Bandwidth Estimation Techniques1.0 Introduction2.0 The Method of Open-Circuit Time Constants(1)(2)(3)(4)(5)(6)(7)(8)2.1 Some Observations and Interpretations2.2 Accuracy of OCt's(9)(10)(11)2.3 Other Important Considerations2.4 Some Useful FormulasFIGURE 1. Incremental model for open-circuit resistance calculations(12)(13)(14)(15)(16)(17)2.5 A Design ExampleFIGURE 2. First-cut design (biasing not shown)FIGURE 3. Incremental model of first-pass design(18)(19)(20)(21)(22)(23)FIGURE 4. Second pass: cascode amplifier(24)(25)(26)(27)(28)(29)(30)(31)(32)FIGURE 5. Third pass: cascode amplifier w/ output source follower(33)(34)(35)(36)(37)(38)(39)(40)(41)(42)(43)(44)FIGURE 6. Fourth pass: cascode amplifier w/ two source followers(45)(46)(47)(48)(49)(50)(51)(52)(53)(54)(55)(56)(57)(58)(59)2.6 Summary of Open-Circuit Time Constants3.0 The Method of Short-Circuit Time Constants3.1 Introduction3.2 Background of the Method of Short-Circuit Time Constants(60)(61)(62)(63)(64)(65)(66)3.3 Some Observations and Interpretations3.4 Accuracy of SCt's3.5 Other Important ConsiderationsFIGURE 7. Cascode amplifier3.6 Summary and Concluding Remarks4.0 For Further Reading5.0 Risetime, Delay and Bandwidth5.1 Introduction5.2 Delay of Systems in Cascade(67)FIGURE 8. Illustrative impulse response(68)(69)(70)(71)(72)(73)5.3 Risetime of Systems in Cascade(74)(75)(76)(77)(78)(79)(80)5.4 A (very short) Application of the Risetime Addition Rule5.5 Bandwidth-Risetime Relations(81)FIGURE 9. RC low-pass filter and step response(82)(83)(84)(85)(86)(87)(88)(89)(90)(91)5.6 Open-Circuit Time Constants, Risetime and Bandwidth(92)(93)(94)6.0 Summary7.0 Appendix: Amplifier Equations Quick ReferenceEq. 2Eq. 3Eq. 4Eq. 5Eq. 6Eq. 7Eq. 8Eq. 9Eq. 10Eq. 11Eq. 12Eq. 13Eq. 14Eq. 15Eq. 16Eq. 17Eq. 18Eq. 19Eq. 20Eq. 21Eq. 22Eq. 23Eq. 24Eq. 25Eq. 26Eq. 27Eq. 28Eq. 29Eq. 30Eq. 31Eq. 32Eq. 33T. H. Lee EE214Bandwidth Estimation Techniques©1995 Thomas H. Lee, rev. October 13, 2004; All rights reserved Page 1 of 35Bandwidth Estimation Techniques1.0 IntroductionFinding the –3 dB bandwidth of an arbitrary linear network can be a difficult problem in general. Consider, for example, the standard recipe for computing bandwidth:1) Derive the input-output transfer function (using node equations, for example)2) Set s = jω;3) Find the magnitude of the resulting expression;4) Set the magnitude = 1/2 of the “midband” value; and5) Solve for ωIt doesn't take a great deal of insight to recognize that explicit computation (by hand) of the –3 dB bandwidth using this method is generally impractical for all but the simplest systems. In particular, the order of the denominator polynomial obtained in step 1 above is equal to the number of poles (natural frequencies), which in turn equals the number of degrees of freedom (measured, say, by the number of initial conditions one may indepen-dently specify), which in turn equals the number of independent energy storage elements (e.g., L or C), which in turn can be as large as the number of energy storage elements (phew!). Thus, a network with n capacitors might require the equivalent of finding the roots of an nth-order polynomial. If n exceeds just four, no algebraic closed form solution exists. Even if n = 2, it might be labor-intensive to obtain the final numerical result.Now, machine computation is cheap and getting cheaper all the time, so perhaps the anal-ysis of networks doesn’t present much of a problem. However, we are interested in devel-oping design insight so that if a simulator tells us that there is a problem, we have some idea of what to do about it. We therefore seek methods that are reasonably simple to apply, yet conveys the desired insight, even if it yields answers that might be approximate. Simulators can then be used to provide final quantitative verification.Two such approximate methods are open- and short-circuit time constants. The former provides an estimate of the high-frequency rolloff while the latter yields an estimate of the low-frequency rolloff point. These methods are valuable because they identify which ele-ments are responsible for the bandwidth limitation. This information alone is often suffi-cient to suggest what modifications should be tried next.2.0 The Method of Open-Circuit Time ConstantsThe method of open-circuit time constants (OCτ's), also known as zero value time con-stants, was developed in the mid-1960's at MIT. As we shall see, this powerful technique allows us to estimate the bandwidth of a system almost by inspection, and sometimes with surprisingly good accuracy. More important, and unlike typical circuit simulation pro-grams, OCτ's identify which elements are responsible for bandwidth limitations. The great value of this property in the design of amplifiers hardly needs expression.T. H. Lee EE214Bandwidth Estimation Techniques©1995 Thomas H. Lee, rev. October 13, 2004; All rights reserved Page 2 of 35To begin development of this method, let us consider all-pole transfer functions only. Such a system function may be written as follows:Vos()Vis()-------------aoτ1s 1+()τ2s 1+()... τns 1+()-----------------------------------------------------------------------=(1)where the various time constants may or may not be real.Multiplying out the terms in the denominator leads to a polynomial we shall express as:bnsnbn 1–sn 1–... b1s 1++++ (2)where the coefficient bn is simply the product of all of the time constants and b1 is the sum of all of the time constants. (In general, the coefficient of the sj term is computed by form-ing all unique products of the n time constants taken j at a time and summing all n!/j!(n-j)! such products.)We now assert that, near the –3 dB frequency, the first-order term typically dominates over the higher-order terms so that (perhaps) to a reasonable approximation, we have:Vos()Vis()-------------aob1s 1+-----------------≈aoτii 1=n∑⎝⎠⎜⎟⎜⎟⎛⎞s 1+-------------------------------= (3)The bandwidth of our original system in radian frequency as estimated by this first-order approximation is then simply the reciprocal of the effective time constant:ωh1b1-----≈1τii 1=n∑-------------ωhest,== (4)Before proceeding further, we should consider the conditions under which our neglect of the higher-order terms is justified, so let us examine the denominator of the transfer func-tion near our estimate of ωh. For the sake of simplicity, we start with a second-order poly-nomial with purely real roots.Now, at