= Electromechanical conversion power balance
<Electromechanical conversion power balance> applies <conservation of energy> to an <electric machine>'s electrical and mechanical ports, internal storage and losses. In an <electric motor> sign convention, with electrical input positive and mechanical output positive,
$$
p_e=p_m+p_{\rm loss}+\frac{dE_{\rm internal}}{dt}.
$$
The internal term can include a changing <magnetic field> and <rotational kinetic energy>. Omitting it is justified only at the relevant steady operating point or after averaging over a cycle with no net storage change.
An original constant-field <direct current> motor model has winding <electrical resistance> $R$, <inductance> $L$, opposing induced <electromotive force> $e$, and current $i$:
$$
v=Ri+L\frac{di}{dt}+e.
$$
Multiplication by $i$ gives
$$
vi=Ri^2+\frac{d}{dt}\!\left(\tfrac12Li^2\right)+ei.
$$
For an ideal conversion coupling, $e=k_e\omega$ and electromagnetic <torque> $\tau_e=k_t i$ give $ei=\tau_e\omega$ when the numerical constants agree in coherent SI units. The induced electrical term is converted <power>, not a resistive loss. DOE's motor analysis derives current from applied minus opposing induced <electric potential difference> and relates <torque> to field and armature current.<DOE Fundamentals Handbook Electrical Science>[Volume 2, Module 6, DC Motor Theory, printed pp.4–6, equations 6-1–6-3 and speed response]{ref}
Suppose $v=24\,\mathrm V$, $i=2\,\mathrm A$, $R=1\,\Omega$, $L=0.5\,\mathrm H$ and $di/dt=4\,\mathrm{A\,s^{-1}}$. Then $e=20\,\mathrm V$ and
$$
48\,\mathrm W=4\,\mathrm W+4\,\mathrm W+40\,\mathrm W.
$$
The terms are winding heating, increasing magnetic <energy>, and mechanical conversion. If $10\,\mathrm W$ is additionally going into rotational acceleration and mechanical losses are neglected, only $30\,\mathrm W$ reaches the shaft load at that instant. Calling all $40\,\mathrm W$ useful shaft output would omit a second storage term.
A proposed experiment records terminal <electric potential difference>, <electric current>, shaft speed and load <torque> during acceleration and after settling. The discrepancy between electrical input and load output can include storage as well as heating. For an <electrical generator>, reversing the port directions gives the corresponding mechanical-input balance; the physical <energy> accounting is unchanged.
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