## Monthly Archives: March 2014

## THEORETICAL DESIGN

Almost all superconducting NMR and MRI magnets are solenoids. The reason for that is the relative simplicity and ease of manufacture and design of solenoids, compared with, for instance, extended dipoles. Although the generation of the RF field could be simpler with a transverse background field, the difficulty of manufacture of a high-background field magnet would far outweigh any advantage in the RF coil. The construction of a high-homogeneity solenoid proceeds in three parts: a winding array is designed, based solely on the analysis of the axial variation of the field of a solenoid; the magnet is wound and the spatial variation of its actual field is measured; and the unwanted errors in the field arising from manufacturing imperfections are removed by shimming.

The center field of a solenoid is given by

Bo = XoJao ln|[a + (a2 + в2)1/2]/[1 + (1 + в2 )1/2l} (2)

B(z) = B0 + (d2B/dz2 )z2/2 + (d4B/dz4 )z4/4! + (d6B/dz6 )z6/6! + ■■■ |

(3) |

NMR Magnets

The analysis of weak solutions imposes several requirements on the magnet. In order to obtain usable signal-to-noise ratios, a large volume of sample must be used. This immediately demands good field uniformity so that variations in background field strength do not give rise to different frequencies, which would, of course, mask the small chemical shifts being sought. High field strength is desired as detailed above. Even if these requirements are met, the dilution of the sample may be such that repeated pulses are required. The final signal-to-noise that can be obtained from a number of pulses is proportional to VN. A run may take many hours or even days to accomplish. During that time not only must the spatial homogeneity of the background field be excellent, but the magnitude of the field also must be constant, or at least must change only very slightly. (The reason that any change is permitted is that a frequency lock can be used to adjust the frequency of the RF to match a slow and slight change in the background field.)

To summarize, the NMR magnet should have high field strength and great uniformity, and the field must be stable.

MRI Magnets

The essential principles for MRI magnets are identical to those for NMR magnets, but the volumes of homogeneity are where J is the overall winding current density, a0 is the inner radius, a is the ratio of outer to inner radii, and 3 is the ratio of length to inner diameter (5). Because SI units are used throughout, is the permeability of free space, 4ет X 107 H/m, a0 is in meters, and B0 is in Tesla.

The field strength decreases at points on the axis away from the center of the solenoid. The axial variation of field strength on the z axis is expressible as a Taylor’s series

Only even terms appear because the center of the solenoid coincides with the origin.

Figure 1 illustrates the geometry of a thin solenoid, of radius a0 and extending a length z0 to the right of the origin. For such a thin solenoid, the derivatives of the field at the origin are as follows:

Bo = 5/Vzo(«o+zo) 1/2 dB/dz = —1/i0ia2(a2 +Zo)_3/2 d2B/dz2 = — ^/x0i3z0a2(a2 + z2)~5/2 dsB/dzs = —^/л0іа2(3а2 — 12z2)(a2 + z2)~7’/2

z

0 |

Figure 1. Geometry of a thin solenoid showing the coordinate system used to define current geometries.

where i is the sheet current density in amp-turns per meter and a0 and z0 are as illustrated in Fig. 1.

The field of a solenoid symmetric about the center plane has even symmetry and no odd derivatives. So, by evaluating the even derivatives of the field at the center, the axial variation of field generated by a solenoid can be calculated to an accuracy determined by the number of derivatives used and by the distance from the center. The derivatives can be treated as coefficients of a Cartesian harmonic series so that

Bz = B0 + ^z2 + b 4Z4 + b6z6 + •••

in which

&2 = Moi[3zoa0(«0 + z2) 5/2l/4

(5) |

b4 = M0i[(45a0z – 60z3)(a2 + z2)-9/2]/48

b6 = —/г0 i[(5a0z — 20a2z3 + 8z5 )(a0 + z0 )-13/2]/1440

For coils of odd symmetry, such as shim coils described later, the corresponding harmonics are

(6) |

b1 = M0i[(a2 + z2) 3/2]/2

b3 = /j,0i[(3a2 — 12z2 )(a2 + z0)—7/2]/12

Notice that the magnitude of any harmonic coefficient is mediated by the denominator of the expressions that each include the term (a§ + z0)(n+1/2), where n is the order of the harmonic. Thus, the generation of high-order harmonics requires coils with large values of current (ampere-turns) or small radius. This is significant in the construction of shim coils, as is noted later.

Associated with an axial variation of field is a radial variation, arising from radial terms in the solution of the Laplace scalar potential equation. For instance, even-order axial variations are accompanied by axisymmetric radial variations (6) of the form

(7) |

B2(z, x,y) = b2[z2 – (x2 +y2)

B4(z, x,y) = b4[z4 – 3(x2 +y2) + §(x2 +^2)2]

These equations show that if b2 or b4 are zero there will be no axisymmetric radial variation of field.

Figure 2 illustrates a set of nested solenoids. Solenoid 1 gives rise to nonzero values of the harmonic coefficients b2, b4, b6, etc. If dimensioned correctly, solenoid 2 by contrast can produce equal values for some or all these coefficients but with opposite polarity. Then at least b2 and b4 will have net zero values, and the first uncompensated harmonic to appear in the expression for axial field variation will be the sixth order. A minimum, but not necessarily sufficient, condition is that as many degrees of freedom are needed in the parameters of the coils as there are coefficients to be zeroed.

j – (At/m) |

a |

0 |

This method can be extended to as many orders as desired. In most high-resolution NMR magnets, the required uniformity of the field at the center is achieved by nulling all orders up to and including the sixth. That is, the solenoid is of eighth order. In the design of the solenoids, no odd order appears, of course. The first residual harmonic will have a very small value close to the center, although at greater axial distances, the field will begin to vary rapidly. Thus, the design of a high – homogeneity solenoid requires only the calculation of the field or the field harmonics on axis, and those harmonics may be easily calculated using only Cartesian coordinates.

## SOME RELEVANT TERMS

Before we proceed to discuss linear antennnas, we need to define and discuss certain terms in accordance with the Institute of Electrical and Electronics Engineers (IEEE) standard definitions of antenna terminology.

Power Radiated, Radiation Intensity and Radiation Resistance

Electromagnetic waves, by virtue of their transverse nature, propagate in a direction perpendicular to the plane containing the electric field E and magnetic field H. The instantaneous Poynting vector P, which is a measure of the power density associated with the electromagnetic wave, is given by

(1) |

where P, E, and H are instantaneous Poynting vector in watts per square meter, electric field in V/m, and magnetic field in amps per meter.

The total power P crossing a sphere enclosing the source (antenna/scatterer) at its center is obtained by integrating the power density over the sphere and is given by

(2) |

P = Wn-dS = Wda

where W is the instantaneous power crossing the sphere per unit area held perpendicular to the direction of the flow, n is the positive outwardly drawn at the point of incidence, and dS is the unit area arbitrarily oriented at the point of incidence. With exp(ja>t) variation assumed, the average power density is given by the time-average Poynting vector Pav:

(3) |

Wav(u, v, w) = —Re(E x H*)

The average radiated power is given by

p – _ * av — — |

H *) da |

(4) |

iff |

The radiation intensity U is defined by the product of power density Prad and the square of the far-field range (r) and is expressed as

U = r2 P. |

(5) |

rad |

## Energy awareness and control of power consumption

The number of existing and prospecting applications has been steadily growing after the development of the WSN paradigm. Regrettably, the energy density of the batteries did not follow the same trend, and the energy harvesting systems can power only a limited class of devices, usually with limited capabilities [10]. For this reason energy consumption modeling and reduction has attracted the interest of both the academic and the industrial worlds. The next sections are devoted to the exploration, modeling, characterization and analysis of the power consumption of a WSN node in relation to specific application. In this section a brief and non-exhaustive review of methods to reduce the power consumption of the nodes is presented.

Due to the limited computational capabilities of the WSN node its load is often limited to trivial computation. The greatest part of energy is spent by the peripherals, especially by the radio module. Thus, a lot of power-saving mechanisms exploit the energy consumption reduction of the node peripherals. In this regard, both passive and active approaches are possible. Passive power conservation mechanisms reduce the energy consumption of a sensor node by turning – off its transceiver interface module when there is no communication activity [18]. Moreover, additional energy savings may also be achieved by optimizing the performance of the processor in an active state changing its operational frequency [19]. In fact, using a processing unit with variable processors speed (VPS), it is possible to decrease its power consumption decreasing the supply voltage and the clock frequency. Exploiting the VPS it is desirable to design a scheduling system, capable to select a suitable supply voltage and relative frequency clock for each task. Dynamic Voltage Scheduling (DVS) is one of these mechanisms able to provide such behavior without degrading the overall performance of the node [19]. Dynamic Power Management (DPM) is another technique to increase the lifetime of a sensor node [20]. DPM acts similarly to DVS, but instead of scaling the clock frequency it can dynamically turnoff the components of the sensor node and wake them up when needed. At microcontroller level this transition of states it is represented by different power mode that shutdown the CPU, memory or additional internal peripherals. It is worth to say that each transition of state takes a certain amount of time and consequent energy consumption as reported in Figure 3. In each power mode, also called low power mode (LMP), different peripherals are incrementally turned off. Each transition from the idle state to a LPM has a fixed cost, indicated in Figure 3 as b0, which is usually negligible. However the energy cost for waking up the microcontroller from a low power mode increases with the depth of the low power modes. For this reason it is important to reduce the number of state transitions, conveniently balancing the scheduling mechanism without using aggressive power down strategies.

Figure 3. Low Power Modes (LPMs) transitions and costs |

Active power conservation mechanisms differ from passive ones in that they achieve a reduction of the energy consumption by avoiding undesired events like collisions, or exploiting energy-aware routing protocols. For instance adjusting the transmission power may help minimizing the probability of occurrence of a collision, an event leading to higher power consumption due totherelateddetectionandretransmission activities. MultipleAccesswith Collision Avoidance [21] (MACA) and Multiple Access with Collision Avoidance Wireless [22] (MACAW) are two different MAC layer channel access protocols, aimed at avoiding or minimizing the collision rate by using a particular handshake signaling. Conversely, Power Controlled Multiple Access [23] (PCMA) is a MAC protocol that can achieve power-controlled transmission and thus collision avoidance, originally proposed for ad-hoc networks but suitable to WSNs as well.

Operating at PHY level and exploiting the frame filtering technique, it is possible to achieve a substantial reduction in energy consumption. Usually receivers perform the channel clear assessment (CCA) in order to check for incoming packets or avoid collision. The IEEE 802.15.4 standard defines three possible methods to perform the assessment:

• Energy above threshold. If the energy detected is above a fixed threshold the CCA shall report a busy medium.

• Carrier sense only. This method checks for a signal with modulation and spreading characteristics of the IEEE 802.15.4. In this case the signal may be above or below the threshold.

• Carrier sense with energy above the threshold. This is a combination of the previous methods checking both signal characteristics and energy.

Once the CCA reports a busy channel the receiver may start its RX phase to obtain the packet content. It is clear that in a dense network, where a lot of transmissions occur, there are many chances to detect a transmission. In this case most of the packets sent on the network are not intended to the receiver itself but to others receivers, generating unintentional package reception. Each package reception is an energy expensive procedure and for this reason it may be reduced at the minimum avoiding unintentional package. In [24] the authors exploit the characteristic of the Texas Instruments CC2520 RF transceiver of executing specific operationsonce; thepacketheaderis received. Inparticular, theauthors modified the firmwareof theRF transceiver tnerderto irineeraninterrripfwheathe packet hoader rcaortc asrecipientan ad^reosdtfff^centfromnsFcen. Inthfrcacethe hd i^i^adrreiharrr^nt^;oan interrupatocec MCU thatporns off the i akiomoduhe caeing the ei^t^i^c^i^icarrc^edi^o roceivethepacnethaylodd. The enecjco CDnrumptionotCCA, reoeitoS freme deoppmg, unintenCioncl ondlvtaerir^c^alkedr^c^Vc isrupreeeneedinnigure f^.^nf^ubF^rd1i^c;Oxia repierenhation is shocioeerrheeccnario whvee annemtentionaidaeh eoe isdeceived. eeSact efterrecevtioniheMPU ko esuoUpesferm any tasiptih na>dFsi^^ с^Са^іссППі^і^іСсє^;^ Vss^£^i^£^tier:. П, v^l"^r^r^a]:^^y^eed]asea^;^r^rigire^(^r^orm^s^e^. Fihdre h. Foepre вп^сЛєє nce wharetdeRT moduletriggefraninieemptun theMCU, savino theenervy cost: Sor ike ned^csc^if receptiak

Time (ms) (a) |

Time (ms) (c) |

Time (ms) (b) |

Time (ms) (d) |

Figure 4. Power consumption of MCU (solid line) and radio transceiver (dotted line) during: (a) channel clear assessment, (b) frame filtering, (c) broadcast reception, and (d) broadcast reception and processing. [24]. Copyright © 2012 EmanueleLattanziandAlessandro Bogliolo, distributedunder the Creative Commons Attribution License.

At routing level several mechanisms were proposed in last years to increase the lifetime of the WSN, increasing the lifetime of the nodes’ batteries [3-5].Most of the proposed techniques take

into account nodes powered by batteries but the routing strategies may change taking in consideration nodes energetically sustained by their environment changing. In this scenario the routing mechanism has to be dynamically selected taking in consideration not the total energy available on the batteries but the energy available for each node at specific time (i. e. the power available). In [25] the authors demonstrate that power-constrained WSNs, as nodes powered by energy harvesting, can be represented as flow networks and that the optimization of the energetic sustainability of the workload can be cast into an instance of maxflow. Starting from this consideration, Lattanzi et al, in [26], propose a non-deterministic routing table that can be actually applied at the sensor nodes in order to achieve the maxflow theoretical optimum. In this case, since the information of the available power is accessible only to the relative node, the nodes have to cooperate to solve the maxflow problem.

## Theoretical Basis

As ions cross the cell membranes, a tiny current source exists inside the body. The body generally consists of fluids, muscles, fat, and other organ tissues which all act as good conductors of electricity. The body is bounded by a skin interface and hence is referred to as a volume conductor. Theoretically, the potential field established by the flow of ionic currents in the body tissues appears at all points within and on the surface of the body. Because of the large differences in the resistances between the environment and the body tissues these voltage potential fields are not found beyond the body surface. Hence, there is no current flow from the body to free space, or put another way our bodies do not supply electrical energy to the outside world.

There are literally billions of individual cells in the heart that depolarize during the cardiac cycle. The sequence of depolarization from the various structures within the heart was schematized in Fig. 4. As one proceeds down to the cellular level, there are only groups of cells depolarizing at any given moment. It is possible to represent those groups of cells which are simultaneously depolarizing as an equivalent source. A common representation of the equivalent cardiac source is that of a dipole with a time-varying magnitude, orientation, and position within the body. This representation is a first – order model useful for understanding the generation of the main waves of the ECG recorded from a bipolar lead with electrodes on the body surface.

The equivalent heart source can be expressed as a threedimensional ‘‘heart vector’’ H in a Cartesian coordinate system. When two electrodes are placed on the body surface, another ‘‘lead vector’’ L is formed by connecting the two electrodes. The voltage recorded between two electrodes from the cardiac source vector is directly proportional to the dot product, H ■ L = H L cos(0). More specifically, the voltage between the two electrodes is given by the component of H in the direction of L. This concept is referred to as the lead field theory. To visualize the lead field concept better, we can use the principle of reciprocity. Briefly, this old network theorem states that when a voltage source at one location within a circuit produces a current between two nodes in the circuit, then one can inject the same measured current in these two nodes to duplicate the original voltage at its original location. Figure 7 demonstrates how this can be applied to the ECG. The top panel shows an outline of a torso with a drawing of the heart in its approximate anatomical position. The two dots represent two body surface electrodes with current flowing between them. The lines crossing through the torso and heart represent the flow of current or the ‘‘reciprocal lead field.’’ Note that the heart is in the densest region of the current lines. Components of the heart vector lying along the lead axis, i. e., parallel to the lead field, produce the largest relative voltages. In the lower panel the recording electrodes are placed along the right side of the torso. The current lines do not substantially cross through the heart, and hence this lead axis, i. e., parallel to the lead field, would be a poor one for recording the ECG. This concept of reciprocally energizing the recording leads and mentally visualizing the extent of the current lines which pass through the heart is useful in understanding the lead field concept and determining whether two electrodes placed on various body sites will record a large amplitude ECG.