# The Miller Effect, Part 3: The Frequency-Dependent Miller’s Theorem

Miller’s theorem assumes that the amplifier is ideal, that the amplifier with voltage gain of Kv 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 Zf is where in general form the frequency-dependent voltage gain is Then for coefficients n1 , n2 of N (s ) and d1 , d2 of D (s ), Finally, referring to Part 2 for the substitutions of the general single-stage amplifier, 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 of a frequency-independent amplifier.

The more familiar input Miller-effect impedance 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.

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 –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 because it removes the bridging impedance and thus separates input and output nodes. It can also lead to an unexpected paradox, in Part 4.

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