Systems may operate with mixed energy domains, including mechanical work, electric work, etc., and the conservation of energy still governs dynamics. A transducer converts energy/signals between energy domains. Based on the magnitude of energies involved, we may highlight their energy-converting behavior or their signal-converting behavior.
These transducers couple linear and rotational motion, and thereby converts energy between two mechanical forms. These include pulley systems and rack and pinon systems.
$$ \dot\theta=n\dot x\Rarr nT=F $$
Insert Illustration
These transducers couple linear mechanical motion and electrical current. These include solenoids and linear motors. In principle, both translational and the below rotational electromechanical transducers can operate with AC voltage, using the governing equations to relate the root-mean-squared values of the voltage and current signals.
$$ \dot x=\alpha e_{12}\Rarr i=\alpha F,~e_{12}i=\dot xF $$
Insert illustration
These transducers couple rotational mechanical motion and electrical current. These include rotary electric motors and generators (which are structurally identical but designed for opposite directions of power transmission).
$$ \dot\theta=\alpha e_{12}\Rarr i=\alpha T,~e_{12}i=\dot\theta T $$
Insert illustration
Current through a wire generates a magnetic field about that wire according to a right-handed convention. When current carrying wire is coiled, the magnetic flux of each coil is superimposed, such that for a coil of radius, $r$, ambient permeability, $\mu$, number of turns, $n$, and current, $i$, the magnetic field strength at the coil’s center, $B$, is given by the following.
$$ B=\frac\mu{2\pi r}ni $$