Experiments and model of intermodulation distortion in a rutile resonator with YBCO endplates

P1: JRX Journal of Superconductivity: Incorporating Novel Magnetism (JOSC) pp1007-josc-474031 October 15, 2003 13:0 Style file version June 22, 2002 Journal of Superconductivity: Incorporating Novel Magnetism, Vol. 16, No. 5, October 2003 ( C °2003) Experiments and Model of Intermodulation Distortion in a Rutile Resonator with Y-Ba2-Cu3-O7−δEndplates J. Mateu,1C. Collado,2O. Men´ endez,2and J. M. O’Callaghan2 Received September 30, 2002 We have derived general equations to calculate intermodulation distortion in resonators with high temperature superconducting (HTS) films which are not restricted to a specific resonator shape and may be used whenever the fields in a resonator generate currents on the surface of one or more HTS films. These equations are applied to rutile-loaded cavities with one or two 10 ×10 mm2Y-Ba2-Cu3-O7−δendplates and are used to extract parameters characterizing the nonlinearities of these films from intermodulation measurements. Even though the films have similar small-signal performance, we have found large variation in the strength of their nonlinearities. KEY WORDS: nonlinear measurements; dielectric cavity; intermodulation measurements; superconductors. 1. INTRODUCTION High Temperature Superconducting (HTS) thin films are being used in narrowband microwave filters due to their low losses and small volume. The use of HTS materials in this application might be limited by their nonlinearities, which give rise to intermodulation distortion (IMD) and other undesired effects [1]. Although most of the experimental work on HTS intermodulation has been done on planar patterned devices [2–4], this type of characterization might not be representative of the properties of the whole HTS film because the nonlinear properties of the test devices are dominated by the high current densities at the edges of the pattern, where the properties of the film may be affected by the etching process. In this work we describe a technique that might contribute to overcome these difficulties, which is based on the use of a rutile resonator to make a small-area characterization of the intermodulation properties of unpatterned HTS films. 1CTTC-Centre Tecnol `ogic de Telecomunicacions de Catalunya, Edifici NEXUS, C/Gran Capit`a 2–4, 08034 Barcelona, Catalunya, Spain. 2Universitat Politecnica de Catalunya, Campus Nord UPC-D3, C/Jordi Girona 1, 08034 Barcelona, Spain. 2. A GENERAL FORMULATION FOR HTS NONLINEARITIES We assume that, at high currents, the electric field at the surface of the superconductor is the sum of a linear contribution due to the surface impedance of the film, plus an additional electric field −→ eNL caused by the nonlinearities [5, 6] which can be expressed as −→ eNL(js)=aNL(js)E js+∂ ∂t£bNL(js)E js¤, (1) where E jsis the surface current density, aNL(js) represents the nonlinear resistive term, and bNL (js) the nonlinear reactive term. Both terms depend on the magnitude of the surface current jsand have to be zero for js=0 to make −→ eNL vanish at arbitrarily small currents. An alternative form of describing the nonlinearities in the HTS consists in quantifying the change in surface impedance as a function of the surface current density (or magnetic field), i.e., in frequency domain we would have E ENL =1Rs(Js)E Js+j1Xs(Js)E Js [3], where 1Rs(Js)+j1Xs(Js) is the variation in surface impedance with respect the small signal value. While finding the equivalence between this formulation and the one in Eq. (1) is straightforward (see Fig. 873 0896-1107/03/1000-0873/0 C °2003 Plenum Publishing Corporation P1: JRX Journal of Superconductivity: Incorporating Novel Magnetism (JOSC) pp1007-josc-474031 October 15, 2003 13:0 Style file version June 22, 2002 874 Mateu, Collado, Men´ endez, and O’Callaghan Fig. 1. Comparison between aNL(t)js(t) in time domain (solid line in rightmost graph) and the deviation of the surface resistance from its small-signal value (1Rsproportional to the amplitude of the sinusoid drawn with dotted line in the rightmost graph). 1), we prefer that in Eq. (1) since it gives all the details of the time-domain function, whereas 1Rs,1Xsonly refer to its first harmonic. Another existing formulation for HTS nonlinearities that can be related to that in Eq. (1) is the one described in [7, 8], according to which the penetration depth depends on the current density. This formulation has been extensively used to justify the generation of IMD in HTS films, and our previous work [9], shows how it can be related to the nonlinear reactive term bNL(js) in Eq. (1). Equation (1) is thus a very general mathematical description of HTS nonlinearities that can be particularized for various models and should account for the generation of both harmonics and intemodulation products. With little loss in generality, in this work we assume that aNL,(j s ), bNL,(j s ) can be expressed as aNL(js)=1Rα|js|,bNL(js)=1Lα|js|α, where α, 1Rα,1Lαare parameters that characterize the strength of the nonlinearity in the HTS. Equation (1) will then read −→ eNL =1Rα|js|αE js+1Lα d dt ¡|js|αE js¢.(2) 3. INTERMODULATION IN HTS RESONATORS In this section, we will derive general equations for the power of the third-order intermodulation products coupled out of a resonator due to nonlinearities in the HTS. The analysis applies to any resonator where the field configuration of the resonant mode is such that it generates currents on the surface of an HTS material. The dielectric-loaded cavity resonators used for Rsmeasurement of HTS films are an obvious example, but the analysis could also be applied to a disk resonator or to planar, patterned HTS resonators. 3.1. Electric Field due to HTS Nonlinearities In any resonator the current at the surface of the HTS film can be written as E js(Er,t)=A(t)f(Er)ˆu(Er), (3) were ˆu(Er) is a unit vector, f(Er)ˆu(Er) describes the spatial dependence of the current density, and A(t) its time dependence. For the case of an intermodulation experiment A(t) will consist of the sum of two sinusoidal signals with amplitudes j1and j2: A(t)=j1cos ω1t+j2cos ω2t.(4) Combining Eqs. (2)–(4) we find that the electric field at the surface of the superconductor can be written as −→ eNL(Er,t)=½1Rα|A(t)|αA(t)+1Lα d dt (|A(t)|αA(t))¾ ×|f(Er)| αf(Er)ˆu(Er).(5) Note that the spatial dependence of −→ eNL (the term |f(Er)|αf(Er)ˆu(Er) in Eq. (5)) differs from that of the current E js. From Eq. (5) we can calculate the electric field at 2ω1−ω2: E E2ω1−ω2=[1Rα+j(2ω1−ω2)1Lα]2C2ω1−ω2|f(Er)|α ×f(Er)ˆu(Er), (6) where 2C2ω1−ω2is the intermodulation produced by the function |A(t)|αA(t)at2ω 1−ω 2 and has the units of (A/m)α+1, i.e. C2ω1−ω2=lim T→∞ 1 TZT 2 −T 2 |A(t)|αA(t)e−j(2ω1−ω2)tdt.(7) Figure 2 shows the dependence of the electric field amplitude at 2ω1−ω2and at 2ω2−ω1as a function of the surface current amplitude j1(i.e., C2ω1−ω2 and C2ω2−ω1vs. j1). When j1=j2both C2ω1−ω2and C2ω2−ω1are proportional to jα+1 1, so a logarithmic plot of C2ω1−ω2versus j1or C2ω2−ω1versus j1has a slope of α+1. When j1¿j2,C2ω1−ω2∝j2 1j2and C2ω2−ω1∝ j1j2 2so, if j2is kept constant, the slopes in Fig. 2 are 2:1 and 1:1, respectively, regardless of the value of α.In the opposite regime, when j1Àj2,C2ω1−ω2∝jα 1jα−1 2 and C2ω2−ω1∝jα−1 1jα 2so, in a logarithmic scale, the magnitude of the electric field at the intermodulation frequency changes with slopes of αand α−1 with the current amplitude j1when j2is fixed. The dependence of C2ω1−ω2and C2ω2−ω1on j1and j2described here accounts for the experimental results described in the following sections of this work and for those reported in [10]. P1: JRX Journal of Superconductivity: Incorporating Novel Magnetism (JOSC) pp1007-josc-474031 October 15, 2003 13:0 Style file version June 22, 2002 Experiments and Model of Intermodulation Distortion in a Rutile Resonator 875 Fig. 2. Dependence of the electric field at the intermodulation frequencies (2C2ω1−ω2and 2C2ω2−ω1) on the amplitude of the surface current j1. When the two current amplitudes j1,j2are kept equal to each other, the fraction C2ω1−ω2/jα 1j2only depends on the order of the nonlinearity α. Hence, instead of C2ω1−ω2(j1,j2,α), we will use the parameter T 2ω1−ω2(j1,j2,α)≡2C2ω1−ω2(j1,j2,α) jα 1j2 (8) in subsequent equations. Figure 3 shows the dependence of T 2ω1−ω2on αfor balanced current amplitudes. In summary, Eqs. (6)–(8) give a general set of equations relating the IMD electric field at the surface of the superconductor with the amplitude of the surface current density. These equations are not reFig. 3. Dependence of T 2ω1−ω2on the order of the nonlinearity αfor balanced current amplitudes. In this case, the amplitude of the electric field at the surface of the HTS at the intermodulation frequency will be proportional to T 2ω1−ω2(α)jα+1 1 stricted to a particular shape of resonator and should be valid whenever aNL(js)=1Rα|js|αand bNL(js)= 1Lα|js|α. 3.2. Resonant IMD Magnetic Field on the HTS Surface From the discussion above, we have determined that the IMD electric field on the HTS surface will be of the form E E2ω1−ω2=E2ω1−ω2|f(Er)|αf(Er)ˆu(Er), (9) with E2ω1−ω2=[1Rα+j(2ω1−ω2)1Lα] ×T 2ω1−ω2(j1,j2,α)jα 1j2.(10) If ω1≈ω2≈2ω1−ω2this field will couple to the same mode at which ω1and ω2resonate and will generate a magnetic field at 2ω1−ω2whose spatial distribution has to be the same than that of E js(Er,t)in Eq. (3), i.e., E H2ω1−ω2=H2ω1−ω2f(Er)ˆv(Er), (11) where ˆv(Er) is a unit vector on the HTS surface perpendicular to ˆµ(Er). The magnetic field on the HTS at 2ω1−ω2is thus fully known if we can determine its amplitude H2ω1−ω2. To do that we consider the power generated at the surface of the HTS at 2ω1−ω2: P=1 2ZS E E2ω1−ω2×E H∗ 2ω1−ω2dE S(12) and relate it to the resonator Q. This power will be either dissipated inside the resonator or coupled outwards, so it has to satisfy P=(2ω1−ω2)W QL , (13) where Wis the energy stored in the resonator at 2ω1−ω2and QLis the loaded quality factor. If we normalize this energy with respect to the field amplitude |H2ω1−ω2|2we get W0=W/|E H2ω1−ω2|2and P=1 2ZS E E2ω1−ω2×E H∗ 2ω1−ω2dE S =(2ω1−ω2)W0 QL H∗ 2ω1−ω2H2ω1−ω2.(14) Using Eqs. (9)–(11) in Eq. (14) we find H2ω1−ω2: P1: JRX Journal of Superconductivity: Incorporating Novel Magnetism (JOSC) pp1007-josc-474031 October 15, 2003 13:0 Style file version June 22, 2002 876 Mateu, Collado, Men´ endez, and O’Callaghan E H2ω1−ω2(Er)=1 2[1Rα+j(2ω1−ω2)1Lα] T 2ω1−ω2(j1,j2,α)jα 1j2 QL (2ω1−ω2)W0 ZS {| f(Er)|αf(Er)f(Er)dS}f(Er)Er.(15) Note that the term 1 2RS|f(Er)|αf(Er)f(Er)dS, is a sort of coupling coefficient between the spatial distribution of the electric field on HTS (|f(Er)|αf(Er)) and that of the magnetic field of the resonant mode in the structure ( f(Er)). If we express this term as 0α, Eq. (15) can be written as E H2ω1−ω2(Er)=[1Rα+j(2ω1−ω2)1Lα] T 2ω1−ω2(j1,j2,α)jα 1j2 0α QL (2ω1−ω2)W0 f(Er)Er.(16) Note that we are assuming that the intermodulation products are only generated by the mixing of the fundamental signals. This means that we are not considering higher order effects that could contribute to E H2ω1−ω2like for example the mixing of higher order spurious signals. 3.3. IMD Power Coupled Out of the Resonator In this section we calculate the power of the intermodulation product coupled out of the structure as a function of the available power of the sources at fundamental frequencies ω1and ω2(P0,ω1and P0,ω2, respectively) and the external coupling factors. We first analyze the case of a one-port resonator. In this resonator the coupling factor κis the ratio of the power coupled to the load (PL) to the power dissipated in the resonator (Pd) [11], so PL=κPd=κ(2ω1−ω2)W Q0 =κ(2ω1−ω2)W0 Q0 |H2ω1−ω2|2, (17) where |H2ω1−ω2|is given by Eq. (16). We now have to relate j1and j2in Eq. (16) with the power available from the sources at ω1and ω2 (P0,ω1and P0,ω2). To do this we use the equation that relates dissipated power (Pd) with available power (P0) in a one-port resonator: Pd=4P0κ (1 +κ)2(18) so, for ω1 4P0,ω1κ (1 +κ)2=ω1W0 Q0 |j1|2(19) and similarly for ω2. Using Eqs. (16), (18), and (19) for ω1and ω2, and assuming ω1≈ω2≈2ω1−ω2≈ω0,weget P L=(4P0,ω1)α(4P0,ω2)µQ0 ω0W0¶α+2·κ (1 +κ)2¸(α+2) |[1Rα+j(2ω1−ω2)1Lα]T 2ω1−ω2(j1,j2,α)0α|2. (20) For a two-port resonator we use the input and output coefficient κin and κout [11]. In this case the power coupled to the load is PL=κout Pd=κout (2ω1−ω2)W0 Q0 |H2ω1−ω2|2(21) and, similarly to Eq. (19), j1,j2can be related to the power available from the sources through 4P0,ω1κin (1 +κin +κout)2=ω1W0 Q0 |j1|2(22) and the resulting equation for PLis PL=(4P0,ω1)α(4P0,ω2)µQ0 ω0W0¶α+2 κoutκα+1 in (1 +κout +κin)2(α+2) |[1Rα+j(2ω1−ω2)1Lα]T 2ω1−ω2(j1,j2,α)0α|2. (23) Equations (20) and (23) are the general equations that relate the measured intermodulation power with the parameters of the film 1Rα,1Lα, and α. In these equations 1Rαand 1Lαonly affect the level of the intermodulation products, but not the slope of their variation with the source power. On the other hand, α affects both the IMD level and the slope. This can also be seen from the equations of the intermodulation electric field (Eqs. (4), (6), and (7)), and from those of the intermodulation magnetic field (Eqs. (8) and (16)). Note that we are assuming that the surface current amplitudes j1,j2in Eq. (16) (or, equivalently, the tangential component of the magnetic fields on the HTS surface Hω1,Hω2) are identical to those that would be on the HTS if it was completely linear (see Eqs. (19) and (22)). In other words, we do not take intoaccountthecompression effectthat occursat high P1: JRX Journal of Superconductivity: Incorporating Novel Magnetism (JOSC) pp1007-josc-474031 October 15, 2003 13:0 Style file version June 22, 2002 Experiments and Model of Intermodulation Distortion in a Rutile Resonator 877 power levels, when a significant fraction of the power that is injected to the resonator at ω1and ω2is transferred to other frequencies (2ω1−ω2,2ω 2−ω 1 , and others) by the effect of the nonlinearities, thereby reducing the amplitude of the fields and currents at ω1 and ω2. 4. ANALYSIS OF A RUTILE-LOADED CAVITY In this section we will apply the general analysis described in the preceding one to the dielectricloaded resonator cavity of Fig. 4. We will analyze a TE011 rutile resonator similar to the one described in [12], but the analysis may also be applied to similar resonators with other dielectric materials (like sapphire) which are commonly used in microwave HTS characterization. We have chosen to use a rutile dielectric because it avoids high fringing fields in the normal-metal housing of the resonator, thus limiting the effect of housing loss in the overall Qof the resonator despite the small dimensions of the cavity. The TE011 mode used has the advantage of avoiding “wrap around” effects which—as described in [13]— enhance the nonlinearities in the device and make the IMD very strongly dependent on the properties of narrow areas of the HTS film. In these conditions, the current density on the HTS endplate in an intermodulation experiment can be written as E js(ρ,t)=(j1cos ω1t+j2cos ω2t)f(ρ)ˆ φ, (24) where ˆ φis the unit vector in the azimuth direction and f(ρ) describes the radial dependence of the TE011 mode which, according to [14], is f(ρ)=         β ξ1 J1(ξ1ρ)ρ≤a β ξ2 J0(ξ1ρ) F0(ξ2ρ)F1(ξ2ρ)b>ρ>a , (25) where βis the z-direction propagation constant, ξ1and ξ2are the ρ-direction wave numbers (inside and outside the dielectric, respecti vely), F0(ξ2ρ)=I0(ξ2ρ)+K0(ξ2ρ)I1(ξ2b)/K1(ξ2b), F1(ξ2ρ)=−I 1 (ξ 2 ρ)+K 1 (ξ 2 ρ)I1(ξ2b)/K1(ξ2ρ) being J0,I1,I2,K0,K1the corresponding Bessel and Hankel functions. From this current distribution and analyzing the cavity as described in the previous section, we obtain E H2ω1−ω2(ρ)=[1Rα+j(2ω1−ω2)1Lα]T 2ω1−ω2 (j1,j2,α)jα 1j20α QL (2ω1−ω2)W0 f(ρ)ˆρ, (26) where 0α=π"¯¯¯¯ β ξ1¯¯¯¯ α+2 IJ2+α+¯¯¯¯ β ξ2 J0(aξ1) F0(aξ2)¯¯¯¯ α+2 IF2+α# (27) and IJ 2+α=Rα 0|J 1(ξ 1ρ)| 2+αρdρ,IF 2+α=Rb a|F 1 (ξ 2ρ)|2+αρdρ. The resulting power coupled out the cavity at 2ω1−ω2is obtained from Eq. (20). Once we know the E H2ω1−ω2generated by one HTS endplate, we can easily extend our study to a cavity having two nonlinear endplates. In this case, the integral in Eq. (14) has to be done over the two HTS endplates. As a result the value of H2ω1−ω2is the sum of the contribution of each endplate, and can be calculated by applying Eq. (26) to each of them. 5. MEASUREMENTS AND DATA FITTING Measurements were made with the resonator shown in Fig. 4, which has a 3-mm high rutile rod (4-mm diameter) of εr∼110 at 77 k, sandwiched between two 10 ×10 mm2endplates. The resonant frequency of the TE011 mode is close to 8 GHz. To make the IMD measurements, the input port is fed with two signals at frequencies f1and f2, both within the passband of the resonator. The power of these sources is either varied simultaneously (balanced measurements), or is changed in only one of the sources, leaving the second tone power constant (unbalanced measurements). Coupling is made by an adjustable coupling loop. The intermodulation products are measured with a one-port setup (see Fig. 5) in which a spectrum analyzer (SA) is used to measure the signals going out of the resonator. The SA is kept isolated from the generators producing the signals at f1and f2so that, if the resonator is critically coupled, the power of the signals at f1and f2 Fig.4. DielectricresonatorstructureusedforIMDcharacterization of HTS films. P1: JRX Journal of Superconductivity: Incorporating Novel Magnetism (JOSC) pp1007-josc-474031 October 15, 2003 13:0 Style file version June 22, 2002 878 Mateu, Collado, Men´ endez, and O’Callaghan Fig. 5. Block diagram of the one port measurement setup. reaching the SA is very small, and the IMD of the SA does not alter the measurements of the power of the intermodulation products at 2 f1−f2and 2 f2−f1. Furthermore, in this arrangement the spectral purity of the sources is not as critical as it is in transmission measurements, where the intermodulation products are easily masked by the phase noise of the sources if |f1−f2|is not sufficiently large. All the measurements made for characterization were done with the resonator immersed in liquid nitrogen. However, comparative measurements were also made in a closed-cycle cryostat to ensure that thermal effects were not dominant in the generation of intermodulation products. In these measurements, if the power applied to the resonator was sufficiently large, we observed variations of the resonant frequency which we attributed to heating effects on the rutile [12]. This variations were small if the source power was sufficiently low. In these conditions the IMD products measured compared well (within 0.5 dB) with those made in liquid nitrogen immersion. A second technique to check for thermal effects is to change the frequency of the envelope of the signal coupled into the resonator (|f1−f2|/2). We have performed such measurements by changing |f1−f2| between 1 and 40 KHz, and found consistent results (i.e., variations with |f1−f2|, within a 0.5 dB). Four different endplates have been used in these measurements. Three of them (samples HTS1, HTS2, and HTS3) are 700-nm thick Y-Ba2-Cu3-O7−δ (YBCO) films grown on 0.5-mm MgO substrates by a commercial supplier (Theva), and a fourth one made of copper. The HTS films were used as delivered by the supplier, with no further processing on our part. We made a round-robin series of small signal Qmeasurements using all the HTS endplates to rule out a large spread in Rsamong the films. The average Rsat 77 K was 0.3 mÄand we can guarantee that the spread is less than 25%, but our capability to assure this is limited by the loss of the rutile, so the real spread in Rs mightbelower.Nonlinearmeasurementswheremade with the combination of endplates shown in Table I. Table I. Endplates Used in the Nonlinear Resonator Measurements (T=77 K) Set Films QLQ0 1 HTS1–Cu 8000 16000 2 HTS2–Cu 6500 16000 3 HTS1–HTS217500 80000 4 HTS3–Cu 8000 16000 5 HTS2–HTS316000 77000 Figure 6 shows the results of the IMD measurements of sets 1 and 2, which reveal very strong differences in the intermodulation performance of samples HTS1and HTS2. For example, the intermodulation products obtained from balanced measurements at 0 dBm in set 1 are 20 dB higher than those of set 2, which cannot be justified by the difference in QL. By applying Eq. (20) to set 1 and set 2 and assuming that resistive nonlinearities are negligible [2,4] we have extracted the nonlinear parameters of HTS1and HTS2. The resulting values are α1=0.2, 1Lα1=5×10−16 Hm0.2/A 0.2and α2= 1, 1Lα2=3.5×10−19 Hm/A, respectively. Figure 7 shows the resulting bNL(js) for each of the two sets in the range of current densities used in our experiments. Fig. 6. Results of measurements and fittings (lines) for set 1 (a) and set 2 (b). Circles: measured power of intermodulation products at 2ω1−ω2as a function of the source power when the power of both sources was increased simultaneously. Squares and diamonds represent the measured power of the intermodulation products at 2ω1−ω2and 2ω2−ω1, respectively, as a function of the source power at ω1when the power of the source at ω2was kept constant at −2 dBm in set 1 and 4 dBm in set 2. The difference between the two source frequencies is 20 KHz. P1: JRX Journal of Superconductivity: Incorporating Novel Magnetism (JOSC) pp1007-josc-474031 October 15, 2003 13:0 Style file version June 22, 2002 Experiments and Model of Intermodulation Distortion in a Rutile Resonator 879 Fig. 7. Plot of bNL(js) for set 1 and set 2 for the range of source power used in the measurements. Once HTS1and HTS2are characterized we have extended our experiments to a resonator having both HTS1and HTS2as endplates (set 3 in Table I). Figure 8 shows the results of the balanced and unbalanced IMD measurements made with this set versus the theoretical results predicted using the values of α and 1Lαfound above, from sets 1 and 2. The IMD of set 4 (HTS3and the Cu endplate) was below the sensitivity of our measurement setup (-95 dBm for an available power of 10 dBm at the resonator port). Coupling in this set was adjusted to obtain the same circulating power than that in set 1 and set 2. This indicates that the HTS nonlinearities of the three samples are very different, and that there is no Fig. 8. Results of measurements and calculations for set 3. Circles: measured power of intermodulation products at 2ω1−ω2as a function of the source power when the power of both sources was increased simultaneously. Squares and diamonds represent the measured power of the intermodulation products at 2ω1−ω2and 2ω2−ω1, respectively, as a function of the source power at ω1when the power of the source at ω2was kept constant at 3 dBm. The lines in the figure are results of calculations performed with parameters obtained from set 1 and set 2. The difference between the two source frequencies is 20 KHz. contribution of the measurement setup to the IMD measurements of sets 1 and 2. Measurable levels of IMD were obtained by combining HTS3and HTS2in set 5, and we obtained results that are consistent with the other sets (larger IMD than set 2). Characterization of the nonlinearities in sample HTS3requires improvements in our measurement setup that we are currently addressing. 6. CONCLUSIONS We have developed a general technique to calculate intermodulation distortion in resonators with HTS materials. We have applied this technique to a rutile resonator to use it for characterization of the nonlinear properties of unpatterned 10 ×10 mm2HTS films. We have found that three state-of-the-art HTS samples can have widely different IMD performance, while having a moderate spread in Rs. We encourage other groups to try this technique so that data can be accumulated for a larger number of HTS samples, and offer our collaboration to supply further details on the resonator and measurement setup. Possible improvements in this technique are related to the possibility of extending the validity of the mathematical formulation used for the HTS nonlinearities to fit the wide range of experimental results found in the literature. The formulation in this work is based on the assumption that the same power law can be applied for the resistive and reactive nonlinearities. While this is a severe restriction, it allows to use our technique in the numerous works that claim that, at low powers, the intermodulation products in HTS devices are due to a current dependence of the penetration depth, since this makes the reactive nonlinearities dominant [7,8]. 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