The actual HOT and COLD trip temperatures vary with component and device tolerances. Therefore, to verify that the required charging limits are still met across all conditions, the TS network should be analyzed using worst-case analysis. This is the error analysis method implemented in the current-based TS calculator in the TI Charger GUI. In this note, worst-case analysis means calculating the most extreme HOT and COLD trip temperatures using the minimum and maximum values of the TS network parameters.

Use the following procedure to perform worst-case analysis on the TS network.

1. Identify the worst-case values of the TS network parameters.

   • NTC R₂₅ tolerance
   • NTC beta tolerance
   • R_S and R_P resistor tolerances
   • TS bias current variation (min/typ/max)
   • TS threshold variation (min/typ/max)
2. Calculate the NTC resistance at the trip points for the worst-case parameter values using Equation 9. To derive Equation 9, convert the selected TS threshold voltage into an equivalent TS network resistance.
   Equation 5. $V_{TH}=I_{BIAS}{R}_{eq}$
   Equation 6. $R_{eq}=\frac{V_{TH}}{I_{BIAS}}$

   Using the compensated TS network equation:

   Equation 7. $R_{eq}=R_P\parallel \left(R_S+R_{NTC}\right)$

   Solving for R_NTC:

   Equation 8. $R_{NTC}=\frac{R_{eq}\left(R_P+R_S\right)-R_P{R}_S}{R_P-R_{eq}}$
   Equation 9. $R_{NTC}=\frac{\frac{V_{TH}}{I_{BIAS}}\left(R_P+R_S\right)-R_P{R}_S}{R_P-\frac{V_{TH}}{I_{BIAS}}}$

   The worst-case R_NTC equations are:

   Equation 10. $R_{NTC,max}=\frac{\frac{V_{TH,max}}{I_{BIAS,min}}\left(R_{P,min}+R_{S,min}\right)-R_{P,min}{R}_{S,min}}{R_{P,min}-\frac{V_{TH,max}}{I_{BIAS,min}}}$
   Equation 11. $R_{NTC,min}=\frac{\frac{V_{TH,min}}{I_{BIAS,max}}\left(R_{P,max}+R_{S,max}\right)-R_{P,max}{R}_{S,max}}{R_{P,max}-\frac{V_{TH,min}}{I_{BIAS,max}}}$
3. Convert the calculated R_NTC trip point resistance to temperature using the R-T table in the NTC datasheet. If the calculated R_NTC falls between two rows, linear interpolation can be used to calculate the corresponding temperature. If the calculated R_NTC falls within the error range for two rows, choose the worst-case temperature.
   • (T₁, R₁)
   • (T₂, R₂)
   Equation 12. $\frac{T-T_1}{T_2-T_1}=\frac{R_{NTC}-R_1}{R_2-R_1}$
   Equation 13. $T=T_1+\frac{R_{NTC}-R_1}{R_2-R_1}\left(T_2-T_1\right)$

   Repeat for the remaining cases to determine the worst-case trip temperature range.

   If an R-T table is not available, the beta equation given in Equation 14 can be used to calculate the trip temperature using the worst-case values for beta and R₂₅.

   Equation 14. $T={\left(\frac{1}{T_0}+\frac{1}{\beta }\ln \left(\frac{R_{NTC}}{R_{25}}\right)\right)}^{-1}$

   Where T is in kelvins

   Worst-case minimum trip temperature:

   Equation 15. $T={\left(\frac{1}{T_{25}}+\frac{1}{{\beta }_{min}}\ln \left(\frac{R_{NTC,max}}{R_{25,min}}\right)\right)}^{-1}$

   Worst-case maximum trip temperature:

   Equation 16. $T={\left(\frac{1}{T_{25}}+\frac{1}{{\beta }_{max}}\ln \left(\frac{R_{NTC,min}}{R_{25,max}}\right)\right)}^{-1}$