1. LEAD ACID BATTERY MODELING IEEE ESSB Summer 2016 Meeting
Frank X. Garcia
12 June 2016

3. Presentation Objectives
Explain lead acid cell operation at the atomic level
Present a Randles circuit model approximation of a lead acid cell
Examine computer simulations trending using Randles circuit model

4. Outline Battery Overview Randles circuit model Components
Electrochemistry
Double layer capacitance
State of health parameters
Randles circuit model
Single cell approximation
Failure modes analysis
Computer Simulations

5. Battery Components A battery is an energy storage device
Converts chemical energy into electrical energy during a discharge, and can also
Store electrical energy during a recharge
The major components of a lead acid battery include:
ELECTROLYTE - aqueous conductor of ions between the negative and positive plates, H2SO4 with H20
(38% concentration of sulfuric acid)
ANODE - terminal post connected to the negative plates
CATHODE- terminal post connected to the positive plates
NEGATIVE PLATES - lead grids filled with pure, spongy Pb
POSITIVE PLATES - lead grids filled with PbO2
SEPARATOR - non-conductive material that separates the negative and positive plates and prevents them from shorting
The negative battery post terminal connects to a spongy lead (Pb) electrode grid
The positive battery post terminal connects to a porous lead dioxide (PbO2) electrode grid
Electrodes are arranged in a parallel matrix cluster of interspersed positive and negative electrodes, separated and immersed in electrolyte fluid, in which same voltage polarity electrodes are attached to each other
Electrode matrix cluster creates one cell. Multiple electrode clusters are connected in series to form a battery.
Electrolyte maintains chemical balance by maintaining system neutrality. Electrolyte contains sulfuric acid (H2SO4) in water (H20) to form an ionic media, in which ions pass in-between electrodes.
Both plates are immersed in electrolyte fluid
Separators are porous and conductive (???) and keep electrodes from shorting to each other.

6. Electrochemistry - New electrodes inserted into electrolyte
Cathode +
- Anode
Cathode +
H2O H+
SO4-2
Pb+2 -
- Anode
PbO2
Pb0
H2O H+
SO4-2 H+ H+ H+
SO4-2
SO4-2 H+
New electrodes inserted into electrolyte
Reduction Oxidation

7. Electrochemistry Reduction Oxidation - @ T = 25C - Anode0V
-0.36 V
1.69V
Double Layer
Capacitances
Diffusion Layer s*
* Discussed Later
Vcell = 1.69V – (-0.36V) = 2.05V
Cathode +
H2O H+
SO4-2
Pb+2 -
- Anode
@ T = 25C
Reduction
Pb+4 + 2O-2(s) + 4H+(aq) + SO4-2(aq) + 2e PbSO4(s) + 2H2O(aq)
Energy released E0 = 1.69 eV
Oxidation
Pb(s) + SO4-2(aq) PbSO4(s) + 2e-
Energy released E0 = eV

8. Charge Cycle: Electrochemical process is reversed
Electrochemistry
Discharge Cycle
Charge Cycle: Electrochemical process is reversed
Vcell + - - I
Iload -
H2O
H2O H+
SO4-2 - - -
H2O
SO4-2
SO4-2
Pb+2
Pb+2
Reduction Oxidation

9. Double Layer Capacitance
Ions adsorbed to the surface of the electrode held by electrostatic force
Positive ions too large to penetrate electrode metal surface
Only electrons can travel through the metal conduction bands
Solvated ions encapsulated by water molecules can migrate through diffusion layer
A second layer of solvated ions create the outer layer forming capacitive double layer
Double layer has an ionic density gradient which allow Ionic migration in the diffusion layer
Bulk region maintains an equal concentration of electrolyte

10. State of Health Parameters
Float voltage
Set to recommended range for optimal battery life
Life expectancy decreases as float voltage increases
Float current
A high float current indicates aging battery
Negative post temperature
Indefinite 10o C temperature rise decrease battery life by 50%
Electrolyte Specific Gravity
Sulfuric acid (H2SO4) concentration can indicate state of charge
Fully charged: 1.26 to 1.3 specific gravity
Admittance/Impedance/Resistance Trending
Admittance will decrease as the battery ages
Impedance and Resistance will increase as battery ages

11. Complete Randles Circuit Model
Rm+
Rm-
Lumped, metallic resistance of the posts, bus bars, grids, plates, and metallic joints at the cathode, Rm+, and the anode, Rm-
Cdl+
Cdl-
Double layer capacitance at the interface between the positive plate surface and the electrolyte, Cdl+, and the negative plate surface and the electrolyte interface, Cdl-. Cdl depends on the effective surface area of each plate, and the battery state of charge, and it is a combination of the series connection of a double layer capacitor and a diffusion layer capacitor.
Rct+
Rct-
Equivalent charge transfer resistance at the interface between the positive plate and the electrolyte due to the ionization losses of Pb, Rct+, and the interface between the negative plate and the electrolyte, Rct-. These are resistive paths that are tunneled through Cdl+ and Cdl-, respectively.
Zdiff+
Zdiff-
Diffusion layer impedance in the diffusion layer near the positive plate, Zdiff+, and the negative plate, Zdiff-. These elements are modeled using the Warburg Diffusion Element, which has a constant phase at all frequencies and a magnitude inversely proportional to the frequency. Refer to Appendix A for a review of the Warburg Diffusion Element.
Cbulk
Equivalent bulk capacitance of the electrolyte material. Note: Cbulk >> Cdl+, Cdl-.
 Rbulk
Equivalent bulk resistance of the electrolyte material
1.69V 
Voltage potential at the cathode relative to the bulk electrolyte, stemming from the Reduction release of energy
-0.36V 
Voltage potential at the negative plate relative to the bulk electrolyte, stemming from the Oxidation release of energy

12. Randles Circuit Model Simplification
Step 1
Ignore Cbulk since its magnitude is much larger than Cdl- and Cd+
Step 2
Add the series resistances Rm+, Rm- and Rbulk
Step 3
Add the 2 voltage potentials V +0.36V = 2.05V

13. Randles Circuit Model Simplification
Note 1
Most models in the literature combine the half-cell circuits resulting in Cdl_eq, Rct_eq and Zdiff_eq
Note 2
Many models in the literature often ignore Zdiff

14. Failure Modes Analysis
Description
Battery Admittance
Corrosion
Negative strap corrosion
Plate lugs, plate strap/lead corrosion
Positive grid corrosion (grid expansion)
Part of aging process
Decreases
Dry-out
Electrolyte Reduction due to overcharging
Premature capacity loss
Shedding of positive active material due to heavy cycling
Sulphation
Negative electrode irreversible sulphation (lead sulfate crystallizing on plate due to undercharging)
Electrolyte Stratification
Electrolyte settles to the bottom of the cell when not in use or not fully charged
Loss of Compression
VRLA glass mat

15. Randles Circuit Parameters Progression
Listed research shows that Randles circuit parameters can represent battery aging affects and failure modes.
A SINGLE INTEGRATED CIRCUIT APPROACH TO REAL CAPACITY
ESTIMATION AND LIFE MANAGEMENT OF VRLA BATTERIES
N.D. Scott
Guardian Link Ltd, United Kingdom
Scott NO., Schooling S. A breakthrough in on"line
VRLA battery testing. Proc.ERA battery conference 98-
Schooling S, Welstead P, 'et aI. A systems
identification approach to the analysis of VRLA batteries.
Proc. INTELEC 1999
Champlin KS., Bertness K. A fundamentally new
approach to battery, performance analysis using
DFRAIDFIS technology. Proc. INTELEC 2000
Randles circuit parameters progression over the cell lifetime or discharge

16. Complete Randles Model
Computer Simulations
Strategy
Examine degradation of a cell using complete Randles Circuit Model
Simulate Nyquist plots of baseline, +20% and +40% impedance due to cell aging
Synthesize component values to determine parametric changes
Circuit Model (one cell)
Schematic
Designation
Equivalent,
Complete Randles Model R3
Rm+ + Rm- R1
Rct+ C1
Cdl+ R2
Rct- C2
Cdl- X1
Zdiff+ (Warburg Constant Phase Element, cpe) X2
Zdiff- (Warburg Constant Phase Element, cpe) C3
Cbulk R4
Rbulk

17. Computer Simulations Zim Zre Nyquist Plots Component Synthesis
Baseline impedance: 1
+20% Impedance: 2
Component Synthesis
Schematic
Designation
Randles
Equivalent
Change
From Baseline R3
Rm+ + Rm-  R1
Rct+ C1
Cdl+  R2
Rct- C2
Cdl- C*
Zdiff+
Zdiff- * - C3
Cbulk R4
Rbulk 1 2
Zim
Figure 1 reference and +20% aged/degraded impedance curve
Battery Aged/Degraded by 20%
Nyquist plot of reference battery impedance with an aged/degraded 20% increased battery impedance.
Smaller curve represents the reference battery impedance and the wider curve is the degraded impedance curve.
Most synthesized battery component values appear to track the impedance spectra. So if the battery degrades, then its impedance increases, and battery model values would trend accordingly. Battery model resistor components should increase and capacitor values should decrease.
5kHz Hz
Zre
* C – constant of the Warburg element,
 - constant phase of the Warburg element

18. Computer Simulations Zim Zre Nyquist Plots Component Synthesis
Baseline impedance: 1
+40% Impedance: 2
Component Synthesis
Schematic
Designation
Randles
Equivalent
Change
From Baseline R3
Rm+ + Rm- + R1
Rct+ C1
Cdl+ - R2
Rct- C2
Cdl- C*
Zdiff+
Zdiff- * - C3
Cbulk R4
Rbulk 1 2
Zim
Figure 1 reference and +20% aged/degraded impedance curve
Battery Aged/Degraded by 20%
Nyquist plot of reference battery impedance with an aged/degraded 20% increased battery impedance.
Smaller curve represents the reference battery impedance and the wider curve is the degraded impedance curve.
Most synthesized battery component values appear to track the impedance spectra. So if the battery degrades, then its impedance increases, and battery model values would trend accordingly. Battery model resistor components should increase and capacitor values should decrease.
5kHz Hz
Zre
* C – constant of the Warburg element,
 - constant phase of the Warburg element

19. Summary
Randles circuit model approximates electrochemistry of a lead acid cell
Simplified Randles circuit model reduces analysis accuracy
Trending in Randles element values add visibility to battery state of health
Baseline immitance using Discrete Frequency Immitance Spectroscopy
Synthesize Randles circuit model battery elements
Analyze battery degradation by comparing element value changes from a known reference