Advanced Microwave and Millimeter Wave Technologies Devices, Circuits and Systems Part 12

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Advanced Microwave and Millimeter Wave Technologies: Semiconductor Devices, Circuits and Systems 432 Up to now the polarization of the incident wave has not be considered However, the induced current in the surface of the receiving antenna is determined by those incident wave components “parallel“ to the polarization of the receiving antenna1 That is to say, the wave field produced by the antenna in the location of the antenna can be expressed as:   E 12  E12  e 2T 12 ,12   where E12 is the actual value of the field and e 2T is normalized vector indicating the polarization of the wave transmitted by the antenna When identifying the type of  polarization of the antenna itself in reception for the normalized vector: e R  21 , 21  , thus, the effective value of the component of the incident field that matches the type of polarization of the antenna at the reception will be as (Márkov & Sazónov, 1978):   E R12  E12  e 2T 12 , 12   e 1R  21 ,  21  (6) Note that for standard polarization vectors is satisfied that:   e 2T 12 , 12   e2T 12 , 12   1;   e R  21 ,  21   e2 R  21 ,  21   Therefore E R12 (6) is the real value of electric involved in the process of reception Note that if the antennas and were equal and with identical directions of pointing( 12   21 y 12   21 ), thus:     e 2T 12 ,12   e2 R 12 ,12   e 1T  21 ,21   e1R  21 ,21    will happen when: e 1T  21 ,21   e1R  21 ,21  , which the polarization of the transmitting antenna and the receiving one are the same but with opposite sense (seen from a common reference system), that is to say, an identical polarization seen from transmitting point of view Therefore, we can write the expression (6) as: where the maximum value E12  E R12   e 12 , 12   e1  21 ,  21  (7) Polarization of an antenna is defined from transmission point of view However, although the reception point of view is opposite to the transmission one, the polarization of an antenna is defined equally Electrodynamic Analysis of Antennas in Multipath Conditions 433 without distinction in the polarization vectors whether it is an antenna transmission or reception Similarly we can write the actual value of the field incident at the antenna from the antenna 1: E 21  E R 21   e1  21 , 21   e2 12 ,12  (8) Given that the denominators of expressions (7) and (8) are equal, and substituting these in (5), we obtain: Z A1  Z1 Z A2  Z  I12   I    21     E E  R12  RRad1  DMax1  FC1  21 ,  21   R 21  RRad  DMax  FC 12 ,12  (9) Analysing both members; the relationship ( I12 E R12 ) depend only on the characteristics of the antenna The field incident on the antenna with the same polarization induced currents on the antenna On the other hand the factors of the left member of (9) depend exclusively on the characteristics of the antenna Similarly, it appears that all the factors of the righthand side of (9) depend exclusively on the characteristics of the antenna Since the above analysis there is anyone restriction on the type of antenna used (in general, antennas and are different), the obvious conclusion is: Z A1  Z1  I12  C    E R12  RRad1  DMax1  FC1  21 ,  21  (10) where C is a constant that has the same value for all antennas Therefore, C can be obtained by replacing in (10) the values of the parameters of any antenna; in particular the Hertz’s dipole Figure shows a dipole antenna formed by a thin conductor of length L, with an impedance Z R connected at its terminals, impinging a wave of linear polarization parallel to the dipole Fig Antenna type dipole in reception Advanced Microwave and Millimeter Wave Technologies: Semiconductor Devices, Circuits and Systems 434 This will induce a current I z  along the conductor The e.m.f de , induced a small segment of length dz will be: de  E  sen    dz (11) As for the conductor circulate the current I z  , a power will be delivered to the antenna: dP  de  I  z   E  sen    I  z   dz The total power delivered to the antenna is:  P  E  sen    I  z   dz L In the terminals (see Figure 1) is obtained: P  I In2   ZA  ZR  And thus: I In2   ZA  ZR    E  sen   I   z   dz L In the case of a Hertz dipole with length l   we can presume the current uniform: I z   I Ent and, therefore:  I In2   ZA  ZR   E  sen   I Ent  l Or what is the same: I Ent I sen    l  12  E E R12 ZA  ZR The radiation pattern of Hertz's dipole is known FC    sen   , its directivity is of: DMax  1.5 Moreover, the radiation resistance is of: RRad  80 l   ; so, substituting these data in (10): C  120   where the value C obtained for the dipole Hertz is unique for all antennas, and it can be replaced in (10) to obtain the expression of the current at the terminals of the any antenna Electrodynamic Analysis of Antennas in Multipath Conditions 435 ( I Ind ), induced by that component of the field of the incident wave ( E R ) with a polarization equal to the receiving antenna itself and arriving at the antenna in the direction defined by angles  and  : I Ind  E R  RRad  DMax    FC  ,     ZA  ZR 120 (12) According to Figure the induced e.m.f at the terminals of the antenna can be determined by: e Ind  I Ind  Z A  Z R  E R   RRad  DMax   FC  ,    120 (13) Taking into account the expressions (12) and (13), the values of the current and the induced e.m.f at the terminals of the antenna have a dependency on the direction of arrival of the incident wave, expressed by FC  ,   , which is the radiation pattern of the antenna in transmission The expression (13) can be written like this: e Ind  ,   e Ind Max  FC  ,   Then eInd Max expresses the value of induced e.m.f when the wave arrives from the direction of maximum reception of the antenna, and FC  ,   represents the normalized radiation pattern of the field of the antenna in reception mode, which is equal to that characteristic of their antenna transmission On the other hand, the coefficient of directivity is function of the radiation pattern Therefore, it confirms that its value is the same regardless of the antenna works as transmitters or receivers Similarly, in the case of linear antennas, the effective length: l Ef   DMax  RRad   120 (14) that depends on the coefficient of maximum directivity, the radiation resistance, and will have the same value in transmission and reception Substituting in expression (13), it becomes: eInd  E R  l Ef  FC ( ,  ) where the product: E R  l Ef  e Ind Max is the maximum value of the induced e.m.f (when FC  ) It is important to stress the significance of the effective length of the antenna; that is to say, this is a length such, when multiplied with the incident field intensity (the polarization equal to the antenna itself, incident in the direction of maximum reception), Advanced Microwave and Millimeter Wave Technologies: Semiconductor Devices, Circuits and Systems 436 gives us the maximum value of the induced e.m.f Known the induced e.m.f in the antenna, with reference to Figure we can determine the voltage at the input terminals of the receiver, supposed this is connected directly to the antenna, by: VR  ZR  e Ind ZA  ZR (15) Considering the presence of a transmission line (the characteristic impedance Z ) between the antenna (impedance Z A ) and receiver (impedance Z R ), it defines the reflection coefficient at the antenna input (Pozar, 2004): A   and at the receiver input:  R  VR  Z A  Z0 Z A  Z0 Z R  Z0 ; the expression (15) will transformed: Z R  Z0   R    A  exp   L     R   A  exp  j    L   e Ind (16) where     j is the propagation constant of the transmission line (which includes the attenuation constant  and the phase constant  ), and L is the length of the line In case of low frequency receivers ( f  30 MHz ) the condition: Z R  Z A is usually applied; thus (replacing in 15) is obtained: V R  e Ind ; so, maximum voltage to the receiver's input Moreover, matching the transmission line to the antenna (  A  ) and Z R  Z ,  R  and expression (16) also gives us the maximum input voltage of the receiver: VR  exp   L   eInd , where the factor: exp   L   takes into account transmission losses along the line On the other hand, in case of higher frequency it is more difficult to provide high power amplifiers, so the purpose is to maximize the real power delivered by the antenna to receiver If the receiver is directly connected to the antenna, the power supplied to the receiver is: PR  I Ind  RR  E R2  2  R R   ZA  ZR  RRad  DMax  FP  ,   120 where R R is the real part of the input impedance of the receiver Maximum transmitting power forces to an impedance matching between the receiver and antenna ( Z R  Z A ) Applying this condition in the above expression and using the classical expression of the efficiency of an antenna: RRad   A  RIn , we obtain (Balanis, 1982): Electrodynamic Analysis of Antennas in Multipath Conditions PR  E R2  437 2  A  DMax   FP  ,   120  When using a transmission line of low losses between the receiver and the antenna and impedance matching between the receiver and the line, the power supplied by the antenna to the line will be: PL  E R2    Z  R A  A  DMax 2   D      F  ,   E      A Max  F p  ,   p R 2 120 120 4  Z A  Z0 A part of power will flow to the receiver: PR  PL  exp (2  L)  E R2     D 2     exp (2  L)  A Max  F p  ,   120 4 (17) and the other part: PCons L  PL  1  exp 2L  , will be consumed by the line In expression (17) we can see that through the appropriate orientation of the antenna, by matching the direction of maximum reception of the antenna with the direction of arrival of the wave we get FP  and the received power reaches its maximum value by adjusting the direction:  PR Max    A  exp 2L A  E R2 2   DMax 120 4 (18) Considering this expression; factor E R2 120 represents the module of the Poynting vector of the incident wave to the antenna If we multiply this factor by the physical area of the antenna, the power incident on the antenna is obtained: PInc  E R2  AGeom 120 (19) The antenna is not able to fully grasp the incident power that really is: PCap  ER2   E2    DMax   R  AEf 120  4  120 (20) 2  DMax 4 (21) where: AEf  Advanced Microwave and Millimeter Wave Technologies: Semiconductor Devices, Circuits and Systems 438 It will be called the effective area of the antenna; namely the area through which the antenna fully captures the incident power density The relationship: A  AEf PCAP  PInc AGeom (22) is called the coefficient of the utilization of the surface of the antenna The expression (18) now we can write:  PR Max    A  exp  2L A    A  PInc    A  exp  2L A  PCap Note that the antenna does not use all the power captured, but a fraction called useful captured power: PCapUse   A  PCap (23) the other part: 1   A   PCap is the power losses in the antenna as heat during the reception Between the antenna and the transmission line is not always met the condition of impedance matching, therefore only part of the useful captured power useful is delivered to the line:  PL    A  P CapÚtil (24) and, only the fraction given by (17) is delivered to the receiver In Figure shows schematically the flow of power from the wave that propagates in free space and arrives to the antenna, to the receiver From the analysis we can summarize the conditions for optimal reception: coincidence of the polarization itself of the antenna with the polarization of the incident wave (This ensures E R maximum); orientation the antenna to the direction of arrival of the wave ( FP  ); effective area (or effective length) maximum of the antenna (which depends on its physical characteristics); high efficiency (  A  ); proper impedances matching between the antenna and feed line (  A  ); low losses of the feed line (  L  ) In practice these conditions are met satisfactorily, so the power at the receiver is close to optimal value: POpt  ER2 E2   A  AEf  R   GMax 120 120 4 (25) This expression clearly reveals the role of the maximum gain at the reception The value of G Max of any antenna indicates either the number of times that the power delivered to the receiver exceeds that delivered by an isotropic radiator ( G Max  ) under the same conditions of external excitation, coupling and losses of the transmission line In a similar Electrodynamic Analysis of Antennas in Multipath Conditions 439 way we can say that the coefficient of directivity DMax of an antenna (as was noted earlier, has the same value in transmission and reception), at the receiving antenna indicates the number of times that the power captured by the antenna exceeds that delivered by an isotropic radiator Finally, keep in mind that the presence of induced current in the receiving antenna also determines an effect known as secondary radiation We must emphasize the fact that, in general, the directional characteristic of the secondary radiation does not match the directional characteristic for the transmitting antenna PLoss(1A) PCap PCap use   A  PCap E E R2 S S PP   120R 120 PInc  R E  AGeom 120 PCap   A  PInc PRe f   A  PCap use PCons L  1  exp(2  L) PL PR  PL  exp(2  L) PL  (1   A )  PCap use to receiver Fig Power flow from the wave in free space to the receiver This is explained by that shape of the induced current distribution on the elements of the antenna is not equal to that found when the excitation takes place at the terminals of the antenna However, the total power of secondary radiation can be calculated from: PSec Rad  I Ind  PRad  ER2 RRad D 2   Max  Fp  ,   2  ZA  ZR 120 (26) In this expression the factor: FP  ,   defines the directional pattern of the antenna during the reception; while PSec Rad is the total power of secondary radiation in all directions of space This phenomenon has a special interest in antennas that act as passive elements, where generally Z R is the impedance of a ( Z R  ) or a pure reactive element ( Z R  jX R ) Particularly this treatment can be extended to objects that not really fulfil the mission of the antennas, that serving (intentionally or not) as reflectors of radio waves In these cases, as can be shown easily from (26), it is possible to reach a power of secondary radiation which is times larger than the optimum power of reception given by the formula (25) The analysis of antennas in reception mode, leads to a set of conclusions of great importance First we establish that many of the properties of the antennas are the same as transmission as reception, which simplifies its research, since it is not necessary to determine these Advanced Microwave and Millimeter Wave Technologies: Semiconductor Devices, Circuits and Systems 440 properties in both regimes Thus, the impedance of the antenna, its directional pattern, its directivity, efficiency, and gain are the same in both schemes of work The expressions obtained (mainly induced e.m.f) in the receiving antenna (13), are useful in tasks of calculation and design of antennas in general There are two parameters that are used in the study of the receiving antennas (aperture antennas mainly); the coefficient of utilization of the surface of the antenna and the effective area Antennas receiving mode in multipath conditions It is said that an antenna operates under multipath conditions when in it impinge radio waves arriving from different directions Figure show the multipath phenomenon BS 1 3 n 2 MS Fig Phenomenon of multipath propagation In Figure it can be seen: transmitter and receiver antennas, rays that define the different propagation paths from transmitter to receiver antenna, and the scattering elements (buildings and cars), which are called scatterer Propagation environments, together with the communications system can be divided into: indoor and outdoor The theory of radio channels is a rather broad topic not covered here, but from point of view of the antenna, we will present (only from the point of view spatial) similar to the patterns that characterize the radiation of the antennas (Rogier, 2006) This way, according to the angular distribution of power that reaches the antenna, we can present them as omnidirectional, and with some directionality Then, the shape of the angular distribution of power that characterizes the channel depends on the position of the antenna inside the environment of multipath propagation Figure shows some examples 900 1200 90 600 180 120 00 210 3300 180 60 330 120 60 0 180 0 180 210 90 90 120 60 210 3300 00 210 330 270 270 270 270 (a) (b) (c) (d ) Fig Angular distribution of the power reaching the antenna by multiple pathways, a) omni directional channel, b) dead zone channel, c) directional channel, d) multidirectional channel 456 Advanced Microwave and Millimeter Wave Technologies: Semiconductor Devices, Circuits and Systems The polymer/CNTs composites were fabricated using two different techniques, cf (Thomassin et al., 2008) The first one consists in melt-blending the polymer matrix, poly(-caprolactone) (PCL), with CNTs using a DSM microextruder The second one, called the “coprecipitation” technique, is the solubilization of PCL in an organic solvent (tetrahydrofurane, THF, especially well suited to PCL) in the presence of the required amount of CNTs After 30 seconds of ultrasonic treatment in order to break the CNTs bundles, the mixture is poured into heptane, which is a poor solvent for PCL The polymer then instantly precipitates and the CNTs are trapped in it At this stage the samples are in solid form, cf FIG 3, they have not yet been foamed, they are simply referred to as “solid samples” throughout the chapter Fig Picture of a nanocomposite solid sample 3.2 Foamed samples Solid samples prepared by the methods described in the previous section were foamed using supercritical CO2 They were first pressurized at about 200 bars at 60°C for 3h in order to saturate the sample with CO2 The pressure was then rapidly released in a few seconds leading to the foaming of the sample A picture of a foamed nanocomposite fabricated this way is shown on FIG (already cut into pieces prior to its characterization), cf (Thomassin et al., 2008) Fig Picture of a nanocomposite foamed sample (already cut into pieces prior to its characterization) Foamed Nanocomposites for EMI Shielding Applications 457 Characterization methods A two-step diagnostic method to find the polymer-CNTs nanocomposites best fitted for shielding applications was developed, cf (Molenberg et al., 2009); solid samples of polymerCNTs blends were first characterized using a microstrip one-line method, cf section 4.2, the best candidates were used to fabricate foams that were then characterized using a waveguide line-line method, cf section 4.1 Even if different methods were used to characterize the various samples, they were all based on the measurement of their scattering parameters (Sii) using a Vector Network Analyzer (VNA) The dielectric constant and conductivity of the samples were then extracted from these parameters Those measurements were made in the 8-40 GHz frequency band The foams were measured using waveguides, while the solid samples were measured using microstrips This is due to their respective geometries, thin flat solid samples would not fill the waveguides enough to make precise measurements while foams are too porous and thick to be reliably measured in a microstrip configuration 4.1 Line-line waveguide configuration technique The Line-Line (LL) method is based on the extraction of the propagation constant (), cf FIG 5, from the measurement of the scattering parameters (Sij) of two transmission lines of different lengths, here two waveguides having their inner volumes entirely filled with the sample under test, cf (Huynen et al., 2001), (Saib et al., 2006) and (Pozar, 2005) Fig Line-line method, from (Huynen et al., 2001) and experimental set-up for the LineLine method, waveguide, transitions and coaxial cables to the VNA From the scattering parameters of both lines, their transfer matrices TL1 and TL2 can be extracted After a few mathematical operations, a matrix TL can be calculated It is diagonal and has the form  e L TL    0   e   L (1) Advanced Microwave and Millimeter Wave Technologies: Semiconductor Devices, Circuits and Systems 458 And from the propagation constant, the dielectric constant and conductivity can be easily determined c    r    j    and    0 ''  j  ' '' (2) It should be noted that the magnitude of diagonal element e-L in equation (1) corresponds to the attenuation undergone by the signal over a thickness L of material As is proportional to , it further confirms that a high conductivity is required for obtaining a good absorption Only a simple coaxial SOLT (short-return-through-line) calibration of the VNA is required and the final values of permittivity and conductivity depend only on the length difference L It is therefore especially well suited to waveguide measurements, because there is no precision waveguide calkit available in our laboratory Such a calkit would have been necessary for more complex and accurate calibrations The comparison between the simple SOLT calibration method and a LRM (Line-Reflect-Match) method, a more precise technique using a reference calkit, is illustrated on Fig With an LRM calibration, the reference planes of the VNA are brought after the coaxial-microstrip transitions, so that the VNA measures the scattering parameters of the line under test only With a SOLT calibration, the VNA reference planes are placed before these transitions and their influence on the measurements of each line is not eliminated Nevertheless, the SOLT calibration is sufficient to make sure that the TL matrix is diagonal, therefore ensuring the validity of the LL method Fig Schematics of the measurement setup where A and B correspond to the coaxial-tomicrostrip transition, and DUT to the device-under-test, from (Saib et al., 2004) 4.2 One-line characterization in microstrip configuration technique The One-Line method is based on the measurement of the scattering parameters (Sii) of only one line (different from the LL method described in the previous section), a microstrip line in this case The sample to be characterized is used as substrate to fabricate a microstrip transmission line A thin copper ribbon is glued to the top of the sample, serving as microstrip and a piece of aluminum tape is glued on the bottom of the solid sample, to form Foamed Nanocomposites for EMI Shielding Applications 459 the ground plane, cf Fig 7(right) From the S matrix, the ABCD chain matrices can be calculated for a transmission line of length L and of characteristic impedance Zc, cf (Saib, 2004) (3)  ch  L Z sh  L     ABCD    Z sh   L   C C    ch   L    The propagation constant  can then be extracted and the dielectric constant and conductivity can be determined, using equation (2) This method is valid only if a precise LRM calibration of the VNA has been done The reference planes must be put after the transitions, and the reference impedances in these planes must be set to 50 , which is ensured using an LRM calibration An Anritsu precision microstrip calkit and the corresponding 3680K Anritsu sample holder (including the transitions) were used for the microstrip measurements, cf Fig Fig Experimental set-up for the One-Line method applied to a microstrip topology, entire set-up (left), Anritsu sample holder (right) with the microstrip visible on top of the sample (substrate) Modelling, design and optimization 5.1 Simple electrical model It must first be noted that Carbon Black (CB) nanoparticles, i.e relatively spherical carbon platelets, are considered here instead of CNTs to simplify geometrical considerations As can be seen on conductivity-versus-frequency plots resulting from nanocomposites measurements, cf section and (Saib et al., 2006), the measured conductivity tends to for very low frequencies This means that the conductive nanoparticles not form a direct conductive pathway for the electrons from one side of the sample to the other side However, at relatively high frequency the conductivity becomes significant, indicating the presence of capacitive couplings between the nanoparticles Taking these observations into 460 Advanced Microwave and Millimeter Wave Technologies: Semiconductor Devices, Circuits and Systems account, a simple electrical model was developed, cf (Saib et al., 2006) This model is represented on Fig The nanocomposite is placed between a ground plane and a microstrip line, to form a microstrip transmission line, the actual configuration of the (solid) samples during the measurements, cf section 5.2 and 6.1 Fig Simple electrical model explaining the frequency dependence of the conductivity of the nanocomposites, from (Saib et al., 2006): (a) first approximation, the CB nanoparticles are spherical, well dispersed inside the polymer matrix and not in physical contact with one another The arrow indicates the dominant direction of the electric field (b) Second approximation, the CB particles form a regular network of conductive grains inside an insulating polymer matrix (c) Equivalent admittance per unit length of microstrip line (d) Corresponding two-layer microstrip transmission line The CB nanoparticles are supposed to be spherical and well dispersed inside the matrix There is also supposed to be no direct physical contact between them, i.e they not touch one another, cf Fig 8(a) This random distribution is approximated by a regular homogeneous network of purely conductive grains inside a completely insulating polymer matrix Therefore, the electrical equivalent of each grain is a simple resistor and there are strong capacitive couplings between all the grains Since the electric field distribution between the top and bottom conductors of the microstrip line is quasi uniform and perpendicular to those conductors, only capacitors and resistors that are parallel to that direction have to be considered, cf Fig 8(b) This corresponds to a two-layered microstrip transmission line, cf Fig 8(d), with an insulating top layer and a conductive bottom layer (conductivity = CL) This line has an electrical equivalent admittance per unit length that corresponds to a simple resistor-capacitor (RT and CT) series circuit, cf Fig 8(c) The simulated results obtained using this simple model are confronted with measurement results in section 6.3 5.2 Optimization of the topology of the samples Fig Schematic diagram the three-layered foam, with 1< 2 < 3 Foamed Nanocomposites for EMI Shielding Applications 461 In order to improve the reflectivity of the shielding material, it would be interesting to create a foam having a gradient of dielectric constant inside the material, analogous to the stealth paint protecting some spy aircrafts from radar detection A way to achieve this gradient is to fabricate multilayered foams, each layer having a different, increasing dielectric constant, cf Fig The measurement results for a three-layered nanocomposite fabricated as previously described are shown and discussed in section 6.4 Results and discussion 6.1 Solid samples Microstrip transmission lines were fabricated, using the nanocomposite thin films as substrates By connecting these lines to a 2-port vector network analyzer (VNA), their scattering parameters were measured and the dielectric constant and conductivity of the samples were then extracted, using the One-Line method described in section 4.2 The results presented in this section were obtained for composites based on different polymer materials: PS (polystyrene), PCL (poly(-caprolactone)), PVC (polyvinyl chloride) and PMMA (polymethyl methacrylate), they all had a 2% CNT content As can be seen on Fig 10 and Fig 11, the PMMA-based sample exhibited the lowest dielectric constant but not highest conductivity, while the PCL-based composite had the highest conductivity but also the highest dielectric constant Fig 10 Measured dielectric constant of four solid nanocomposite samples versus frequency (composites based on different polymer matrices) 462 Advanced Microwave and Millimeter Wave Technologies: Semiconductor Devices, Circuits and Systems Fig 11 Measured conductivity of the solid nanocomposite samples versus frequency (composites based on different polymer matrices) Once a polymer matrix is selected, an optimal content of CNTs to add into the matrix must be determined It can be shown that the addition of CNTs increases the conductivity of the nanocomposite but also increases its dielectric constant, which is not desired Since foamed samples have lower dielectric constants than solid ones with the same composition, the exact content of CNTs should be chosen after measurement of foamed samples rather than solid ones, cf section 6.2 It should also be noted that other chemical process parameters have a non-negligible influence Those are beyond the scope of the present chapter, but can be found in (Thomassin et al., 2007) 6.2 Foamed samples PCL was selected as polymer matrix because it exhibited the highest conductivity in its solid form, cf Fig 11 Its dielectric constant was also the highest one, cf Fig 10, but foaming the nanocomposite should significantly decrease it The foamed samples were inserted in four waveguides of different dimensions, covering the to 40 GHz band, cf Fig Their scattering parameters were measured using the Line-Line method described in section 4.1 As mentioned in the previous section, a foamed material has a lower dielectric constant than a solid one of the same chemical composition This is shown on Fig 12, the black and red curves correspond to PCL samples with no CNT content, and the dielectric constant of the solid sample is twice that of the foam It can be explained by the porosity of the foams and the subsequent presence of air inside the sample, the air having a dielectric constant of Foamed Nanocomposites for EMI Shielding Applications 463 Fig 12 Measured dielectric constant of solid and foamed PCL nanocomposite samples, all having a similar CNT content On the other hand, the conductivity of foams is usually higher than that of solid samples, mainly because the CNT are forced into the side walls of pores in the foamed material and this way form a more regular network This effect is shown on Fig 13 for a foam and a solid sample having a close but not equal CNT content, both PCL composites Fig 13 Measured conductivity of solid and foamed PCL/CNTs nanocomposite samples, both having a similar CNT content 464 Advanced Microwave and Millimeter Wave Technologies: Semiconductor Devices, Circuits and Systems The conductivity of foamed nanocomposites increases significantly with the CNT content, as can be seen on Fig 14(left) and the Shielding Effectiveness also increases in the same proportions, cf Fig 14(right) Therefore, in order to have a good SE, it appears, from these results, that the CNT content must be as high as possible Fig 14 Measured conductivity of foamed PCL nanocomposite samples with increasing CNT contents (left) and the corresponding Shielding Effectiveness (right) But the addition of CNTs has an adverse effect on the Reflectivity because the dielectric constant increases significantly with CNT content, cf Fig 15 To measure R, a metallic sheet (or Perfect Electrical Conductor – PEC) is added on the output interface, cf Fig 9, and the sample with the PEC sheet is then characterized (Line-line method, same as samples without PEC) Part of the incident power is absorbed inside the material, the remaining power then reaches the PEC where it is totally reflected and, another part of the power is absorbed on the way back Because of the PEC, the SE in this configuration is theoretically infinite The power reflected back to the port of the VNA, cf Fig 6, is the combination of the power directly reflected back at the input interface and the power reaching the metallic plate and being reflected back (the remaining power that has not been absorbed by the material on the way back) Fig 15 Measured dielectric constant of foamed PCL nanocomposite samples with increasing CNT contents (left) and the corresponding Reflectivity (right) Foamed Nanocomposites for EMI Shielding Applications 465 In conclusion, foamed nanocomposites have both a higher conductivity and a lower dielectric constant than solid samples The first are therefore better suited for shielding applications than the latter The optimum CNT content has to be the result of a compromise between Shielding Effectiveness and Reflectivity, i.e between conductivity and dielectric constant 6.3 Comparison between results using the electrical model and from measurements According to the electrical model developed in section 5.1, at low frequency, since there are no physical contacts between the grains, the equivalent capacitor, cf Fig 8(c), is a virtual ‘open-circuit’; therefore there is no current flowing through the composite At high enough frequency, the equivalent capacitor is a virtual short circuit and the conductivity becomes constant Its value depends then only on the conductivity of the conductive layer of Fig 8(d) In other words, at low frequency when ( CT)-1 >> RT, the conductivity is very low At high frequency, when ( CT)-1