Quadrature coils

The variation of resistance is proportional to the amplitude of the part of the flux crossing the coil in quadrature with the current in the solenoid. 

The compact MRI system has an advantage for rf coil design because solenoid coils can be used in most applications. The solenoid coil has about three times better SNR than that of the saddle-shaped coil (14). Even if the saddle-shaped or birdcage coil is used in the quadrature mode, the solenoid coil will still have better SNR because an SNR gain of only about 1.4 times is obtained in that mode. 

All experiments were performed in a 1.9-T horizontal bore magnet (Oxford Instruments, Oxford, UK) with a dear bore diameter of 31 cm. Magnetic field gradients were produced by a 12-cm id water-cooled gradient set (Resonance Research, Billerica, MA, USA), capable of a maximum output of 300 mT m-1, and were driven by Techron 7700 amplifiers (Techron Inc., Elkhart, IN, USA). Rf excitation was accomplished using either a quadrature driven birdcage coil (Morris Instruments, Ottawa, ON, Canada), or an 8-tum laboratory-built solenoid coil, driven by an ENI LPI-10 1000 W amplifier or a Matec Model 525 class-C amplifier. 

Fig. 1. Fetai head hoiders. (a) Rat Fioider. (b) Rabbit Fioider, with heads in situ, (c) Both holders were designed to fit within the 72 mm quadrature T/R coil and directly into the anesthetic port of the standard Broker mouse cradle enabling fast and accurate repositioning of samples in between imaging acquisitions.

A cylindrical structure fitting the geometry of the magnet bore is the saddle coil. It can provide homogeneous B fields at moderate NMR frequencies (25 MHz) for larger volumes (up to 30 cm diameter). However, its sensitivity is lower by a factor of /3 compared to a solenoid. But a factor of >/2 can be recovered, if two saddle coils are arranged in quadrature for independent detection of the jc- and y-components of the induced signal. 

Furthermore, we will focus on the magnetic field and the electromotive force at the receiver of the two-coil induction probe. Substituting eq. 2.37 into 2.27 the magnetic field Hz can be represented as a sum of two components, namely the quadrature component which is shifted in phase by 90° with respect to the primary magnetic field, Hq, or the current in the source, and the inphase component which is shifted in phase by 0° or 180° with respect to the primary field, and we have ... 

In this chapter we will derive an expression for the vertical component of the magnetic field on the axis of a borehole when the source of the primary field is a vertical magnetic dipole and the formation has an infinite thickness. Special attention will be paid to the analysis of frequency responses of quadrature and inphase components of the field, including their asymptotic behavior. The influence of various parameters of a geoelectric section will also be investigated. Such questions as the influence of finite dimensions of coils, displacement of the induction probe wdth respect to the borehole axis, the role of magnetic permeability and dielectric constant will be studied. 

Calculations demonstrate that the influence of frequency and conductivity of a formation on the magnitude of the ratio Q S/So is practically the same as in the previous case. At the range of small values of parameter (T2/xa the relative contribution of currents induced in the bed constitutes about 80% while for a value of 02lMXi = 0.64 the contribution of the formation is equal to 70% but the ratio Q S/So essentially increases. For this reason with an increase of the frequency the depth of investigation of a two-layered medium by a two-coil induction probe does not change until the signal from the formation is greater or at least comparable with that caused by induced currents in the borehole. Also the natural limitation of a further increase of frequency is related with a nonunique interpretation, inasmuch as the spectrum of the quadrature component has a maximum. 

As follows from eqs. 4.126 the second term of the quadrature component and the leading term of the inphase component of the magnetic field, H, do not depend on the probe length, nor on the geoelectrical parameters of the borehole and the invasion zone. Therefore, regardless of the separation of the coils measuring these quantities we can essentially increase the depth of investigation on the induction probe. 

For illustration values of quadrature and inphase components of for various displacements e = ro/a, as the two-coil induction probe is located parallel to the borehole axis are given in Table 4.8. The ratio of conductivities = 1/16. 

As follows from numerical analysis, in this case five angular harmonics describe the field with high accuracy for all considered values of a//ii where h is the skin depth in the borehole. It is appropriate to notice that the influence of displacement on inphase and quadrature component of the field increases with an increase of frequency. At the same time within this range of frequencies the inphase component is less sensitive to displacement than is the quadrature component. For example, even if a/h = 1.6 we have In/iJ(e = 0.5)/In/i (e = 0) = 1.04, while Q/i (e = 0.5)/Qh e = 0) = 1.51. It is explained by the fact that within a wide range of frequencies the density of charges arising at the interface between the borehole and the formation is shifted in phase by 90° with respect to the current in the transmitter. Correspondingly, we can expect that the quadrature component of the field for a two-coil probe will be mainly subjected to the influence of eccentricity. 

We assume that, for most cases which are of great practical interest in induction logging, data presented in this table coincide with the magnetic field of a direct current. Therefore, by measuring the inphase component with a three-coil induction probe or with a probe of two coils, parameter s and, respectively, coefficient of the probe are defined. This enables us to calculate, the apparent conductivity by making use of quadrature component data. 

We will investigate frequency responses of quadrature and inphase components of the field measured by the receiver coil of the induction probe. Examples of the responses are presented in Figs. 5.9-5.22. Analysis of results of calculations allows us to outline the main features of field behavior, such as ... 

In order to overcome these difficulties let us consider an analytical method of determination of probe parameters. As is well known the quadrature component of the electromotive force arising at the receiver of a two-coil induction probe due to induced currents in a medium is ... 

Suppose that the induction probe consists of s transmitter and t receiver coils. Then it can be presented as a system of st two-coil induction probes, and the quadrature component of the electromotive force induced in measuring coils of this probe can be written as ... 

Until now it was assumed that interaction between currents is absent, i.e. all currents induced in a conducting nonuniform medium, regardless of the distance from the source, are shifted in phase by 90°. For this reason electromotive forces induced in measuring coils of a probe are in phase with each other, and they are added and subtracted in the same way as scalars. If only one component of the electromotive force, for instance the quadrature component, is measnred it is subjected to the same operations as scalars, regardless of whether currents are shifted in phase by 90°, or the internal skin effect manifests itself and due to it at every point of a medium there are both quadrature and inphase components of the induced current. 

As was demonstrated in Chapter 3 for the quadrature component of the magnetic field of a two-coil induction probe we have ... 

As an example Table 7.27 contains values of ratio of quadrature component of electromotive force when the probe is located on the borehole axis to that corresponding to a uniform medium with conductivity of the formation t = 0.8). As is seen from the table the skin effect does not practically affect the radial responses of the three-coil probe provided that the thickness of the skin effect, h, is related with the borehole radius as ... 

Now let us consider the calibration curve for a three-coil probe (Fig. 7.40). For comparison values of quadrature components for both two- and three-coil probes are given in Table 7.32. 

Besides, improvement of the vertical response of a multi-coil induction probe due to measurements at higher frequencies allows us to apply induction logging in a more resistive medium, inasmuch as the ratio between the quadrature component of the electromotive force and that of the primary field increases. 

First, let us assume that a two-coil induction probe, which has coil dimensions that are much smaller than its length, is located in a uniform medium and the skin depth is much greater than the separation between coils. Then, as was shown in Chapter 2, for the quadrature component of the electromotive force we have ... 

As was shown above, in a nonuniform medium apparent conductivity, aa, depends on the distribution of resistivity and the probe length. For interpretation it is appropriate to present results of a solution of the forward problem as well as experimental data as a function of apparent conductivity, the probe length and parameters of a medium. Knowing the probe length and the electromotive force of the primary field, it is a simple matter to calculate the probe coefficient and transform the quadrature component of the electromotive force into the apparent conductivity. In accord with eq. 7.12 the coefficient of the coil probe can be written as ... 

We will assume that coils of a two-coil probe present themselves as layered ones placed on a nonconducting base. Then, in accord with results obtained in Chapter 4 for the quadrature component of the electromotive force at the range of small parameters we have ... 

For a multi-coil probe located in a uniform medium with conductivity a, the quadrature component of the electromotive force in measuring coils is defined from relation ... 

In fact, the integral response, as well as the differential one, defining a signal in receiver coils due to induced currents in an arbitrary cylindrical layer with a constant resistivity, present the basic element of these calculations. However, the presence of caverns, deviation from radial distribution of resistivity because of nonuniform penetration of a borehole filtrate into a formation, its finite thickness are factors which can influence the focusing features of multi-coil induction probes. In order to eliminate the influence of these factors and to increase the depth of investigation, regardless of the geoelectric section, we will consider in this chapter another approach, based on the use of a two-coil probe and a simultaneous measurement at two or more frequencies if the quadrature component is measured. 

At the low-frequency part of the spectrum with an increase of the probe length the field asymptotically tends to that in a uniform medium with the formation resistivity. Minimum of curves of apparent conductivity, Oa/o ii is related with the fact that quadrature component of the field becomes equal to zero. For relatively shallow penetration of borehole filtrate into the formation (02/01 2) for a > 16 the influence of the invasion zone on the signal value for a two-coil probe does not exceed 25%. 

We will investigate vertical responses of a two-coil probe at the range of small parameters (Figs. 10.27-10.28). It is natural that the influence of the surrounding medium increases with an increase of its conductivity and a decrease of the formation thickness. For comparison curves of apparent conductivity, when the source is the vertical magnetic dipole, are given in Fig. 10.18. Here apparent conductivity is introduced as Oal[Pg.585]

A similar situation occurs when the rotor pole axis is at right angles to the axis of the stator coils. Here the magnetic reluctance is at its maximum value due to the widest part of the air gap facing the stator coils. The complete reactance in this position is called the quadrature axis synchronous reactance Xsq . Deducting Xa results in the quadrature axis reactance X. ... 

The situation is different when the rotor poles are at right angles to the stator coils. There is no induction in the fleld circuit and the reluctance is high, being almost the same as for the steady state condition. In this situation the corresponding quadrature axis transient reactance X approximately equals the reactance Xq. Cylindrical rotors of two-pole high speed generators have a nearly uniform rotor diameter and almost constant air gap all around the periphery. Hence the reactance X is almost equal to X. ... 

Some generators have the damper bars connected to a ring at either end of the pole structure, which provides some damping action from the quadrature axis. This provides a set of short-circuited coils in the quadrature axis, which are air cored and able to repel the flux that is attempting to enter their region. 

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