Small-Signal Power in a Resistor

A resistor's power depends on two things at once: voltage and resistance. Zoom in close to one operating point and the curved power surface looks flat. That flat tangent plane predicts small changes with simple arithmetic.

Drag on the surface to move the point, or click a red marker to jump to it.

How it works

Tangent planes and linear approximation

In one sentence

If $P$ is differentiable at $(V_0, R_0)$ , then $P(V, R) \approx P(V_0, R_0) + P_V(V_0, R_0)(V - V_0) + P_R(V_0, R_0)(R - R_0)$ , with an error that shrinks faster than the distance to $(V_0, R_0)$ : the tangent plane is the best linear approximation.

General formula

{\htmlData{sym=0}{P}}({\htmlData{sym=1}{V}}, {\htmlData{sym=2}{R}}) = \frac{{\htmlData{sym=1}{V}}^2}{\htmlData{sym=2}{{R}}}, \qquad \Delta {\htmlData{sym=0}{P}} \approx {\htmlData{sym=3}{\frac{\partial P}{\partial V}}}\,{\htmlData{sym=5}{\Delta V}} + {\htmlData{sym=4}{\frac{\partial P}{\partial R}}}\,{\htmlData{sym=6}{\Delta R}} = \frac{2{\htmlData{sym=1}{V}}}{\htmlData{sym=2}{{R}}}\,{\htmlData{sym=5}{\Delta V}} - \frac{{\htmlData{sym=1}{V}}^2}{{\htmlData{sym=2}{R}}^2}\,{\htmlData{sym=6}{\Delta R}}

Each partial derivative is a slope in one direction. Moving a little in both directions adds the two changes: slope times step, twice.

What the symbols mean

$P$
Power
Watts turned into heat in the resistor.
$V$
Voltage
The push across the resistor, in volts.
$R$
Resistance
How hard the resistor resists current, in ohms.
$\frac{\partial P}{\partial V}$
Sensitivity to voltage
Extra watts per extra volt, with R held fixed.
$\frac{\partial P}{\partial R}$
Sensitivity to resistance
Change in watts per extra ohm, with V held fixed. It is negative.
$\Delta V$
Small voltage change
A nudge in voltage away from the operating point.
$\Delta R$
Small resistance change
A nudge in resistance away from the operating point.

At the operating point $(V_0, R_0)$ the tangent plane is $P \approx P_0 + P_V\,(V - V_0) + P_R\,(R - R_0)$ .
Its two tilts are the partial derivatives: $P_V = 2V/R$ (more voltage, more power) and $P_R = -V^2/R^2$ (more resistance, less).
For small nudges $\Delta V$ and $\Delta R$ , the plane's prediction is the differential $dP$ , and it is very close to the true change.
Far from the point the surface curves away from the plane, so the approximation only works nearby.

Learn the math

DifferentialsCalc III · Lesson 22, Partial Derivatives Tangent Planes and Linear ApproximationsCalc III · Lesson 25, Applications of Partial Derivatives