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4.4: Surface-Coupled Systems - A Permanent Polarization Synchronous Machine

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    With field sources modeled by surface charges or surface currents, it is natural to generalize the approach taken in Sec. 4.3 to the description of a wide class of complex electromechanically kinematic systems. The technique involves breaking the region of interest into source-free subregions that have uniform properties and hence can be described by the transfer relations of Sec. 2.16. Sources are then relegated to boundaries between subregions and are taken into account in the boundary conditions used to splice fields together. It is the objective in this section to illustrate the systematic approach that can be taken with such models by developing the lumped-parameter mechanical and electrical terminal relations for the rotating machine shown in Fig. 4.4.1.

    clipboard_e9a940683a0f9bda99db43b64090b6a3e.png
    Fig. 4.4.1 Cross-sectional view of permanent polarization rotating machine

    The rotor consists of a material having polarization density that is uniform and permanent:

    \[ \overrightarrow{P} = P_o [ \overrightarrow{i}_r \, cos(\theta - theta_r) - \overrightarrow{i}_{\theta} \, sin(\theta - \theta_r)] = Re P_o (\overrightarrow{i}_r - j \overrightarrow{i}_{\theta}) e^{-j(\theta - theta_r)} \tag{1} \]

    Field coordinates are \((r,\theta)\) while \(\theta_r (t)\) is the rotor axis. Thus, the polarization density is uniform and directed collinear with the rotor axis at the angle \(\theta_r (t)\). The region between the rotor (with radius \(R\)) and the stator (radius \(R_o\)) is an air gap. Stator electrodes shown in the figure have respective potentials \(\pm v(t)\) and are imbedded in a dielectric having permittivity \(\varepsilon_s\). The length of the device in the \(z\) direction,\(l\), is considered large compared to the radial dimensions.

    Within the rotor, there is no free charge density. Moreover, because the permanent polarization is uniform and hence has no divergence, Gauss' law (Eq. 2.3.27) reduces to

    \[ \Delta \cdot \varepsilon_o \overrightarrow{E} = 0 \tag{2} \]

    Within the rotor, as well as in the air gap and in the surrounding dielectric of the stator, the fields are Laplacian. The transfer relations of Sec. 2.16 are directly applicable to describing the bulk fields

    Boundary Conditions: The potential at \(r = R_o\) is constrained to be \(\pm v(t)\) on the respective portions of the stator surface covered by the electrodes. The potential between the electrodes on the dielectric surface at \(r = R_o\) is approximated by the continuous linear distribution shown in Fig. 4.4.2.

    clipboard_eddbcbf24d1322cdfb54d1de120304c28.png
    Fig. 4.4.1.

    In Fig. 4.4.1, the notation \((a)...(d)\) is used to denote positions adjacent to interfaces between regions. (This convention is introduced in Sec. 2.20.) Thus, the potential distribution of Fig. 4.4.2 is both \(\phi^a\) and \(\phi^b\). In anticipation of the Laplacian solutions used to describe the bulk fields in cylindrical geometry, the potential of Fig. 4.4.2 is now expanded in a Fourier series (see Sec. 2.15for a discussion of Fourier series):

    \[ \phi^a = \phi^b = \sum_{m = - \infty}^{+ \infty} \tilde{\phi}_m^a (t) \, e^{-jm \theta}; \quad \tilde{\phi}_m^a = \frac{2v(t)}{m \pi} \frac{sin(m \theta_o)}{ \theta_o m} sin (\frac{m \pi}{2}) \label{1} \]

    In the following it is assumed that the dielectric surrounding the rotor is of sufficient radius compared to \(R_o\), that fields decay to zero before reaching the outer surface of the dielectric.

    At the rotor air-gap interface the tangential \(\overrightarrow{E}\) and hence the potential must be continuous. Thus the Fourier amplitudes are related by

    \[ \tilde{\phi}_m^c = \tilde{\phi}_m^d \label{2} \]

    In addition, Gauss' law (Eq. 2.10.21a) and Equation \ref{1} require that

    \[ \overrightarrow{n} \cdot \varepsilon_o \, [] E [] = - \overrightarrow{n} \cdot [] P [] \rightarrow \varepsilon_o E_r^c - \varepsilon_o E_r^d = Re (P_o e^{j \theta_r}) e^{-j \theta} \label{3} \]

    This latter expression relates the Fourier amplitudes by

    \[ \varepsilon_o \tilde{E}_{rm}^c - \varepsilon_o \tilde{E}_{rm}^d = \frac{P_o}{2} \Big [ \delta_{1m} e^{j \theta_r} + \delta_{-1m} e^{-j \theta_r} \Big ] \label{4} \]

    where \(\delta_{nm}\), Kronecker's delta function, is unity for \(n = m\) and is otherwise zero.

    Bulk Relations: The transfer relations, Eqs. (a) of Table 2.16.2 with \(k = 0\), are now used to represent the fields at the boundaries. In the stator dielectric surrounding the electrodes \((r > R_o)\),\(\alpha \rightarrow \infty\) and \(\beta \rightarrow R_o\) while \(\varepsilon \rightarrow \varepsilon_s\) :

    \[ \varepsilon_s \tilde{E}_{rm}^a = \varepsilon_s f_m ( \infty, R_o) \tilde{\phi}^a \label{5} \]

    In the air gap \((R_o > r > R) \, \), \(\alpha \rightarrow R_o, \, \beta \rightarrow R\) and \(\varepsilon \rightarrow \varepsilon_o\) so that

    \[\begin{bmatrix} \varepsilon_o \tilde{E}_{rm}^b \\ \varepsilon_o \tilde{E}_{rm}^c \end{bmatrix} = \varepsilon_o \begin{bmatrix} f_m (R,R_o) & g_m (R_o,R) \\ g_m (R,R_o) & f_m (R_o,R) \end{bmatrix} \begin{bmatrix} \tilde{\phi}_m^b\\ \tilde{\phi}_m^c \end{bmatrix} \label{6} \]

    Finally, within the rotor \((r < R)\) the relations are used with \(\alpha = R, \,\) \(\beta \rightarrow 0\) and \(\varepsilon \rightarrow \varepsilon_o\) :

    \[ \varepsilon_s \tilde{E}_{rm}^d = \varepsilon_s f_m (0, R) \tilde{\phi}_m^d \label{7} \]

    The boundary conditions given by Eqs. \ref{2} and \ref{4} and the bulk relations of Eqs. \ref{5}, \ref{6} and \ref{7} comprise six expressions that can be used to determine the Fourier amplitudes \((\tilde{\phi}_m^c, \, \tilde{\phi}_m^d, \, \tilde{E}_{rm}^d, \, \tilde{E}_{rm}^a, \, \tilde{E}_{rm}^b)\) with the driving amplitudes \((\tilde{\phi}_m^a, \, \tilde{\phi}_m^b)\) given by Equation \ref{1}. The solution for any one of the amplitudes is usually much easier than this statement makes it seem, but nevertheless it is worthwhile to have the objective of the model in view before proceeding further.

    Torque as a Function of Voltage and Rotor Angle \((v, \theta_r)\): The rotor is enclosed by a surface at the radial position \((c)\) in the air gap. The method using the Maxwell stress to compute the torque is as outlined in connection with Eq. 4.2.3. With the fields represented by Fourier series, Eq. 2.15.17 reduces the average of the shear stress over the enclosing surface to a summation on the products of the Fourier amplitudes:

    \[ \tau_z = R (2 \pi R l) \langle D_r^c E_{\theta}^c \rangle_{\theta} = 2 \pi R^2 l \sum_{m = - \infty}^{+ \infty} \, (\varepsilon_o \tilde{E}_{rm}^c)^6{*}( \frac{jm}{R} \tilde{\phi}_m^c) \label{8} \]

    Substitution for \(\varepsilon_o \tilde{E}_{rm}^c\) from Equation \ref{6} b introduces the stator field, which is given by Equation \ref{1}, and the same field \(\tilde{\phi}_m^c\) as already appears in Equation \ref{8}. On physical grounds it is expected that this latter "self-field" term should not make a contribution. This is indeed the case, because \(f_m\) is an even function of \(m\) so that terms in \(|\tilde{\phi}_m^c|^2\) cancel out of the sum. The mth term is cancelled by the -mth term. Thus, Equation \ref{8} reduces to

    \[ \tau_z = 2 \pi R^2 l \sum_{m = - \infty}^{+ \infty} \, \varepsilon_o g_m (R,R_o)(\tilde{\phi}_m^b)^{*}( \frac{jm}{R} \tilde{\phi}_m^c) \label{9} \]

    and all that is required to determine the torque is an evaluation of \(\tilde{\phi}_m^c\)

    With this objective, substitution of Eqs. \ref{6} b and \ref{7} into Equation \ref{4} with Equation \ref{2} used to replace \(\tilde{\phi}_m^d\) with \(\tilde{\phi}_m^c\) gives an expression that can m be solved for \(\tilde{\phi}_m^c\):

    \[ \tilde{\phi}_m^c = \frac{ \frac{P_o}{2} [ \delta_{1m} e^{j \theta_r} + \delta_{--1m} e^{-j \theta_r}] \, - \varepsilon_o g_m (R,R_o) \tilde{\phi}_m^b }{ \varepsilon_o [ f_m (R_o,R) \\\, - f_m (0,R) ]} \label{10} \]

    This expression and Equation \ref{1} in turn can be used to evaluate the torque, Equation \ref{9}. (Again, because \(g_m\) and \(f_m\) are even in \(m\), the self-field terms sum to zero):

    \[ \tau_z (v, \theta_r) = \frac{-4 R l g_1 \, (R,,R_o)}{ f_1 (R_o,R) - f_1 (0,R)} \, \frac{sin \, \theta_o}{\theta_o} \, v(t) P_o \, sin \, \theta_r \label{11} \]

    In a lumped parameter model for the device, with \(v(t)\) and \(\theta_r(t)\) functions of time determined by the external electrical and mechanical constraints, this relation represents the electrical-to-mechanical coupling. The reciprocal mechanical-to-electrical coupling completes the model.

    Electrical Terminal Relations: To describe the electrical terminals, the total charge \(q\) on the respective electrodes is required, again as a function of the terminal variables \((v,\theta_r)\). The charge on the upper electrode is

    \[ \begin{align} q = l \, \int_{- \frac{\pi}{2} + \theta_o}^{\frac{\pi}{2} - \theta_o} \, (\varepsilon_s E_r^a - \varepsilon_o E_r^b) R_o d \theta &= l \, \int_{- \frac{\pi}{2} + \theta_o}^{\frac{\pi}{2} - \theta_o} \sum_{m = - \infty}^{+ \infty} (\varepsilon_s \tilde{E}_{rm}^a - \varepsilon_o \tilde{E}_{rm}^b) e^{-jm \theta} R_o d \theta \nonumber \\ &= l R_o \, \sum_{m = - \infty}^{+ \infty} \frac{2}{m} (\varepsilon_s \tilde{E}_{rm}^a - \varepsilon_o \tilde{E}_{rm}^b) sin \, m (\frac{\pi}{2} - \theta_o) \nonumber \end{align} \label{12} \]

    The electric flux normal to the outer and inner surfaces of the electrode are computed from Eqs. \ref{5} and \ref{6} a, respectively:

    \[ \varepsilon_s \tilde{E}_{rm}^a - \varepsilon_s \tilde{E}_{rm}^b = \varepsilon_s f_m (\infty, R_o) \tilde{\phi}^a - \varepsilon_o f_m (R, R_o) \tilde{\phi}^b_m - \varepsilon_o g_m (R_o, R) \tilde{\phi}^c_m \label{13} \]

    The amplitudes \((\tilde{\phi}^a_m,\tilde{\phi}^b_m)\) are given in terms of \(v(t)\) by Equation \ref{2}, while \(\tilde{\phi}^c_m\) is given by Equation \ref{10}. Thus Equation \ref{13} is evaluated in terms of \((v,\theta_r)\):

    \[ q = C_s v(t) - A_r P_o \, cos \, \theta_r (t) \label{14} \]

    where \(C_s\), the stator self-capacitance, is independent of \(\theta_r\) and is

    \[ C_s = \frac{4 l R_o}{ \pi} \sum_{m = - \infty \\ odd}^{+ \infty} \frac{ sin \, m (\frac{\pi}{2} - \theta_o)}{m^2} \frac{sin \, m \theta_o}{m \theta_o} \, sin(\frac{m \pi}{2}) \Bigg [ \varepsilon_s f_m (\infty, R_oo) - \varepsilon_o f_m (R,R_o) + \frac{\varepsilon_o g_m (R_o,R) g_m (R,R_o)}{ f_m (R_o,R) - f_m (0,R)} \Bigg] \label{15} \]

    and \(A_r\) is a constant having the units of area

    \[ A_r = \frac{2 l R_o g_1 (R_o,R)}{f_1 (R_o,R) - f_1 (0,R) } cos \, \theta_o \label{16} \]

    The required electrical terminal relation is Equation \ref{14}.

    For reasons that stem from the approximations made in the field description, the model represented by Eqs. \ref{11} and \ref{14} is not self consistent. At the dielectric air-gap interface between electrodes, the potential is continuous, but \(\overrightarrow{n} \cdot [] \overrightarrow{D} []\) is not. In physical terms, this means that the fields are as though segmented electrodes existed at \(r = R_o\) in these transition regions having the linear potential distribution of Fig. 4.4.2 and supporting a surface charge that can be computed from Equation \ref{13}. This charge is not included in Equation \ref{14} and might for some purposes be ignored. But, if the mechanical and electrical terminal relations are used as stated, the electromechanical system, which after all does not include energy dissipating elements, is given a model that does not conserve energy. In fact, once the torque is known, energy conservation formalisms introduced in Sec. 3.5 not only provide an alternative to computing the electrical terminal relations, but lead to a self-consistent model and a recognition that Equation \ref{15} can be considerably simplified.

    In terms of lumped parameters, the system can be pictured as having the terminal pairs of Fig. 4.4.3. The electrical terminal pairs are interconnected so that \(v_1= -v_2= v\) and by symmetry, \(q_1 = - q_2 = q\). Thus, the incremental energy conservation equation is

    \[ \delta w = 2v \delta q - \tau_z d \theta_r \label{17} \]

    clipboard_e3209e24b32135d9803c7fa5d86232a72.png
    Fig. 4.4.1.
    clipboard_ef6ecc53dd35fc58f9e0b250796d38764.png
    Fig. 4.4.4. State space integration contour

    Not accessible through the external electrical terminals is the electric energy storage due to the permanent polarization. In Equation \ref{17} it is understood that \(P_o\) is held fixed. Transformation to a hybrid energy function \(w^{"}(v,P_o,\theta_r)\) is made by replacing \(v \delta (2 q) \rightarrow \delta (2qv) - 2q \delta v\)\) and defining \(w^{"} = 2qv-w\), so that

    \[ \delta w^{''} = 2q \delta v + \tau_z d \theta_r \label{18} \]

    This expression is integrated on the state-space contour shown in Fig. 4.4.4. First, with the rotor at \(\theta_r = \pi/2\), the polarization is brought up to its final state. Then the voltage is raised. Finally, with \(P_o\) and \(v\) held fixed, the rotor is turned to the angle \(\theta_r\) of interest. With the rotor at \(\theta_r = \pi/2\), the net charge induced on the upper electrode because of the polarization is zero. Hence, the net charge on the upper stator electrode is computed from Equation \ref{13}, but with \(\varepsilon_o E_r^b\) determined as if the rotor were not present. From Equation \ref{6},

    \[ \varepsilon_o \tilde{E}_{rm}^b = \varepsilon_o f_m (0,R_o) \tilde{\phi}_m^b \label{19} \]

    Hence, Equation \ref{12} gives

    \[ q = C_s v; \quad C_s = \frac{4 l R_o}{\pi} \sum_{m = - \infty \\ odd}^{\infty} \frac{sin \, m (\frac{\pi}{2} - \theta_o)}{m^2} \frac{sin \, m \theta_o}{m \theta_o} sin (\frac{m \pi}{2}) [ \varepsilon_s f_m ( \infty, R_o) - \varepsilon_o f_m (0,R_o)] \label{20} \]

    In view of Eqs. \ref{20} and \ref{11}, the integration of Equation \ref{18} on \(v\) and then on \(\theta_r\) leads to

    \[ w^{''} = 2 [\frac{1}{2} C_s v^2] + \bigg [ \frac{4 R l g_1 (R,R_o)}{f_1 (R_o,R) - f_1(0,R)} \frac{sin \, \theta_o}{\theta_o} \bigg] vP_o cos\, \theta_r \label{21} \]

    Finally, because \(w^{"} = w^{"}(v,P_o ,\theta_r)\), the required terminal charge follows as

    \[ q = \frac{1}{2} \frac{\partial{w^{''}}}{\partial{v}} = C_s v - A_r P_o cos\, \theta_r \label{22} \]

    where

    \[ A_r = \frac{-2Rlg_1 (R,R_o)}{f_1 (R_o,R) - f_1 (0,R)} \frac{ sin \, \theta_o}{\theta_o} \label{23} \]

    and \(C_s\) is given by Equation \ref{20}. Simplification of Equation \ref{15} leads to Equation \ref{20}, but for the reasons discussed, Eqs. \ref{16} and \ref{23} differ by the factor \([sin \, \theta_o/ \theta_o]/cos \, \theta_o\). The use of Eqs. \ref{22} and \ref{23} for the electrical terminal relation has the advantage that the model is then self-consistent in its representation of energy flow. The same advantage would exist if the energy relations were used to compute the electrical torque from the electrical terminal relations. This more conventional technique would make use of Equation \ref{14} and an integration of Equation \ref{18} in the sequence, \(P_o, \theta_r\) and \(v\). To carry out the second leg of this integration without making a contribution requires that symmetry be used to argue that there is no electrical torque even though the rotor is polarized.


    This page titled 4.4: Surface-Coupled Systems - A Permanent Polarization Synchronous Machine is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by James R. Melcher (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.