Internal Resistance - r, Voltage Loss, and Efficiency from Two Readings

An internal resistance calculator turns two voltage readings of any DC source — its open-circuit voltage and its loaded terminal voltage — into the resistance hiding inside the cell, the voltage that resistance steals, and the share of power that reaches the load. If a battery that reads 12.6 V on a multimeter drops to 11 V when the headlights turn on, this calculator shows which fraction of the missing voltage is being lost inside the source rather than in the wiring.

• • Battery health checks: Compare a fresh cell with a worn one to see how much r has grown and whether the battery is still usable for a high-current application.
• • Lab power supply characterization: Measure how close a bench DC supply comes to ideal behavior by reading open-circuit voltage, then loaded voltage at a known current.
• • Solar panel and PV module testing: Estimate the source resistance of a small panel from its no-load voltage and a loaded operating point on its I–V curve.
• • Physics homework and lab write-ups: Verify V = ε − I·r with real numbers for an introductory circuits or electromagnetism problem set.

Every practical DC source — alkaline cell, lithium pack, lead-acid battery, bench supply, or USB port — behaves like an ideal EMF ε in series with a small r. That r is what makes the terminal voltage sag under load, and it is the single number that best describes how 'stiff' the source is.

You do not need to disassemble the source to find r. Two voltage measurements and one current are enough, so the same form works for a 1.5 V AA cell and a 12 V car battery. As Wikipedia's internal resistance entry notes, every real electrochemical or DC source has a small resistance inside it that causes the terminal voltage to drop as current rises.

If you only need V, I, or R for an ideal resistor in a circuit, the Ohm's Law Calculator solves that case directly with the same V = I·R relationship this calculator extends to non-ideal sources.

The internal resistance model treats any DC source as an ideal EMF ε in series with a small resistor r. The terminal voltage observed at the load is ε minus the I·r drop that occurs inside the source.

• V_no_load (ε): Open-circuit voltage measured with no load — the EMF of the source in volts.
• V_load: Terminal voltage measured while the load is drawing current, in volts.
• I_load: Steady current the load draws from the source during the loaded reading, in amperes.

Once you have r, the rest of the results follow from ordinary circuit algebra. The voltage drop is the difference between no-load and loaded readings; the internal heat is I²r; the load power is V_load × I_load; and efficiency is the load's share of total power delivered.

At 100 A the same 0.04 Ω source wastes a third of its power as heat — the headline reason high-current battery packs need very low r. According to Wikipedia's electromotive force entry, the terminal voltage of any non-ideal DC source equals its EMF minus the drop across r, so V = ε − I·r is the right model for batteries, lab supplies, and solar cells alike.

After computing internal loss and load power, the Work, Energy, and Power Calculator helps confirm the energy balance in watts and joules for the same circuit.

These four ideas are the minimum vocabulary you need to read this kind of problem and to interpret what the calculator tells you.

These four concepts reappear in every circuit you will meet that has a real power source. The model is intentionally simple so that textbook problems and bench measurements can use the same numbers.

If you need the underlying resistivity of the cell's plates or tabs to estimate r from geometry, the Conductivity to Resistivity Calculator converts between conductivity σ and resistivity ρ for the same material.

Three measurements and a click are enough. Take them in the order shown so the loaded voltage is read at a stable current rather than while the load is still ramping up.

1. 1 Measure the open-circuit voltage: Disconnect the load. Set your multimeter to DC volts and read the voltage across the source terminals. Record this as V_no_load.
2. 2 Connect the load and measure loaded voltage: Attach the load and let the current settle for a few seconds. Read the terminal voltage while the load is still drawing steady current.
3. 3 Read or compute the load current: Use a clamp meter, an inline ammeter, or compute I = V_load / R_load if the load is a known resistor. Enter the current in amperes.
4. 4 Enter the three values: Type V_no_load in volts, V_load in volts, and I_load in amperes. The defaults of 12 V, 11.4 V, and 2 A show a realistic lead-acid result immediately.
5. 5 Read r and the diagnostic outputs: Read r in ohms, the voltage lost inside the source, the power dissipated inside r, and the share of total power that reaches the load.

On a 12 V car battery the open-circuit voltage is 12.6 V. With the headlights on, the alternator is offline and the terminal voltage drops to 12.0 V while the lamps draw about 5 A. Plugging V_no_load = 12.6, V_load = 12.0, and I_load = 5 into the calculator gives r = 0.12 Ω — a healthy battery even at high current.

When the source is a charged capacitor rather than a battery, the Capacitor Charge Time Calculator models the same RC behavior with time constant τ = r·C instead of a steady-state reading.

Treating r as a number you measure — rather than a vague complaint about weak batteries — gives you concrete decisions to make.

• • Detect battery wear early: A cell whose r has doubled is the first sign of aging. A repeated measurement turns 'battery feels weak' into 'r has climbed from 0.05 Ω to 0.18 Ω, time to retire this cell from high-current use.'
• • Predict voltage sag before it bites: Once you know r, you can predict terminal voltage at any current using V = ε − I·r. The calculator gives you the slope; you draw the line.
• • Choose cells and supplies on real numbers: Two 12 V batteries can look identical on a multimeter yet behave very differently at 20 A. Side-by-side r values tell you which one actually delivers under load.
• • See where the energy is going: The internal heat loss I²r is power you paid for and never used. The efficiency output makes that wasted energy visible.
• • Validate lab models with one equation: A measured r lets you back-calculate EMF from V_load and I_load, the standard textbook check that V = ε − I·r is the right model for your source.

The same three numbers give you all of those answers at once. That is what makes a calculator-based approach more useful than a single number scribbled on a battery.

Once you know the internal loss in watts, the Watt Converter helps translate that number across power units if you are budgeting a battery pack in milliwatts or megawatts.

Several physical and chemical effects push r up or down. Knowing which one you are looking at tells you whether the change is permanent or something you can engineer away.

• • The single-resistor model assumes r is constant, but in a real cell r depends on current, state of charge, and temperature. Treat the result as a useful average over the load you measured rather than a universal constant.
• • AC methods such as 1 kHz impedance spectroscopy give a different, frequency-dependent r. The DC method used here corresponds to the slope of the V–I line at the chosen operating point and may not match the AC impedance exactly.

These caveats do not make the method wrong — they just tell you when to take the number with a grain of salt. A consistent measurement procedure usually matters more than swapping calculators.

According to HyperPhysics (Georgia State University), r of a source is the slope of its voltage-versus-current line, found experimentally by measuring the no-load voltage and the loaded voltage at a known current.

If the source behaves more like a charged capacitor than a battery, the Capacitance Calculator gives you the capacitance and stored energy that drive the same voltage-versus-time behavior in the limit.