A generalized amplifier and the Miller-effect paradox
In this article (continuing from “Open-Circuit or Nodal Time Constants?”), the last generalized circuit is derived, is the most abstract, and can be applied widely. With it, we are led into deeper consideration of the familiar Miller effect, and in particular to the lesser-known effect at the output node. Beware of the output Miller transform of a bridging capacitance when applied to amplifiers such as the cascode stage. An interesting “inverse Miller effect” leads to a paradox in amplifier dynamics and the uncovering of another longstanding error in time-constant calculation.Generalized Single-Stage Amplifier
Previous analyses can be abstracted to the more general single-stage amplifier, shown below.
The transfer function is found by applying basic circuit laws, beginning with KCL at the output node:
where transconductance-amplifier output current is defined as positive coming out of the amplifier. Then the amplifier current on the right side of the equation is going into the amplifier and out of the output node. This equation boils down to
This is the open-loop forward-path voltage gain. Applying KCL at the input node,
Combining the two KCL equations,
The closed-loop transimpedance is in a form that makes the feedback blocks explicit for the feedback gain formula:
The forward path consists of two parallel paths, an active path, Ga and a passive path, Gp:
To give the general stage a more concrete instantiation, let Zf = 1/s·Cf, and Zi and ZL be shunt RCs:
The two forward-path transmittances are
The closed-loop gain can be expressed as a voltage gain and the amplifier reconfigured as a voltage amplifier by thevenizing the Norton input of ii and Zi so that they are instead an input source voltage of Zi·ii in series with Zi. Then if the transimpedance is expressed for the closed loop,
while the voltage gain of the loop is
The left grouping of b is as terms separated according to OCTCs per capacitance; the right grouping is by node resistances and their time constants. The pole-zero placement is shown below. (This is a root contour and not a root locus plot; a circuit parameter other than quasistatic loop gain is varied. Although loop gain increases as Rm → 0 Ω, it is not the loop gain and also directly affects the value of the RHP zero.)
Examination of the a and b coefficients of D(s) leads to identification of three time constants, as shown in the table below. D(s) takes the form,
where the time constants are given in the following table. None of the three are OCTCs, though τL, in the right grouping as RL·(Cf + CL), appears in both a and b but this does not necessarily mean that it is an OCTC.
(|| is a math operator, not a topological descriptor.) The damping of the generalized single-stage amplifier is
As Kv increases, k increases and the minimum damping also increases. Increasing Kv separates real poles. Maximum bandwidth occurs when both poles are as high in frequency as possible which occurs when they are equal ζ = 1 ξ = k = 1 τL = τa = τp. When the time constants are equated, the resulting condition is
which is not realizable, for under the condition that τa = τp, . For real circuits, τa < τp. The cascaded CE analysis of the previous article demonstrated this.Frequency-Dependent Miller’s Theorem
Miller’s Theorem applies in a frequency-dependent way to the generalized stage. Applied at the output node,
Then for coefficients n1, n2 of N(s) and d1, d2 of D(s),
Finally (for the record), this turns into the ponderous
The Kv/(1 + Kv) output Miller effect multiplier is apparent in both the linear term of Cf and in the left factor. The ideal Miller effect of
is complicated by the frequency-dependent rational function that spoils the ideal Zof0.
The more familiar input Miller effect is
The denominator is the same as that of Zof but the numerator is D(s) instead of N(s). The ideal Miller’s theorem appears at the left as
For both input and output nodes, the use of Miller’s Theorem to calculate capacitances for bandwidth or inductive peaking should take into account that the ideal formulas lack additional equivalent elements that can complicate the analysis. The frequency-dependent rational factors in Zif and Zof include hf-gyrated Zi elements.
To summarize, Miller’s Theorem consists of two Miller transforms, an input and an output transform. An amplifier with an inverting voltage gain of –Kv between input and output ports with Zf bridging input and output port nodes can be transformed into an unbridged amplifier with equivalent Miller impedances across input and output ports of
This pair of transforms is in itself a useful circuit theorem for simplifying circuits under analysis because it removes the bridging impedance and thus separates input and output nodes. It can also lead to some unexpected paradoxes.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, Zif = 1/s·Cf but the output C-multiplier, (1 + Kv)/Kv = (1 + 1/Kv) → ∞, causing Zof → 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 increases with decreasing Kv. Then collector node capacitance goes to infinity and bandwidth to zero, so it would seem. In actuality, the opposite occurs and bandwidth is maximized by Kv = 0. The paradox is resolved upon closer inspection of the collector time constant,
where the contribution to τL of the part of the time constant caused by Cf is
The base-to-collector voltage gain of the CE, after the generalized single stage model, is
Then substituting Kv into τLf,
As RL → 0 Ω, τLf → Rm·Cf, or contributes a pole factor at (s·(Rm·Cf) + 1). This pole factor combines with the RHP zero factor contributed by the base-to-collector passive path, Gp, to form an all-pass filter;
which has no effect on the transfer function magnitude but contributes phase delay. Each pole and zero contributes delay of the same amount which totals –π/2 (–90°) at –1/Rm·Cf.
Therefore, as the output Miller capacitance increases with decreasing Kv, RL decreases and τLf actually decreases and approaches Rm·Cf. The passive forward path through Cf becomes a significant factor in the overall effect of the output Miller effect, causing the overall response to be that of a single-pole, single- zero all-pass filter.
The above analysis applied only to τLf, though for properly-compensated CB dynamics, CL ≈ 0 pF. Then τL ≈ τLf, and the above resolution of the output Miller paradox for Kv << 1 is resolved. For a significant CL, total
as Kv → 0, RL → 0 Ω and τL → Rm·Cf. The effect of CL on CB-stage dynamic compensation still applies, but it does not detract from resolution of the paradox.
The inverse Miller-effect paradox also leads to another conclusion that is not always observed in calculating collector-node capacitance for inductive peaking calculations or bandwidth estimation. It seems reasonable to apply the output Miller transform to Cc and add it to CL to obtain the total collector capacitance, Co. By this reasoning, (1 + 1/Kv) is the Miller multiplier to Cc.
However, from direct derivation of the time constants at the collector, whether in the textbook CE stage without RE, the general single-stage BJT model, or the generalized single stage, the output Miller multiplier of Cc (or Cf) does not occur though the input Miller effect always occurs for the input (base) node. The reason it is lacking for the output node is seen in the resolution of the paradox: RL affects both the time constant and Kv, and the output Miller multiplier does not appear in τL. Whenever Kv varies with RL, Cc has no Miller multiplier. The output Miller paradox demonstrates the importance of the passive forward path in amplifiers and that it is not always possible to neglect the RHP zero it contributes without introducing inaccuracy.
The linear pole coefficient (b) can be factored in several ways that summarize different ways that bandwidth might be approximated. From the general single-stage BJT model, first observe that a “dual” of Miller’s Theorem falls out from a different factorization of
The factorization, , views Rbc from the base, with the Miller multiplier applied to the base resistance and the collector resistance, RL, in series with it. The alternative factorization, , is a collector view, where a kind of “dual Miller multiplier”, (1 + Ki), is applied to the collector resistance with Rb added to it. Ki is a meaningful current gain that is often used in fast amplifier design because stages are usually driven by current sources with a Norton equivalent input resistance of Rb.
Now consider the linear pole coefficient of the generalized single-stage amplifier and factor it in three different ways by collecting terms according to RL, Ri, and the capacitances:
In the first two equations, the terms are differentiated by node. The Miller multiplier has Kv for input (base) referred Cf and Ki for output (collector) referred Cf. One might be inclined to use for capacitance in inductive peaking at the output node, but the correct value is found in the OCTC equation associated with RL of CL. Cf forms its own pole with Rbc that can be expressed equivalently as referred by the Miller multiplier to either input or output node.Closure
The Miller effect was originally constrained to apply to the base (input) capacitance. When applied to output capacitance, it is in effect irrelevant because Kv and the output time constant are both affected by the resistance, RL, of the time constant. It is more theoretically sound to calculate the pole of the OCTC Cbc (or Cf) separate from the output pole, τL = RL·CL. Inductive peaking in the output (collector or drain) circuit applies to CL and not an equivalent Miller output capacitance.