(ET) Battery capacity calculation

["Rod Hower" <Rod Hower ametek com>]

This was on another list, thought you would be interested.

Lee Hart <leeahart earthlink net> wrote:
>> Peukert *is* valid for changing discharge currents. The E-Meter
>> uses it this way.

Emil Naepflein wrote:
> This is wrong or at least not very exact. You can use Peukert for
> changing discharge currents, but the result will be pessimistic
> because the battery will regain capacity under no load or small
> load because only so much active mass has been transformed that
> is equivalent to the derived Ahs and the rest of the active mass
> is still available.

Emil, are you familiar with the E-meter? In my experience, it actually
does provide a fairly accurate indication of battery capacity (once it
is programmed to match the battery's actual characteristics). It is
certainly more accurate than the usual voltage-based or
coulombmetric-based state-of-charge meters.

The E-meter keeps track of the number of amphours you have removed from
the battery, and estimates the charge remaining *if you were to keep
drawing current following the same load pattern*. Obviously, if you
reduce your load current, you can draw out more amphours. As you do
this, the E-meter recalculates a new value for the remaining time.

For example, suppose I am driving home. It normally takes me another 30
minutes. But the E-meter tells me I only have 20 minutes of time left at
the present rate. So I slow down to a speed where it would take 40
minutes to get home. The E-meter's 'time remaining' display changes to
45 minutes; now I know I can get home before the pack goes dead.

Or, assume the same example, but I am watching the SOC and amphour
displays instead. I know that it took 20ah to get to work, so I need
20ah to get back home. I also know that 40ah is "dead" (20% SOC) at the
speeds I normally drive. But as I leave work, I see that the SOC is 40%;
not 50%. Thus I know I have to drive slower or I'll run out. So I slow
down, and the currents are less. The E-meter uses Peukert's equation on
an incremental basis to calculate SOC. When I get home, I will have
removed 50ah from the pack (possible because of the lower currents), and
the E-meter will still show 20% SOC (because the pack could deliver more
amphours at the lower discharge rate).

Here is a description of the E-meter's algorithm:
----------------------------------------------------------------------
Practical Application of Peukert's Equation -- by Steve Kahle

One of the most interesting new features implemented in the E-meter is
the treatment of Peukert's
Equation. Peukert describes the effect of varying discharge rates on a
battery, but he neglected to
document a comprehensive description of both charge and discharge
currents needed to determine the
state-of-charge of the battery. CECO engineers have tried to complete
the description of the cycle in a
practical way that will be described below.

Discharging

Using a user selectable capacity (C) and a user selectable Peukert
coefficient (N), the E-meter calculates a
20 hour discharge rate (I20) and capacity (Cp) based on Peukert's
equation. Then a multiplier (M) is
calculated from the ratio of Cp/C. Two state-of-charge capacity values
are tracked:

             Crem = C + (I x time)         and         Cprem = Cp + (Ip x
time)

             I20 = C/20 hr

             Cp = (I20) N x 20 hr

             M = Cp / C

Discharge rates equal to or less than the 20 hr rate are multiplied by
the factor (M) to normalize them to
Cp. Then they are used as the effective discharge rate (-Ip). Example:

             C = 200 Ahr             N = 1.25

             I20 = 200 Ahr / 20 hr = 10 A

             Cp = (10 A)^1.25 x 20 hr = 355.6 Ahr

             M = 355.6 Ahr / 200 Ahr = 1.778

Therefore a 20 hr discharge at 10 A would deplete the 100% of the
battery capacity using Crem or Cprem
at the same rate.

Discharge rates greater than the 20 hr rate are calculated using
Peukert's equation and have a greater
effect on capacity, reducing the time the load may be supplied. Example:

             C = 200 Ahr             N = 1.25          I = 30 A

             time = (Crem - C) / I = (0 Ahr - 200 Ahr) / 30 A = 6.67 hrs
       (this is
time to deplete the battery)

             Ip = (30 A)^1.25 = 70.2 A

             time = (Cprem - Cp) / Ip = (0 Ahr - 355.6 Ahr) / 70.2 A = 5.06
hrs
(Peukert time to deplete)

As the discharge rate goes up, the greater effect Peukert's equation
has.

Charging

When I goes from discharge to charge, a multiplier (U) is calculated
from the ratio of Cprem / Crem.
Charging current is then the observed current (I) multiplied by (U) and
then multiplied by the battery's
efficiency (CEF) to calculate an effective charge rate (+Ip). Further
Cprem is limited while rising by Crem
as an upper limit. Example:

             C = 200 Ahr             I = +30 A         Crem = 100 Ahr
        Cprem = 148 Ahr        CEF = 90%

             U = 148 Ahr / 100 Ahr = 1.48

             +Ip = (30 A) x 1.48 x 0.90 = 39.96 A

This way as the battery is charged both Crem and Cprem will reach full
at the same time.

Summary

Using this combination of formulas to determine a Peukert's capacity
(Cp), a 20 hr discharge rate (I20),
multipliers for discharge (M) and charge (U), with given data
representing a 20 hr capacity (C), charge
efficiency (CEF) and Peukert's coefficient (N), the E-meter can
calculate an accurate state-of-charge on a
wide variety of batteries and applications. The break point at the 20 hr
discharge rate with the (M)
multiplier used at or below the 20 hr rate removes the portion of the
performance curve where Peukert's
equation diverges from reality. The charge multiplier (U) used with the
battery's own efficiency (CEF)
finishes the other half of the cycle that Peukert didn't choose to
address.
--------------------------------------------------------------------
"Never doubt that a small group of committed people can change the
world. Indeed, it's the only thing that ever has!" -- Margaret Meade
--
Lee A. Hart  814 8th Ave N  Sartell MN 56377  leeahart_at_earthlink.net