Miller’s theorem assumes that the amplifier is ideal, that the amplifier with voltage gain of *K*_{v} has infinite bandwidth. When it does not, the Miller transform becomes more complicated. Miller’s Theorem applies in a frequency-dependent way to the generalized stage in The Miller Effect, Part 2: The Generalized Single-Stage Amplifier. Applied at the output node, the equivalent *Miller output impedance* of *Z*_{f} is

where in general form the frequency-dependent voltage gain is

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

Finally, referring to Part 2 for the substitutions of the general single-stage amplifier, this turns into the ponderous

The *K*_{v} /(1 + *K*_{v}) output Miller effect multiplier is apparent in both the linear term of *C*_{f} and in the left factor. The ideal Miller effect of

is complicated by the frequency-dependent rational function that spoils the ideal *Z*_{of0} of a frequency-independent amplifier.

The more familiar input Miller-effect impedance 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*_{of} include hf-gyrated *Z*_{i} elements.

**Summary**

To summarize, Miller’s Theorem consists of two *Miller transforms*, an input and an output impedance transform. An amplifier with an inverting voltage gain of –*K*_{v} between input and output ports with *Z*_{f} 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 because it removes the bridging impedance and thus separates input and output nodes. It can also lead to an unexpected paradox, in Part 4.