# 5.13: Electroquasistatic Induction Motor and Tachometer

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A configuration for establishing basic notions concerned with electric induction interactions is shown in Fig. 5.13.1, where a thin sheet having surface conductivity as moves uniformly in the z-direction with the velocity \(U\). \(^1\) At a distance d above the sheet, a traveling wave of potential is imposed by means of electrodes, while the potential a distance d below is constrained by a solid electrode to be constant. The objective in this section is to determine the dependence of the electrical shear force tending to carry the sheet in the \(z\) direction on the frequency w, the relative material and wave velocities, and the electrical surface conductivity. Later, the configuration is used to make a tachometer. In actual construction, the sheet might be wrapped around on itself to form a rotating shell.

The active volume breaks into two regions joined by the conducting sheet. Thus, an analytical model simply involves the combination of transfer relations for the free space regions, and the boundary conditions for the sheet. The transfer relations of Table 2.16.1, Eqs. (a) becomes

\[\begin{bmatrix} \hat{D}_x^{a} \\ \hat{D}_x^{b} \end{bmatrix} = \begin{bmatrix} -\varepsilon_o k \, coth(kd) & \frac{\varepsilon_o k}{sinh(kd)} \\ \frac{-\varepsilon_o k}{sinh(kd)} & \varepsilon_o k \, coth(kd) \end{bmatrix} \begin{bmatrix} \hat{V}_o \\ \hat{\phi}^{b} \end{bmatrix} \label{1} \]

where the surface potentials have been identified as those of the electrodes and sheet, and the variables refer to the upper region with superscripts as defined in Fig. 5.13.1. From Eq. 5.12.4 with \(\partial{()}/\partial{y}\) and \([] J_x [] = 0\) (the regions adjacent to the sheet are insulating),

\[ \sigma_s k^2 \hat{\phi}^b + j( \omega - kU) (\hat{D}_x^b - \hat{D}_x^c) = 0 \label{2} \]

where it is recognized that the net surface charge density on the sheet is \((\hat{D}_x^b - \hat{D}_x^c)\).Finally, the description is completed by the transfer relations for the lower region, again provided by Table 2.16.1:

\[\begin{bmatrix} \hat{D}_x^{c} \\ \hat{D}_x^{d} \end{bmatrix} = \begin{bmatrix} -\varepsilon_o k \, coth(kd) & \frac{\varepsilon_o k}{sinh(kd)} \\ \frac{-\varepsilon_o k}{sinh(kd)} & \varepsilon_o k \, coth(kd) \end{bmatrix} \begin{bmatrix} \hat{\phi}^b \\ 0 \end{bmatrix} \label{3} \]

Incorporated in the potentials on the right are the boundary conditions that \(\phi^b = \phi^c\) and \(\phi^d = 0\).These three expressions can be viewed as five equations for the unknowns \(\phi^b\) and \((D_x^a, D_x^b, D_x^c, D_x^d)\). Before further manipulation is undertaken, it is advisable to look forward to the required variables.

## Induction Motor

Summation of shear stresses on the sheet (see Eq. 4.2.2) shows that the space-average force density in the \(z\) direction is

\[ \langle T_z \rangle_z = \frac{1}{2} R_e j k \hat{\phi}^b [ \hat{D}_x^b - \hat{D}_x^c]^{*} \label{4} \]

The total complex surface charge density required in Eq. \ref{4} follows from the subtraction of Eqs. \ref{1}b and \ref{3}a:

\[ \hat{D}_x^b - \hat{D}_x^c = \frac{-\varepsilon_o k}{sinh(kd)} \hat{V}_o + 2 \varepsilon_o k \, coth(kd) \hat{\phi}^b \label{5} \]

and substitution of this expression into Eq. \ref{4} further reduces the surface force density to

\[ \langle T_z \rangle_z = \frac{-\varepsilon_o k^2}{2 sinh(kd)} R_e j k \hat{\phi}^b \hat{V}_0^{*} \label{6} \]

The complex sheet potential is found by again using Eq. \ref{5}, but this time to eliminate \(\hat{D}_x^b - \hat{D}_x^c\) from Eq. \ref{2}:

\[ \hat{\phi}^b = \frac{j S_e \hat{V}_o}{2 \, sinh \, kd (1+ j S_e \, coth \, kd)} \label{7} \]

where \(S_e\) is product of the angular frequency \((\omega - kU)\) measured by an observer moving with the material velocity \(U\) and the relaxation time constant \(2 \varepsilon_o / k \sigma_s\):

\[ S_e \equiv \frac{2 \varepsilon_o (\omega - kU)}{k \sigma_s} \label{8} \]

The surface force density follows by substituting Eq. \ref{7} into \ref{6}:

\[ \langle T_z \rangle_z = \frac{\varepsilon_o k^2 \hat{V}_o \hat{V}_o^{*}}{4 sinh^2(kd)} \frac{S_e}{(1 + S_e^2 \, coth^2 kd)} \label{9} \]

This result is analogous to one obtained for a magnetic induction machine in Sec. 6.4. It exhibits a maximum which is determined by the frequency in the frame of the moving sheet relative to the effective relaxation time. That is, the optimum or largest electric surface force density is

\[ \langle T_z \rangle_z = \frac{\varepsilon_o k^2 \hat{V}_o \hat{V}_o^{*}}{8 sinh^2(kd)}; \, S_e = tanh (kd) \label{10} \]

Again, this result fits the general description of a "shearing" type of electromechanical energy converter given in Sec. 4.15. The surface force density takes the form of an electric stress \(\varepsilon_o (k V_o)^2 /2\) multiplied by factors reflecting the geometry and charge relaxation phenomena. The factor \((sinh \, kd)^{-2}\) represents the Laplacian decay of the fields from the excitation to the sheet and then back again.

A sketch of the dependence of \( \langle T_z \rangle_z\) on \(S_e\) is shown in Fig. 5.13.2. The physical origins of this curve are understood by interpreting Eq. \ref{7}. At very low material-frame frequencies, \(S_e \rightarrow 0\) and \( \hat{\phi}^b \rightarrow 0\). The sheet behaves as a perfect conductor, supports no tangential electric field intensity, and hence no electrical force in the \(z\) direction.

In the opposite extreme, the frequency is large compared to the reciprocal relaxation time for system of sheet and adjacent regions of free space, and the amount of surface charge induced on the sheet becomes small. This follows from Eqs. \ref{5} and \ref{7}. The optimum of Fig. 5.13.2 represents the compromise between the extremes of \(S_e\) small, and hence the wrong lag angle, and \(S_e\) large and hence reduced sheet surface charge.

Fig. 5.13.2 Dependence of time-average surface force density normalized to \((\varepsilon_o |kV_o|^2 \, sinh^2 \, kd)\) as a function of frequency in moving frame of reference, normalized to relaxation time. \(S_e\) is defined by Eq. \ref{8} \((kd = 1)\).

## Electroquasistatic Tachometer

It is the induced force upon the moving, semi-insulating sheet that is emphasized so far. The reverse effect of the motion on the field is emphasized by the slightly revised configuration of Fig. 5.13.3. Instead of a traveling wave, the imposed potential is now a standing wave. Points of zero amplitude retain fixed positions along the \(z\) axis. For the purpose of detecting the material velocity \(U\), a pair of electrode segments is positioned in the grounded wall just below the moving sheet. The time variation of charge induced on these segments gives rise to a current, \(i\), which is measured by means of external circuitry. Each segment is one half-wavelength, and positioned so that, in the absence of material motion, there is as much positive as negative surface charge induced on a segment surface. Thus, the electrodes are designed so that there is no output current in the absence of a material motion. But, with motion, the fields are skewed so that there is a net charge induced on each output segment. The result, an output signal \(v_o\) reflecting the material velocity \(U\), is now going to be computed.

There is considerable analogy between the interaction studied here in the context of charge relaxation, and the magnetic diffusion example of Sec. 6.4. To make a practical device for measuring the rotational velocity of a shaft, the sheet pictured in Fig. 5.13.3 would be closed on itself, with the standing wave of imposed potential and the output segments perhaps arranged as in Fig. 5.13.4. By contrast with the conventional drag-cup tachometer, the sheet material in the device studied in this section would be made from semi-insulating material, rather than a metal.

Fig. 5.13.3. A device for measuring the velocity \(U\) is made by exciting from above with a standing wave of potential and measuring the induced current on an electrode pair below the sheet.

Fig. 5.13.4. Adaptation of the planar configuration of Fig. 5.13.3 to measure rotational velocity of shell of slightly conducting material.

The fields from a standing wave of excitation potential are simply the superposition of two of the traveling waves analyzed already. That is, the excitation can be written as

\[ \phi^a = Re \hat{V}_o \, cos(kz)e^{j \omega t} = Re \frac{\hat{V}_o}{2} \, (e^{-jkz} + e^{jkz}) e^{j \omega t} \label{11} \]

The surface charge induced on the equipotential plane below the moving sheet is desired. It is assumed that the current, i, is measured through a sufficiently small resistance that the output eleatrodes remain at essentially zero potential. Thus, the output electrode surface charge is simply \(D_x^d\) and is found from Eq. \ref{3}b, as the superposition of the responses to the two traveling-wave components of the drive identified by Eq. \ref{11}:

\[ \hat{D}_x^d = \frac{- \varepsilon_o k}{sinh (kd)} ( \hat{\phi}_+^b + \hat{\phi}_{-}^b) \label{12} \]

The potential amplitudes called for with Eq. \ref{12} are given by evaluating Eq. \ref{7} with \(\hat{V}_o \rightarrow \hat{V}_o/2\) and \(k\) first positive and then negative:

\[ \hat{\phi}_{\pm} = j S_{e \pm} \hat{V}_o/4 \, sinh(kd) [ 1 + j S_{e \pm} coth(kd)] \label{13} \]

\[ S_{e \pm} \equiv 2 \varepsilon_o (\omega \mp kU)/k \sigma_s \nonumber \]

The combination of Eqs. \ref{12} and \ref{13} give the space-time dependence of the charge induced on the lower surface:

\[ D_x^d = - Rej \frac{\varepsilon_o k \hat{V}_o}{4 \, sinh^2 (kd)} \Big \{ \frac{S_{e+} e^{-jkz}}{1 + j S_{e+} coth(kd)} + \frac{S_{e-} e^{jkz}}{1 + j S_{e-} coth(kd)} \Big \} e^{j \omega t} \label{14} \]

The net charge on the right electrode is now computed by integrating the surface charge over its area, from \(z = 0\) to \(z = \pi/k\) and over the width \(w\) of the electrode in the \(y\) direction. The required current is the time rate of change of the net charge on the electrode, and therefore given by

\[ \hat{i} = j \omega \hat{q} = - \frac{j \omega W \varepsilon V_o}{2 \, sinh^2 (kd)} \Big [ \frac{S_{e+}}{1 + j S_{e+} coth(kd)} - \frac{S_{e-}}{1 + j S_{e-} coth(kd)} \Big ] \label{15} \]

As required, the net charge on the electrode vanishes in the absence of a material motion. To bee thedependence of the output current on the material velocity, Eq. \ref{15} is expanded, using the definition of \(S_{e+}\) from Eq. \ref{13}:

\[ |\hat{i}| = I_o \frac{ \Big [ \frac{2 \varepsilon_o \omega}{ k \sigma_s} \, coth(kd) \Big ] (kU/ \omega)}{\sqrt{[1 + S_{e+}^2 coth^2 (kd)] [1+ S_{e-}^2 coth^2(kd) ]}} \label{16} \]

where

\[ I_o \equiv \frac{\omega \varepsilon_o |\hat{V}_o| w}{sinh(kd) \, cosh(kd)}; \, S_{e \pm} \equiv \frac{2 \varepsilon_o \omega}{k \sigma_s} (1 \mp \frac{kU}{\omega}) \nonumber \]

With the excitation frequency large compared to \(kU\), the dependence of \(S_{e \pm}\) on \(U\) is weak, and Eq. \ref{16} shows that the output current is then a linear function of the material velocity. The general dependence of \(|\hat{i}|\) on the ratio of sheet velocity to wave phase velocity, w/k, is illustrated in Fig. 5.13.5.\(^2\)

Fig. 5.13.5. Dependence of output signal on material velocity \(U\) relative to phase velocity \((\omega/k)\) for tachometer of Fig. 5.13.3. Parameter is \(2 \varepsilon_o \omega coth(kd)/ k \sigma_s\)

1. For description of a somewhat similar device, see S. D. Choi and D. A. Dunn, "A surface-ChargeInduction Motor," Proc. IEEE 59, No. 5, 737-748 (1971).

2. For a similar approach to measuring fluid velocity, see J. R. Melcher, "Charge Relaxation on a Moving Liquid Interface," Phys. Fluids 10, 325-331 (1967).