Basic Electronics, Chapter 7: Challenge 2 solution (comparators and hysteresis)

Formulas: Vin(switch) = Vref x (1 + R1/Rf) - Vout x (R1/Rf)
          rising threshold  VH = Vref x (1 + R1/Rf) - Vl x R1/Rf          (output LOW before switching, Vl = 0)
          falling threshold VL = Vref x (1 + R1/Rf) - Vh x R1/Rf          (output HIGH before switching, Vh = Vcc)
          hysteresis = VH - VL = (Vh - Vl) x R1/Rf
          NTC:  R = R0 x exp( B x (1/T - 1/T0) )     T = 1 / ( 1/T0 + ln(R/R0)/B )     Vs = 3.3 x Rb / (Rt + Rb)

(a) 5 V rail-to-rail comparator, Vref = 2.5 V, R1 = 10 k, Rf = 220 k:   R1/Rf = 0.04545
      VH = 2.5 x 1.04545 - 0 = 2.614 V
      VL = 2.614 - 5 x 0.04545 = 2.614 - 0.227 = 2.386 V
      hysteresis = 5 x 0.04545 = 0.227 V

(b) Fan: ON at 40 C, OFF at 35 C; NTC 10 k (B = 3950) on top, 10 k below, 3.3 V.  Vs rises with temperature, so a non-inverting
    comparator (Vs through R1 to the + input, Vref on the - input) turns the fan on when it gets warm.
    1. Resistances:  R(40 C) = 10 k x exp(3950 x (1/313.15 - 1/298.15)) = 10 k x exp(-0.6347) = 5.301 k
                     R(35 C) = 10 k x exp(3950 x (1/308.15 - 1/298.15)) = 10 k x exp(-0.4300) = 6.506 k
    2. Divider voltages:  Vs(40 C) = 3.3 x 10 / (5.301 + 10) = 2.157 V        Vs(35 C) = 3.3 x 10 / (6.506 + 10) = 1.999 V
       VH = 2.157 V (rising, fan ON),  VL = 1.999 V (falling, fan OFF)        hysteresis = 0.157 V
    3. R1 / Rf = hysteresis / Vcc = 0.157 / 3.3 = 0.0477    ->    Rf = 10 k / 0.0477 = 210 k  ->  E12 value 220 k  (R1/Rf = 0.04545)
    4. Vref from VH = Vref x (1 + R1/Rf):   Vref = 2.157 / 1.04545 = 2.063 V
       Divider from 3.3 V: ratio 2.063 / 3.3 = 0.625.  12 k (top) and 20 k (bottom, two 10 k in series):
       Vref = 3.3 x 20 / 32 = 2.0625 V
    5. Actual thresholds:   VH = 2.0625 x 1.04545 = 2.156 V        VL = 2.156 - 3.3 x 0.04545 = 2.156 - 0.150 = 2.006 V
       Temperatures:  VH: Rt = 10 k x (3.3/2.156 - 1) = 5.305 k   ->  39.98 C       (fan ON at 40.0 C)
                      VL: Rt = 10 k x (3.3/2.006 - 1) = 6.448 k   ->  35.2 C        (fan OFF at 35.2 C)
    The thermistor divider has a source resistance (NTC in parallel with 10 k, about 3.5 k at 40 C) that adds to R1 and so shifts the
    hysteresis; put a follower between the divider and R1, as the build does, or make R1 and Rf much larger.
    The 12 k / 20 k reference draws 3.3 / 32 k = 0.1 mA; it is "stiff" because it only feeds a high-impedance input.

(c) Stage A with Rf = 47 k:   R1/Rf = 10 / 47 = 0.2128
      VH = 1.65 x 1.2128 = 2.001 V          VL = 2.001 - 3.3 x 0.2128 = 2.001 - 0.702 = 1.299 V          hysteresis = 0.702 V
      LDR resistance at the switch points (LDR on top, 10 k below):
        rising  Vs = 2.001 V:  R = 10 k x (3.3 / 2.001 - 1) = 6.49 k     (LED ON below 6.5 k)
        falling Vs = 1.299 V:  R = 10 k x (3.3 / 1.299 - 1) = 15.4 k     (LED OFF above 15.4 k)
      (with Rf = 100 k it was 8.18 k and 12.2 k, a hysteresis of 0.33 V)
      A 50 mV peak-to-peak flicker is well inside both dead bands (0.33 V and 0.70 V), so neither choice can switch falsely because of it.
      The 47 k design has the wider dead band: the light must change by a ratio of 15.4 / 6.49 = 2.4 : 1 between ON and OFF
      (against 1.5 : 1 for 100 k), so it is more robust against noise and slow drifts, but less sensitive to small light changes.

Checklist
  - Hysteresis = Vcc x R1/Rf; choose it a few times larger than the noise.
  - Vref = VH / (1 + R1/Rf) when Vl = 0.
  - Positive feedback goes to the + input; check which input the signal and which the reference use.
  - A high-impedance divider needs a buffer, or its source resistance must be added to R1.