Programming for Robotics 1, Chapter 2: Challenge 1 solution (bus times and pull-up resistors)

Formulas: I2C:  bits = 9 x (2 + register bytes + data bytes)       t = bits / f           (9 bit times per byte: 8 data bits and the ACK; START and STOP ignored)
          SPI:  t = 8 x bytes / f                                   UART (8N1):  t = 10 x bytes / baud
          pull-up:  Rp(min) = (Vcc - VOL) / IOL        Rp(max) = t(rise) / (0.8473 x C)        t(rise) = 0.8473 x Rp x C

(a) A register-pointer byte and 6 data bytes: bits = 9 x (2 + 1 + 6) = 81.
        100 kHz:   t = 81 / 100 000 = 0.810 ms
        400 kHz:   t = 81 / 400 000 = 202.5 us
        three sensors at 400 kHz:  3 x 202.5 = 607.5 us, which is 607.5 / 10 000 = 6.075 % of the 10 ms period of a 100 Hz loop.
    (At 100 kHz the same three reads would take 2.43 ms, 24.3 % of the period: the faster clock is worth having.)

(b) SPI at 2 MHz, one command byte and 6 data bytes: 7 bytes x 8 bits = 56 bits,  t = 56 / 2 000 000 = 28.0 us.
    UART at 57 600 baud, 6 bytes:  60 bits,  t = 60 / 57 600 = 1.0417 ms.
    So for the same 6 bytes: SPI 28 us, I2C at 400 kHz 202.5 us, UART 1.04 ms. SPI is the fastest here because it has no address byte, no ACK bits and a clock that is not limited to 400 kHz.

(c) 3.3 V, IOL = 3 mA, VOL = 0.4 V, C = 150 pF, fast mode (rise time at most 300 ns).
        Rp(min) = (3.3 - 0.4) / 0.003 = 966.7 ohm
        Rp(max) = 300 ns / (0.8473 x 150 pF) = 2360 ohm
    The E12 values inside the window are 1.0 k, 1.2 k, 1.5 k, 1.8 k and 2.2 k (2.7 k is too big: its rise time would be 343 ns).
    The geometric middle of the window is sqrt(966.7 x 2360) = 1510 ohm, so 1.5 k keeps the most margin on both sides:
        t(rise) = 0.8473 x 1500 x 150 pF = 190.6 ns (limit 300 ns)          current when pulled low = 2.9 V / 1500 = 1.93 mA (limit 3 mA)
    With C = 400 pF at 400 kHz:  Rp(max) = 300 ns / (0.8473 x 400 pF) = 885 ohm, which is BELOW Rp(min) = 967 ohm. No resistor satisfies both limits.
    The fixes are to shorten the wires or remove devices (less capacitance), or to slow the bus: at 100 kHz the rise time may be 1000 ns,
        Rp(max) = 1000 ns / (0.8473 x 400 pF) = 2950 ohm, and 2.2 k works:  t(rise) = 0.8473 x 2200 x 400 pF = 745.6 ns.

The same numbers recomputed in Python (the code was run, and the output below is what it printed):

CODE:
def i2c_time_s(n_read, f_hz, n_reg_bytes=1):
    return 9 * (2 + n_reg_bytes + n_read) / f_hz


print("(a) I2C, 6 bytes: %d bits; 100 kHz %.3f ms; 400 kHz %.1f us; three sensors at 400 kHz %.1f us = %.3f %% of 10 ms" % (
    9 * (3 + 6), 1000 * i2c_time_s(6, 100000), 1e6 * i2c_time_s(6, 400000), 3e6 * i2c_time_s(6, 400000),
    100 * 3 * i2c_time_s(6, 400000) / 0.010))
print("(b) SPI 2 MHz, 1 + 6 bytes: %.1f us;  UART 57600 baud, 6 bytes: %.4f ms" % (1e6 * 8 * 7 / 2e6, 1000 * 10 * 6 / 57600))
vcc, vol, iol, c, tr = 3.3, 0.4, 0.003, 150e-12, 300e-9
r_min = (vcc - vol) / iol
r_max = tr / (0.8473 * c)
print("(c) Rp(min) = %.1f ohm, Rp(max) = %.1f ohm" % (r_min, r_max))
for rp in (1000, 1200, 1500, 1800, 2200, 2700):
    ok = r_min <= rp <= r_max
    print("    %4d ohm: tr = %.1f ns, I(low) = %.2f mA  %s" % (rp, 1e9 * 0.8473 * rp * c, 1000 * (vcc - vol) / rp, "inside" if ok else "outside"))
c2 = 400e-12
print("    C = 400 pF at 400 kHz: Rp(max) = %.1f ohm, below Rp(min) = %.1f: no value works" % (tr / (0.8473 * c2), r_min))
print("    C = 400 pF at 100 kHz (tr 1000 ns): Rp(max) = %.1f ohm; 2200 ohm gives tr = %.1f ns" % (1000e-9 / (0.8473 * c2), 1e9 * 0.8473 * 2200 * c2))
END CODE
OUTPUT:
(a) I2C, 6 bytes: 81 bits; 100 kHz 0.810 ms; 400 kHz 202.5 us; three sensors at 400 kHz 607.5 us = 6.075 % of 10 ms
(b) SPI 2 MHz, 1 + 6 bytes: 28.0 us;  UART 57600 baud, 6 bytes: 1.0417 ms
(c) Rp(min) = 966.7 ohm, Rp(max) = 2360.4 ohm
    1000 ohm: tr = 127.1 ns, I(low) = 2.90 mA  inside
    1200 ohm: tr = 152.5 ns, I(low) = 2.42 mA  inside
    1500 ohm: tr = 190.6 ns, I(low) = 1.93 mA  inside
    1800 ohm: tr = 228.8 ns, I(low) = 1.61 mA  inside
    2200 ohm: tr = 279.6 ns, I(low) = 1.32 mA  inside
    2700 ohm: tr = 343.2 ns, I(low) = 1.07 mA  outside
    C = 400 pF at 400 kHz: Rp(max) = 885.2 ohm, below Rp(min) = 966.7: no value works
    C = 400 pF at 100 kHz (tr 1000 ns): Rp(max) = 2950.5 ohm; 2200 ohm gives tr = 745.6 ns
END OUTPUT

Check on method: count the address bytes and the register byte as well as the data bytes (the 2 + 1 in the bit count), and use 9 bit times per I2C byte but 10 per UART byte and 8 per SPI byte. And a pull-up has two limits, not one: too small and the chip cannot pull the wire low, too large and the wire rises too slowly. When the window closes, the answer is a slower bus, not a cleverer resistor.