the zero lies atObserve that the pole and zero frequencies associated with the input and outputimpedances are not necessarily respectively identical, nor are they respectivelyidentical to those of the voltage transfer relationship.

Example 6.3. The operational amplifier (op-amp) circuit shown inFigure 6.17(a) exploits the resistance, to implement shunt–shunt globalfeedback. The signal source is a voltage, whose Thévenin resistance isand the load termination is a resistance of value The simplified dominantpole equivalent circuit of the op-amp is given in Figure 6.17(b), where sym-bolizes the positive and frequency-invariant open loop gain, is the effectiveinput resistance, is the effective input capacitance, and is the Théveninequivalent output resistance of the op-amp. Determine expressions for the openloop voltage gain, of the circuit, the 3-dB bandwidth, of the openloop circuit, the loop gain, the closed loop voltage gain of theentire amplifier, and the 3-dB bandwidth, of the closed loop. Also,derive approximate expressions for the low frequency closed loop driving point


input and output impedances, and respectively. The approxi-mations invoked should reflect the commonly encountered op-amp situation oflarge open loop gain, large input resistance, and small output resistance.

Solution 6.3.

Comment. There are several ways to approach this problem. For example, theproblem solution can be initiated by taking the conductance, associated withthe resistance, as the feedback parameter. The gain for which opencircuits the feedback path, can be evaluated, as can the return ratio and null returnratio with respect to To this end, note that the normalized return ratio is theimpedance “seen” by with zero source excitation, while the normalized nullreturn ratio is the impedance seen by with the output voltage response nulled.Equations (6.46)–(6.49) can then be applied to address the issues of this problem.Alternatively, the feedback parameter can be taken as the short circuit interconnectbetween the feedback resistor, and either the input or the output port of theop-amp, whereupon (6.71)–(6.73) can be invoked. The strategy adopted herewithentails the replacement of resistance by its two port equivalent circuit. As isto be demonstrated, this strategy unambiguously stipulates the analytical natureof the open loop gain, the loop gain, and even the feedforward factor associatedwith

Figure 6.18(a) repeats the circuit displayed in Figure 6.17(a) but addition-ally, it delineates the voltage (with respect to ground) and current, andat the input port of the feedback resistance, as well as the voltage and current,

and at the feedback input port. Clearly,

1

which suggests that the feedback resistance, can be modeled as the two portnetwork offered in Figure 6.18(b). When coalesced with the op-amp model ofFigure 6.17(b), this two port representation allows the circuit of Figure 6.18(a)to be modeled as the equivalent circuit shown in Figure 6.18(c). In the latter

200 Chapter 6

structure, the output port of the op-amp has been modeled by a Norton equiv-alent circuit to facilitate analytical computations. The signal source circuit hasalso been replaced by its Norton equivalent circuit, where

The model in Figure 6.18(c) illuminates the presence of global feedbackin the form of the current generator, across the network input port. Itfollows that the feedback parameter (symbolized as P in earlier discussions) is

The model at hand also shows that feedforward through the feedbackresistance is incurred by way of the current generator, at the output port.If the feedback term is set to zero, the resultant model is the open loop equivalentcircuit submitted in Figure 6.19(a). It is important to understand that althoughthis structure is an open loop model, its parameters nonetheless include the

2


resistance, which accounts for feedback subcircuit loading of the amplifierinput port, resistance, which incorporates output port loading caused bythe feedback network, and the generator, which emulates feedforwardphenomena associated with the feedback subcircuit. A straightforward analysisof the circuit in Figure 6.19(a) delivers

and

It follows that the open loop transimpedance, say is expressible as

202 Chapter 6

where the magnitude of the zero frequency value of the open loop transim-pedance is

3 Because the open loop voltage gain is seen to be

where the magnitude of the zero frequency open loop voltage gain is

and

is the open loop 3-dB bandwidth.4 Since global feedback prevails, the normalized null return ratio is zero.

The circuit appropriate to the determination of the normalized return ratio isoffered in Figure 6.19(b), wherein with reference to Figure 6.18(c), thefeedback generator is supplanted by an independent current source, the Nor-ton source current, is nulled, and the controlling voltage, for the feedbackgenerator, is replaced by its phase inverted value, Since the original sig-nal voltage source has been replaced by its Norton equivalent circuit, both theNorton signal source and the feedback generator are current sources incidentwith the amplifier input port. This renders simple the computation of the nor-malized return ratio; in particular, an inspection of the circuit in Figure 6.19(b)confirms

whence the loop gain of the amplifier is

where the zero frequency loop gain is

5 The preceding analytical stipulations render a closed loop transimpedanceof


which is expressible as

where the magnitude of the zero frequency amplifier transimpedance is

and

is the closed loop 3-dB bandwidth. It follows that the closed loop voltage gain is

Comment. For most operational amplifier networks like that depicted inFigure 6.17(a), the zero frequency loop gain, is muchlarger than one. This means that the zero frequency closed loop gain collapses tothe well-known relationship,

Moreover, the open loop op-amp gain, is invariably much larger than theresistance ratio. and the op-amp input and output resistances easily satisfythe inequalities, and Thus, the zero frequencyloop gain closely approximates

In turn, the closed loop bandwidth of the circuit becomes

6 The closed loop input impedance can be found through use of (6.58),while (6.59) applies to a determination of the closed loop output impedance.An inspection of Figure 6.19(a) provides a low-frequency open loop inputresistance of

and an open loop output resistance of

204 Chapter 6

The results of the third and fourth computational steps above confirm

and

Thus, the approximate low frequency closed loop I/O impedances are

and

Comment. Although the low frequency output resistance of the circuit inFigure 6.18(a) is somewhat larger than the low frequency input resistance, boththe input and the output resistances are small owing to very large open loopop-amp gain. Although the circuit is commonly used as a voltage amplifier, thesmall I/O resistance levels make the amplifier more suitable for transimpedancesignal processing.

References

[1]

[2]

[3]

[4]

[5]

[6]

A. Armstrong, “Some recent developments in the audion receiver”,Proceedings of the IRE, vol. 3, pp. 215–247, 1915.

J. Choma, Jr., “NPN operational amplifier”, United States Patent,Number 4,468,629, August 1984.

W. G. Beall and J. Choma, Jr., “Charge-neutralized differential ampli-fiers”, Journal of Analog Integrated Circuits and Signal Processing, vol.1,pp. 33–44, September 1991.

A. B. Grebene, Bipolar and MOS Analog Integrated Circuit Design.New York: Wiley–Interscience, ch. 11, 1984.

A. B. Kahng and S. Muddu, “An analytical delay model for RLC inter-connects”, IEEE Transactions on Computer-Aided Design of IntegratedCircuits and Systems, vol. 16, pp. 1507–1514, December 1997.

G. Palumbo and J. Choma, Jr., “An overview of single and dual loopanalog feedback; Part II: design examples”, Journal of Analog IntegratedCircuits and Signal Processing, vol. 17, pp. 195–219, November 1998.


[7]

[8]

[9]

E. Säckinger and W. Guggenbühl, “A high swing, high impedance MOScascode circuit”, IEEE Journal of Solid-State Circuits, vol. 25, pp. 289–298, February 1990.

R. L. Geiger and E. Sánchez-Sinencio, “Active filter design using opera-tional transconductance amplifiers: a tutorial”, IEEE Circuits and DevicesMagazine, pp. 20–32, March 1985.H. Khorramabadi and P. R. Gray, “High-frequency CMOS continuoustime filters”, Proceedings of IEEE, vol. 3, pp. 1498–1501, May 1984.

D. Johns and K. Martin, Analog Integrated Circuit Design. New York:John Wiley and Sons, Inc., ch. 15, 1997.

T. H. Lee, The Design of CMOS Radio-Frequency Integrated Circuits.United Kingdom: Cambridge University Press, ch. 15, 1998.

J. F. Duque-Carrillo, “Control of the common-mode component inCMOS continuous-time fully differential signal processing”, Journal ofAnalog Integrated Circuits and Signal Processing, vol. 4, pp. 131–140,September 1993.J. K. Roberge, Operational Amplifiers. New York: John Wiley and Sons,Inc., 1985.G. Palumbo and J. Choma, Jr., “An overview of single and dual loop ana-log feedback; part I: basic theory”, Journal of Analog Integrated Circuitsand Signal Processing, vol. 17, pp. 175–194, November 1998.

J. N. Babanezhad and R. Gregorian, “A programmable gain/loss cir-cuit”, IEEE Journal of Solid-State Circuits, vol. 22, pp. 1082–1090,December 1987.H. W. Bode, Network Analysis and Feedback Amplifier Design.New York: D. Van Nostrand Co., Inc., 1945.N. Balabanian and T. A. Bickart, Electrical Network Theory. New York:John Wiley and Sons, Inc., ch. 3, 1969.R. G. Meyer and R. A. Blauschild, “A wide-band low-noise mono-lithic transimpedance amplifier”, IEEE Journal of Solid-State Circuits,vol. SC-21, pp. 530–533, August 1986.Y. P. Tsividis, “Design considerations in single-channel MOS analogintegrated circuits”, IEEE Journal of Solid-State Circuits, vol. SC-13,pp. 383–391, June 1978.

Y. P. Tsividis and P. R. Gray, “An integrated NMOS operational ampli-fier with internal compensation”, IEEE Journal of Solid-State Circuits,vol. SC-11, pp. 748–753, December 1976.

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

206 Chapter 6

[21]

[22]

[23]

[24]

G. Kron, Tensor Analysis of Networks. New York: John Wiley & Sons,1939.

G. Kron, “A set of principals to interconnect the solutions of physicalsystems”, Journal of Applied Physics, vol. 24, pp. 965–980, August1953.

R. A. Rohrer, “Circuit partitioning simplified”, IEEE Transactions onCircuits and Systems, vol. 35, pp. 2–5, January 1988.

J. Choma, Jr., “Signal flow analysis of feedback networks”, IEEETransactions on Circuits and Systems, vol. CAS-37, pp. 455–463, April1990.

Documents

GENERALIZED FEEDBACK CIRCUIT ANALYSISmsn/nitiphat.pdf · GENERALIZED FEEDBACK CIRCUIT ANALYSIS Scott K. Burgess and John Choma, Jr. Department of Electrical Engineering–Electrophysics,