  Using the first law of thermodynamics and the Parseval-Plancheral theorem, it
is shown that every DC-DC converter must trade-off size, heat and conducted
electromagnetic noise. It is therefore fundamentally impossible to
simultaneously reduce all three of these characteristics to their respective
theoretical minimum values. A figure of merit is introduced which holistically
captures the performance of an arbitrary DC-DC converter, this is called the
converter's non-ideality and it has a target value of zero. It is derived using
the first law of thermodynamics and is shown to be dependent on the efficiency
and the root-mean-squared total harmonic distortion of the output voltage.
Interestingly, it is also shown that: boost conversion is impossible without
energy storage; ideal rectifiers convert all of the input power spectral
density into DC (and introduces more noise in the process); the input current
of any DC-DC converter scales with the gain of the device squared. Using an
arbitrary DC-DC converter, the culprit of this inherent trade-off is shown to
be the act of switching itself. Switching creates harmonics which need to be
filtered or transformed into heat in order to get a pure DC voltage at the
output of the converter. Even with ideal sub-systems, whether or not resonant
conversion is employed, the result stands. The conclusion is that spreading of
the switching noise is a primary goal in attempting to reach the impossible.
