This short book is a systematic, detailed treatment of dynamic circuit analysis from a design-oriented viewpoint, and fills in more of the theoretical basis of dynamics than in my recently-published book, Transistor Amplifiers . It starts by introducing the big picture of the methods of dynamic circuit analysis from a historic development perspective on page 1, as shown below.
This flowchart summarizes what is essentially in the book. The general goal is to start with a circuit diagram, find its transfer function in the complex-frequency domain of the form
and determine from it the poles and zeros of the circuit. Then some properties of the transfer function, such as bandwidth and risetime, are approximated from the transfer function.
One challenge is factorization of the polynomials of the numerator and denominator. If they are cubic (with an s3 term) or higher, factorization is not attempted, though there is a math formula for finding the roots of cubic polynomials. It does not inspire much intuitive understanding of dynamics, however! Not all is lost in that circuits constrain polynomials to have some properties that aid in determining where their poles and zeros are, approximately. These various methods comprise about half of the analysis, with the Cochrun-Grabel method, Extra Element Theorem, n EET, and others listed in the above chart and developed in the book. They are ways of avoiding high-degree polynomial factorization. Central to this effort is the quest for the open-circuit time constants (OCTCs), and this can often be done by inspection of the circuit. (Christophe Basso’s recent book, Linear Circuit Transfer Functions , (Wiley), Book Review: Linear Circuit Transfer Functions, reviewed on Planet Analog , covers the topic using the EET, with detailed examples.)
About a third of the book finds OCTCs from general circuits such as the common-emitter (CE) and common-base (CB) amplifier stages. Undergraduate active-circuits textbooks invariably work out the CE circuit with grounded emitter, though in actual engineering, emitter resistance, RE , is often present. This floats the base-emitter junction from ground, making the analysis more difficult. Happily, once the OCTCs are found for general circuits, their formulas form with them a template for further use, thereby avoiding having to work out tedious formulas multiple times. OCTCs of the general, single-stage BJT amplifier stage, with base, emitter, and collector series resistors as shown below, is one circuit template.
The OCTCs are derived for the two capacitances of the BJT T model, Cc (or C μ ) across the b-c junction, and Ce (or Cπ ) across the b-e junction. (The results of BJT analysis also apply to FETs with a change of notation.) The textbook CE stage (RE = 0 Ω ) is worked out as a starting exercise leading to the generalized single-stage amplifier and cascaded CE stages.
In the section “OCTCs and the Base Node Time Constant”, a commonly-occurring yet often misleading approximation of the Cc time constant, usually dominated by the Miller Effect, is not based on the resistance across the base-collector nodes but at the base node to ground. The nodal resistance is not the open-circuit b-c resistance, and leads to an error in the pole location. In some cases, it is not in error by much, but conceptually it can mislead students into thinking that it is the nodal and not the open-circuit port resistances that lead to pole determination.
The linear pole coefficient ( α 1 of the α 1 x s term in the transfer-function denominator) for the general BJT stage above is
where Rb is the base node resistance (to ground), K υ is the base-to-collector voltage gain (of the Miller effect), and Rbe is the resistance across the b-e junction. The OCTC bandwidth formula requires the TC groupings according to capacitances, as in the left-side expression:
The terms of τc are designated as time constant components (not separate TCs). When the OCTCs are rearranged and separated by node instead, then the base and collector node TCs are
The difference in the stage bandwidth that results from the OCTCs versus the nodal TCs when the root-sum-of-squares bandwidths, expressed as time constants (τbw = 1/ ωbw ), is
The τbw differ in the last term under the radical in that τcc ≠ τce .
As for bandwidth formulas, there is more than one. Three are derived in the book. The OCTCs, τe and τc , are
Stage bandwidth can be found from the OCTC bandwidth formula approximation as
There are yet other ways to derive an approximate bandwidth formula. Of the three in the book, the OCTC bandwidth formula is based on the linear pole term coefficient, which is the sum of the OCTCs. Expressed as a time constant,
The most accurate approximation of the three results from truncating the pole polynomial by approximating it as a quadratic polynomial and deriving the quadratic bandwidth formula ,
where ζ is the pole damping, with pole angle of φ = cos–1 ( ζ ). The approximation is based on negligible contribution to pole locations by higher-degree terms in the denominator of the transfer function, and this is often valid.
The approximate quadratic bandwidth formula , derived in the MIT circuits textbook, Electronic Principles: Physics, Models, and Circuits (Wiley, 1969) by Paul E. Gray and Campbell Searle, is
where α and b are the quadratic ( α x s2 ) and linear (b x s ) term pole coefficients.
Two book sections that address the complexity of real circuit design are the sections “Pole Separation” and “Stage Interaction”. BJT circuits without external reactances or feedback have only real poles. Pole separation is a method of dynamics problem reduction because it separates dynamic effects into less-interactive frequency regions of poles and zeros. Each can be analyzed separately so that it is not uncommon for low-, mid-, and high-frequency analyses to be distinct.
Similarly, reduction of reactance interactions that cause pole shifting can be partially achieved by circuit modularization , which is based on the use of unilateral circuits: circuits that allow causal propagation from input to output but not in reverse, such as buffer amplifiers. Modularization is a design technique intended to minimize the transfer-function polynomial factorization problem by reducing the circuit to a product of independent lower-degree polynomials.
To avoid analytical paralysis, interactions are avoided or controlled by separating circuits into stages by design. By decomposing an amplifier into stages, each having a known transfer function and method for bandwidth analysis, and by controlling adjacent stage interactions through control of their input and output impedances, amplifier dynamic design becomes manageable by having been reduced through decomposition into multiple smaller design problems. This is the general strategy of amplifier dynamic design. Although the ideal separation of stages is that of isolation , it is exchanged in optimized amplifier design for controlled stage interaction by designing successive stages to have desired input impedances. These are usually capacitive or shunt RC impedances that become parameters in the transfer function of the preceding stage.
Finally, though not at all exhaustively, Miller’s Theorem is worked out by including the frequency dependence of the amplifier. This results in two Miller transforms ,
This pair of transforms is in itself a useful circuit theorem for simplifying circuits because it removes the bridging impedance across the amplifier and thus separates input and output nodes. It can also lead to an unexpected paradox, the “inverse Miller-effect” paradox.
Ideally, the incremental CE collector voltage of a cascode amplifier is zero and CE Kv = 0. When this is substituted into the Miller formulas, then Zi = 1/s x Cf (1 + Kv ), but the output C-multiplier of Zo , (1 + Kv )/Kv = (1 + 1/Kv ) → ∞ as Kv goes to zero, causing Zo → 0 Ω , a short circuit caused by output Miller capacitance that is infinite. As Kv becomes less than 1, an “inverse Miller effect” occurs at the output whereby the input and output nodes exchange roles and the Miller Cf multiplying effect increases with decreasing Kv . Then collector node capacitance goes to infinity and bandwidth to zero, so it would seem. Actually, the opposite occurs and bandwidth is maximized by Kv = 0. The paradox is resolved upon closer inspection of the collector time constant.
Some other section topics are: “Equivalence of Hybrid- π and T BJT Models”, “Design-Oriented Pole-Zero Determination” (“…we need not attempt n th-degree polynomial factorization in all its mathematical generality because these polynomials apply to physical circuits with additional constraints on the coefficients …”), “The Β Transform”, “The μ Transform”, “ Β (s ) Transform”, “Base-Emitter OCTC in the High-Frequency Region”, “Impedance EET (ZEET)”, “Blackman’s Impedance Theorem (BZT)”, and a quizzical “The μ (s ) Transform?”.
These are a few snippets of what is in this high-density-content book. It is for anyone who is serious about wanting a more rigorous understanding of circuit dynamics, and especially for the sake of designing analog circuits without dreading the dynamics aspects.