# 科学传播 Scientific Communication MA262-15

$$T_{x_{i}, x_{l+1}} \cong 2^{b+1}\left|x_{k}-x_{k+1}\right| .$$
For a sinusoidal input, the input levels are given by:
$$x_{k}=a \operatorname{Sin}\left(\omega k T_{s}+\theta\right) ; x_{k+1}=a \operatorname{Sin}\left(\omega(k+1) T_{s}+\theta\right) .$$
where $a$ is the amplitude of the input sinusoid and $\theta$ is the initial phase, assumed random, since it is not synchronized with the sampler in general. The total number of logic transitions from Eq. (16) is then given by:
$$T_{x_{k}, x_{t+1}} \equiv 2^{b+1} a\left|\operatorname{Sin}\left(\omega T_{s}\right)\right| \operatorname{Sin}\left(\omega(k+1 / 2) T_{s}+\theta\right) \mid .$$

## MA260-10 COURSE NOTES ：

negative resistance. Therefore, $R_{E q}$ can be approximated as:
$$R_{E q} \approx \frac{-2}{\operatorname{Re}\left[G_{m, e f f}\right]-\operatorname{Re}\left[g_{\pi}\right]}$$
Equation (2) suggests that $R_{E q}$ degrades rapidly as $\operatorname{Re}\left[\mathrm{g}{\pi}\right]$ approaches $\operatorname{Re}\left[\mathrm{G}{\mathrm{m}, \mathrm{eff}}\right]$, and has a distinct corner frequency which will be revisited later. An expression for the negative to positive transition frequency of $R_{E_{q}}\left(\omega_{t r a n}\right)$ can be calculated by considering the condition $\operatorname{Re}\left[G_{m_{,} e f f}\right]-\operatorname{Re}\left[g_{\pi}\right] \leq 0$, which results in:
$$\omega_{t r a n}=\sqrt{\frac{\left(1+\frac{r_{b}}{r_{\pi}}\right)\left(g_{m}-\frac{1}{r_{\pi}}\right)}{r_{b} C_{\pi}^{2}}}$$
Assuming $r_{b} / r_{\pi} \ll 1, r_{\pi}=\beta / g_{m}$ and $\beta \gg 1$ where $\beta$ is the base to collector current gain, then equation (3) can be simplified to:
$$\omega_{\text {tran }} \approx \sqrt{\frac{\omega_{T}}{r_{b} C_{\pi}}}$$