The fundamental formula for EMF (electromotive force) in a practical DC power source is ε = V + Ir, which can also be written as ε = I(R + r). In this equation, ε is the EMF in volts, V is the terminal voltage measured under load, I is the circuit current in amps, r is the source's internal resistance, and R is the external load resistance. Despite the name, EMF is not a mechanical force; it is the maximum potential difference a source generates when zero current is flowing (open-circuit condition).
To understand this physically, imagine a water pump. The EMF is the maximum pressure the pump can generate when the output valve is completely closed. The internal resistance (r) is a built-in kink in the pump's own output hose. The terminal voltage (V) is the actual pressure you measure at the end of the hose when water is flowing. The faster the water flows (higher current), the more pressure is lost to the internal kink, dropping your usable terminal voltage.
The Core Formula for EMF and Symbol Definitions
Before plugging numbers into a calculator, you must understand the exact physical quantity each symbol represents. Misidentifying terminal voltage for EMF is the most common mistake hobbyists make when sizing battery packs for high-draw loads like motors or heating elements.
| Symbol | Quantity | SI Unit | How to Measure on the Bench |
|---|---|---|---|
| ε (or E) | Electromotive Force (EMF) | Volts (V) | Measure voltage across the source terminals with no load connected (open circuit) using a high-impedance digital multimeter (DMM). |
| V | Terminal Voltage | Volts (V) | Measure voltage across the source terminals while the circuit is active and drawing current. |
| I | Current | Amperes (A) | Measure inline with a DMM in series, or use a clamp meter / shunt resistor for high-current loads. |
| r | Internal Resistance | Ohms (Ω) | Calculated by measuring the voltage drop under a known load, or measured directly with a specialized AC milliohm meter. |
| R | External Load Resistance | Ohms (Ω) | The equivalent resistance of the connected circuit (e.g., a resistor, motor winding, or PCB trace). |
Real-World EMF and Internal Resistance Data
The formula for EMF assumes r is a fixed value, but in reality, internal resistance varies wildly based on chemistry, state of charge (SoC), and temperature. Below is a data-dense reference table of common power sources. Use these baseline values for initial circuit simulations and voltage drop calculations.
| Power Source / Chemistry | Nominal Voltage | True EMF (Fresh / 100% SoC) | Typical Internal Resistance (r) | Max Continuous Current |
|---|---|---|---|---|
| AA Alkaline (e.g., Duracell Coppertop) | 1.50 V | 1.60 V - 1.65 V | 150 mΩ - 300 mΩ | ~0.5 A |
| 18650 Li-ion (e.g., Samsung 30Q) | 3.60 V | 4.20 V | 15 mΩ - 25 mΩ | 15 A |
| 12V SLA / AGM (e.g., Optima RedTop) | 12.00 V | 12.70 V - 12.90 V | 2 mΩ - 5 mΩ | 100+ A (Cranking) |
| CR2032 Lithium Coin Cell | 3.00 V | 3.20 V - 3.30 V | 10 Ω - 30 Ω | 0.015 A (15 mA) |
| 9V Alkaline Carbon-Zinc | 9.00 V | 9.50 V | 1.5 Ω - 2.5 Ω | ~0.1 A |
Note: Data sourced from cadex Electronics' Battery University impedance archives and manufacturer datasheets. Internal resistance increases significantly as the cell discharges and as ambient temperature drops below 10°C.
Rearranged Forms for Circuit Debugging
On the bench, you rarely know every variable. You usually know your battery's EMF and your load, and you need to find the resulting current or terminal voltage. Here are the algebraic rearrangements of ε = I(R + r) solved for each variable:
- Solve for Current (I):
I = ε / (R + r)
Use when sizing a fuse or checking if a load will trip a BMS overcurrent limit. - Solve for Terminal Voltage (V):
V = ε - Ir
Use to verify if a microcontroller's LDO will receive enough headroom voltage during peak transmission bursts. - Solve for Internal Resistance (r):
r = (ε - V) / I
Use to diagnose a degraded battery pack. If r is 2x the datasheet spec, the cells are nearing end-of-life. - Solve for Load Resistance (R):
R = (ε - Ir) / I
Use when designing a dummy load or sizing a current-limiting resistor for an LED array.
Worked Examples with Unit Tracking
Abstract formulas are useless without rigorous unit tracking. The most frequent cause of calculation errors is failing to convert milliamps to amps or milliohms to ohms before multiplying. Below are two bench-realistic scenarios.
Problem 1: Diagnosing a Sagging Lead-Acid Battery
Scenario: You are testing a 12V sealed lead-acid (SLA) battery for a DIY solar gate opener. With no load, your multimeter reads an open-circuit EMF (ε) of 12.70 V. You connect a 2.0 Ω heating element as a dummy load. A clamp meter reads a steady current (I) of 6.20 A. What is the internal resistance (r) of the battery?
- Identify knowns and convert units:
ε = 12.70 V
R = 2.0 Ω
I = 6.20 A - Select the rearranged formula:
r = (ε - V) / I
Wait, we don't have V directly, but we know V = I × R. Let's use the expanded form:
ε = I(R + r) - Substitute and solve for r:
12.70 V = 6.20 A × (2.0 Ω + r)
12.70 V = 12.40 V + (6.20 A × r)
12.70 V - 12.40 V = 6.20 A × r
0.30 V = 6.20 A × r
r = 0.30 V / 6.20 A
r = 0.0484 Ω (or 48.4 mΩ) - Sanity Check: A healthy 12V SLA battery should have an internal resistance around 5 mΩ to 10 mΩ. At 48.4 mΩ, this battery has high internal resistance, indicating sulfation or age. It will struggle to crank a motor.
Problem 2: Preventing ESP32 WiFi Brownouts
Scenario: An ESP32-WROOM-32 dev board draws a peak current of 240 mA during WiFi transmission bursts. It is powered by a single 18650 Li-ion cell with an EMF of 4.15 V and an internal resistance of 0.020 Ω. The onboard AMS1117-3.3 LDO requires a minimum input voltage of 3.80 V to maintain a clean 3.3V rail. Will the terminal voltage drop below the LDO's dropout threshold?
- Identify knowns and convert units:
ε = 4.15 V
I = 240 mA = 0.240 A (Crucial conversion!)
r = 0.020 Ω
V_dropout = 3.80 V - Select the formula for Terminal Voltage:
V = ε - Ir - Calculate the voltage drop across internal resistance:
V_drop = 0.240 A × 0.020 Ω = 0.0048 V (4.8 mV) - Calculate final Terminal Voltage:
V = 4.15 V - 0.0048 V = 4.1452 V - Conclusion: 4.1452 V is well above the 3.80 V LDO requirement. The 18650 cell's low internal resistance easily handles the 240 mA burst without causing a brownout. (Note: If we had used a CR2032 coin cell with r = 15 Ω, the drop would be 3.6 V, instantly killing the ESP32).
Assumptions, Limits, and Common Unit Traps
The formula ε = V + Ir is a linear approximation. It works beautifully for quick bench calculations, but you must understand its boundaries to avoid designing a circuit that fails in the field.
When the Formula Applies (and When It Doesn't)
This formula applies strictly to DC steady-state circuits. It assumes the internal resistance is purely resistive. In reality, batteries and power supplies have internal impedance, which includes capacitive and inductive reactance. If your load is switching at high frequencies (like a 20 kHz PWM motor driver or a switching buck converter), the instantaneous voltage sag will be dictated by the battery's AC impedance and the ESL (Equivalent Series Inductance) of your PCB traces, not just the DC internal resistance.
The Unit Mistakes That Break the Math
When calculating EMF, a single unit mismatch will yield an answer that is off by a factor of 1,000. Review this comparison table before finalizing your design math:
| Common Datasheet Value | Incorrect Math Input | Correct Math Input (SI Base Units) | Resulting Error Magnitude |
|---|---|---|---|
| Current: 500 mA | I = 500 | I = 0.5 | Calculated voltage drop is 1000x too large. |
| Resistance: 25 mΩ | r = 25 | r = 0.025 | Calculated voltage drop is 1000x too large. |
| Capacity: 3000 mAh | Using mAh as Amps | Not a variable in the EMF formula | Fundamental misunderstanding of charge vs. current. |
Induced EMF: The Generator and Inductor Formula
While ε = V + Ir governs chemical batteries, the formula for EMF in generators, transformers, and inductors is entirely different. This is governed by Faraday's Law of Induction. If you are designing a DIY wind turbine, a flyback transformer, or calculating inductive kickback from a relay coil, you must use the induced EMF formula:
ε = -N (dΦ / dt)
- ε = Induced EMF (Volts)
- N = Number of turns in the coil
- dΦ / dt = Rate of change of magnetic flux (Webers per second)
The negative sign represents Lenz's Law, indicating that the induced EMF opposes the change in current that created it. This is why a simple 12V relay coil can generate a 200V spike when switched off, instantly frying an Arduino GPIO pin if you forget to wire a reverse-biased flyback diode across the coil terminals. For a deeper dive into magnetic circuits, consult the Georgia State University HyperPhysics reference on Faraday's Law.
Whether you are calculating the voltage sag of a LiPo pack under a 30A drone motor load or sizing the flyback diode for an inductive solenoid, always start by identifying which type of EMF you are dealing with. Track your units rigorously, measure your real-world open-circuit voltage with a calibrated meter, and remember that a battery's internal resistance is a dynamic variable, not a fixed constant.
