The undervoltage condition for this design is when the battery voltage, VBATT, is 10-V. The voltage seen at the pin of the comparator must not exceed 5.7-V as specified in the Recommended Operating Condition table of the TLV902x and TLV903x High-Precision Dual and Quad Comparators Data Sheet.

To determine how much the battery voltage should be reduced to stay below the 5.7-V input voltage limit, consider the maximum battery voltage of 48-V. At this value, the voltage must be divided down by a factor of 8.42. For this example, we round up to 9 to be safe.

This means that the following equation must be true to yield a maximum non-inverting input voltage of approximately 5.33-V.

Next, use the external hysteresis tool posted on this E2E thread. This excel tool is based on the Inverting Comparator with Hysteresis Circuit and Non-inverting comparator with hysteresis circuit amplifier cookbooks. Due to the configuration of the tool, we need to create a Thevenin Equivalent source and resistance for VBATT, R6, and R7 to match Vin and R1 in the hysteresis tool circuit on the right side of the image below.

Therefore, the equivalent R1 for the tool is the parallel combination of R6 and R7. R8 is equal to the R2 output value, R3 is the selected 10-kΩ pullup, and R4 and R5 are consistent for the tool and the design. The Thevenin equivalent undervoltage value would be the undervoltage threshold value divided by 9. This would be approximately 1.11-V for the 10-V undervoltage value. In this design, we have a desired undervoltage reset (V_RES) of 12-V when recovering from an undervoltage condition. This V_RES level of 12-V also has to be divided by 9, translating to a 1.333-V input into the tool. V_CC, V_DIV, and V_PU are all equal to the reference voltage established by the TL431 device of 3.29-V.

Using the values provided by the tool, we know that the parallel combination of R7 and R6 must equal R1. This means that the following equation must be true.

Using the previous equation and the following equation, solve for the ideal values of R7 and R6.

For this design, the ideal values of R7 and R6 are 76.38kΩ and 611.01kΩ, respectively. For this design, the voltage will most likely be decreasing. This means that the section of the hysteresis transfer curve highlighted by the following figure is important when determining a standard resistor value for R6.

If a standard resistor value less than 611.01kΩ is selected for R6, then the voltage seen at the non-inverting pin of the CH1 comparator will be slightly higher than desired. In other words, the voltage drop across R6 will be smaller as its resistance decreases. With a fixed, divided voltage seen at the non-inverting pin, it is important to select a resistance slightly larger than 611.01kΩ for R6. This would result in a high-to-low transition slightly before the desired 10-V value (further to the right on the x-axis in the previous figure). This is preferable because lithium-ion batteries must not be depleted below the rated voltage value. This is why a slightly higher than theoretical 1% standard resistor value of 619kΩ was selected for R6. See the impact of this resistor selection on the following voltage transfer curve.

For this design, the standard 1% values of R7 and R6 are 76.8kΩ and 619kΩ, respectively.

VTL must also be divided down to match the 1.111-V undervoltage threshold. Even though the tool provides values for R4 and R5, both of these resistances do not impact the positive feedback. This means the ratio of the two resistances is all that needs to be maintained. This is why we are able to select R4 and R5 to be 164kΩ and 100kΩ, respectively.