There is a paradoxical “inverse Miller effect” that occurs in transistor amplifiers. When *K*_{v}
→
0 with a bridging capacitance of *Z*_{f} = 1/*s* x *C*_{f}, then the Miller-effect equivalent *C*_{o} = *C*_{f} x (1 + 1/*K*_{v} ) 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 *K*_{v} = 0. When this is substituted into the Miller formulas, then *Z*_{if} = 1/*s* x *C*_{f} x (1 + *K*_{v}), but the output C-multiplier, (1 + *K*_{v})/ *K*_{v} = (1 + 1/*K*_{v} )
→
∞
, causing *Z*_{of}
→
0 Ω, a short circuit caused by output Miller capacitance that is infinite. As *K*_{v} becomes less than 1, an “inverse Miller effect” occurs at the output whereby the input and output nodes exchange roles and the Miller *C*_{f} increases with decreasing *K*_{v}. Then collector node capacitance goes to infinity and bandwidth to zero, so it would seem. Actually, the opposite occurs and bandwidth is maximized by *K*_{v} = 0. The paradox is resolved upon closer inspection of the collector time constant,

where the time-constant contribution to

*τ*_{L} caused by

*C*_{f} is

The base-to-collector voltage gain of the CE, after the generalized single stage model, is

Then substituting *K*_{v} into
*τ*_{Lf},

As *R*_{L}
→
0
Ω
,
*τ*_{Lf}
→
*R*_{m} x *C*_{f}, or contributes a pole factor at (*s* x (*R*_{m} x *C*_{f}) + 1). This pole factor combines with the RHP zero factor contributed by the base-to-collector passive path, *G*_{p}, 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^{o}) at the frequency magnitude of 1/*R*_{m} x *C*_{f}.

Therefore, as the output Miller capacitance increases with decreasing *K*_{v}, *R*_{L} decreases and
*τ*_{Lf} actually decreases and approaches *R*_{m} x *C*_{f}. The passive forward path through *C*_{f} 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
*τ*_{Lf}
, though for properly-compensated CB dynamics, *C*_{L}
≈
0 pF. Then
*τ*_{L}
≈
*τ*_{Lf}
, and the above resolution of the output Miller paradox for *K*_{v} << 1 is resolved. For a significant *C*_{L} , total

as *K*_{v}
→
0, *R*_{L}
→
0 Ω and
*τ*_{L}
→
*R*_{m} x *C*_{f}. The effect of *C*_{L} 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 *C*_{c} and add it to *C*_{L} to obtain the total collector capacitance, *C*_{o}. By this reasoning, (1 + 1/*K*_{v}) is the Miller multiplier to *C*_{c}. 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 *R*_{E} and *R*_{B}, or the generalized single stage of Part 2, the output Miller multiplier of *C*_{c} (or *C*_{f}) 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: *R*_{L} affects both the time constant and *K*_{v}, and the output Miller multiplier does not appear in
*τ*_{L}. Whenever *K*_{v} varies with *R*_{L}, *C*_{c} 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, *R*_{L} + *R*_{b} x (1+*K*_{v}), views *R*_{bc} from the base, with the Miller multiplier applied to the base resistance and the collector resistance, *R*_{L}, in series with it. The alternative factorization, *R*_{b} + *R*_{L} x (1+*K*_{i}), is a collector view, where a “dual” or output-side Miller multiplier, (1 + *K*_{i}), is applied to the collector resistance with *R*_{b} added to it. *K*_{i} 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 *R*_{b}.

Now consider the linear pole coefficient of the generalized single-stage amplifier and factor it in three different ways by collecting terms according to *R*_{L}, *R*_{i}, 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 *K*_{v} for input (base) referred *C*_{f} and *K*_{i} for output (collector) referred *C*_{f}. One might be inclined to use [*C*_{L} + (1+*K*_{i})x*C*_{f}] for capacitance in calculation of inductive peaking at the output node, but the correct value is found in the OCTC equation associated with *R*_{L} of *C*_{L}. *C*_{f} forms its own pole with *R*_{bc} 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, *C*_{c}, to load the collector node and, it would seem, decrease bandwidth by forming a time constant, *R*_{L} x [*K*/(1 + *K*)]x*C*_{c}, with the collector-node resistance, *R*_{L} and output-side Miller-transformed [*K*/(1 + *K*)] x *C*_{c}. But this is not actually a circuit time constant because *K* itself varies with *R*_{L}, 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.)