Reciprocal lead field

In addition to these general impedance simulations, the sensitivity of an electrode arrangement to impedance variations in the bladder region has been assessed. For this, the principle of the reciprocal lead field described by [12] and adapted to COMSOL by [11] has been used. To quantify to which extent a small volume contributes to the measured overall impedance, a second, virtual current source is introduced by making the voltage measurement electrodes current carrying. The current density vector field by this virtual current source is called the reciprocal lead field and specifies the sensitivity of the voltage measurement electrodes. The measured impedance can be expressed as... 

The sensitivity to the impedance of a small volume is maximized if the dot product of injected current density and reciprocal lead field is maximized. As the sensitivity field we... 

In many cases, there is partial solid solubility between the pure components of a binary system, as in the Pb-Sn phase diagram of Figure 11.5, for example. The solubility limits of one component in the other are given by solvus lines. Note that the solid solubility limits are not reciprocal. Lead will dissolve up to 18.3 percent Sn, but Sn will dissolve only up to 2.2 percent Pb. In Figure 11.5, there are two two-phase fields. Each is bounded by a distinct solvus and liquidus line, and the common sofidus line. One two-phase field consists of a mixmre of eutectic crystals and crystals containing Sn solute dissolved in Pb solvent. The other two-phase field consists of a mixture of eutectic crystals and crystals containing Pb solute dissolved in Sn solvent. 

It follows from the principle of reciprocityy described by Hermann von Helmholtz in 1853 [Helmholtz, 1853], that the lead field is identical to the electric current field that arises in the volume conductor if a unit current, called reciprocal current iRy is fed to the lead. 

Because of reciprocity, the field of lead vectors is the same as the current field raised by feeding a reciprocal current of 1A to the lead. 

The lead field may be illustrated either with lead vectors in certain locations in the volume conductor or as the flow lines of the distribution of the reciprocal current in the volume conductor. This is called the lead current field. In the latter presentation, the lead field current flow lines are oriented in the direction of the sensitivity, and their density is proportional to the magnitude of the sensitivity. 

To analyze the situation with a tetrapolar electrode system in contact with, for example, a human body, we must leave our simplified models and turn to lead field theory (see Section 6.4). The total measured transfer impedance measured is the ratio of recorded voltage to injected current according to Eq. 6.39. The impedance is the sum of the impedance contributions from each small volume dv in the measured volume. In each small volume, the resistance contribution is the resistivity multiplied by the vector dot product of the space vectors (the local current density from a unit reciprocal current applied to the recording electrodes) and (the local current density from a unit current applied to the true current carrying electrodes). With disk-formed surface electrodes, the constrictional resistance increase from the proximal zone of the electrodes may reduce sensitivity considerably. A prerequisite for two-electrode methods is therefore large band electrodes with minimal current constriction. 

It was a long way to go before the genesis of the surface potential differences caused by action potentials deep in the thorax was understood. It was the work of Einthoven and the lead concept that paved the way. It was Burger and van Milaan who introduced the lead vector, making it possible to find the direction to go. Richard McFee replaced the lead vector by the lead field, defined as the electric field set up in the body by a unit current applied to the pick up electrode pair. Otto Schmitt reintroduced the old Helmholtz concept about reciprocity and introduced the concept of transfer impedance already known from the use of four-electrode technique. And it was David Geselowitz who finally put it in the elegant mathematical form. A certain similarity with the Faraday—Maxwell intellectual process runs in our minds. 

From considerations of reciprocity, Z is equal of the lead field, that is the electric field in the heart arising from unit current injected into the lead, V (McFee and Johnston, 1953). Therefore,... 

This part of our chapter has shown that the use of the two variables, moduli and phases, leads in a direct way to the derivation of the continuity and Hamilton-Jacobi equations for both scalar and spinor wave functions. For the latter case, we show that the differential equations for each spinor component are (in the nearly nomelativistic limit) approximately decoupled. Because of this decoupling (mutual independence) it appears that the reciprocal relations between phases and moduli derived in Section III hold to a good approximation for each spinor component separately, too. For velocities and electromagnetic field strengths that ate nomrally below the relativistic scale, the Berry phase obtained from the Schrddinger equation (for scalar fields) will not be altered by consideration of the Dirac equation. 

To understand the principle of operation of important non-reciprocal (see below) microwave devices, consider what occurs when a plane-polarized microwave is propagated through a ferrite in the direction of a saturating field Ht. The wave can be resolved into two components of equal amplitude but circularly polarized in opposite senses, i.e. into a right-polarized and a left-polarized component. These two components interact very differently with the material, leading to different complex relative permeabilities H r+ = n r+ - j/r" i and /r - = /T- — j/r"-, as shown in Fig. 9.40. Because of the... 

Umklapp process In the interaction of a continuous wave (photon, electron, etc.) with the lattice, the quasi-momentum of the wave is conserved, modulo a vector in the reciprocal lattice. The introduction of these quanta of momentum leads to the Umklapp process. In many macroscopic treatments the matter is treated as a continuous medium and Umklapp processes are neglected. In our treatment, Umklapp processes are included in the coulombic interactions (calculation of the local field), but implicitly omitted in the retarded interactions, since we dropped the term (cua/c)2 in (1.64). 

For pure liquids, the Debye equation suggests that the molar polarization should be a linear function of the reciprocal temperature. Furthermore, one should be able to analyze relative permittivity data for a polar liquid like water as a function of temperature to obtain the dipole moment and polarizability from the slope and intercept, respectively. In fact, if one constructs such a plot using data for a polar solvent, one obtains results which are unreasonable on the basis of known values of p and ocp from gas phase measurements. The reason for the failure of the Debye model in liquids is the fact that it neglects the field due to dipoles in the immediate vicinity of a given molecule. However, it provides a reasonable description of the dielectric properties of dilute polar gases. In liquids, relatively strong forces, both electrostatic and chemical, determine the relative orientation of the molecules in the system, and lead to an error in the estimation of the orientational component of the molar polarization. 

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