ChargerGauge

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Let:

  • VV be the open-circuit, aka idle, voltage of a batteries bank
  • S(V)S(V) be the ampere scale function of value from 0% to 100% with respect to the idle voltage, S(V t 1)S(V t 0)=1C t 0 t 1I tdtS(V_{t_1})-S(V_{t_0})=\frac{1}{C} \int_{t_0}^{t_1} I_t d t where CC is the total ampere capacity of the batteries bank and V 0V_0 means having empty voltage. For example of simplicity, it could be S(V)=V207.2S(V) = \frac{V-20}{7.2} then 20 is the empty voltage and 27.2 is the full voltage and C=600= 0 I tdtC=600 = \int_{0}^{\infty} I_t d t is a batteries bank of capacity 600 ampere-hours.
  • XX the voltage seen by the charger, for example X=27.2X=27.2
  • rr be the resistance by the charging circuit and the internal resistance of the batteries bank, so the charging current is I=XVrI=\frac{X-V}{r}, for example r=0.025r=0.025 ohm

What is the needed time TT to charge the batteries bank from idle voltage V 0V_0 to V 1V_1?

Because CS(V)dV=I tdtC S'(V) d V= I_t d t, it follows dt=CS(V)I tdV=rCS(V)XVdVd t = \frac{C S'(V)}{I_t} d V =\frac{r C S'(V)}{X-V} d V, therefore T=rC V 0 V 1S(V)XVdVT=r C \int_{V_0}^{V_1} \frac{S'(V)}{X-V} d V

By the above example, say V 0=26.6V_0=26.6 and V 1=27.0V_1=27.0, then

T=0.0256007.2 26.6 27.0dV27.2V=0.0256007.2ln(27.226.627.227.0)=2.29T=\frac{0.025 \cdot 600}{7.2} \cdot \int_{26.6}^{27.0} \frac{d V}{27.2-V}=\frac{0.025 \cdot 600}{7.2} \cdot \ln(\frac{27.2-26.6}{27.2-27.0})=2.29 hours

What is the total wh of the batteries bank when it is full?

V tI tdt=CVS(V)dV=6007.2 20 27.2VdV=14160\int V_t I_t d t=C \int V S'(V) d V=\frac{600}{7.2} \int_{20}^{27.2} V d V=14160 wh

The better the battery, the more vertical about the curve S(V)S(V) or horizontal if X-axis and Y-axis flipped for the curve. If it is centralized at the voltage V ratingV_{rating} then V tI tdt=CVS(V)dVCV rating\int V_t I_t d t=C \int V S'(V) d V \approx C V_{rating}