How this instrument works
Power dissipation is the share of electrical power a component converts into heat rather than passing along as useful output. A resistor is built entirely for this job — dropping voltage in a divider, biasing a transistor, or simply absorbing power on purpose — so all of the power that reaches it counts as dissipation. A motor is different: most of the electrical power it draws leaves the shaft as torque, and only the fraction lost to its own winding resistance turns to heat. P = I²R isolates exactly that resistive loss, indifferent to what the rest of the device is doing — it only cares how much current is forced through how many ohms.
The shape of the formula comes from two facts holding at once. Power is always current times the voltage driving it, P = I·V. Inside a resistor, that voltage is itself the product of the current and the resistance — Ohm's law, V = I·R. Fold the second fact into the first and current stops appearing just once; it turns up twice, first as the moving charge itself, second time hidden inside the voltage that same motion built across the resistor. That double billing is the entire reason the exponent lands on 2 rather than 1, and it's why current, more than voltage, decides how hot something runs.
The one assumption baked into the formula is that R holds still while I changes. At DC and at the low frequencies of ordinary building wiring, that's close enough to exact to build a system around. It stops being exact at radio frequency or fast switching edges, where current crowds toward a conductor's outer surface — the skin effect — shrinking the area carrying current and pushing the effective resistance above the value a multimeter would read at DC. It also assumes the resistance isn't itself a function of current, which rules out diodes, transistor junctions, and anything else that doesn't obey Ohm's law to begin with.
- Type the load current into the Current box — a stall-test reading, a datasheet spec, or whatever a clamp meter shows right now, not a supply's maximum rating.
- Enter the Resistance field in ohms — the winding, wire, or resistor's value at the operating temperature you care about, since resistance itself can shift with heat.
- Read Power dissipated for the wattage that component must shed continuously as heat for as long as that current keeps flowing.
- Re-run the numbers at a higher, worst-case current — a stall, a short, a jam — since a modest rise there produces a disproportionate jump in this result.
Worked example — a stalled gearmotor winding at 2 A
A small DC gearmotor in a robot arm carries an armature winding measuring 10 Ω end to end. Block the output shaft and run a stall test: the winding settles at a steady 2 A. Square that current — 4 — and multiply by 10 Ω, giving 40 W: heat produced inside a few grams of copper wire and enamel insulation, with no airflow past it because a motionless rotor drives no cooling fan.
Push the jam a little harder and stall current climbs to 4 A instead of 2 A — double, not quadruple. What the winding sheds does not merely double to 80 W; squaring 4 and multiplying by the same 10 Ω gives 160 W, a full four multiples of the first figure. That is why motor controllers include a current-limit or stall-timeout circuit rather than trusting a slow thermal cutoff: enamel insulation starts to char within seconds at that heat load, long before a bulk temperature sensor elsewhere in the housing would notice anything wrong.
Questions
Why does dissipation quadruple when current only doubles?
Because current appears twice in the formula — once directly, and once again as the size of the voltage drop it creates across the resistance, V = IR. Doubling I doubles both factors of that product at once, so P = I²R rises by a factor of four. A winding or resistor carrying twice its normal current isn't running 'twice as hot' in any simple sense; it's carrying four times the heat load in the same mass of material.
Is power dissipation the same as the power a motor draws from its supply?
No. Total electrical power into a motor is P = VI at its terminals, and under normal running most of that becomes mechanical work turning the shaft. Power dissipation is only the fraction lost to the winding's own resistance, P = I²R, which becomes heat instead of torque. Stall the shaft and the mechanical output drops to zero, so essentially all the electrical power that was going into motion now has nowhere to go but into I²R heat.
Can a component overheat while staying inside its voltage rating?
Yes, easily — voltage rating and thermal limit are separate specifications. A resistor or winding rated for a generous working voltage will still cook itself if the current through it climbs high enough, because heating tracks I²R, not the applied voltage alone. A shorted turn, a jammed motor, or a partially shorted winding can pull current well past normal while the supply voltage never exceeds its rated figure.
Does I²R still describe the heating at radio frequency or fast switching?
Only approximately. At high frequency, current crowds toward a conductor's outer surface — the skin effect — shrinking the cross-section it actually flows through and raising the effective AC resistance above the DC value this formula assumes. A wire that runs cool carrying 60 Hz current can run measurably hotter at the same RMS current a few hundred kilohertz higher, because R itself is no longer the figure printed on the spool.
How do I check whether a resistor can survive this much dissipation?
Compare the calculated wattage against the part's printed power rating, then apply the manufacturer's derating curve for your ambient temperature — most axial resistors hold their full rating only at 25°C free air, tapering toward zero near their maximum case temperature. A part computed at 40 W needs a genuine 40-plus-watt package with somewhere for the heat to go; a small carbon-film resistor at that dissipation fails in seconds.
Why do stall conditions matter more than normal running conditions here?
Because current is usually highest exactly when a motor or actuator is prevented from moving — a jammed gear, a locked rotor, a stalled pump — and I²R heating climbs with the square of that peak figure. A motor that comfortably dissipates a few watts while spinning freely can dissipate ten times that or more the instant it stalls, which is why controllers add current limiting or thermal cutoffs instead of sizing only for the normal running case.