Measuring the dielectric constant ε_r of Teflon, Polyester, FR4, G10, or FR4 with an RF generator and a good bolometer. The goal is to measure the dielectric constant ε_r (ω), also called Dk, of a substrate with a good degree of accuracy. Several standardised systems can be found in literature, each with specific pros and cons. IPC (https://www.ipc.org) has 13 different methods to determine Dk and Df, also called tan(δ). ASTM (https://www.astm.org) and NIST (https://www.nist.gov) provide a wide range of testing methods. Many OEM and Universities use their own specially developed measurement methods. The results of one testing method may not be consistent with those of another, even though the same material was used. There is no Perfect Measurement Method. To the end of measuring ε_r (ω), the following was used: Microstrip technology – which will actually be employed to build filters and impedance matching networks. A geometry that is “closed”, easily repeatable and has a definite size: an O-Ring. This will avoid any fringe effect which may make measuring the actual size difficult and increase dispersion by radiation. The input and output ports are formed by gaps, as this produces a negligibly small capacitance and ensures the best available Q as well as very well-defined and easily repeatable resonance dips. Cons include some degree of signal attenuation (between ~-27dB and ~-16dB). However, this method will not measure the dissipation factor tan(δ) = Df. Permittivity. Permittivity is a physical quantity describing the behaviour of a dielectric material in the presence of an electric field. In particular, it is a measure of how much the molecules oppose an external electric field. Permittivity affects the propagation of electric fields. The value of an electric field at a distance r, produced by a single charge q, equals to: 1.1 |E|=q/(4πε_0 r^2 ) where ε_0= 8.854⋅〖10〗^(-12) Farads/metre and is the permittivity of vacuum (that is, in absence of atoms). The molecules inside any material will oppose the electric field modifying it according to the following formula: 1.2 |E|=q/(4πε_r ε_0 r^2 ) That is, an ε_r is added (a dimensionless number, always greater than 1 which makes the electric field E lower than what it would be in a vacuum). In total 1.3 ε = ε_r ε_0 ε_r is referred to as dielectric constant. Relative permittivity has two components, a real one and an imaginary one: 1.4 ε_r = ε_r’ – jε_r” ε_r’ is associated to the dielectric constant and ε_r” is associated to the dissipation factor (Df) of the material. 1.5 ε_r = Dk = ε_r’/ε_0 1.6 tan(δ) = Df = ε_r”/ ε_r’ Rogers (a company producing PCB substrates) specifies that between about 100 MHz and 300 GHz most of the interactions between the electric field and the substrate material are caused by the motion and rotation of the elementary dipoles within the substrate (see https://www.e-doodles.it/en/practical-measurements-of-dielectric-constant-and-loss-for-pcb-materials-at-high-frequency/) The motion of the dipoles contributes to ε_r (Dk) Molecular friction due to the rotation of the dipoles contributes to tan(δ) or Df. Microstrip The microstrip transmission line is a type of planar technology. As it is extremely easy to produce as well as economical, it is often preferred over other, more power-efficient types of signal transmission technology ensuring lower dispersion, such as waveguide or stripline. The microstrip is structured like a regular printed circuit. It is based on a dielectric substrate of thickness h, dielectric constant ε_r and loss factor tan(δ). The back side of the substrate is covered by a ground plane, while the front displays a line of thickness t, width w and length L. The study of the characteristics of the electromagnetic field is complicated by the presence of two different dielectrics: the substrate and air. The lines of the electric field extend beyond the line on the front of the PCB, propagating in the air. Fig. 2.1 – Two different dielectric constants: ε_0 for air and ε = ε_r ε_0 for the substrate To solve this problem, the term ε_reff is introduced and called effective permittivity, which makes it possible to imagine conductors as immersed in a homogeneous medium capable of supporting the TEM propagation mode. Depending on the values of w and t, effective permittivity always assumes values within the range 2.1 1<ε_reff<ε_r and clearly also depends on the type of substrate used. This is because effective permittivity is affected by the permittivity of the substrate, as described below: 2.2 ε_eff=[(ε_r+1)/2+(ε_r-1)/2 K] where 2.3 K=1/√(1+12h/w) and is true for w>h. Conversely, (this formula will prove helpful later on): ε_r=(2ε_eff+k-1)/(1+k) Both theory and practice prove that ε_r decreases as the frequency increases; this means the function is〖 ε〗_r (ω), although it will vary according to the substrate used (for example, variations are very rapid for FR4 but much, much slower for Teflon). Fig. 2.2 Trendline of Dk (ε_r) for frequencies between 0.1GHz and 10GHz (Rogers) for RO4003C 20mil (0.508mm) laminate The function 〖 ε〗_r (ω) is very important to determine the adequate size of microwave filters and matching lines. – Resonator ring A broad range of different methods can be used to measure the ε_r (ω) of a PCB (there are at least 20), each with pros and cons. In this case, since the goal is measuring the dielectric constant (or rather, its variations) between 2 and 10GHz, the best solution is using a resonator ring, as it’s very simple to build as well as reliable. The signal is injected and extracted through two very brief gaps in the PC tracks, which have been added to avoid overloading the resonant circuit and shift its resonance frequency. A closed ring with no gaps eliminates much of the fringe effect. Resonator rings ensure minimal or negligible loss by radiation, so that ε_r can be calculated more accurately. The results refer to very narrow band ranges/widths (resonance); this avoids issues connected to spurious modes. Fig. 3.1 – Gerber file image of the resonator ring with a 40mm diameter and 0.4mm gap The substrate used was Teflon but from an unknown manufacturer, which makes the electric parameters uncertain. The dielectric was 0.75mm thick and the copper deposited on both sides was 35micron thick. As the substrate was Teflon, a good starting value for the dielectric constant ε_r is 2.1-2.5. Fig. 3.2 – The O-ring created on the Teflon substrate The O-ring measures exactly 40mm in diameter to the centre of the 2.29mm line and has two ports, each located at 180°. The thickness of the line is based on the assumption that the Teflon has a ε_r of 2.3 which, coupled with the 0.75mm thick substrate, would result in an impedance of 50ohm. However, even if ε_r were slightly different, the resonance and consequently the calculation of ε_r would produce no detectable errors. The ring will resonate (the parameter S_21 will show response peaks) at multiples of lambda, where lambda is the wavelength corresponding to the circumference of the ring. By applying the signal to one of the ports and a bolometer to the other (or connecting the resonator ring to a VNA), the first frequency (n=1) will cause the signal to divide, run along to two paths and merge again on the output port, summing up in phase. At this frequency S_21 peaks. As the frequency rises, the value of the output signal drops drastically (the two signals no longer arrive in phase) until the second frequency is reached (n=2), then it rises again reaching another peak. The same happens until the third frequency is reached (n=3), and so on. What are the frequencies? If the dielectric constant of the substrate were 1: 3.1 l=nλ 3.2 it follows that λ=l/n 3.3 but λ=C_0/f 3.4 hence, from 3.2 and 3.3 f_n=n C_0/l In this specific case, C_0= 2.99792458×〖10〗^8m/s (lightspeed in vacuum) l = 0.12566m (40mm x 3.14) The result is (for n = 1 -> 6) n=1 f_1= 2.386GHz n=2 f_2=4.771GHz n=3 f_3=7.157GHz n=4 f_4= 9.543GHz n=5 f_5= 11.928GHz n=6 f_6= 14.314GHz These are the resonance frequencies of a substrate with ε_r = 1. However, we know that ε_r is clearly not equal to 1. By using a generator and a bolometer covering the entire range in question and connecting them to the two ports (in any order), the first (n=1) peak of S_21 can be determined. The frequency is then doubled and the exact point of S_21 close to this almost double frequency (n=2) measured. Why almost? Because ε_r drops as the frequency rises, causing S_21 peaks to shift. The same applies to the triple frequency and so on. With Teflon this discrepancy is minimal, approximately equal to the errors introduced with this measurement method. It will be very difficult to detect and even more difficult to measure. With G10 and FR4 this phenomenon will be much more evident and easier to measure. Fig. 3.3 – The six resonance peaks With the O-ring shown by figure 3.2, the resonances have been found at the following frequencies: n=1 f_1= 1.631GHz n=2 f_2= 3.261GHz n=3 f_3= 4.889GHz n=4 f_4= 6.515GHz n=5 f_5= 8.115GHz n=6 f_6= 9.722GHz In order to relate these to the nominal frequencies calculated earlier, it can be observed that 3.5 λ_0=C_0/f_0 3.6 λ_0=C_0/(f√(ε_reff )) 3.7 it can be inferred that █( @ f)_0=f√(ε_reff ) 3.8 hence ε_reff=(█(f_(o(n))@¯(f_n )))^2 Therefore: n f_(o(n)) f_n ε_reff n=1 f_01= 2.386GHz f_1= 1.631GHz ε_reff = 2.140 n=2 f_02= 4.771GHz f_2= 3.261GHz ε_reff = 2.141 n=3 f_03= 7.157GHz f_3= 4.889GHz ε_reff = 2.143 n=4 f_04= 9.543GHz f_4= 6.515GHz ε_reff = 2.145 n=5 f_05= 11.928GHz f_5= 8.115GHz ε_reff = 2.161 n=6 f_06= 14.314GHz f_6= 9.722GHz ε_reff = 2.168 Last step: using 2.3 and 2.4, ε_r (ω) values can finally be obtained. n=1 f_01= 2.386GHz f_1= 1.631GHz ε_reff = 2.140 ε_r = 2.58 n=2 f_02= 4.771GHz f_2= 3.261GHz ε_reff = 2.141 ε_r = 2.59 n=3 f_03= 7.157GHz f_3= 4.889GHz ε_reff = 2.143 ε_r = 2.59 n=4 f_04= 9.543GHz f_4= 6.515GHz ε_reff = 2.145 ε_r = 2.59 n=5 f_05= 11.928GHz f_5= 8.115GHz ε_reff = 2.161 ε_r = 2.61 n=6 f_06= 14.314GHz f_6= 9.722GHz ε_reff = 2.168 ε_r= 2.62 ε_r has been approximated to the first two decimals, which is more than satisfactory considering the method used. To conclude, the assumption: ε_r = 2.6 proved true for a – more than satisfactorily wide – range between 1GHz and 10GHz. It is worth noting that ε_reff is the true global dielectric constant, i.e. the constant that reflects the fact that propagation actually occurs partly in the dielectric and partly in the air above the microstrip, where ε_r= 1. 4. Conclusions In sum: A printed circuit was built as shown by fig. 3.1. The diameter may vary between 20 and 80 millimetres, covering lower frequencies around 1 GHz or higher frequencies around 20GHz. The n frequencies at which the circuit resonates are calculated. Measurements are taken with a good bolometer. It will suffice to measure the relative maximum, the absolute value in dB is not necessary. Using a VNA (Vector Network Analyser) will yield more accurate measurements, as the attenuation introduced by the two gaps produces fairly low signal values. However, the gaps must be used to ensure a sufficiently high Q and, consequently, a sufficiently narrow resonance. From these measurements and the previously determined frequency values, ε_reff values can now be calculated. From ε_reff values, the corresponding ε_r values can be obtained. A Gerber file is provided to build a 40mm o-ring on Teflon.

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