# The Miller Effect, Part 4: The Inverse Miller-Effect Paradox

There is a paradoxical “inverse Miller effect” that occurs in transistor amplifiers. When Kv → 0 with a bridging capacitance of Zf = 1/s x Cf , then the Miller-effect equivalent Co = Cf x (1 + 1/Kv ) at the output node goes to infinity, and, it would seem, bandwidth goes to zero. But just the opposite happens.

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 Zif = 1/s x Cf x (1 + Kv ), 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. Actually, the opposite occurs and bandwidth is maximized by Kv = 0. The paradox is resolved upon closer inspection of the collector time constant, where the time-constant contribution to

τL 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 x Cf , or contributes a pole factor at (s x (Rm x 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 (–90o ) at the frequency magnitude of 1/Rm x Cf .

Therefore, as the output Miller capacitance increases with decreasing Kv , RL decreases and τLf actually decreases and approaches Rm x 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 x Cf . The effect of CL on CB-stage dynamic compensation still applies and does not upset the 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, the general single-stage BJT model that includes RE and RB , or the generalized single stage of Part 2, the output Miller multiplier of Cc (or Cf ) does not occur though it 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 also 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, shown below, a “dual” of Miller’s Theorem falls out from a different factorization of where the BJT amplifier quasistatic current gain is The factorization, RL + Rb x (1+Kv ), 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, Rb + RL x (1+Ki ), is a collector view, where a “dual” or output-side 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 (shunt) 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: input node, Ri: output node, RL: OCTCs, capacitors:

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 [CL + (1+Ki )xCf ] for capacitance in calculation of 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

Base-collector (or gate-drain) capacitance causes the bridging capacitance, Cc , to load the collector node and, it would seem, decrease bandwidth by forming a time constant, RL x [K /(1 + K )]xCc , with the collector-node resistance, RL and output-side Miller-transformed [K /(1 + K )] x Cc . But this is not actually a circuit time constant because K itself varies with RL , and the bandwidth-limiting time constant appears only on the base (or gate) side of the circuit. (The inverse Miller-effect paradox is presented in Transistor Amplifiers , pp. 354 – 356, or for a more systematic, step-by-step treatment of circuit dynamics, including why OCTCs are needed in calculating dynamics parameters, see Circuit Dynamics: A Design-Oriented Analysis , both by the author, D. Feucht, at www.innovatia.com.)

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