Experiment 12: Cells in Series and Parallel - Theory & Concepts

Experiment 12: Cells in Series and Parallel

$Learning Objectives$

Identify cell elements and distinguish emf, loaded terminal voltage, internal resistance, capacity, stored energy, state of charge, and C-rate with correct units.
Derive equivalent emf and internal resistance for series, parallel, and series-parallel arrangements using explicit polarity and matched-cell assumptions.
Predict load current, voltage sag, internal heating, efficiency, current sharing, and the consequences of reversed or unequal cells.
Connect voltage and current meters safely and compare measured results with ideal predictions while accounting for resolution, burden, contact resistance, and changing cell state.
Plan and document an instructor-approved low-voltage investigation that avoids short circuits, incompatible cells, unsafe meter placement, and unsupported high-current tests.

Electrochemical cells convert chemical energy into electrical energy. Connecting cells changes the voltage, current capability, capacity, internal resistance, and total stored energy of the source. This topic develops the theory needed to analyze real battery packs before applying it to a low-voltage laboratory activity.

$Target Learning Outcome$

Compare cells in series, parallel, and series-parallel arrangements by relating their polarity, equivalent emf, internal resistance, current capability, capacity, energy, and practical safety requirements.

1. Electrochemical Sources

$Electrochemical Cell$

An electrochemical cell is a device that converts chemical energy into electrical energy through oxidation and reduction reactions occurring at separate electrodes.

$Battery$

A battery is an assembly of one or more electrochemical cells connected and packaged to provide electrical energy. Everyday language often calls a single cell a battery, but the terms are not technically identical.

$Electrode$

An electrode is a conductive region where an oxidation or reduction reaction exchanges electrons with the external circuit.

$Electrolyte$

An electrolyte is an ion-conducting material that permits charge transport inside the cell while electrons travel through the external circuit.

$Separator$

A separator is an electrically insulating, ion-permeable layer that prevents direct electronic contact between the electrodes while allowing ionic conduction.

$How a discharging cell completes a circuit$

During discharge, chemical reactions drive electrons from the negative terminal through the external load toward the positive terminal. Inside the cell, ions move through the electrolyte to preserve charge balance. The external electron flow and internal ionic flow are both necessary for continuous current.

A cell stops delivering useful current when reactants are depleted, reaction products obstruct transport, internal resistance becomes excessive, or the terminal voltage falls below the load requirement.

$Primary and secondary cells$

A primary cell is intended mainly for one discharge cycle and is not normally recharged. A secondary cell uses reactions that can be reversed by an external charging source. Rechargeability depends on chemistry and construction, not merely on applying a reverse voltage.

2. Electrical Quantities of a Cell

$Electromotive Force$

Electromotive force, represented by $E$ or $\mathcal{E}$ , is the energy supplied by a source per unit charge. Despite its name, emf is a voltage, not a mechanical force.

Electromotive force

Emf is the source energy supplied for each coulomb of charge transferred through the circuit.

E = \frac{W_{\text{source}}}{q}

Variables

Symbol	Description	Unit
$E$	electromotive force	V or J/C
$W_{\text{source}}$	energy supplied by the source	J
$q$	charge transferred	C

$Open-Circuit Voltage$

Open-circuit voltage is the terminal voltage measured when no intentional external load is connected. Because a voltmeter draws very little current, this value is often used as an experimental approximation of emf.

$Terminal Voltage$

Terminal voltage $V_{\text{t}}$ is the potential difference available across the external terminals while the cell is operating under a specified condition.

$Internal Resistance$

Internal resistance $r$ is the effective opposition to current inside a real cell. It represents combined electrochemical, ionic, contact, and material losses that cause heating and voltage drop under load.

$Ideal and real source models$

An ideal voltage source maintains constant terminal voltage regardless of current. A real cell is better represented by an ideal emf $E$ in series with an internal resistance $r$ .

With the circuit open, $I\approx0$ , so $V_{\text{t}}\approx E$ .
During discharge, the internal drop $Ir$ reduces terminal voltage.
At larger current, voltage sag and internal heating become more significant.
Internal resistance is not perfectly constant; it changes with chemistry, temperature, state of charge, age, and current history.

Terminal voltage during discharge

The available terminal voltage equals emf minus the internal voltage drop while the cell supplies current.

V_{\text{t}} = E - Ir

Variables

Symbol	Description	Unit
$V_{\text{t}}$	terminal voltage under load	V
$E$	cell emf	V
$I$	discharge current	A
$r$	internal resistance	$\Omega$

Current supplied to a resistive load

The external resistance and internal resistance form one series path in the real-source model.

I = \frac{E}{R+r}

Variables

Symbol	Description	Unit
$R$	external load resistance	$\Omega$

Internal power loss

Internal resistance converts part of the source energy into heat inside the cell.

P_{\text{internal}} = I^2r

Variables

Symbol	Description	Unit
$P_{\text{internal}}$	rate of internal energy dissipation	W

$Short-circuit current is not a safe experiment$

The simple model predicts $I_{\text{sc}}=E/r$ when $R=0$ . Because $r$ can be small, the current may be dangerously large and may cause heating, leakage, fire, venting, or permanent damage. Never measure short-circuit current by placing an ammeter directly across a cell.

3. Capacity, Energy, and Discharge Rate

$Capacity$

Capacity is the amount of charge a cell can deliver under specified discharge conditions. It is commonly expressed in ampere-hours rather than coulombs.

Charge represented by ampere-hour capacity

One ampere-hour corresponds to 3600 coulombs of charge.

Q = 3600C_{\text{Ah}}

Variables

Symbol	Description	Unit
$Q$	deliverable charge	C
$C_{\text{Ah}}$	rated capacity	Ah

$Energy Rating$

Energy rating is the approximate electrical energy a cell or battery can deliver. It is commonly expressed in watt-hours.

Approximate stored electrical energy

Nominal voltage multiplied by rated capacity gives a convenient energy estimate.

W_{\text{Wh}} \approx V_{\text{nom}}C_{\text{Ah}}

Variables

Symbol	Description	Unit
$W_{\text{Wh}}$	nominal stored energy	Wh
$V_{\text{nom}}$	nominal cell or pack voltage	V

$State of Charge$

State of charge is the fraction of usable capacity remaining relative to a defined fully charged condition.

$C-Rate$

C-rate expresses current relative to rated capacity. A discharge at $1C$ would ideally use the rated capacity in approximately one hour, while $0.5C$ would ideally take approximately two hours.

C-rate

Discharge current is normalized by the rated ampere-hour capacity.

\text{C-rate} = \frac{I}{C_{\text{Ah}}}

Variables

Symbol	Description	Unit
$I$	charge or discharge current	A
$C_{\text{Ah}}$	rated capacity	Ah

$Why rated capacity is conditional$

Actual capacity depends on discharge current, cutoff voltage, temperature, age, chemistry, and rest periods. The simple runtime estimate $t=C_{\text{Ah}}/I$ is useful for comparison but is not a guarantee of actual service time.

Series-Aiding Cell Path

Series cells form one current path. Their emfs add algebraically by polarity and their internal resistances add because the same current passes through each cell.

Cell 1: orient polarity to aid the string

Cell 2: same current passes through the cell and internal resistance

Additional series cells: continue the same current path

Load: receives the string terminal voltage after internal drops

4. Cells Connected in Series

$Series-Aiding Connection$

A series-aiding connection joins the positive terminal of one cell to the negative terminal of the next so that the cell emfs act in the same direction.

$Electrical behavior of a series string$

The same current passes through every cell in a series string. Series connection primarily increases voltage. For matched cells, the ampere-hour capacity remains approximately equal to one cell, while the total watt-hour energy increases because the pack voltage increases.

Equivalent emf of series-aiding cells

Cell emfs add algebraically according to their polarity orientation.

E_{\text{s}} = E_1 + E_2 + \cdots + E_n

For $n$ identical aiding cells,

E_{\text{s}} = nE

Variables

Symbol	Description	Unit
$E_{\text{s}}$	equivalent series emf	V
$n$	number of identical cells	dimensionless

Equivalent internal resistance of series cells

The internal resistances add because the same current passes through each cell.

r_{\text{s}} = r_1+r_2+\cdots+r_n

For $n$ identical cells,

r_{\text{s}}=nr

Variables

Symbol	Description	Unit
$r_{\text{s}}$	equivalent series internal resistance	$\Omega$

Load current from identical series cells

The increased emf and increased internal resistance both influence the load current.

I_{\text{s}} = \frac{nE}{R+nr}

Variables

Symbol	Description	Unit
$I_{\text{s}}$	current supplied by the series string	A

$Capacity and energy of matched series cells$

For $n$ identical cells, each rated $E$ volts and $C_{\text{Ah}}$ ampere-hours:

Pack voltage is approximately $nE$ .
Pack ampere-hour capacity is approximately $C_{\text{Ah}}$ .
Pack energy is approximately $nEC_{\text{Ah}}$ watt-hours.
Every cell carries the full pack current.
The weakest cell limits the usable series-string capacity.

$Series-Opposing Connection$

A series-opposing connection places at least one cell polarity opposite to the others, causing its emf to subtract from the aiding emfs.

$Consequences of a reversed series cell$

A reversed cell reduces net voltage and may be forced into reverse charging by the remaining cells. Reverse charging can damage the cell, especially when the reversed or depleted cell is the weakest member of the string.

Matched Parallel Cell Current Sharing

Matched cells share the same two nodes. Branch currents add at the load; unequal emf or internal resistance can drive unequal sharing or circulating current.

Ideal matched-cell model

Each cell has the same emf and comparable state of charge
All branches share the same terminal voltage
Equivalent internal resistance is lower and branch currents share the load

Real mismatch to watch

Different open-circuit voltages can drive circulating current
Different internal resistance causes unequal branch current
Chemistry, capacity, temperature, age, and protection must be compatible

5. Cells Connected in Parallel

$Parallel Cell Connection$

A parallel connection joins all positive terminals to one common node and all negative terminals to another common node.

$Electrical behavior of matched parallel cells$

Matched cells in parallel operate at approximately the voltage of one cell. Parallel connection primarily increases total capacity and current capability while reducing equivalent internal resistance. Ideally, the cells share the load current.

Equivalent emf of matched parallel cells

Identical cells connected across the same two nodes retain the emf of one cell.

E_{\text{p}} = E

Variables

Symbol	Description	Unit
$E_{\text{p}}$	equivalent parallel emf	V

Equivalent internal resistance of parallel cells

Internal resistances combine using the parallel-resistance relationship.

\frac{1}{r_{\text{p}}}=\frac{1}{r_1}+\frac{1}{r_2}+\cdots+\frac{1}{r_n}

For $n$ identical cells,

r_{\text{p}}=\frac{r}{n}

Variables

Symbol	Description	Unit
$r_{\text{p}}$	equivalent parallel internal resistance	$\Omega$

Load current from identical parallel cells

Reduced equivalent internal resistance improves terminal-voltage regulation under substantial load.

I_{\text{p}} = \frac{E}{R+r/n}

Variables

Symbol	Description	Unit
$I_{\text{p}}$	total current supplied by the parallel group	A

$Capacity and energy of matched parallel cells$

For $n$ identical cells, each rated $E$ volts and $C_{\text{Ah}}$ ampere-hours:

Pack voltage is approximately $E$ .
Pack capacity is approximately $nC_{\text{Ah}}$ .
Pack energy is approximately $nEC_{\text{Ah}}$ watt-hours.
Ideally, each cell supplies approximately $I_{\text{total}}/n$ .
Reduced current per cell can reduce voltage sag and internal heating.

$Circulating Current$

Circulating current is current that flows between paralleled sources because their open-circuit voltages are unequal, even when the external load is small or disconnected.

$Directly paralleling unequal cells$

Do not directly parallel cells with different chemistries, nominal voltages, states of charge, capacities, ages, temperatures, or damage conditions. The higher-voltage cell may force current into the lower-voltage cell, producing heating, unwanted charging, leakage, or failure.

6. Current Sharing and Cell Mismatch

$Why parallel current may not divide equally$

Equal current sharing requires closely matched open-circuit voltages, internal resistances, temperatures, wiring resistances, and states of charge. A branch with lower total resistance or slightly higher emf tends to carry more current.

Unequal sharing may accelerate aging because the most heavily loaded cell experiences greater heating and deeper cycling. Practical battery systems may use matched cells, balancing circuits, fuses, current-limiting elements, and a battery-management system.

Branch current for two nonidentical parallel cells

A node-voltage model can determine how unequal sources share a common load.

I_1 = \frac{E_1-V_{\text{t}}}{r_1}

I_2 = \frac{E_2-V_{\text{t}}}{r_2}

I_{\text{load}}=I_1+I_2

Variables

Symbol	Description	Unit
$I_1, I_2$	currents contributed by the individual cells	A
$E_1, E_2$	individual cell emfs	V
$r_1, r_2$	individual internal resistances	$\Omega$

$Matched does not mean perfectly identical$

Manufacturing tolerance and aging make exact equality impossible. Safe parallel operation depends on keeping differences sufficiently small and using protection appropriate to the chemistry, voltage, capacity, and possible fault current.

7. Series-Parallel Battery Packs

$Series-Parallel Arrangement$

A series-parallel arrangement forms equal series strings and connects those equal-voltage strings in parallel.

$Meaning of NsNp notation$

Battery configurations are often described as $N_{\text{s}}S N_{\text{p}}P$ .

$N_{\text{s}}$ is the number of cells in series in each string.
$N_{\text{p}}$ is the number of equal series strings connected in parallel.
Total cell count is $N_{\text{s}}N_{\text{p}}$ .
Voltage is mainly set by $N_{\text{s}}$ .
Capacity and current sharing are mainly set by $N_{\text{p}}$ .

Equivalent properties of a matched NsNp pack

These ideal relationships assume identical cells and balanced current sharing.

E_{\text{pack}}=N_{\text{s}}E

r_{\text{pack}}=\frac{N_{\text{s}}r}{N_{\text{p}}}

C_{\text{pack}}=N_{\text{p}}C_{\text{Ah}}

W_{\text{pack}}\approx N_{\text{s}}N_{\text{p}}EC_{\text{Ah}}

Variables

Symbol	Description	Unit
$E_{\text{pack}}$	nominal pack emf	V
$r_{\text{pack}}$	equivalent pack internal resistance	$\Omega$
$C_{\text{pack}}$	pack capacity	Ah
$W_{\text{pack}}$	nominal pack energy	Wh

$Only equal-voltage strings should be paralleled$

Never directly parallel series strings having different numbers of cells or substantially different voltages. The voltage difference can drive a severe equalizing current between strings.

8. Power Delivery and Efficiency

Power delivered to the external load

The useful electrical power received by a resistive load is determined by its current and voltage.

P_{\text{load}}=IV_{\text{t}}=I^2R

Variables

Symbol	Description	Unit
$P_{\text{load}}$	power delivered to the load	W

Source efficiency in the simple internal-resistance model

Efficiency is the fraction of generated power delivered to the external resistance.

\eta = \frac{P_{\text{load}}}{P_{\text{load}}+P_{\text{internal}}} =\frac{R}{R+r}

Variables

Symbol	Description	Unit
$\eta$	fractional source efficiency	dimensionless

$Load resistance and voltage regulation$

When $R$ is much greater than $r$ , current is modest, terminal voltage stays close to emf, and efficiency is high. When $R$ approaches $r$ , current and internal heating increase while terminal voltage falls significantly. At $R=r$ , the simple model predicts maximum power transfer to the load, but only half of the generated power reaches the load; the other half is lost internally.

9. Interactive Source-Load Simulation

$How to use the circuit simulation for this topic$

Use the supply-voltage control as the equivalent emf of a selected cell arrangement. Change the resistance to represent the external load and observe the resulting current and power. The simulation illustrates source-load relationships but does not model chemistry, capacity fade, cell imbalance, or time-dependent discharge.

12 V

6 Ω

4 Ω

R_{eq}=R_1+R_2,\quad I_1=I_2=I_T

Equivalent R

10.00 Ω

Total current

1.20 A

Total power

14.40 W

Branch I1

1.20 A

Branch I2

1.20 A

Resistor power sum

14.40 W

KVL residual

0.000000 V

$Simulation investigations$

Compare one-cell voltage with twice that voltage to represent two ideal cells in series.
Keep voltage fixed and increase external resistance; observe the reduction in current and power.
Compare the simulated ideal result with the real-cell expression $I=E/(R+r)$ .
Explain why adding internal resistance always lowers load current and terminal voltage.
Identify which variables the circuit simulation omits from a real battery pack.

10. Important Practical Facts

$Factors that change cell performance$

Cell behavior depends strongly on:

Temperature: low temperature commonly increases internal resistance; excessive temperature accelerates degradation and may create safety risks.
State of charge: emf and internal resistance vary through the discharge cycle.
Age and cycle history: repeated use changes capacity and resistance.
Discharge current: high current produces greater voltage sag and heating.
Rest and recovery: terminal voltage may partially recover after a load is removed.
Chemistry: nominal voltage, safe limits, charging method, energy density, and failure modes differ among cell types.

$Battery Management System$

A battery management system is an electronic system that monitors and protects a battery pack by measuring quantities such as cell voltage, current, and temperature and by controlling charging, discharging, balancing, and fault response.

$Cell Balancing$

Cell balancing is the process of reducing differences in state of charge or voltage among series-connected cells so that no individual cell reaches unsafe limits before the others.

$Series strings require cell-level monitoring$

The pack voltage can appear acceptable even when one series cell is overcharged, deeply discharged, reversed, or failing. High-energy rechargeable packs therefore require appropriate cell-level monitoring and protection.

Safe Voltage and Current Measurement Topology

For an instructor-approved low-voltage resistive-load test, the ammeter belongs in series with the load while the voltmeter connects across the load or source terminals.

Confirm cell polarity, chemistry, state, meter lead jacks, and measurement range

Insert the ammeter in series so only the intended load current passes through it

Connect the voltmeter in parallel across the quantity being measured

Energize through the specified current-limiting load; record readings and disconnect safely

$Measured Versus Ideal Circuit Behavior$

With the load disconnected, a high-input-resistance voltmeter reads open-circuit terminal voltage, which approximates emf only within the simple source model. Under load, terminal voltage changes because of internal resistance and electrochemical effects. Take comparable measurements at similar state of charge and temperature, and record the meter range and resolution. Lead/contact resistance and ammeter burden can affect loaded measurements.

Percentage Difference From an Ideal Prediction

A comparison metric using a nonzero predicted value as reference; it is not a complete uncertainty budget.

\%\,\text{difference}=\frac{|X_{\text{measured}}-X_{\text{predicted}}|}{|X_{\text{predicted}}|}\times100\%

Variables

Symbol	Description	Unit
$X_{\text{measured}}$	Measured quantity	-
$X_{\text{predicted}}$	Nonzero ideal-model prediction in the same units	-

$Meter Placement and Cell Safety$

Never connect the current-input terminals of a meter directly across a cell; that creates a very low-resistance path and can damage the meter or cell. Do not intentionally short-circuit cells, reverse-charge a rechargeable cell, mix chemistries or markedly different capacities, or parallel cells with materially different open-circuit voltages. Follow the instructor-approved cell type, load, protection, and manufacturer limits.

11. Laboratory Application

$Pre-laboratory checks$

Use low-voltage cells approved by the instructor.
Confirm that both cells have the same chemistry, nominal voltage, and similar state of charge.
Inspect for leakage, swelling, corrosion, excessive heat, or damaged insulation.
Measure open-circuit voltage before any parallel connection.
Confirm ammeter and voltmeter terminal placement and range selection.
Use a current-limiting load; never short the cells.

$Theory-guided investigation$

STEP-BY-STEP

Measure the open-circuit voltage of each cell separately and compare the values.
Predict the series-aiding voltage from the algebraic sum of the individual cell voltages.
Connect the cells in series aiding, measure the combination voltage, and compare it with the prediction.
Connect a specified load and measure current for one cell and for the series pair.
Only when the cells are approved as closely matched, connect them in parallel using correct polarity.
Measure parallel-combination voltage and load current for the same external resistance.
Compare all observations with the ideal and real-source models.
Disconnect the parallel group after testing and report any heating, unstable reading, or unexpected voltage difference.

Investigate Cell Connections Safely

Measure individual and series-cell behavior, gate the parallel test on instructor approval and close matching, and report observations without short-circuit testing.

Use low-voltage cells approved by the instructor, correct polarity, confirmed meter ranges and terminals, and a current-limiting load. Never connect an ammeter directly across a cell or measure short-circuit current.

Start with instructor-approved low-voltage cells → Check polarity and meter ranges; record each cell's open-circuit voltage; Check polarity and meter ranges; record each cell's open-circuit voltage → Predict the series-aiding voltage from the individual cell voltages; Predict the series-aiding voltage from the individual cell voltages → Connect cells in series aiding; measure combination voltage and compare with prediction; Connect cells in series aiding; measure combination voltage and compare with prediction → Use the specified current-limiting load; measure current for one cell and the series pair; Use the specified current-limiting load; measure current for one cell and the series pair → Are the cells closely matched and approved for parallel connection?; Are the cells closely matched and approved for parallel connection? — Yes → Connect in parallel with correct polarity; measure voltage and load current at the same resistance; Are the cells closely matched and approved for parallel connection? — No → Skip the parallel test; do not connect cells that are not approved as closely matched; Connect in parallel with correct polarity; measure voltage and load current at the same resistance → Disconnect the parallel group after testing; Disconnect the parallel group after testing → Compare observations with ideal and real-source models; note heating or unstable readings; Skip the parallel test; do not connect cells that are not approved as closely matched → Compare observations with ideal and real-source models; note heating or unstable readings; Compare observations with ideal and real-source models; note heating or unstable readings → Report voltage, load-current, comparison, and any unexpected behavior; Report voltage, load-current, comparison, and any unexpected behavior → End

Start with instructor-approved low-voltage cells: terminator
Check polarity and meter ranges; record each cell's open-circuit voltage: process
Predict the series-aiding voltage from the individual cell voltages: process
Connect cells in series aiding; measure combination voltage and compare with prediction: process
Use the specified current-limiting load; measure current for one cell and the series pair: process
Are the cells closely matched and approved for parallel connection?: decision
Connect in parallel with correct polarity; measure voltage and load current at the same resistance: process
Disconnect the parallel group after testing: process
Skip the parallel test; do not connect cells that are not approved as closely matched: process
Compare observations with ideal and real-source models; note heating or unstable readings: process
Report voltage, load-current, comparison, and any unexpected behavior: document
End: terminator

$Expected theoretical trends$

For a load resistance much larger than cell internal resistance:

Two matched cells in series produce approximately twice one-cell voltage and nearly twice one-cell current through the same load.
Two matched cells in parallel produce approximately one-cell voltage and nearly one-cell total current through a large fixed load.
Parallel cells share the total current, so each cell carries less current than one cell would carry alone.
Differences between prediction and measurement may result from internal resistance, cell mismatch, contact resistance, meter loading, temperature, and changing state of charge.

Key Takeaways

A cell converts chemical energy into electrical energy through coupled electrode reactions and ionic transport.
Emf is energy supplied per unit charge, while terminal voltage is the voltage available at the external terminals.
A real cell can be modeled as an ideal emf in series with internal resistance.
During discharge, $V_{\text{t}}=E-Ir$ , and internal heating is $I^2r$ .
Ampere-hour capacity measures charge, while watt-hour rating measures approximate stored energy.
Series-aiding cells add voltage and internal resistance; matched series cells retain approximately one-cell ampere-hour capacity.
Matched parallel cells retain one-cell voltage, add capacity, reduce equivalent internal resistance, and share current.
Unequal cells in parallel can produce circulating current, while a reversed series cell can be forced into reverse charging.
An $N_{\text{s}}S N_{\text{p}}P$ pack uses series cells to set voltage and parallel strings to set capacity and current capability.
Cell voltage, internal resistance, capacity, and safety limits depend on chemistry, temperature, state of charge, age, and current.
Simulations can clarify source-load relationships, but real batteries also require chemical, thermal, aging, and protection considerations.
Safe laboratory work requires matched cells, correct polarity, current-limiting loads, proper meter connections, and zero short-circuit tests.

$References$

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