Ohm's Law
The single equation that ties voltage, current and resistance together — and explains everything from BMS shunt sizing to why grid-scale BESS voltage sags under load.
Musk's Rocket Science · Part 1 — Electricity Basics · Chapter 3 · 14 min read
In 1825 a German high school physics teacher named Georg Simon Ohm began a quiet, obsessive series of experiments in his spare time. Using nothing more exotic than lengths of wire, a crude battery and a galvanometer he built himself, he systematically varied wire length, thickness and material, and measured how current responded. He published in 1827 — and the German scientific establishment largely ignored or ridiculed the work as too simplistic to be meaningful. It took nearly a decade for its significance to be recognised.
Today that “too simple” relationship is arguably the most-used equation in electrical engineering, and it carries his name. It is also the payoff for Chapters 1 and 2: once you know what voltage, current and resistance individually mean, Ohm’s Law is not really a new fact to memorise — it is almost an obvious consequence of definitions you already have.
3.1 — The equation
V = I · R
Voltage across a component equals the current through it multiplied by its resistance.
Because this is a genuine algebraic equation, it rearranges into three equivalent forms depending on which quantity you are solving for.
Table 3.1 — The three forms of Ohm's Law
| Solve for | Equation | Use when you know… |
|---|---|---|
| Voltage | V = I × R | Current and resistance |
| Current | I = V / R | Voltage and resistance |
| Resistance | R = V / I | Voltage and current |
In plain English
Many textbooks teach this with a triangle: V at the top, I and R at the bottom. Cover whichever letter you want to solve for, and the triangle shows you the other two. Side by side means multiply (V = I × R); one over the other means divide (I = V/R, or R = V/I).
It is a memory trick, not new physics — but a genuinely useful one.
3.1.1 — Graphing the law
Because V = IR is a straight-line equation — the form y = mx, with resistance playing the role of the slope — plotting voltage against current for a fixed resistance always produces a straight line through the origin. Different resistances produce different slopes.
3.2 — Worked examples
Worked example 3.1 — Sizing a BMS current-sense shunt resistor
A BMS measures pack current indirectly, by reading the tiny voltage dropped across a precision shunt resistor placed in the main current path. Suppose the design target is a full-scale reading of 5.0mV at the pack’s rated 200A continuous current.
R = V / I = 0.0050V / 200A = 25 μΩ
A genuinely tiny resistance — deliberately so. A shunt with too much resistance would waste power as heat (P = I²R) and disturb the very current it is trying to measure; too little, and the voltage signal becomes too small to measure accurately against sensor noise.
That trade-off is why precision current shunts are a specialised, calibrated component in every BMS and EMS current-sensing chain, not an arbitrary resistor pulled from a bin.
Worked example 3.2 — Why motor stall current is dangerous
An electric bus’s DC traction motor has a winding resistance of R = 0.05Ω, driven from a 700V commercial-EV-class battery pack. While spinning normally the motor generates a back-EMF that opposes the supply and limits current to a safe, moderate value.
But if the motor is stalled — wheels blocked, or at the very instant of pulling away from a dead stop under full load — no back-EMF is generated, and the winding resistance alone limits current:
I = V / R = 700 / 0.05 = 14,000 A
An enormous spike compared with normal cruising draw, capable of destroying the motor windings or the inverter’s power semiconductors within milliseconds. This is exactly why traction inverters include fast current limiting, and why a BMS carries two-stage discharge over-current protection. Ohm’s Law says this spike is not a fault condition to be surprised by — it is an expected consequence of the physics that the power electronics must be designed to survive.
3.3 — Ohm's Law and electrical safety, revisited
Chapter 2 gave you a table of shock severity by current, and a table of typical human skin resistance. Ohm’s Law is the missing link between them: for any fixed voltage, a person’s resistance — and their PPE — directly determines how much current actually flows, and therefore how dangerous the situation is. This matters directly to anyone working around BESS enclosures, EV charging cabinets or utility switchgear.
Worked example 3.3 — The same voltage, very different outcomes
Fixed DC bus voltage at a BESS site: V = 400V.
Full rated PPE (insulated gloves + dielectric boots): R ≈ 1,000,000Ω → I = 0.4 mA
Same technician, degraded and wet PPE: R ≈ 5,000Ω → I = 400/5,000 = 80 mA
The first is below the perception threshold from Chapter 2’s safety table. The second is approaching the fibrillation-risk zone. The voltage never changed. Only resistance did — and that alone was the difference between a non-event and a potentially fatal incident.
Important
This is precisely why BESS and EV charging infrastructure safety procedures mandate rated PPE, lockout-tagout, and dry working conditions before any work near high-voltage DC buses — not superstition, but Ohm’s Law in direct practical application. Raising the resistance in the current’s path is one of the most effective ways to reduce the danger of a fixed-voltage system.
The resistance figures here are illustrative and vary widely with contact area, moisture and skin condition. Treat them as intuition-building, not as a safety margin to rely on, and never as a substitute for the site’s own electrical safety procedures.
3.4 — When Ohm's Law breaks: Ohmic and non-Ohmic
Ohm’s Law works beautifully for resistors, wires and most simple conductors, where resistance stays constant regardless of voltage or current. These are Ohmic devices, and their V–I graph is always a straight line. But many real components are non-Ohmic: their effective resistance changes with operating conditions.
Table 3.2 — Ohmic and non-Ohmic examples across the energy stack
| Type | Behaviour | Examples |
|---|---|---|
| Ohmic | Resistance constant regardless of voltage or current; straight-line V–I graph | Copper busbars, transmission conductors at constant temperature, resistive heating elements |
| Non-Ohmic | Resistance changes with operating point; curved V–I graph | PEM electrolyser cells, fuel cells, power semiconductors (SiC and GaN diodes and transistors), battery cells under heavy load |
Technical framing
For non-Ohmic devices V = IR still holds at any single instant — but R is no longer a fixed constant; it becomes a function of V or I. Engineers handle this using dynamic resistance, the local slope of the curve at a specific operating point, or by using the full non-linear device model in circuit or process simulation — exactly how electrolyser stack sizing and fuel cell system design are actually done in practice.
3.5 — Real batteries aren't perfect: internal resistance
So far we have treated a battery as an ideal, unwavering voltage source. Real cells are not that generous. Every cell has a small internal resistance arising from its chemistry and construction. A real battery string is better modelled as a perfect voltage source — its EMF, or electromotive force — in series with a small internal resistance r.
V_terminal = EMF − I · r
As current draw increases, more voltage is lost internally, and the voltage available at the terminals sags below the resting open-circuit value.
Worked example 3.4 — Voltage sag in a grid-scale BESS string
A high-voltage BESS string built from 470 LiFePO4 cells in series has an open-circuit EMF of about 1,500V. Each cell has an internal resistance around 2.0mΩ — a realistic spec for a well-built 100Ah-class LiFePO4 cell. In series, 470 cells give a string-level internal resistance of roughly:
r = 470 × 2.0mΩ ≈ 0.94 Ω
At a 100A discharge current, a realistic 1C-class rate for grid dispatch:
Voltage sag = I × r = 100 × 0.94 = 94 V
Terminal voltage under load = 1,500 − 94 = 1,406 V
Noticeably below the string’s resting voltage, purely due to internal resistance, with no capacity actually lost.
Why this matters later
This is precisely why internal resistance is a headline spec on every cell datasheet, and why tracking it over a cell’s lifetime is one of the best early warning signs of ageing. Internal resistance rises as a cell degrades, meaning more voltage sag, more wasted heat and less usable power at high current — often before rated capacity visibly drops.
At grid scale, rising string-level internal resistance directly erodes a BESS asset’s ability to hit its contracted power rating during dispatch, which is why utility-scale operators track it as a key state-of-health indicator. It is also why our balancing and capacity-testing procedure tracks impedance alongside capacity, and why cell manufacturers set a retirement threshold at double the initial internal resistance.
3.6 — Quick reference
Ohm's Law cheat sheet
| Form | Solves for | Remember |
|---|---|---|
| V = I × R | Voltage | More current or more resistance ⇒ more voltage needed |
| I = V / R | Current | More voltage ⇒ more current; more resistance ⇒ less current |
| R = V / I | Resistance | The slope of a component’s V–I graph |
Chapter summary
- ✓Ohm’s Law: V = IR — discovered by Georg Simon Ohm in 1827, initially dismissed, now the most-used equation in electrical engineering.
- ✓Three equivalent forms: V = IR, I = V/R, R = V/I — use whichever matches the two quantities you already know.
- ✓For a fixed resistance, plotting V against I gives a straight line through the origin; the slope is the resistance.
- ✓It directly explains real engineering decisions: sizing a BMS current-sense shunt, predicting a traction motor’s stall current, and why the same voltage can be harmless or lethal depending purely on resistance.
- ✓Ohmic devices — busbars, transmission conductors — have constant resistance and a straight V–I line. Non-Ohmic devices — PEM electrolyser cells, power semiconductors, batteries under load — have resistance that changes with operating conditions.
- ✓Real battery strings have internal resistance, so terminal voltage sags under load: V_terminal = EMF − I·r. This is normal and predictable, not capacity loss — though rising internal resistance over time is a useful indicator of cell ageing and a key state-of-health metric for grid-scale BESS assets.
Test your understanding
- 1A 48V data centre DC bus is connected across a 0.8Ω load bank. How much current flows?
- 2You measure 3A flowing through a component with 9V across it. What is its resistance?
- 3Why does a BMS current-sense shunt need to be a specific, calibrated low resistance rather than an arbitrary resistor?
- 4Sketch what the V–I graph would look like for two BESS busbars, one with 2mΩ resistance and one with 8mΩ, on the same axes. Which line is steeper, and why?
- 5A BESS string has an EMF of 1,500V and an internal resistance of 0.8Ω. What is its terminal voltage when supplying 150A to the grid?
- 6Challenge: using the shunt example in section 3.2, what shunt resistance would you need for a full-scale reading of 5.0mV at a higher pack rating of 400A instead of 200A?
Frequently asked questions
What is Ohm's Law?+
Ohm’s Law states that the voltage across a component equals the current through it multiplied by its resistance: V = I × R. It rearranges into I = V/R and R = V/I. Georg Simon Ohm published it in 1827, where it was initially dismissed as too simplistic; it is now the most-used equation in electrical engineering.
How do you size a BMS current-sense shunt resistor?+
Divide the target full-scale sense voltage by the rated current: R = V/I. For a 5.0mV reading at 200A continuous, R = 0.0050/200 = 25 μΩ. The value is a deliberate trade-off — too much resistance wastes power as I²R heat and disturbs the current being measured, too little leaves a signal too small to read accurately against sensor noise. That is why shunts are calibrated precision components rather than arbitrary resistors.
What is the difference between Ohmic and non-Ohmic devices?+
Ohmic devices — copper busbars, transmission conductors at constant temperature, resistive heating elements — have constant resistance and a straight-line V–I graph. Non-Ohmic devices — PEM electrolyser cells, fuel cells, SiC and GaN power semiconductors, battery cells under heavy load — have resistance that changes with the operating point, producing a curved graph. V = IR still holds instant by instant, but R becomes a function of V or I.
Why does battery voltage drop under load?+
Because every real cell has internal resistance. A battery behaves as an ideal voltage source (its EMF) in series with a small internal resistance r, so V_terminal = EMF − I·r. A 470-cell LiFePO4 BESS string at 2.0mΩ per cell has about 0.94Ω of internal resistance, so a 100A discharge sags it by 94V — reading 1,406V instead of its resting 1,500V, with no capacity actually lost.
Why is rising internal resistance a sign of battery ageing?+
Internal resistance increases as a cell degrades, which means more voltage sag under load, more energy wasted as heat, and less usable power at high current. It often rises measurably before rated capacity visibly drops. At grid scale, rising string-level internal resistance directly erodes a BESS asset’s ability to hit its contracted power rating during dispatch, which is why operators track it as a key state-of-health indicator and cell manufacturers set a retirement threshold at roughly double the initial value.
Why is a motor’s stall current so high?+
A spinning motor generates a back-EMF that opposes the supply and limits current. When stalled — wheels blocked, or at the instant of pulling away under full load — there is no back-EMF, so only the winding resistance limits current. An electric bus traction motor with a 0.05Ω winding on a 700V pack gives I = 700/0.05 = 14,000A, enough to destroy windings or inverter semiconductors in milliseconds. This is why traction inverters include fast current limiting and a BMS carries two-stage over-current protection.
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Musk's Rocket Science is an original educational series explaining the physics and chemistry behind everyday electricity and battery technology. Figures and worked examples use standard physical constants and representative real-world values for illustration.