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Electromagnetic Dissipation

The heat generated by electromagnetic fields is often the controlling feature of an engineering design. Semiconductors inevitably produce heat, and the distribution and magnitude of the heat source is an important consideration whether the application is to computers or power conversion. Often, the generation of heat poses a fundamental limitation on the performance of equipment. Examples where the generation of heat is desirable include the heating coil of an electric stove and the microwave irradiation of food in a microwave oven.

Ohmic conduction is the primary cause of heat generation in metals, but it also operates in semiconductors, electrolytes, and (at low frequencies) in semi-insulating liquids and solids. The mechanism responsible for this type of heating was discussed in Sec. 11.3. The dissipation density associated with Ohmic conduction is E E.

An Ohmic current can be imposed by making electrical contact with the material, as for the heating element in a stove. If the material is a good conductor, such currents can also be induced by magnetic induction (without electrical contact). The currents induced by time-varying magnetic fields in Chap. 10 are an example. Induction heating is an MQS process and often used in processing metals. Currents induced in transformer cores by the time-varying magnetic flux are an example of undesirable heating. In this context, the associated losses (which are minimized by laminating the core) are said to be due to eddy currents.

Ohmic heating can also be induced by "capacitive" coupling. In the EQS examples of Sec. 7.9, dielectric heating is caused by the currents associated with the accumulation of unpaired charges.

Whether due to magnetic induction or capacitive coupling, the generation of heat is described by the dissipation density Pd = E E identified in Sec. 11.3. However, the polarization and magnetization terms in the conservation theorem, (11.2.7), can also be responsible for energy dissipation. This occurs when the (electric or magnetic) dipoles do not align instantaneously with the fields. The polarization and magnetization constitutive laws differ from the laws postulated in Sec. 11.3.

As an example suggesting how the polarization term in (11.2.7) can represent dissipation, picture the artificial dielectric of Demonstration 6.6.1 (the ping-pong ball dielectric) but with spheres that are highly resistive rather than perfectly conducting. The accumulation of charge on the poles of the spheres in response to the application of an electric field is described by a rate, rather than a magnitude, that is proportional to the field. Thus, we would expect P/ t rather than P to be proportional to E. With a coefficient representing the properties and geometry of the spheres, the polarization constitutive law would then take the form

equation GIF #11.88

If this law is used to express the polarization term in the conservation law, the second term on the right in (11.2.7), a positive definite quantity results.

equation GIF #11.89

As might be expected from the physical origins of the constitutive law, the polarization term now represents dissipation rather than energy storage.

When materials are placed in electric fields having frequencies so high that conduction effects are negligible, losses due to the polarization of dipoles become the dominant heating mechanism. The artificial diamagnetic material considered in Demonstration 9.5.1 suggests how analogous losses are associated with the dynamic magnetization of a material. If the spherical particles comprising the artificial diamagnetic material have a finite conductivity, the induced dipole moments are not in phase with an applied sinusoidal field. What amounts to Ohmic dissipation on the particle scale is accounted for on the macroscopic scale by a modified constitutive law of magnetization.

The most common losses due to magnetization are encountered in ferromagnetic materials. Hysteresis losses occur because of the coercion required to obtain alignment of ferromagnetic domains. We will end this section with the relationship between the hysteresis curve of Fig. 9.4.6 and the dissipation density.

Energy Conservation for Temporally Periodic Systems

Many practical situations involve fields that vary with time in a periodic fashion. The sinusoidal steady state is the most common example. If the energy conservation law (11.0.8) is integrated over one period T, the energy storage term makes no contribution.

equation GIF #11.90

As a result, the time average of the conservation law states that the time average of the input power goes into the time average of the dissipation. The time average of the integral form of the conservation law, (11.1.1), becomes

equation GIF #11.91

This expression, which assumes that the dynamics are periodic but not necessarily sinusoidal, gives us two ways to compute the total energy dissipation. Either we can use the right-hand side and integrate the power dissipation density over the volume, or we can use the left-hand side and integrate the time average of S da over the surface enclosing the volume.

Consider the sinusoidal steady state as a particular case. If P and M are related to E and H by linear differential equations, an approach can be taken that is familiar from circuit theory. The phase and amplitude of each field at a given location are represented by a complex amplitude. For example, the electric and magnetic field intensities are written as

equation GIF #11.92

A complex vector (r) has three complex scalar components x (r), y (r), and z(r). The meaning of each is the same as the meaning of a complex voltage in circuit theory: e.g., the magnitude of x(r), |x(r)|, gives the peak amplitude of the x component of the electric field varying cosinusoidally with time, and the phase of x(r) gives the phase advance of the cosine time function.

In determining the time averages of products of quantities that are in the sinusoidal steady state, it is helpful to make use of the time average theorem. With * designating the complex conjugate,

boxed equation GIF #11.25

This can be shown by using the identity

equation GIF #11.93

Induction Heating

In this case, the heating is represented by Ohmic conduction and Pd given by (11.3.3c). The examples from Chaps. 7 and 10 involving conductors of finite conductivity offer the opportunity to apply this relation to the evaluation of the right-hand side of (4). If the same total time average power is calculated using the left-hand side of this expression, it may seem that Ohm's law is not required. However, remember that this law is also reflected in the field quantities used to calculate S.

Example 11.5.1. Induction Heating of the Thin Shell

The thin conducting shell of Fig. 11.5.1, in a field Ho(t) applied collinear with its axis, was described in Example 10.3.1. Here the applied field is in the sinusoidal steady state

equation GIF #11.94

floating figure GIF #14
Figure 11.5.1 Circular cylindrical conducting shell in imposed axial magnetic field intensity Ho(t)

According to (10.3.9), the complex amplitude of the response, the magnetic field inside the shell, is

equation GIF #11.95

where m = fraction GIF #25 o a.

The complex amplitude of the surface current density circulating in the shell follows from (10.3.8).

equation GIF #11.96

Because the current density is uniform over the radial cross-section of the shell, the dissipation density can be written in terms of the surface current density K = E.

equation GIF #11.97

It follows from the application of the time average theorem, (6), that the total time average dissipation is

equation GIF #11.98

where l is the shell length. To complete the derivation based on an integration of the density over the volume of the conductor, this expression can be evaluated using (10).

equation GIF #11.99

The same result is found by evaluating the time average of the Poynting flux density integrated over a surface that is just outside the shell at r = a. To see this, we again use the time average theorem, (6), and recognize that the surface integral amounts to a multiplication by the surface area of the shell.

equation GIF #11.100

To evaluate this expression, (10) is used to determine E

equation GIF #11.101

Evaluation of (14) then gives

equation GIF #11.102

which is the same result as found by integrating the dissipation density over the volume, (13).

The dependence of the time average power dissipation on the normalized frequency is shown in Fig. 11.5.2. At very low frequencies, the induced current is not large enough to have an appreciable effect on the imposed field. Thus, the electric field is proportional to the time rate of change of the applied field, and because the dissipation is proportional to the square of E, the power dissipation increases as the square of . At high frequencies, the induced current can be no more than that required to shield the imposed field from the region inside the shell. As a result, the dissipation reaches an asymptotic limit.

floating figure GIF #15
Figure 11.5.2 Time average power dissipation density normalized to po as defined with (13) as a function of the frequency normalized to the magnetic diffusion time defined with (9).

Which of the two approaches is best for finding the total power dissipation? The answer depends on what field information is available. Certainly, the notion that the total heat generated can be found by integrating over a surface that is completely outside the heated material is a fundamental consequence of Poynting's theorem.

Dielectric Heating

In the sinusoidal steady state, we can identify the power dissipation density associated with polarization by finding the time average

equation GIF #11.103

In view of the time average theorem, (6), this becomes

equation GIF #11.104

If the polarization P does not follow the electric field E instantaneously, yet the material is still linear and isotropic, the complex vector can be related to by a complex susceptibility. Or, instead, the complex displacement flux density vector is related to by a complex dielectric constant.

equation GIF #11.105

Here is the complex permittivity with real and imaginary parts ' and -", respectively.

Evaluation of (18) using this constitutive law gives

equation GIF #11.106

Thus, " represents the electrical dissipation associated with the polarization process.

floating figure GIF #16
Figure 11.5.3 Definition of angle defining the loss tangent tan( ) in terms of the real and the negative of the imaginary parts of the complex permittivity.

In the literature, the loss tangent tan is often used to represent dissipation. It is the tangent of the phase angle of the complex dielectric constant defined in terms ' and " in Fig. 11.5.3. Thus,

equation GIF #11.107

From this definition, it follows from Eulers formula that

equation GIF #11.108

Given the complex amplitude of the electric field, D is

equation GIF #11.109

If the electric field is Eo cos ( t), then D is | | Eo cos ( t - ). The electric displacement lags the electric field by the phase angle .

In terms of the loss tangent defined by (21), the time average electrical dissipation density of (20) becomes

equation GIF #11.110

Usually the loss tangent and ' are measured. In the following example, we compute the complex permittivity from a model of the polarizable medium and find the electrical dissipation on a macroscopic basis. In this special case we have the option of finding the time average loss by considering each of the dipoles on a microscopic basis. This is not generally possible, because the interactions among dipoles that are neglected in this example are usually too complicated for an analytic treatment.

Example 11.5.2. An Artificial Lossy Dielectric

By putting together examples considered in Chaps. 6 and 7, we can illustrate the origins of the complex permittivity. The artificial dielectric of Example 6.6.1 and Demonstration 6.6.1 had "molecules" consisting of perfectly conducting spheres. As a result, the polarization was pictured as instantaneously in step with the applied field. We consider now the result of having spheres that have finite conductivity.

The response of a single sphere having a finite conductivity and permittivity surrounded by free space is a special case of Example 7.9.3. The response to a sinusoidal drive is summarized by (7.9.36), where we set a = 0, a = o, b = , and b = . All that is required from this solution for the potential is the moment of a dipole that would give rise to the same exterior field as does the sphere. Comparison of the potential of a dipole, (4.4.10), to that given by (7.9.36a) shows that the complex amplitude of the moment is

equation GIF #11.111

where e (2o + )/. If mutual interactions between dipoles are ignored, the polarization density P is this moment of a single dipole multiplied by the number of dipoles per unit volume, N. For a cubic array with a distance s between the dipoles (the centers of the spheres), N = 1/s3. Thus, the complex amplitude of the electric displacement is

equation GIF #11.112

Combining this result with the moment given by (25) yields the desired constitutive law in the form = , where the complex permittivity is

equation GIF #11.113

The time average power dissipation density follows from this expression and (20).

equation GIF #11.114

The dependence of the power dissipation on frequency has the same form as for the induction heating example, Fig. 11.5.2. At low frequencies, the surface charges induced at the north and south poles of each sphere are completely determined by the external field. Thus, the current density within the sphere that makes possible the accumulation of these surface charges is proportional to the time rate of change of the applied field. At low frequencies, the dissipation is proportional to the square of the volume current and hence to the square of the time rate of change of the applied field. As a result, at low frequencies, the dissipation density increases with the square of the frequency.

As the frequency is raised, less surface charge is induced on the spheres. Although the amount of charge induced is inversely proportional to the frequency, there is a compensating effect because the volume currents are responsible for the dissipation, and these are proportional to the time rate of change of the charge. Thus, the dissipation density reaches a saturation value as the frequency becomes very high.

One tool used to form a picture of atomic, molecular, and domain physics is dielectric spectroscopy. Using this approach, the frequency dependence of the complex permittivity is used to gain insight into the microscopic structure.

Magnetization, like polarization, can also be the source of dissipation. The time average dissipation density due to magnetization follows by taking the time averge of the third and fourth terms on the right in the basic power theorem, (11.2.7). Combined, these terms give

equation GIF #11.115

For small-signal applications, this source of dissipation is dealt with by introducing a complex permeability such that = . The role of the complex permeability is similar to that of the complex permittivity. The artificial diamagnetic material of Example 9.5.2 and Demonstration 9.5.1 can be used to exemplify the concept. Instead of perfectly conducting spheres that give rise to a magnetic moment instantaneously induced antiparallel to the applied field, spherical shells of finite conductivity would be used. The dipole moment induced in the individual spherical shells would be deduced following the same approach as in Sec. 10.4. The resulting dipole moment would not be in phase with an applied sinusoidally varying magnetic field. The derivation of an equivalent complex permeability would follow from the same line of reasoning as used in the previous example.

Hysteresis Losses

Under periodic conditions in magnetizable solids, B and H are related by the hysteresis curve described in Sec. 9.4 and illustrated again in Fig. 11.5.4. What time average power dissipation is implied by the hysteresis?

As before, B and H are collinear. However, neither is now a single-valued function of the other. Evaluation of (29) is accomplished by breaking the cycle into two parts, each involving a single-valued relationship between B and H. The first is the upswing "trajectory" from A C in Fig. 11.5.4. Over this half-cycle, which takes B from BA to BC, the trajectory is H+(B). With B taken as BA when t = 0, it follows from (11.4.4) and (11.4.5) that

equation GIF #11.116

This is the area under the curve of H versus B between A and C in Fig. 11.5.4, traversed on the "upswing." A similar evaluation for the "downswing," where the trajectory is H-(B), gives

equation GIF #11.117

The time average power dissipation, (29), then is the sum of these two contributions divided by T.

equation GIF #11.118

Thus, the area within the hysteresis loop is the energy dissipated in one cycle.

floating figure GIF #17
Figure 11.5.4 With the application of a sinusoidal magnetic field intensity, a steady state is reached in which the hysteresis loop shown in the B-H plane is traced out in the direction shown. The dashed area represents the energy density associated with upward traversal from A to C. The dotted area inside the loop represents the energy density dissipated per traversal of the loop.

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