Cells in Series

When you need more power from a battery-operated device, manufacturers often place multiple cells inside in a line — think of the four AA batteries in a television remote. This arrangement is called cells in series. In a series connection, the positive terminal of one cell is connected directly to the negative terminal of the next. The result is that the electromotive forces (EMFs) of each cell add together, giving a larger overall voltage that can drive more current around the circuit. This is exactly why a 9 V battery contains six 1.5 V cells stacked end-to-end inside its casing. However, there is a trade-off. Every real cell has an internal resistance — a small resistance inside the cell itself, arising from the resistance of the electrolyte and electrodes. When cells are connected in series, their internal resistances also add together, just like resistors in series. This increased internal resistance limits the maximum current the battery can supply and causes a larger voltage drop inside the battery when current flows. Understanding both effects — increased EMF and increased internal resistance — is essential for designing circuits that work efficiently and safely. Series cells are used whenever a higher terminal voltage is needed than a single cell can provide. The key principle is simple: EMFs add arithmetically, and so do internal resistances. Whether the cells are identical or have different ratings, the same rule applies. Mastering this concept allows you to predict exactly how a battery bank will behave in any circuit.

Key takeaways
  • What 'series' means for cells
  • Adding EMFs
  • Adding internal resistances
  • Terminal voltage and current
  • Aiding vs. opposing cells

What 'series' means for cells

Cells are in series when they are connected end-to-end so that current has only one path to follow through all of them. The positive terminal of cell 1 connects to the negative terminal of cell 2, and so on. Because charge flows through every cell in turn, each cell does work on that charge, and the total work done per unit charge — the total EMF — is the sum of the individual EMFs.

Adding EMFs

EMF (symbol ε, measured in volts) is the energy supplied per coulomb of charge by the cell's chemical reaction. If you have n cells with EMFs ε₁, ε₂, …, εₙ in series, the total EMF is ε_total = ε₁ + ε₂ + … + εₙ. Think of it like climbing stairs: each cell lifts the charge up by its own voltage step, so the total height gained equals all the steps added together. If all cells are identical with EMF ε, then ε_total = nε. Cells must be connected aiding (+ to −) for this addition to work; a reversed cell subtracts its EMF.

Adding internal resistances

Each cell also has an internal resistance r. Because the cells are in series, current must pass through every internal resistance one after another — exactly like resistors in series in an external circuit. The total internal resistance is therefore r_total = r₁ + r₂ + … + rₙ. For n identical cells each with internal resistance r, r_total = nr. This is the hidden cost of series connection: the battery's ability to deliver current without losing voltage internally decreases as more cells are added.

Terminal voltage and current

When the battery is connected to an external resistance R, the current in the circuit is I = ε_total / (R + r_total). The terminal voltage — the voltage actually available at the battery's output terminals — is V_terminal = ε_total − I · r_total. This is always less than ε_total when current is flowing, because some energy is 'used up' driving current through the internal resistances. The terminal voltage equals ε_total only when no current flows (open circuit). This is why batteries feel warm during heavy use: power is being dissipated inside them.

Aiding vs. opposing cells

If a cell is accidentally inserted the wrong way round (or deliberately reversed), its EMF opposes the others. In that case, the reversed cell's EMF is subtracted: ε_total = ε₁ + ε₂ − ε₃ (for a reversed cell 3). Its internal resistance still adds, however, because current still has to pass through it. This situation reduces the total EMF while increasing internal resistance — a doubly bad outcome — which is why correct polarity matters when replacing batteries.

The Key Equation

εtotal=∑i=1nεiandrtotal=∑i=1nri\varepsilon_{\text{total}} = \sum_{i=1}^{n} \varepsilon_i \quad \text{and} \quad r_{\text{total}} = \sum_{i=1}^{n} r_i
  • \varepsilon_{\text{total}} — Total EMF (V (volts)): The combined electromotive force of all cells in the series arrangement, representing the total energy supplied per coulomb of charge.
  • \varepsilon_i — EMF of the i-th cell (V (volts)): The electromotive force of each individual cell in the series chain.
  • r_{\text{total}} — Total internal resistance (Ω (ohms)): The combined internal resistance of all cells in series; it increases the total resistance of the circuit and reduces terminal voltage under load.
  • r_i — Internal resistance of the i-th cell (Ω (ohms)): The resistance inside each individual cell due to its electrolyte and electrode materials.
  • n — Number of cells (dimensionless): The total number of cells connected in series.

Worked Example

Problem. Three identical cells, each with an EMF of 1.5 V and internal resistance of 0.5 Ω, are connected in series. They are then connected to an external resistor of 6.0 Ω. Calculate: (a) the total EMF, (b) the total internal resistance, (c) the current in the circuit, and (d) the terminal voltage of the battery.

εtotal=4.5 V,rtotal=1.5 Ω,I=0.60 A,Vterminal=3.6 V\varepsilon_{\text{total}} = 4.5\,\text{V},\quad r_{\text{total}} = 1.5\,\Omega,\quad I = 0.60\,\text{A},\quad V_{\text{terminal}} = 3.6\,\text{V}

Answer. Total EMF = 4.5 V; total internal resistance = 1.5 Ω; current = 0.60 A; terminal voltage = 3.6 V.

Figure
Diagram
Three Cells in Series Connected to External Resistor−+Cell 1ε₁=1.5V, r₁=0.5Ω−+Cell 2ε₂=1.5V, r₂=0.5Ω−+Cell 3ε₃=1.5V, r₃=0.5ΩR6.0ΩI = 0.60 Aε_total = 4.5 Vr_total = 1.5 Ω V_terminal = 3.6 V+−
Origins · 1 minAlessandro Volta and how this was discovered
AV
Alessandro Volta
1745–1827 · Italian

Volta invented the voltaic pile in 1800 — the world's first true electrochemical battery. It consisted of alternating discs of zinc and copper separated by cloth soaked in brine, stacked in series. This arrangement produced a steady, controllable electric current for the first time in history, directly demonstrating that series cells produce a cumulative EMF. The unit of electromotive force, the volt (V), is named in his honour. His work laid the foundation for all of modern circuit theory, including the analysis of cells in series. Napoleon Bonaparte was so impressed by Volta's invention that he made him a count and a senator. Volta demonstrated his pile to Napoleon personally in Paris in 1801.

  • Electromotive Force and Internal Resistance — Understanding EMF and internal resistance of a single cell is the essential foundation before studying how these quantities combine when cells are placed in series.
  • Cells in Parallel — The parallel arrangement of cells is the direct counterpart to series; in parallel the EMF stays the same but internal resistance decreases, making comparison with series essential.
  • Resistors in Series — The rule for adding internal resistances of series cells is identical to the rule for adding resistors in series, so mastering one directly supports understanding of the other.
  • Terminal Voltage — Terminal voltage — the voltage delivered to the external circuit — depends directly on the total EMF and total internal resistance of a series battery arrangement.