Low temperature

Intent

• Estimate the ripple an aluminum electrolytic capacitor’s ESR develops in the cold.
• Know what a ceramic capacitor’s temperature code does not tell.
• Read a battery’s terminal voltage in the Rint model, and know why a lithium-ion cell is not charged in the cold.

Theory

Aluminum electrolytic capacitors

The electrolyte’s resistivity rises in the cold, and it often takes an aluminum electrolytic capacitor’s ESR at its lowest rated temperature to 10 to 100 times its room-temperature ESR; the cold also lowers the capacitance [Cornell Dubilier Electronics, Aluminum Electrolytic Capacitor Application Guide]. The part’s data sheet gives its own low-temperature figures. The ripple voltage across the ESR rises in proportion (ripple voltage across the ESR): a converter whose output ripple meets its limit at room temperature can exceed it in the cold.

Ceramic capacitors

A class-2 ceramic capacitor’s code, such as X7R, describes how its capacitance changes with temperature, and implies nothing about voltage coefficients [Analog Devices (Maxim Integrated), Temperature and Voltage Variation of Ceramic Capacitors, or Why Your 4.7 µF Capacitor Becomes a 0.33 µF Capacitor]. The dc voltage alone can leave a small fraction of the nominal capacitance: in the article’s example, a 4.7 µF Y5V capacitor rated 6.3 V keeps 0.33 µF at 5 V at room temperature [Analog Devices (Maxim Integrated), Temperature and Voltage Variation of Ceramic Capacitors, or Why Your 4.7 µF Capacitor Becomes a 0.33 µF Capacitor], and temperature changes it again. Check the capacitance at the dc voltage it runs at, across the temperatures it will see.

Batteries

In the Rint model a battery is its open-circuit voltage behind an internal resistance, and both are functions of the state of charge, the state of health and the temperature [He et al. 2011, Sec. 2.1]. A charging current lifts the terminals above the open-circuit voltage:

$$V = V_b + I_b R_b$$

$V$

output voltage

$I_b$

battery current (charging positive)

$R_b$

battery internal resistance

$V_b$

battery open-circuit voltage

Terminal voltage of a battery in the Rint model✓ verified by 8 tests

Assumptions, convention and source

Assumptions

• battery modelled as its open-circuit voltage behind an internal resistance (Rint model)

Notes

charging lifts the terminals above the open-circuit voltage by the current times the internal resistance, and discharging pulls them below; a charger that holds the terminals at a voltage limit delivers less current when the resistance is higher

Source

He et al. 2011, Sec. 2.1 (The Rint Model)

Derivation

step-by-step sympy derivation

Equation id

bat.v_term · equations.yaml entry

A charger that holds the terminals at a voltage limit therefore delivers less current into a battery whose resistance is higher: take the resistance at the temperature the battery runs at.

At low temperature, a lithium-ion cell’s carbon anode receives lithium ions less readily, which makes it easier for metallic lithium to form on the carbon’s surface, and the battery’s thermal stability decreases. Below the lower charging temperature specified for the cell, it must not be charged with any current [Japan Electronics and Information Technology Industries Association (JEITA), A Guide to the Safe Use of Lithium-Ion Secondary Cells in Notebook-type Personal Computers and Tablet Terminals].

Worked example

A cell being charged, and an output capacitor’s ESR ripple at room temperature and at its low-temperature limit:

Cold example

Symbol	Value	How it is obtained
$V_b$ battery open-circuit voltage	3.7 V	given
$I_b$ battery current (charging positive)	1 A	given
$R_b$ battery internal resistance	100 mΩ	given
$R_\mathrm{ESR}$ equivalent series resistance	50 mΩ	given
${R_\mathrm{ESR}}_{\mathrm{cold}}$ equivalent series resistance	500 mΩ	given
$\Delta i_{L,\mathrm{pp}}$ inductor ripple, peak-to-peak	2 A	given
$V$ output voltage	3.8 V	Terminal voltage of a battery in the Rint model
$\Delta v_\mathrm{ESR}$ ESR ripple, peak-to-peak	100 mV	Ripple voltage across a capacitor's series resistance
${\Delta v_\mathrm{ESR}}_{\mathrm{cold}}$ ESR ripple, peak-to-peak	1 V	Ripple voltage across a capacitor's series resistance

Charged at 1 A through 100 mΩ, the cell’s terminals sit at 3.8 V, above its open-circuit 3.7 V. The capacitor’s ESR develops 100 mV of peak-to-peak ripple at room temperature and 1 V with its ESR ten times higher.

Try it

▶ Open in the equation explorer: Ripple voltage across a capacitor's series resistance
Preset from the example bench-cold

Sweep the ESR across a decade: the ripple follows it.

▶ Open in the equation explorer: Terminal voltage of a battery in the Rint model
Preset from the example bench-cold

Sweep the internal resistance: at the same charging current, the terminals rise with it.

Bench exercise

Measure a converter’s output ripple at room temperature, then with its output capacitor cooled (a cold chamber, or cold spray on the capacitor alone), with the probe’s ground connection kept short (probes). Where the ESR’s part dominates the ripple, divide the ripple by the peak-to-peak ripple current into the capacitor to estimate the ESR, and compare the two temperatures. Watch for condensation as the capacitor warms again. For a battery, log its terminal voltage and current at two temperatures while it discharges at a constant current, and estimate its resistance from a step in the current at each.

Gotchas

• An electrolytic’s ESR at its lowest rated temperature is often 10 to 100 times its room-temperature ESR [Cornell Dubilier Electronics, Aluminum Electrolytic Capacitor Application Guide].
• A ceramic’s code says nothing about its dc voltage [Analog Devices (Maxim Integrated), Temperature and Voltage Variation of Ceramic Capacitors, or Why Your 4.7 µF Capacitor Becomes a 0.33 µF Capacitor].
• A lithium-ion cell below its lower charging temperature must not be charged [Japan Electronics and Information Technology Industries Association (JEITA), A Guide to the Safe Use of Lithium-Ion Secondary Cells in Notebook-type Personal Computers and Tablet Terminals].
• A charger at its voltage limit delivers less current into a battery of higher resistance (terminal voltage).

Go deeper

• Application note Aluminum Electrolytic Capacitor Application Guide — Cornell Dubilier Electronics · retrieved 2026-09-24
  How an aluminum electrolytic capacitor's capacitance, ESR and life change with temperature, ripple current and voltage.
• Technical article Temperature and Voltage Variation of Ceramic Capacitors, or Why Your 4.7 µF Capacitor Becomes a 0.33 µF Capacitor — Analog Devices (Maxim Integrated) · retrieved 2026-09-24
  How a class-2 ceramic capacitor's capacitance changes with temperature and with its dc voltage, and why its code tells only the first.
• Application note A Guide to the Safe Use of Lithium-Ion Secondary Cells in Notebook-type Personal Computers and Tablet Terminals — JEITA · retrieved 2026-09-24
  Why a lithium-ion cell's charging is limited in temperature, voltage and current, and what a charger must do outside those limits.

Quiz

1. An output capacitor has 0.02 ohm of ESR, and the ripple current into it is 3 A peak-to-peak. What ripple does the ESR develop?

   1. 60 mV peak-to-peak
   2. 6 mV peak-to-peak
   3. 0.6 V peak-to-peak
   4. 150 mV peak-to-peak
   Answer and explanation

   Correct answer: A The ESR times the peak-to-peak ripple current: 0.02 ohm times 3 A.

2. Suppose the same capacitor is an aluminum electrolytic whose data sheet shows its ESR rising more than tenfold at its low-temperature limit. What happens to that ripple there?

   1. It stays at 60 mV
   2. It falls, since the capacitor is cold
   3. It rises more than tenfold, past 0.6 V
   4. It disappears
   Answer and explanation

   Correct answer: C The ESR ripple is proportional to the ESR: a limit met at room temperature can be exceeded in the cold.

3. What does a class-2 ceramic capacitor's code, such as X7R, tell you?

   1. Its capacitance at its rated voltage
   2. How its capacitance changes with temperature, and nothing about its dc voltage
   3. That its capacitance does not change
   4. Its ESR
   Answer and explanation

   Correct answer: B The code implies nothing about voltage coefficients: check the capacitance at the dc voltage it runs at.

4. A cell has 3.6 V open-circuit and 0.08 ohm internal resistance. Charged at 2.5 A, where do its terminals sit?

   1. 3.8 V
   2. 3.6 V
   3. 3.4 V
   4. 4.2 V
   Answer and explanation

   Correct answer: A The charging current lifts the terminals by 2.5 A times 0.08 ohm, 0.2 V.

5. A charger holds a cell's terminals at 4.2 V while its open-circuit voltage is 4.0 V. How much current flows with 0.1 ohm of internal resistance, and with 0.4 ohm?

   1. 2 A both times
   2. 0.5 A, then 2 A
   3. 2 A, then 0.5 A
   4. 42 A, then 10.5 A
   Answer and explanation

   Correct answer: C The current is (4.2 V - 4.0 V) over the resistance: a higher resistance takes less current at the same limit.

6. A cell's terminal voltage drops from 3.70 V to 3.52 V when its discharge current steps from 1 A to 3 A. What is its internal resistance?

   1. 0.18 ohm
   2. 0.06 ohm
   3. 1.17 ohm
   4. 0.09 ohm
   Answer and explanation

   Correct answer: D In the Rint model the voltage step over the current step is the resistance: 0.18 V over 2 A.

7. Why must a lithium-ion cell not be charged below its lower charging temperature?

   1. Its open-circuit voltage becomes zero
   2. The charger's voltage limit no longer applies
   3. Its carbon anode receives lithium ions less readily, and metallic lithium forms more easily on the carbon's surface
   4. Its internal resistance becomes zero
   Answer and explanation

   Correct answer: C In the cold the carbon anode takes up lithium ions less readily, which makes it easier for metallic lithium to form on its surface, and the battery's thermal stability decreases.