10.2.3. DC motors#

Electric subsystem in potential regions

\[e = R i + v\]

Electromagnetic induction

\[v_L = -\int_{\ell} \vec{e} \cdot \hat{t} = - \oint_{\partial S} \vec{e} \cdot \hat{t} = \dfrac{d}{dt}\int_{S} \vec{b} \cdot \hat{n} \]

Mechanical sub-system

\[I \ddot{\alpha} = C^{em} + C^{load} \ ,\]

with \(C^{em} = F b \cos \alpha\), and \(F = i_L B a\).

No commutation, neglecting the self-induction. With no commutation, \(i = i_L\), \(v = v_L\) and thus the magnetic flux reads

\[\int_{S} \vec{b} \cdot \hat{n} = - B a b \sin \alpha = - B A \sin \alpha \ ,\]

so that its time derivative and the voltage different at the port reads

\[v_L = \dot{\alpha} B A \cos \alpha \ .\]

Current in the circuit reads

\[i = \dfrac{1}{R} \left( e - v \right) = \dfrac{1}{R} \left( e - \dot{\alpha} B A \cos \alpha \right) \ ,\]

electromagnetic torque

\[C^{em} = i_L B A \cos \alpha = \dfrac{BA}{R} \cos \alpha \left( e - \dot{\alpha} B A \cos \alpha \right)\]

With commutation. \(i_L = i \cdot \text{sign} \{ \cos \alpha \}\), \(v = v_L \cdot \text{sign} \{ \cos \alpha \}\)

\[\begin{split}\begin{aligned} C^{em} = i_L B A \cos \alpha & = \text{sign} \{ \alpha \} \cdot \dfrac{BA}{R} \cos \alpha \left( e - \dot{\alpha} B A \cos \alpha \cdot \text{sign} \{ \alpha \} \right) = \\ & = \dfrac{BA}{R} \left| \cos \alpha \right| \, e - \cos^2 \alpha \dfrac{(BA)^2}{R} \, \dot{\alpha} \ . \end{aligned}\end{split}\]

Multiple windings. With \(N\) windings equally spaced \(\Delta \theta = \frac{\pi}{N}\), \(\alpha_n = \alpha + \frac{n}{N} \pi\), are connected in series, quantities in the DC motor become so regular that can be approximated with their average value on a turn of the motor,

\[v = \dot{\alpha} B A \sum_{n} |\cos \alpha_n| \simeq \dot{\alpha} \frac{2 N BA}{\pi}\]
\[i_L = i = \frac{1}{R}\left( e - \dfrac{2 N BA}{\pi} \dot{\alpha} \right) \]
\[\begin{split}\begin{aligned} C^{em} = i_L B A \sum_k \left| \cos \alpha_k \right| & \simeq \dfrac{2 N}{\pi} BA \dfrac{1}{R} \left( e - \dfrac{2N}{\pi} BA \dot{\alpha} \right) = \\ & = \dfrac{1}{R} \dfrac{2 N BA}{\pi} e - \dfrac{1}{R} \left(\dfrac{2N BA}{\pi} \right)^2 \dot{\alpha} \ . \end{aligned}\end{split}\]

The dynamical system of a brushed DC motor then are

\[\begin{split}\begin{aligned} I \ddot{\alpha} & = C^{load} + C^{em} \\ e & = R i + v \end{aligned}\end{split}\]

Energy balance. Multiplying the mechanical equation by \(\dot{\alpha}\) and circuit equation by \(i\),

\[\begin{split}\begin{aligned} 0 & = \dot{\alpha} \left( I \ddot{\alpha} - C^{load} - C^{em} \right) - i \left( e - R i - v \right) = \\ & = \dot{\alpha} \left( I \ddot{\alpha} - C^{load} - i \dfrac{2 N BA}{\pi} \right) - i \left( e - R i - \dfrac{2 N B A}{\pi} \dot{\alpha} \right) = \\ & = \dfrac{d}{dt} \left( \dfrac{1}{2} I \dot{\alpha}^2 \right) - C^{load} \dot{\alpha} + e i - R i^2 \end{aligned}\end{split}\]

energy balance equation can be written as

\[ \dfrac{d}{dt} \left( \dfrac{1}{2} I \dot{\alpha}^2 \right) - R i^2 = C^{load} \dot{\alpha} + e i \ , \]

where power of external actions on the system and the internal dissipation \(R i^2\) equals the time derivative of the kinetic energy. Here the conversion of electromagnetic power to mechanical power is conservative, except for the dissipation loss in resistors.

todo

  • better on average and different connections

  • add pictures

  • more on energy balance

  • add self-inductance, being

    \[\begin{split}\begin{aligned} v_L = \dfrac{d \psi}{d t} & = \dfrac{d}{d t}\left( \widetilde{N} \phi \right) = \\ & = \dfrac{d}{d t} \left( \widetilde{N} \left( B A \sin \alpha + \phi_d \right) \right) = \\ & = \dfrac{d}{d t} \left( B A \widetilde{N} \sin \alpha \right) + \dfrac{d}{dt} \left( L i \right) = \\ & = \dot{\alpha} \, B A \widetilde{N} \cos \alpha + \dfrac{d}{dt} \left( L i \right) \ . \end{aligned}\end{split}\]

    being \(\phi_d = \dfrac{\widetilde{N} i}{\theta}\) the dispersed flux producing self-induction, and \(L = \dfrac{\widetilde{N}^2}{\theta}\) the self-inductance. KVL equation on the electric circuit gives

    \[e = R i + v = R i + L \dfrac{d}{dt} (i ) + K \dot{\alpha} \ ,\]

    having introduced average quantities for multiple windings. todo define \(\widetilde{N}\) for multiple windings