Standard-mode I^2C requires t_LOW≥ 4.70 μs and t_HIGH≥ 4.00 μs on SCL, t_r≤ 1.00 μs…

Standard-mode I\(^2\)C requires \(t_{\mathrm{LOW}}\ge 4.70\,\mu\mathrm{s}\) and \(t_{\mathrm{HIGH}}\ge 4.00\,\mu\mathrm{s}\) on SCL, \(t_r\le 1.00\,\mu\mathrm{s}\), and \(f_{\mathrm{SCL}}\le 100.0\,\mathrm{kHz}\). A master is programmed with the following four candidate \((t_{\mathrm{LOW}},t_{\mathrm{HIGH}})\) pairs, in microseconds, not including rise and fall times. Rise time \(t_r=400\,\mathrm{ns}\) and fall time \(t_f=200\,\mathrm{ns}\) add to the period as \(T=t_{\mathrm{LOW}}+t_{\mathrm{HIGH}}+t_r+t_f\). Pair \(P_1\): \((4.70,4.00)\). Pair \(P_2\): \((4.00,4.70)\). Pair \(P_3\): \((5.00,5.00)\). Pair \(P_4\): \((4.70,4.00)\) but with \(t_r=1.20\,\mu\mathrm{s}\) and \(t_f=200\,\mathrm{ns}\). (a) For each pair compute \(T\) and \(f_{\mathrm{SCL}}\). (b) For each pair, decide whether Standard-mode is satisfied, listing every violated constraint by name. (c) For the pairs that fail, compute the smallest increase in \(t_{\mathrm{HIGH}}\) (leaving \(t_{\mathrm{LOW}}\), \(t_r\), and \(t_f\) unchanged) that would make every Standard-mode constraint hold, or state that increasing \(t_{\mathrm{HIGH}}\) alone cannot fix the failure. (d) Fast-mode requires \(t_{\mathrm{LOW}}\ge 1.30\,\mu\mathrm{s}\), \(t_{\mathrm{HIGH}}\ge 0.600\,\mu\mathrm{s}\), \(f_{\mathrm{SCL}}\le 400.0\,\mathrm{kHz}\). Decide whether \(P_2\) as given (with the original \(t_r=400\,\mathrm{ns}\), \(t_f=200\,\mathrm{ns}\)) is a legal Fast-mode clock, even though it failed Standard-mode.

The SCL period includes rise and fall; meeting \(t_{\mathrm{LOW}}\) and \(t_{\mathrm{HIGH}}\) does not automatically meet \(f_{\mathrm{SCL}}\).