This Paper written by Bruno Putzeys describes the operation of an Ncore/UcD amplifier.
On page 2 the author presents the following formula, which he previously used to determine the approximate oscillation behaviour an amplifier:
$$\arg(H(2i \cdot \pi \cdot f)) = 0$$
This formula makes sense to me, it seems to be the Barkhausen Oscillation Criterium applied in context (phase shift has to be an integer multiple of \$2 \cdot pi\$ in order for oscillation to occur).
He then states the above formula is quite wrong and delivers an exact formula describing the oscillation behaviour:
$$\arg \left( \lim_{t \to 0^-} \sum_{n=1}^{\infty} \frac { (1 - e^{-2i \cdot \pi \cdot n \cdot h}) \cdot (1 - e^{2i \cdot \pi \cdot n \cdot h}) } { 2 \cdot n } \cdot H(2i \cdot \pi \cdot f \cdot n) \cdot e^{i \cdot f \cdot t}\right)= 0$$
\$h\$ represents duty cycle (from \$0\$ to \$1\$),
\$f\$ represents frequency,
\$H(S)\$ represents the transfer function of the amplifiers control loop.
The fraction $$\frac{ (1 - e^{-2i \cdot \pi \cdot n \cdot h}) \cdot (1 - e^{2i \cdot \pi \cdot n \cdot h}) } { 2 \cdot n } $$ seems represent attenuated harmonics, but I can't see any logical connection to the context. What exactly is it doing? As it dependens on duty cycle, I would guess it has something to do with that?
Another aspect that is puzzling me is the limit. Since \$\lim_{x \to 0^-} e^{ix}\$ tends toward \$1\$, \$lim_{t \to 0^-} e^{i \cdot f \cdot t} \$ should do the same, right? I don't see what effect this is supposed to have.
Can someone help me understand how the author arrived at this equation? Right now it feels like magic to me.