Resonance spectra were obtained to determine the dielectric properties of composite polymer - dielectric films, where the dielectric was MgWO 4, FeWO 4 or Zr(WO 4)2. The resonant frequencies for the films were in the 383 to 47 3 MHz range. The relaxation times , the real and imaginary permittivit ies, and dissipation factor and conductivities for the three materials were determined . 2 Introduction Dielectric composite materials made of epoxy resin as matrix , to which a ceramic filler powder is added to control the dielectric constant of the substrate on which electronic circuits are fabricated , are widely used in electrical engineering . Composites with ceramics filler s of alumina and boron nitride pow ders1, strontium titanate2, barium titanate3, bismuth manganite and lead - titanate4 have been fabricated and their complex dielec tric constant has been studied. In the present work resonance spectra5 were experimental ly obtained , to determine the complex relative permittivity 𝜀𝑟∗ as a function of frequency , of composite polymer -dielectric films , where the dielectric was MgWO 4, FeWO 4 or Zr(WO 4)2. Such films have potential application in planar substrates where the electrical properties of circuits that are built on the substrates depend in part on the permittivity of the latter. Theory The current -voltage relationship of a parallel plate capacitor with air as the dielectric is 𝐼=𝑗𝜔𝐶𝑜𝑉 (1) where 𝜔 is the angular frequency, 𝐶𝑜 the capacitance, and 𝐼 and 𝑉 are the current and voltage phasors. With a dielectric material other than air present between the capacitor electrodes , the complex relative dielectric permittivity 𝜀𝑟∗ , is introduced to account for the lag between the displacement and polarization in the dielectric . The current -voltage relationship can now be given as 𝐼=𝑗𝜔𝜖𝑟∗𝐶𝑜𝑉 (2) 𝐼=𝑗𝜔(𝜖𝑟 ′−𝑗𝜖𝑟′′)𝐶𝑜𝑉 (3) where 𝜖𝑟∗ has a real and imaginary component . The current I then has two components 𝐼= 𝐼𝐿+𝐼𝐷 (4) where the quadrature, displacement current 𝐼𝐷=𝑗𝜔𝜀𝑟′𝐶𝑜𝑉 , and the in phase, loss current 𝐼𝐿= 𝜔𝜖𝑟′′𝐶𝑜𝑉 ( see Fig. 1). 3 Fig. 1 Phasor representation of current, voltage and complex permittivity for dielectric film 𝐼𝐿 contributes to the true power dissipated in the dielectric, and 𝐼𝐷 to the VAr , reactive power supplied to the component over part of a quarter cycle and then returned to the power source over the next quarter . Then δ in Fig. 1 is the phase angle between the total current I and the quadrature current ID and the dielectric dissipation factor tan δ is given by tan𝛿=𝜔𝜀𝑟′′𝐶𝑜𝑉 𝜔𝜀𝑟′𝐶𝑜𝑉 = εr′′ εr′ (5) The capacitor can also be represented by the series combination of a resistor , of resistance R, an inductor of inductance L, and a capacitor of capacitance C. Then f or an AC voltage V across the capacitor the current through the capacitor is given by 𝐼= {1−𝑗(𝜔𝐿 𝑅 − 1 𝜔𝐶𝑅 )}𝑉 𝑅{1+(𝜔𝐿 𝑅− 1 𝜔𝐶𝑅)2 } (6) Considering the RLC circuit, at and around the resonant frequency equations that apply are: 𝑄= 𝜔𝑜𝐿 𝑅 ; 𝑄=1 𝜔𝑜𝑅𝐶 ; 𝑄= 𝜔𝑜 𝐵𝑊 (7) where ωo is the resonant angular frequency , BW the bandwidth in frequency betwee n the 3dB points in the resonance curve , and Q is quality factor of the circuit. Equs ( 6) and ( 7) give 𝐼= {1−𝑗𝑄(𝑓 𝑓𝑜 − 𝑓𝑜 𝑓 )}𝑉 𝑅{1+𝑄2(𝑓 𝑓𝑜− 𝑓𝑜 𝑓 )2 } (8) where f is the cyclic frequency and fo the resonant frequency . 4 Equating the real and imaginary parts of equs ( 3) and ( 8) gives 𝜖𝑟′=𝑄(𝑓𝑜 𝑓−𝑓 𝑓𝑜) 2𝜋𝑓𝐶𝑜𝑅{1+𝑄2(𝑓 𝑓𝑜−𝑓𝑜 𝑓)2 } (9) 𝜖𝑟′′=1 2𝜋𝑓𝐶𝑜𝑅{1+𝑄2(𝑓 𝑓𝑜−𝑓𝑜 𝑓)2 } (10) Then substituting for ϵr′′ and ϵr′ in equ ( 5) gives the dissipation factor tan𝛿=1 𝑄(𝑓𝑜 𝑓−𝑓 𝑓𝑜) (11) Experimental Capacitors of dielectric composite films of polymer -tungstate powder between copper plates were fabricated . The tungstates were 400 mesh powders of MgWO 4, FeWO 4 or Zr(WO 4)2 in a polymer binder. The resulting composite dielectric film consisted of an epichlorohydrin - bisphenol A resin mixed with a telluride and cured with 2,4,6 -Tris(dimethyl -amino - methyl)phenol. The dialectric films were about 0.18 mm thick. The impedance Z, resistance R, and reactance X, of the capacitor as a function of frequency were measured up to 2.5 GHz using a vector network analyzer. At the resonance frequency the reactance is zero and the magnitude of the impedance |𝑍| is equal to the pure resistance R. Results and Discussion Table 1 shows the experimentally determined values of resonant frequency and quality factor for the different films. Reproducibility of resonant frequency between films of the same material was less than 1%. fo ±2 MHz Q MgWO 4 473 25.89 FeWO 4 448 13.18 Zr(WO 4)2 383 9.34 Table 1 Resonant frequency vs quality factor for composite films The results were used in Equs 9 and 10 to plot the variation of the real and imaginary permittivity for each material as a function of frequency (See Fig 2/Table 2 and Fig 3 /Table3 ) 5 Fig 2 variation of 𝜖𝑟′ with frequency for films of polymer -tungstate composites Zr(WO 4)2 f (MHz) 1 100 364 f resonance = 383 𝜖𝑟′ 9.90 10.60 ε' max = 48.57 0 FeWO 4 f (MHz) 1 100 424 f resonance = 448 𝜖𝑟′ 6.91 7.24 ε' max = 44.97 0 MgWO 4 f (MHz) 1 100 463 f resonance = 473 𝜖𝑟′ 3.86 4.04 ε' max = 49.02 0 Table 2 select data on 𝜖′ vs frequency derived from Fig 2 The results of Fig 2 show that the real permittivity is approximately constant between 1 and 100 MHz, for instance rising from 9.90 to 10.60 in this range for the Zr(WO 4)2 film. After 100MHz, for Zr(WO 4)2 , 𝜖𝑟′ increases exponentially with frequency to a maximum of 48.57 at 364MHz before descending to zero at the resonance frequency of 383 MHz. values for 𝜖𝑟′ for FeWO 4 and MgWO 4 are given in Table 2 and within the 1 t o100 MHz range, 𝜖′ is approximately 4, 7 and 10 for the MgWO 4, FeWO 4 and Zr(WO 4)2 films, respectively. 6 Fig 3 𝜖𝑟′′vs frequency for polymer -tungstate composite films Zr(WO 4)2 f (MHz) 1 100 200 f resonance = 383 𝜖𝑟′′ 0.0028 0.32 1.04 92.42 FeWO 4 f (MHz) 1 100 200 f resonance = 448 𝜖𝑟′′ 0.0012 0.13 0.36 91.11 MgWO 4 f (MHz) 1 100 200 f resonance = 473 𝜖𝑟′′ 0.0003 0.036 0.10 96.14 Table 3 select data on 𝜖′′vs frequency derived from Fig 3 At 1 MHz 𝜖𝑟′′ for MgWO 4 is 0.0003 while for Zr(WO 4)2 𝜖𝑟′′is an order of magnitude larger at 0.0028. 𝜖𝑟′′ increases with frequency and at 200MHz it is still an order of magnitude larger for Zr(WO 4)2 compared to MgWO 4. At resonance 𝜖𝑟′′ is of the same order of magnitude for all three materials. As the imaginary permittivity is proportional to losses the largest energy loss occurs in the Zr(WO 4)2 based films. The dissipation factor tan δ is given in Fig 4. It shows that the dissipatio n is highest at resonance but it is spread over a larger frequency band for the Zr(WO 4)2 films and the smallest band for the MgWO 4 ones. 7 Fig 4 dissipation factor tan δ as a function of frequency Zr(WO 4)2 σ (1/Ωm) 3.98 E-10 1.81 E-9 4.82 E-9 1.16 E-8 2.88 E-8 8.86 E-8 5.13 E-7 1.96 E-6 1.19 E-6 f (MHz) 50 100 150 200 250 300 350 384 R 400 FeWO 4 σ (1/Ωm) 1.67 E-10 7.37 E-10 1.86 E-9 4.05 E-9 8.55 E-9 1.91 E-8 5.13 E-8 2.28 E-7 2.26 E-6 2.24 E-6 f (MHz) 50 100 150 200 250 300 350 400 447 R 450 MgWO 4 σ (1/Ωm) 4.66 E-11 2.04 E-10 5.07 E-10 1.08 E-9 2.19 E-9 4.59 E-9 1.09 E-8 3.55 E-8 3.53 E-7 2.53 E-6 2.93 E-7 f (MHz) 50 100 150 200 250 300 350 400 450 473R 500 Table 4 conductivity vs frequency values derived from Fig 3 where R indicates the resonant frequency. Since c onductivity is given by 𝜎=2𝜋𝑓𝜀𝑜𝜖′′where 𝜀𝑜 is the permittivity of free space, the data of Fig 3 were used to calculate the conductivity of the films as a function of frequency (see Table 4) . These show that from 50 MHz up to just before resonance, conductivity is highest for Zr(WO 4)2 and varies from the order of 10-10 to 10-7 Ω-1 m-1 for Zr(WO 4)2 and FeWO 4 and from 10-11 to 10-7 Ω-1 m-1 for MgWO 4. The dielectric relaxation time 𝜏=1/∆𝑓 was obtained from Fig 3 where ∆𝑓 is the frequency difference between the resonant frequency and the frequency where 𝜖′′ falls to half its maximum value. The relaxation time was 48, 56 and 11 1 nanoseconds for the zirconium, iron and magnesium tungstate composite films, respectively. As expected, the longer the relaxation time the sharper the resonance. 8