# 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,

Rearranging,

The closed-loop transimpedance is in a form that makes the feedback blocks explicit for the feedback gain formula:

where

The forward path consists of two parallel paths, an active path, *G _{a}* and a passive path,

*G*:

_{p}

To give the general stage a more concrete instantiation, let *Z _{f}* = 1/

*s*·

*C*, and

_{f}*Z*and

_{i}*Z*be shunt RCs:

_{L}

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 *i _{i}* and

*Z*so that they are instead an input source voltage of

_{i}*Z*·

_{i}*i*in series with

_{i}*Z*. Then if the transimpedance is expressed for the closed loop,

_{i}

while the voltage gain of the loop is

where

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 *R _{m}* → 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 *τ _{L}*, in the right grouping as

*R*·(

_{L}*C*+

_{f}*C*), appears in both

_{L}*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

*k*is the

*interaction factor*and

*ξ*is the OCTC

*separation factor*. Minimum

*ζ*= at

*ξ*= 1 (equal OCTCs). It is plotted below as

*ζ*(

*ξ*,

*k*).

As *K _{v}* increases,

*k*increases and the minimum damping also increases. Increasing

*K*separates real poles. Maximum bandwidth occurs when both poles are as high in frequency as possible which occurs when they are equal

_{v}*ζ*= 1

*ξ*=

*k*= 1

*τ*=

_{L}*τ*=

_{a}*τ*. When the time constants are equated, the resulting condition is

_{p}which is not realizable, for under the condition that *τ _{a}* =

*τ*, . For real circuits,

_{p}*τ*<

_{a}*τ*. The cascaded CE analysis of the previous article demonstrated this.

_{p}**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 *n*_{1}, *n*_{2} of *N*(*s*) and *d*_{1}, *d*_{2} of *D*(*s*),

Finally (for the record), this turns into the ponderous

The *K _{v}*/(1 +

*K*) output Miller effect multiplier is apparent in both the linear term of

_{v}*C*and in the left factor. The ideal Miller effect of

_{f}

is complicated by the frequency-dependent rational function that spoils the ideal *Z _{of}*

_{0}.

The more familiar input Miller effect is

The denominator is the same as that of *Z _{of}* 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 *Z _{if}* and

*Z*include hf-gyrated

_{of}*Z*elements.

_{i}To summarize, Miller’s Theorem consists of two *Miller transforms*, an input and an output transform. An amplifier with an inverting voltage gain of –*K _{v}* between input and output ports with

*Z*bridging input and output port nodes can be transformed into an unbridged amplifier with equivalent Miller impedances across input and output ports of

_{f}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 *K _{v}* = 0. When this is substituted into the Miller formulas,

*Z*= 1/

_{if}*s·C*but the output C-multiplier, (1 +

_{f}*K*)/

_{v}*K*= (1 + 1/

_{v}*K*) → ∞, causing

_{v}*Z*→ 0 Ω, a short circuit caused by output Miller capacitance that is infinite. As

_{of}*K*becomes less than 1, an “inverse Miller effect” occurs at the output whereby the input and output nodes exchange roles and the Miller

_{v}*C*increases with decreasing

_{f}*K*. 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

_{v}*K*= 0. The paradox is resolved upon closer inspection of the collector time constant,

_{v}

where the contribution to *τ _{L}* of the part of the time constant caused by

*C*is

_{f}

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}*C*, or contributes a pole factor at (

_{f}*s*·(

*R*·

_{m}*C*) + 1). This pole factor combines with the RHP zero factor contributed by the base-to-collector passive path,

_{f}*G*, to form an all-pass filter;

_{p}

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/*R _{m}*·

*C*.

_{f}Therefore, as the output Miller capacitance increases with decreasing *K _{v}*,

*R*decreases and

_{L}*τ*actually decreases and approaches

_{Lf}*R*·

_{m}*C*. The passive forward path through

_{f}*C*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.

_{f}The above analysis applied only to *τ _{Lf}*, though for properly-compensated CB dynamics,

*C*≈ 0 pF. Then

_{L}*τ*≈

_{L}*τ*, and the above resolution of the output Miller paradox for

_{Lf}*K*<< 1 is resolved. For a significant

_{v}*C*, total

_{L}

as *K _{v}* → 0,

*R*→ 0 Ω and

_{L}*τ*→

_{L}*R*·

_{m}*C*. The effect of

_{f}*C*on CB-stage dynamic compensation still applies, but it does not detract from resolution of the paradox.

_{L}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*to obtain the total collector capacitance,

_{L}*C*. By this reasoning, (1

_{o}*+ 1/*

*K*) is the Miller multiplier to

_{v}*C*.

_{c}However, from direct derivation of the time constants at the collector, whether in the textbook CE stage without *R _{E}*, the general single-stage BJT model, or the generalized single stage, the output Miller multiplier of

*C*(or

_{c}*C*) 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:

_{f}*R*affects both the time constant and

_{L}*K*, and the output Miller multiplier does not appear in

_{v}*τ*. Whenever

_{L}*K*varies with

_{v}*R*,

_{L}*C*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.

_{c}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

where

The factorization, , views *R _{bc}* from the base, with the Miller multiplier applied to the base resistance and the collector resistance,

*R*, in series with it. The alternative factorization, , is a collector view, where a kind of “dual Miller multiplier”, (1 +

_{L}*K*), is applied to the collector resistance with

_{i}*R*added to it.

_{b}*K*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

_{i}*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*, and the capacitances:

_{i}In the first two equations, the terms are differentiated by node. The Miller multiplier has *K _{v}* for input (base) referred

*C*and

_{f}*K*for output (collector) referred

_{i}*C*. 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

_{f}*R*of

_{L}*C*.

_{L}*C*forms its own pole with

_{f}*R*that can be expressed equivalently as referred by the Miller multiplier to either input or output node.

_{bc}**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 *K _{v}* and the output time constant are both affected by the resistance,

*R*, of the time constant. It is more theoretically sound to calculate the pole of the OCTC

_{L}*C*(or

_{bc}*C*) separate from the output pole,

_{f}*τ*=

_{L}*R*·

_{L}*C*. Inductive peaking in the output (collector or drain) circuit applies to

_{L}*C*and not an equivalent Miller output capacitance.

_{L}“Proof” that the Laplace Transform is Nothing

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