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4.15: Overview of Electromechanical Energy Conversion Limitations

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    This chapter has two broad objectives. On the one hand, examples are chosen to illustrate techniques for using a field description in deducing lumped-parameter models. On the other hand, the examples convey an overview of systems that are electromechanically kinematic while providing a back-ground for understanding the kinematic systems taken up in Chaps 5 and 6 and the coupling to deform-able media developed in later chapters.

    The Maxwell stress acting on a "control volume" enclosing the moving material, introduced in Sec. 4.2 as a convenient way to relate the fields to the total force or torque, is also useful in obtaining a qualitative perception of basic limitations on the energy conversion processes. These volumes are represented in an abstract way by Fig. 4.15.1. The longitudinal direction, denoted by \((l)\), generally represents the direction of material motion. Perpendicular to this is the transverse direction denoted by \((t)\).

    The net magnetic or electric force on the volume in general has contributions from both the transverse and longitudinal surfaces, \(A_l\) and \(A_t\). But, in all of the examples of this chapter, shear stresses rather than normal stresses contribute to the energy conversion process. To exploit this fact, the active volume of the devices has a longitudinal dimension that is large compared to transverse dimensions. For example, in rotating machines, maximum use of the magnetic or electric stress is made by having an "air gap" that is narrow compared to the circumference of the rotor. In the Van de Graaff machine, the same considerations lead to a "slender" configuration with the belt charges producing an electric field \(E_t\) across a narrow gap and the generated field being \(E_l\).

    In all of these "shearing" types of electromechanical energy converters, the mechanical power output takes the form

    \[ P_m = UA_t K \, [] \, \mu H_l H_t \, [] \quad \Bigg | \quad P_m = U A_t K \, [] \, \varepsilon E_l E_t \, [] \label{1} \]

    Here, \(U\) and \(A\) are respectively the material velocity and an effective transverse area, e.g., the rotor surface velocity and area respectively in a rotating machine. The largest possible net contribution of the magnetic or electric shear stress contribution, \([] \, \mu H_l H_t \, [] \) and \([] \, \varepsilon E_l E_t \, [] \) respectively, is obtained if stress contributions to one of the surfaces of the control volume are minimized. Generally, this is accomplished by designing field sources into the volume. The factor \(K\) in Equation \ref{1} reflects geometry,material properties and phase angles. In a synchronous machine, it accounts for the air-gap spacing,the sinusoidal spatial dependence of the excitations and the relative phase of stator and rotor excitations. In the variable-capacitance machine of Sec. 4.13, this factor (which represents the "cut" of the ideal power output that is obtained) is also proportional to the product of the saliency amplitudes on rotor and stator.

    Because of their higher energy conversion density, it is generally recognized that conventional magnetic electromechanical energy conversion systems are more practical than their electric counter-parts. This predisposition has its basis in the extreme disparity between electric and magnetic shear stresses that can be produced under ordinary conditions.

    In conventional magnetic equipment, the limit on the magnetic flux density, set by the saturation of magnetic materials, is in the range of 1-2 tesla (10 -20 kgauss). The electric field intensity in air at atmospheric pressure (over macroscopic dimensions in the range of 1 mm to 10 cm usually of interest) is limited to less than the breakdown strength, \(3 x 10^6 \, V/m\). Thus under conventional conditions, the ratio of powers converted by electric and magnetic devices having the same velocity \(U\), effective area \(A_t\) and factor \(K\) is (from Equation \ref{1}) the ratio of the respective shear stresses.Using as typical numbers, \(B = 1\) and \(E = 10^6 \, V/m\), this ratio is

    \[ \frac{(P_m)_{electric}}{(P_m)_{magnetic}} \simeq \frac{\varepsilon_o E_l E_t}{B_l B_t/ \mu_o} \simeq 10^{-5} \label{2} \]

    clipboard_e1f4014221b08665a2525183d019a3945.png
    Fig. 4.15.1. Abstraction of regions of active electromechanical coupling in magnetic and electric field systems.

    The disadvantage inherent to electric energy conversion devices can be made up by increasing the velocity, the effective area, or the electrical breakdown strength. Now, illustrated by some examples is the way in which rough estimates of the energy converted can be made with Eqs. \ref{1}, provided the factors are evaluated with some appreciation for the underlying engineering limitations.

    Synchronous Alternator

    A large synchronous machine, driven by a turbine in a modern power plant,would have the typical parameters:

    rotor radius \(b \simeq 0.5 m\)

    rotor surface velocity \(U = 2 \pi 60 b = 188 m/sec\)

    rotor length \(l = 7 m\)

    air gap transverse and longitudinal flux densities \(\simeq 1 \, \text{tesla}\)

    These figures are typical of the full-scale generator modeled by the machine shown in Fig. 4.7.1c. An upper bound on the factor \(K\) in Equation \ref{1} to take into account the sinusoidal field distributions on rotor and stator, is reasonably taken as \(1/2\). Thus, from Equation \ref{1}a, the mechanical power requirement (and with reasonable efficiency, therefore the maximum electrical power output) is expected to be approximately

    \[ P_m = (188) [ (2 \pi) (0.5) (7)] (0.5) (1)/ 4 \pi x 10^{-7} = 1.6 x 10^{9} \, watts \label{3} \]

    This is about 50% more than the power rating of existing equipment having roughly the parameters used.

    Superconducting Rotating Machine

    The limit on practical magnetic shear stress set by the saturation of magnetic materials more basically arises from the Ohmic heating limit on current density. Asynchronous machine like that described in Sec. 4.9 but with no magnetic materials is in principle not limited by saturation. But it is limited by the current density consistent with available means for removing the heat from the windings. (A current density of \(3 x 10^6 \, A/m^2\) is projected for the normal conducting armature of the machine shown in Fig. 4.9.2.) The incremental increase in magnetic field associated with increasing the current density once the magnetic materials have been saturated makes conventional operation in this range generally unattractive.

    One way to obtain higher field intensities than are practical using conventional conductors is to make use of superconductors. In time-varying fields, superconductors in fact have losses and are difficult to stabilize. But, for slowly varying and d-c fields they can be used to produce magnetic field intensities greater than the \(1-2 \, tesla\) range of conventional equipment. Under balanced synchronous conditions, the field winding is only subject to d-c fields, while the armature winding carries a-c currents and is subject to a-c fields. Thus, in the machine of Fig. 4.9.2, the rotor winding is super-conducting while the stator is composed of normal conductors. With that machine, the projected (rotor)field is in the range of \(5-6 \, tesla\) and the area \(A_t\) required for a given power conversion accordingly reduced. For example, a two-pole \(60 Hz\) machine having \(B_r = 1 \, tesla\), \(B_e = 5 \, tesla\) and rotor length and radius \(l = 5 m\) and \(R = 0.3 m\), respectively, has an estimated mechanical power input of \(A_tT_{\theta r} R \Omega = (2 \pi l R)(B_r B_{\theta} /2 \mu_o)(R)(2 \pi f) \simeq 2 x 10^9 \, watts\). These are representative of the parameters for a projected \(2000 MVA\) superconducting alternator\(^1\).

    Variable-Capacitance Machine

    In machines exploiting electrical shear stresses, the limit on power converted posed by electrical breakdown can be pushed back by either making the insulation an electronegative gas under pressure, or vacuum.Typical improvements in breakdown strength with increasing pressure above atmospheric are shown in Fig. 4.15.2.\(^2\) In principle the field intensity can be increased to more than \(3 x 10^7 \, V/m\), and hence the electric shear stress can be increased by a factor of more than 100 over that used in calculating Equation \ref{2}.

    clipboard_ebf1a56b7982ce55ca745be262737db0f.png

    Fig.4.15.2. Breakdown strength of common gases as a function of gas pressure for several different electrode combinations.\(^2\)

    The machine shown in Fig. 4.13.1c is designed for operation in vacuum. Here, the mean free path is very long compared to the distance between electrodes. As a result, breakdown results as particles are emitted from the electrode surfaces, accelerating until impacting the opposite electrode where they can produce further catastrophic results. Because the voltage difference between electrodes determines the velocity to which particles are accelerated, break-down is voltage-dependent. Put another way, the breakdown field that can be supported by vacuum is a decreasing function of the gap distance. It also depends on the electrodes. Using steel electrodes having exposed areas of \(20 cm^2\), a typical breakdown strength under practical conditions appears to be \(4 x 10^7\) volts across a 1-mm gap\(^3\).

    The electric machines illustrate how the power conversion density can be increased by dividing the device volume into active subregions. In an electric machine, current densities are small and as a result little conducting material is required to make an electrode function as an equipotential. By making stator and rotor blades (as well as intervening vacuum gaps) thin, it is possible to pack a larger amount of area At into a given volume. The limitation on the thickness and hence on the degree of reticulation that can be achieved in practice comes from the mechanical strength and stability of the rotor. Because of material creep and fracture, centrifugal forces pose a limit on the rotational velocity; but more important in this case, if a blade passes through a high-field region slightly off center, the result can be a transverse deflection that is reinforced by the next pulsation. The tendency for the blades to undergo transverse vibrations as they respond parametrically to the pulsating electric stress on each of their surfaces limits the effective area.

    As numbers typical of the machine shown in Fig. 4.13.1c (where there are six gaps), consider:

    \(R = \text{mean radius of blades} = 0.2 \, m \, \text{blade length} = 0.12 \, m \)

    \(U = \text{mean blade velocity at} \, 30,000 \, rpm = 630 \, m/sec \, \text{(an extremely high velocity)} \)

    \(E = 5 x 10^6 \, V/m\)

    \(A_t = (0.2)(2 \pi)(0.12) = 0.9 \, m^2\)

    Remember that the maximum electric field appears where the electrodes have their nearest approach, so the average field used is considerably less than the maximum possible. According to Equation \ref{1}b with \(K=1\), the power output is then at most \(125 kW\). Actually, the factor \(K\) significantly modifies this rough estimate. According to Fig. 4.13.2b, for \(\xi_o/d = 0.4\) and \(a \lambda /4\) phase,

    \[ K = (3.2) \Bigg [ \frac{d}{2 \lambda} \Big (\frac{\xi_o}{d} \Big )^2 \Bigg ] \label{4} \]

    For \(d/\lambda \simeq 0.1, \, K \simeq 2.5 x 10^{-2}\), and the fraction of the ideal energy conversion is not very large. In-stead of \(125 kW\), the postulated machine is predicted to produce \(3 kW\).

    Electron-Beam Energy Converters

    One class of electric field energy convertors that often have very respectable energy conversion densities make use of electrons themselves as the moving material.The model of Sec. 4.6 is developed with this class of devices in mind. A high-energy conversion density can result from the extremely large electron velocities that are easily obtained. For example,an electron having mass \(m\) and charge \(q\) accelerated to the potential \(\phi\) has the velocity

    \[ U = \sqrt{\frac{2 q \phi}{m}} \label{5} \]

    For the electron, \(m = 9.1 x 10^{-31} \, kg\) and \(q = 1.6 x 10^{-19} \, C\). Thus, an accelerating potential of \(10 kV \) results in a beam velocity of \(6 x 10^7 \, m/sec\)!

    In electron-beam devices, the electric shear stress is not usually limited by electrical breakdown, but rather by the necessity for maintaining columnated electrons in spite of their tendency to repel each other. To inhibit lateral motion of the charges particles due to their space charge, ad magnetic field is commonly imposed in the direction of electron streaming. The Lorentz force, Eq. 3.1.1, then tends to convert any radial motion into an orbital motion, while letting electrons stream in the same direction as the imposed magnetic field.\(^4\)

    Electron beams are typically used to convert d-c electrical energy to high-frequency a-c. In fact,the high beam velocity requires that for a synchronous interaction, the frequency \(f\) is the beam velocity \(U\) divided by the wavelength of charge bunches; \(f = U/ \lambda\). Hence, for a wavelength \(\lambda = 6 \, cm\), the frequency for a traveling-wave interaction with the \(10 \, kV\) beam would be essentially \(f = 6 x 10^7 / 6 x 10^{-2} = 10^9 \, Hz\).The practical limit on how short \(\lambda\) can be while obtaining useful coupling between beam and traveling-wave structure is evident from Sec. 4.6.

    The kinematic picture for the beam is useful for making the electroquasistatic origins of the coupling clear and to identify the nature of the synchronous interaction upon which devices like the traveling-wave tube depend. But, because the electron bunching takes place self-consistently with the coupling fields, it is necessary, in engineering electron-beam devices, to treat the electrons as a continuum in their own right.\(^4\) Such examples are taken up in Chapter 11.

    Both electron-beam devices and synchronous alternators convert mechanical to electrical energy. As a reminder rather than a revelation, note that the synchronous alternator is of far more fundamental importance for human welfare, because when attached to the shaft of a turbine driven by a thermal heat cycle, it is capable of converting low-grade thermal energy to a high-grade electrical form. Its con-version of energy naturally fits into schemes for production of energy from natural basic sources.By contrast, the electron-beam devices only convert d-c electrical energy to a high-frequency electrical form.


    1. J. L. Kirtley, Jr., and M. Furugama, "A Design Concept for Large Superconducting Alternators,"IEEE Power Engineering Society, Winter Meeting, New York, January 1975.

    2. J. G. Trump et al., " Influence of Electrodes on D-C Breakdown in Gases at High Pressure," Electrical Engineering, November (1950).

    3. A. S. Denholm, "The Electrical Breakdown of Small Gaps in Vacuum," Can. J. Phys. 36, 476 (1959).

    4. M. Chodorow and C. Susskind, Fundamentals of Microwave Electronics, McGraw-Hill Book Company,New York, 1964.


    4.15: Overview of Electromechanical Energy Conversion Limitations is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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