4.12: Quasi-One-Dimensional Models and the Space-Rate Expansion

The "narrow-air-gap" model for rotating machines and long-wave models for electromagnetic wave propagation are examples of quasi-one-dimensional models. The following sections illustrate the use of such models in the kinematic description of electromechanical interactions. Extensive use will be made in later chapters of models that similarly exploit a relatively slow variation of distributed quantities in a "longitudinal" direction relative to "transverse" directions.

Fig. 4.12.1.(a) Cross-sectional view of synchronous electric field energy converter with stator and rotor composed of perfectly conducting materials constrained by a time-varying voltage source. The stator geometry is static, while the rotor moves to the right. (b) Interaction represented by time-varying capacitance.(c) Detail of air gap showing components of \(E_z\) to satisfy boundary conditions.

An example is shown in Fig. 4.12.1. Perfectly conducting surfaces having the potential difference \(v(t)\), vary from the planes \(x = 0\) and \(x = -d\) by the amounts \(\xi_s(z,t)\) and \(\xi_r(z,t)\), respectively.What are the fields in the gap? This configuration is the basis for the study of the variable-capacitance machine in Sec. 4.13. Fields in the gap can be approximated by two techniques. If \(\xi_s\) and \(\xi_r\) are small compared to \(d\), the boundary conditions can be linearized, and the fields found approximately. This is the approach used in Sec. 4.3 for describing the salient pole interactions (Eq. 4.3.16). It formally amounts to expanding the fields in an amplitude parameter expansion with the zero-order fields those with \(\xi_s\) and \(\xi_r\) equal to zero, the first-order terms those given by keeping only linear terms in \((\xi_s,\xi_r)\) and so on. Thus, the validity of the model hinges on the amplitudes \((\xi_s,\xi_r)\) being small.

In quasi-one-dimensional models, amplitudes are not necessarily small. Rather, certain spatial rates of change are small. In the configuration of Fig. 4.12.1, the distance \(\lambda\) typifying variations in the \(z\) direction is long compared to the distance \(d\), \(\gamma \equiv (d/ \lambda)^2 << 1\).

The relationship between linearized and quasi-one-dimensional models is illustrated in Fig. 4.12.2.Linearized quasi-one-dimensional models must be consistent with the long-wave limit of the linearized model. In establishing complex models, this fact is often used to motivate the appropriate "zero-order"approximation which is the starting point in developing a quasi-one-dimensional model.

Fig. 4.12.2.Schematic characterization of relationships among three-dimensional,quasi-one-dimensional and linearized models.

Usually, quasi-one-dimensional models are motivated by physical reasoning, with little need for formality. This is partly because higher order terms are seldom used. But, at least once,.it is worthwhile to see how higher order terms are found, and that the approximation used is the lowest order term in an expansion in powers of a space-rate parameter, in the example of Fig. 4.12.1, of \(\gamma = (d/ \lambda)^2\).

The procedure here is analogous to that of Sec. 2.3 on quasistatics. The spatial coordinate \(z\),in which variables evolve slowly, plays the role of time. The physical idea that this slow variation ought to make one field component dominate the other is built into the normalization of variables.If modulations of the electrodes are slowly varying compared to the transverse distance \(d\), each section of the electrodes tends to form a parallel-plate capacitor. With \(E_o\) a typical electric field in the \(x\) direction (the "dominant" field component), \(d\) taken as the typical length in the \(x\) direction,but \(\lambda\) as that length in the \(z\) direction, the appropriate normalization is

\[ \begin{align} E_x &= E_o \underline{E}_x \quad \quad &x = d \underline{x} \nonumber \\ E_z &= E_o (d/ \lambda) \underline{E}_z \quad \quad &z = d \lambda \underline{z} \nonumber \\ \xi_r &= d \underline{\xi}_r , \, \xi_s = d \underline{\xi}_s \quad \quad &\nu = (E_o d) \underline{\nu} \nonumber \end{align} \label{1} \]

In the gap, \(\overrightarrow{E} \) is irrotational and solenoidal. In terms of the normalized variables, these conditions are

\[ \begin{align} &\frac{\partial{E_x}}{\partial{z}} - \frac{\partial{E_z}}{\partial{x}} = 0 \nonumber \\ &\frac{\partial{E_x}}{\partial{x}} = - \gamma \frac{\partial{E_z}}{\partial{z}} \nonumber \end{align} \label{2} \]

where the space-rate parameter \( \gamma \equiv (d/ \lambda)^2\). To complete the formulation in terms of normalized variables,boundary conditions at the scalloped perfect conductors are that the potential difference be \(v(t)\) and the tangential fields vanish:

\[ E_z = - \frac{\partial{\xi_x}}{\partial{z}} E_x (x = \xi_s); \, E_z = - \frac{\partial{\xi}}{\partial{z}} E_x ( x = \xi_r -1); \, \int_{\xi_r - 1}^{\xi_s} E_x dx = v \label{3} \]

Only two of these three expressions are independent.

The normalized field components are now expanded in series of the form

\[ \begin{align} &E_x = E_{xo} + \gamma E_{x1} + \gamma^2 E_{x2} + ... \nonumber \\ &E_z = E_{zo} + \gamma E_{z1} + \gamma^2 E_{z2} + ... \nonumber \end{align} \label{4} \]

Note that only one dimensionless parameter is involved, so for the particularly simple case at hand,there is no ambiguity as to what lengths are most critical.

Substitution of the series of Equation \ref{4} into Eqs. \ref{2} gives a pair of expressions which are polynomial in \(\gamma\). Coefficients of each order in \(\gamma\) must vanish; thus, the zero-order terms involve only the zero-order fields

\[ \begin{align} &\frac{\partial{E_{xo}}}{\partial{z}} - \frac{\partial{E_{zo}}}{\partial{x}} = 0 \nonumber \\ &\frac{\partial{E_{xo}}}{\partial{x}} = 0 \nonumber \end{align} \label{5} \]

but the first order expressions are "driven" by the zero order fields

\[ \begin{align} &\frac{\partial{E_{x1}}}{\partial{z}} - \frac{\partial{E_{z1}}}{\partial{x}} = 0 \nonumber \\ &\frac{\partial{E_{x1}}}{\partial{x}} = - \frac{\partial{E_{zo}}}{\partial{z}} \nonumber \end{align} \label{6} \]

It follows from Eqs. \ref{3} c and \ref{5} b that \(E_{xo}\) is quasi-one-dimensional. It only depends on \((z,t)\):

\[ E_{xo} = E_{xo} (z,t) = \frac{ v}{\xi_s + 1 - \xi_r} \label{7} \]

What has been deduced as the zero-order \(E_x\) is just the voltage divided by the distance between conductors. If variations with z are sufficiently slow, each section of the system forms a plane-parallel capacitor. To find the other component of the zero-order field, note that \(E_{xo}\) is only a function of \((z,t)\), so Equation \ref{5} a can be integrated to obtain

\[ E_{zo} = x \frac{ \partial{E_{xo}}}{\partial{z}} + f(z,t) \label{8} \]

where \(f(z,t)\) is an integration function. This function is determined by substitution of Equation \ref{8} into Equation \ref{3} a:

\[ E_{zo} = x \frac{\partial{E_{xo}}}{\partial{z}} - \frac{ \partial{}}{\partial{z}} (E_{xo} \xi_s) \label{9} \]

Substitution now shows that the tangential field on the lower surface is zero, Equation \ref{3}b is satisfied. The zero-order fields are represented in dimensionless form by Eqs. \ref{7} and \ref{9}.

The first-order fields are predicted by Eqs. \ref{6}, now that the zero-order fields are known. From Eqs. \ref{6} b and \ref{9},

\[ - \frac{\partial{E_{x1}}}{\partial{x}} = - \frac{\partial{E_{zo}}}{\partial{z}} = - x \frac{\partial^2{E_{xo}}}{\partial{z^2}} + \frac{\partial^2{}}{\partial{z^2}} (E_{xo} \xi_s) \label{10} \]

The functional dependence on \(x\) on the right in this expression is explicit, and therefore integration gives

\[ E_{x1} = - \frac{x^2}{2} \frac{\partial^2{E_{xo}}}{\partial{z^2}} - x \frac{\partial^2{}}{\partial{z^2}} (E_{xo} \xi_s) + g(z,t) \label{11} \]

Because the zero-order \(E_x\) already satisfies the boundary condition, \(E_{xo}\) integrates to v across the gap (Equation \ref{3} c), the same integral of Equation \ref{11} must vanish and that serves to determine the integration function \(g(z,t)\). At this point, two terms in the series of Equation \ref{4}a have been found, and they are sufficient to show what is meant by the expansion

\[ E_x = \frac{v}{(1 + \xi_s - \xi_r)} + \gamma \Bigg \{ \frac{\partial^2{E_{xo}}}{\partial{z^2}} \Bigg [ - \frac{x^2}{2} + \frac{1}{6} \frac{\xi_s^3 + (1- \xi_r)^3}{ \xi + (1 - \xi_r)} \Bigg ] + \frac{\partial{}^2}{\partial{z^2}} (E_{xo} \xi_s) (x - \frac{1}{2} [ \xi_s - (1- \xi_r)]) \Bigg \} \label{12} \]

By the definition of \(l\) used in normalizing \(z\), \(\partial^2{E_{xo}/ \partial{z^2}}\) on the order of \(E_{xo}\). Hence, the first term in Equation \ref{12} gives an accurate picture of the field, provided \( \gamma << 1\).

The procedure outlined is mainly of conceptual value. Certainly the quasi-one-dimensional modeling of a complex problem begins with a physically motivated approximation: here, Equation \ref{7}. Because no more than the zero-order solutions are usually required, the formalism of normalizing the variables and identifying dimensionless space parameters is not usually required.

In retrospect, the zero-order fields have a dependence on the transverse direction \((x)\) that is the lowest order polynomial in \(x\) consistent with the boundary conditions. Thus, \(E_{xo}\) varies as \(x^o\) (it is independent of \(x\)); while \(E_{zo}\) can satisfy the boundary conditions only if it includes a linear dependence on \(x\).