10.2.1. Electromechanic systems: first examples with induction#

In this section first examples of electromechanical systems converting mechanical and electromagnetic power and viceversa exploiting electromagnetic induction are discussed. These examples can be interpreted as rudimentary models of motors or generators. Electromagnetic induction is governed by Faraday’s law

\[\begin{split}\begin{aligned} 0 & = \oint_{\partial s_t} \vec{e}^* \cdot \hat{t} + \dfrac{d}{dt} \int_{s_t} \vec{b} \cdot \hat{n} = \\ & = \oint_{\partial s_t} \vec{e} \cdot \hat{t} + \int_{s_t} \partial_t \vec{b} \cdot \hat{n} \ , \end{aligned}\end{split}\]

as derived from integral form of governing equations on arbitrary domain \(s_t\) (todo add link), that can move in space, with the definition of

\[\vec{e}^* = \vec{e} - \vec{b} \times \vec{v}_b \ ,\]

and \(\vec{v}_b\) is the velocity of the point of the boundary of the surface \(\partial s_t\). As shown in the examples of this section, a time-varying flux of the magnetic field may induce:

  • electromotive force resulting in

    • voltage difference at the electric port of an open circuit, \(v = \dfrac{d \psi(\vec{b})}{d t}\),

    • current in a closed loop, \(i = \dfrac{v}{R} = \dfrac{1}{R} \dfrac{d \psi(\vec{b})}{d t}\),

    being the flux \(\psi = N A B \cos \alpha\) of uniform magnetic field across a \(N\)-winding loop of area \(A\) in a plane with unit normal vectro forming an angle \(\alpha\) with the magnetic field;

  • force on conductors, either moving conductors or conductors with electric current, governed by the expression of Lorentz’s force,

    \[\vec{f} = \rho \vec{e} - \vec{j} \times \vec{b}\]

    or in integral form (elementary on the length of the conductor only) with no net charge

    \[d \vec{F} = - i \vec{b} \times \hat{t} \, d \ell \ .\]

10.2.1.1. Simple loop with moving side in a constant and uniform magnetic field#

Mechanical sub-system

\[m \ddot{x} = F^{ext} + F^{em}_x\]

Faraday’s law (with Ohm’s law \(\vec{e}^* = \rho_R \vec{j}^*\), and negligible resistance of the circuit except for the section \(l_R\); it’s possible to use the equivalent form of Faraday’s law, on the second line of the first equation of this section)

\[\begin{split}\begin{aligned} 0 & = \oint_{\partial s_t} \vec{e}^* \cdot \hat{t} + \dfrac{d}{dt} \int_{s_t} \vec{b} \cdot \hat{n} = \\ & = \int_{l_R} \vec{e}^* \cdot \hat{t} + \dfrac{d}{dt} \left( B a x \right) = \\ & = R i + B a \dot{x} \ . \end{aligned}\end{split}\]

Faraday’s experience, or consequence of Lorentz’s force

\[F^{em}_x = i B a = - \dfrac{(B a)^2}{R} \dot{x} \ .\]

Inserting into the mechanical sub-system, even without self-inductance, the electromagnetic effects appears as a damping term

\[\begin{split}\begin{aligned} m \ddot{x} & = F^{ext} - \dfrac{(B a)^2}{R} \dot{x} \\ m \ddot{x} + \dfrac{(B a)^2}{R} \dot{x} & = F^{ext} \\ \end{aligned}\end{split}\]

todo

  • Show force acting on the moving side of the circuit starting from Lorentz’s force

    • discuss the motion of a rod in a magnetic field, without connection to a circuit; discuss electric charge distribution

  • as \(\partial_t \vec{b} = 0\), it’s possible to use potential \(v\) to define the electromagnetic field \(\vec{e} = - \nabla v\)

10.2.1.2. Rotating loop in a constant and uniform magnetic field#

…

\[0 = R i + \dfrac{d}{dt} \left( A B \cos \alpha \right) \ ,\]

…

10.2.1.3. Simple loop in a time-varying magnetic field#

A time-dependent magnetic flux may induce electric current in an electric circuit…

…

Faraday’s law (with Ohm’s law \(\vec{e}^* = \rho_R \vec{j}^*\), and negligible resistance of the circuit except for the section \(l_R\);…)

\[\begin{split}\begin{aligned} 0 & = \oint_{\partial s_t} \vec{e}^* \cdot \hat{t} + \dfrac{d}{dt} \int_{s_t} \vec{b} \cdot \hat{n} = \\ & = \int_{l_R} \vec{e}^* \cdot \hat{t} + \dfrac{d}{dt} \left( B A \right) = \\ & = R i + A \dot{B} \ , \end{aligned}\end{split}\]

having here assumed that the area of the circuit is constant, and the circuit is planar in a plane with unit normal vector aligned with \(\vec{b} = B(t) \hat{z}\)