Analysis of the Nonlinear Dynamics of an Injection-Locked Dual-Mode Oscillator

A detailed investigation of the nonlinear dynamics of injection-locked dual-mode oscillators, intended for a sensor application, is presented. It starts with the analytical study of a cubic-nonlinearity oscillator loaded with two distributed LC resonators. We demonstrate the stable behavior of the two periodic oscillation modes when one or two resonators are affected by the material under test (MUT). We also show the impact of the harmonic content on the oscillation frequencies and the mechanism that gives rise to the oscillatory solutions. Next, a practical transistor-based oscillator is addressed. We initially make use of a novel resonance diagram, able to predict the oscillation coexistence and then perform an accurate calculation of the periodic oscillations through an exhaustive analysis method, based on harmonic balance (HB). In the presence of the input signal, the two locking bands existing for each dielectric constant are simultaneously obtained by tracing the contours of a nonlinear current function. The phenomenon that gives rise to the switching from one oscillation to the other is identified with concepts from bifurcation theory. Finally, the MUT dielectric constant and loss tangent are detected from the central frequencies and bandwidths of the two locking bands using a linear interpolation procedure.


Analysis of the Nonlinear Dynamics of an
Injection-Locked Dual-Mode Oscillator Mabel Pontón , Member, IEEE, Almudena Suárez , Fellow, IEEE, and Sergio Sancho , Senior Member, IEEE Abstract-A detailed investigation of the nonlinear dynamics of injection-locked dual-mode oscillators, intended for a sensor application, is presented.It starts with the analytical study of a cubic-nonlinearity oscillator loaded with two distributed LC resonators.We demonstrate the stable behavior of the two periodic oscillation modes when one or two resonators are affected by the material under test (MUT).We also show the impact of the harmonic content on the oscillation frequencies and the mechanism that gives rise to the oscillatory solutions.Next, a practical transistor-based oscillator is addressed.We initially make use of a novel resonance diagram, able to predict the oscillation coexistence and then perform an accurate calculation of the periodic oscillations through an exhaustive analysis method, based on harmonic balance (HB).In the presence of the input signal, the two locking bands existing for each dielectric constant are simultaneously obtained by tracing the contours of a nonlinear current function.The phenomenon that gives rise to the switching from one oscillation to the other is identified with concepts from bifurcation theory.Finally, the MUT dielectric constant and loss tangent are detected from the central frequencies and bandwidths of the two locking bands using a linear interpolation procedure.

I. INTRODUCTION
D UAL-FREQUENCY oscillators, mostly used in multi- standard communication systems [1], [2], [3], [4], [5], enable the generation of two oscillations at distinct frequencies with the same active core.Though most previous research focuses on their operation in free-running conditions, Goel and Hashemi [6], [7] demonstrate the interest in introducing a locking signal with a high spectral purity to reduce the oscillation phase noise.An injection signal is also used in the dual-oscillator sensor proposed in [8], which takes advantage of the high sensitivity of the locking bands to the material under test (MUT).This sensitivity was demonstrated in [9], where a single-frequency oscillator, instead of a dual one, was considered.In [8], the injection-locked oscillator changes from one oscillation mode to the other with the tuning voltages, which enables sensing at two frequencies.Both permittivity and permeability are obtained with very good experimental results, though all the simulations are carried out in small signals.In turn, the analyses of [9] rely on an empirical estimation of the oscillator quality factor from the locking bandwidth, so they are approximate.In fact, the locking band of a transistor-based oscillator has more complex dependences [10], [11], [12], [13], [14], [15], [16], [17], involving the device nonlinearities and the entire passive linear network.For a more realistic prediction, Sancho et al [18], devoted to a single-frequency injection-locked oscillator, make use of a semianalytical formulation, based on the linearization of the oscillator circuit about its free-running solution.This is carried out by introducing an auxiliary generator (AG) in harmonic balance (HB) and applying finite differences [18].Due to this linearization, the validity of [18] is limited to small input amplitudes.
This work extends [18] with a detailed investigation of the nonlinear dynamics of injection-locked dual-frequency oscillators.Unlike [18], the oscillators will be analyzed without any limiting assumptions about the amplitude of the injection source.We will consider two types of operation: differential, with one oscillation more sensitive to the MUT than the other, and dual, with both oscillations exhibiting a similar sensitivity to the MUT.This should enable the extension of the advantages of sensing at two different frequencies in the case of oscillator sensors.Namely, differential sensors [19], [20], [21], [22], [23] increase the robustness against the environmental parameters that affect the sensor behavior, and dual-band sensors [24], [25], [26], [27], [28] increase the reliability, since the dielectric-constant spectrum of each material is unique.When dealing with liquid mixtures, dualband sensors enable the detection of the volumes of three different components [29], [30].
The investigation will start with an analytical study of the dual-frequency oscillator in free-running conditions.It will be based on a cubic nonlinearity, commonly used in oscillator literature [13], [15], [31], [32] due to its capability to predict the saturation of the oscillation amplitude.The passive structure will consist of two series LC resonators, implemented on microstrip [21], [22], in parallel with the nonlinearity.As will be demonstrated, this configuration is well suited for the dual sensing application, since the two coexistent periodic oscillations are stable in the full range of MUT dielectric constants.In the presence of harmonic content, we will show the dependence of each of the two oscillation frequencies on both resonators, which will have an impact on the performance as a differential sensor.We will also demonstrate the generation of periodic oscillations through a turning-point mechanism, which will prevent the detection of the oscillation coexistence with through a small-signal stability analysis.Then, we will address the more demanding case of a transistor-based oscillator.We will make use of a novel resonance diagram, able to predict the oscillation coexistence, followed by an accurate calculation of the periodic solutions.
In the presence of the injection signal, the two locking bands existing for each dielectric constant will be obtained by making use of the same nonlinear impedance function extracted from HB in free-running conditions.The locked solution curves will be given by the constant magnitude contours of a nonlinear current function [33], [34] depending on that impedance.We will also identify the phenomenon that makes the system switch from one oscillation to the other when varying the input frequency.The demonstration will be based on an extension of the contour methods in [33] and [34] to quasi-periodic regimes and concepts from bifurcation theory [16], [31], [32], [35].Once the central frequencies and bandwidths of the two locking bands are known, the real and imaginary parts of the dielectric constant will be detected through a linear interpolation procedure.In summary, the three main novelties of this work are the proposal of a dual-frequency oscillator sensor with two stable periodic oscillations in the full range of MUT dielectric constants, the prediction of the circuit behavior through a completely new set of accurate analysis methods, compatible with commercial HB, and the in-depth investigation of the switching mechanism, with the aid of bifurcation concepts.
This article is organized as follows.Section II presents an analytical study of the dual frequency oscillator in free-running conditions.Section III addresses the transistor-based oscillator, analyzed in small signal, and steady state.Section IV presents the analysis in injection-locked conditions and the investigation of the mechanism for the oscillation switching.Finally, Section V describes the method to obtain the dielectric constant.

II. ANALYTICAL STUDY OF THE DUAL-MODE OSCILLATOR
The analytical investigation will be based on the oscillator in Fig. 1.The passive network consists of two series LC resonators, implemented on a microstrip, in parallel with resistor R. The nonlinear active element is described with a current-controlled voltage source, v(i) = ai + bi 3 , in consistency with the series resonance [36].The coefficient a < 0 is the small-signal resistance, whereas the coefficient b > 0 enables the saturation of the oscillation amplitude [32].When part of the oscillator circuit, the positive cubic term becomes more pronounced the amplitude increases, causing the growth rate to slow down and eventually reach a saturation point.Each LC resonator is composed of a wide (capacitive) transmission-line section and a narrow (inductive) section.They have the respective length and width l C and W C and l L and W L , obtained through the approximate expressions [37] summarized in the Appendix.For differential operation, the MUT will be placed on top of the capacitive section of one of the resonators only (Resonator 1), so C 1 will depend on the dielectric constant ε M of the MUT, as shown in the Appendix [38], [39].For dual operation, it will be placed on top of the two capacitive sections, so both C 1 and C 2 will depend on ε M .

A. Passive Load Impedance
The passive load impedance Z L will be expressed in terms of the input impedances Z 1 and Z 2 of the two series resonators and resistor R where the superscripts r and i indicate the real and imaginary parts.For simplicity, we will initially neglect the losses of the distributed resonators, so Z k = j Z i k , where k = 1 and 2. The function Z r L (ω) is 0 when Z i 1 = 0 or Z i 2 = 0 and exhibits a maximum at the frequency ω c that fulfills Z i 1 + Z i 2 = 0, given by Z r L (ω c ) = R.In turn, the resonance frequencies satisfy The frequencies satisfying Z i 1 (ω 1 ) = Z i 2 (ω 2 ) = 0 agree with the individual resonance frequencies, ω 1 and ω 2 , of the two series LC resonators, which fulfill To preserve continuity, we must have ∂ Z i L (ω c )/∂ω < 0. To illustrate this, we have considered two identical resonators (with the values shown in the caption of Fig. 1) and placed an MUT with ε M = 3 on the top of the capacitive section of Resonator 1.In Fig. 2, we have traced the functions Z i L (ω), Z i 1 (ω), and Z i 2 (ω) versus ω.The slope of Z i L (ω) is initially positive, then negative and, finally, positive again.

B. Describing-Function Analysis
For |a| < R, the circuit in Fig. 1 exhibits a single dc solution, i = 0. Its stability is analyzed by calculating the roots of the characteristic equation [10], [40], obtained by linearizing the circuit about i = 0.The solution i = 0 is unstable since, for Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.all the ε M values, there is a pair of complex-conjugate poles on the right-hand side of the complex plane (RHS).However, the circuit exhibits two stable steady-state oscillations, instead of one.To show this, we will make use of the describing function [41] associated with v(i) = ai + bi 3 .We will approximate the current circulating through the device as i(t) = I cos(ωt).The describing function, of impedance type, is the ratio between the fundamental component of the voltage v(t) across the device and the current amplitude I , given by: Z N (I ) = a + 3/4bI 2 .Then, the steady-state oscillation conditions are where Z T is the total impedance function and the superscripts r and i indicate the real and imaginary parts.To solve (3), we first obtain the roots of (3b), which agree with the three roots of Z i L = 0, i.e., ω 1 , ω 2 , and ω c .Each of these frequencies is introduced in (3a).Since bI 2 > 0, to obtain a steady-state oscillation we must have a + Z r L (ω) < 0. Because Z r L (ω c ) = R, for |a| < R, this condition will only be fulfilled at ω 1 and ω 2 .When placing the MUT on the top of Resonator 1 and varying ε M , the two oscillation frequencies evolve as shown in Fig. 3 [curves in green, for an analysis at the fundamental frequency (NH = 1)].The frequency ω 2 associated with Resonator 2 (Osc2) remains constant whereas ω 1 (Osc1) undergoes significant variations.The amplitudes of Osc1 and Osc2 are identical and given by I = 2[−a/(3b)] 1/2 , as obtained from (3a) after replacing Z r T (ω 1 ) = 0 and Z r T (ω 2 ) = 0.For comparison, we will consider two different values of a.For a = −49 , we obtain I ∼ = 0.23 A and for a = −10 and I ∼ = 0.11 A. As shown in Section II-D, the impact of the harmonic content will be different in the two cases.

C. Stability of the Periodic Oscillations
The stability of the two steady-state oscillations has been analyzed with the conversion-matrix approach [42], [43].The results for a = −49 and the periodic oscillation at ω 1 are shown in Fig. 4, where the real parts of the dominant poles (denoted by λ r m , where m is a counter) are represented versus ε M .The poles corresponding to the oscillation at ω 2 (not represented) exhibit a similar variation.For all the ε M values, all the poles have a negative real part, except the  Real parts of the dominant poles of the oscillatory solution at frequency ω 1 versus ε M .They are denoted by λ r m , where m is a counter.
pair of complex-conjugate poles at the oscillation frequency (associated with the solution autonomy [16], [31], [35]), which should ideally have a real part equal to 0. The situation is the same for a = −10 .From the results of these analyses, the two periodic oscillations, at ω 1 and ω 2 , are stable for all ε M , which is the main advantage of the structure in Fig. 1.The reason is the absence of secondary Hopf bifurcations [10], [16], [31], [32], [35].These bifurcations (undesired for the intended sensing application) appear intrinsically in other dualfrequency oscillators [4], [5], [6], [7].At a secondary Hopf bifurcation, a pair of complex-conjugate poles of the periodic solution crosses the imaginary axis, which gives rise to the generation/extinction of a quasi-periodic solution, having two incommensurate fundamental frequencies.Note that at the bifurcation point, the amplitude at the generated/extinguished fundamental frequency tends to 0 [16], [44].In the following, we will demonstrate the absence of secondary Hopf bifurcations in the circuit of Fig. 1, using a simplified analysis.
Neglecting the intermodulation terms, we will express the quasi-periodic solution generated from the periodic solution at ω 1 as i(t) = Re{I 1 e jω 1 t + I 2 e jω 2 t }, where I 1 and I 2 are the amplitudes at ω 1 and ω 2 , respectively, and ω 2 is the second (undesired) fundamental frequency.Note that at the secondary Hopf bifurcation, we should have I 2 → 0. The phases can be set to 0 because the frequencies are incommensurate.Using the double input describing function [41], we obtain the steadystate system Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
= a + The subsystem composed of Z r T,1 = 0 and Z r T,2 = 0 only has the solution I 1 = I 2 , which contradicts the condition I 2 → 0 fulfilled at the secondary Hopf bifurcation.Thus, secondary Hopf bifurcations cannot exist.The demonstration is analogous for the oscillation at ω 2 .

D. Impact of the Harmonic Content
To illustrate the effect of the harmonic content, we will initially consider the case of NH = 3 harmonic terms.The independent current is represented as i(t) = I 1 cos ωt + I 3 cos(3ωt + φ), where ϕ is the phase shift between the two frequency components.The steady-state equations are where the integer subscript indicates the harmonic term.Due to the impact of the term −3/4bI 2 1 I 3 sin(φ) in Z i T,1 = 0, the oscillation frequencies, ω o1 and ω o2 , obtained from (5) will, in general, differ from the resonance frequencies ω 1 and ω 2 .Note that with NH = 1, we had ω o1 = ω 1 and ω o2 = ω 2 .In the case of (5), to fulfill Z i T,1 = 0 with I 3 sin φ ̸ = 0 we need Z i L ,1 (ω) ̸ = 0. Thus, Z i L ,1 (ω) must be calculated with the complete expression (1).Unless ω is relatively far from ω c , the imaginary part Z i L ,1 (ω) will depend on the element values of the two resonators, as illustrated in Fig. 2. Therefore, the MUT placed on Resonator 1 will affect the frequency of the two oscillation modes.The above analysis demonstrates that the nonlinearity of the active element gives rise to a coupling mechanism.
In the presence of harmonic content, the number of oscillations can be different from 2. To see this, we will replace the expression of Z i L ,1 (ω), taken from (1), in Z i T,1 = 0, which provides where n 3 (ω 2 ) and d 4 (ω 2 ) are the polynomials in ω 2 of third and fourth degrees, respectively.Equation ( 6) can be rewritten as follows: The above polynomial equation should be solved in combination with the entire system (5).The polynomial p(ω, ε M ) has the real coefficients.As a result, when ε M is varied, any new periodic solutions of (5) will appear in pairs, from degenerate points at which their amplitudes and frequencies will be identical.
The impact of the harmonic content has been analyzed considering NH = 3 and NH = 31 harmonics (Fig. 3).When a = −10 (blue dots), the results for NH = 3 and NH = 31 are almost overlapping with those obtained with the describing function.This is because the steady-state waveforms are nearly sinusoidal.On the other hand, for a = −49 , with both NH = 3 and NH = 31, the solution curves are qualitatively different from those obtained with the describing function [using (3)], which corresponds to NH = 1.For ε M < 3, we obtain only one oscillation, Osc1, at the frequency ω o1 .However, at ε M = 3, two other oscillations: Osc2 and Osc_c, at ω o2 and ω oc , respectively, are generated in a turning-point bifurcation, thus, with identical amplitudes and frequencies, in agreement with our previous demonstration.Using polezero identification [45], [46], we have analyzed the stability of the complete periodic curves for different NHs.In all cases, the oscillations at ω o1 and ω o2 are stable for all the ε M values.This is because they do not undergo any secondary Hopf bifurcations.We note that, for a = −49 , the frequency ω 2 of Osc2 (which should remain constant) varies with ε M in the lower ε M interval.As previously derived, this is due to the impact of the harmonic content, which will be smaller for a higher quality factor and a larger frequency distance between the resonances (larger ε M ).

III. TRANSISTOR-BASED OSCILLATOR
The transistor-based dual-frequency oscillator is shown in Fig. 5(a) and (b).Its active core (inside the dashed rectangle) contains the transistor BFP420, as well as its feedback and termination elements.The line capacitive section affected by the MUT [Fig.5(c)] is modeled with the effective dielectric constant ε eff (ε M , W/ h), as shown in (A.4).For the experimental characterization, we use the generator R&S SMT06 and connect the sensor output to an FSWP8 Phase noise analyzer with the spectrum analyzer function [Fig.5(d)].

A. Resonance Analysis
We have initially analyzed the stability of the dc solution with pole-zero identification [45], [46], and for all the ε M values, there is a single pair of complex-conjugate poles in the RHS.For ε M = 1, we have σ 2 ± jf 2 = 3.686e8 ± j2.109e9 Hz, and for ε M = 30, we have σ 2 ± jf 2 = 3.5464e8 ± j2.1672e9 Hz.Thus, this rigorous stability analysis predicts one oscillation only.To check for the possible existence of other steady-state oscillations, we will perform a computationally undemanding resonance analysis.It detects oscillations that are not due to the instability of the dc solution but are the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.result of an excess of negative resistance at the resonance frequencies (at which the reactance is equal to 0).Examples are the oscillations generated through a turning-point mechanism (like Osc2 and Osc_c in Fig. 3), which cannot be predicted through the stability analysis of the dc solution.Considering the series resonance of the two resonators, we will calculate the total impedance at the branch that connects the active core and the passive linear network.The simulation setup, which includes an AG [16] of the current type, is shown in Fig. 5(a).The ideal filter in parallel, described in Section III-B, is only needed for the nonlinear analyses.In small signal, the real and imaginary parts of this total impedance are where Z N (I ∼ = 0, ω) is the impedance of the active core under a small-signal current excitation (I ∼ = 0).Its imaginary part Z i N (I ∼ = 0, ω) will shift the original resonances, as observed in (8a).However, provided the two resonators have a highquality factor, the high slope versus ω will be (approximately) maintained.
To obtain the resonance diagram, we perform a double sweep in ω and ε M , and trace the two zero-value contours Z r T (ω, ε M ) = 0 and Z i T (ω, ε M ) = 0 in the plane defined by (ω, ε M ).The results when placing the MUT on the top of Resonator 1 are shown in Fig. 6.
The contour Z r T (ω, ε M ) = 0 delimits the region of negative resistance, shadowed in red.In turn, the contour Z i T (ω, ε M ) = 0 passes through all the resonance frequencies (both with Fig. 6. Resonance analysis by tracing the zero-value contours Z r T (ω, ε M ) = 0 and Z i T (ω, ε M ) = 0.
positive and negative slopes versus ω).In consistency with the analyses in Section II, we have two frequencies, ω 1 and where k = 1 and 2. Thus, the two stable periodic oscillations can be expected [36].Due to the frequency dependence of Z i N , the desired resonances at ω 1 and ω 2 do not start from the same value.The resonance at ω 1 , associated with the resonator that hosts the MUT is more sensitive to ε M .On the other hand, Z i N (I ∼ = 0, ω) affects the value of the resonance frequency ω 2 , unlike what happens in the ideal case [see (2)].However, the sensitivity of ω 2 decreases fast with ε M , due to the growing distance between ω 1 and ω 2 .
The above resonance diagram enables a simple check of the possible existence of steady-state oscillations that cannot be predicted with a small-signal stability analysis.However, this analysis is only approximate due to the amplitude dependence of the active core and the impact of the harmonic content.Thus, the steady-state periodic oscillations must be accurately calculated, which will be the object of Section III-B.

B. Steady-State Oscillations
The steady-state oscillations are calculated in HB with the setup shown in Fig. 5(a).We make use of the current-type AG, which has the amplitude I AG and operation frequency ω.The implementation includes an ideal filter in parallel that prevents the influence of the AG at the harmonics kω, where k ̸ = 1.The steady-state oscillation condition is fulfilled when the ratio between the AG voltage, V AG , and current I AG , agreeing with the total loop impedance Z T , is equal to 0 Note that the above function is calculated at the fundamental frequency, but makes use of HB, with as many harmonic terms as required, to obtain the voltage V AG .Here, we will consider NH = 5 harmonics.To obtain the solution curves versus ε M , we will perform a triple sweep in ε M , ω, I AG , and calculate, for each ε M , the intersections between the two zero-value contours Z r T (ε M , I AG , ω) = 0 and Z i T (ε M , I AG , ω) = 0 [47] in the plane (I AG , ω), making use of in-house software.As an example, Fig. 7(a) shows the contours obtained for ε M = 3.The steady-state solutions are given by the intersection points between the two contours.We obtain three intersections, which provide three distinct oscillations: Osc1, Osc2, and Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Osc_c, at the respective frequencies ω 1 , ω 2 , and ω c .When sweeping ε M , this procedure exhaustively provides all the solution curves, corresponding to different oscillation modes.
The solution curves, in terms of the amplitude I AG and frequency ω versus ε M , are shown in Fig. 7(b) and (c).The oscillation at ω o2 only exhibits small amplitude and frequency variations, whereas the other two, at ω o1 and ω oc , exhibit significant changes.The periodic oscillations at ω o1 and ω oc are generated through a turning point, in consistency with the demonstration in Section II.To validate the results of the contour-intersection method, we have carried out two independent simulations: a default oscillator analysis in commercial HB and an AG optimization [16] to fulfill Z T = 0.The results are superimposed with circles in Fig. 7.The default oscillator analysis only provides Osc2.On the other hand, the AG optimization fails to converge from ε M = 5.In comparison, the analysis through contour intersections exhaustively provides all the oscillation modes with no need for any initialization procedures.
The stability of the three periodic solutions has been analyzed versus ε M using pole-zero identification [44], [45].We have found that Osc1 and Osc2 are always stable, whereas Osc_c is always unstable.Fig. 8(a)-(c) shows the pole-zero locus of each of the three solutions when ε M = 3.Note that, as expected, the three periodic solutions have a pair of complex-conjugate poles at the oscillation frequency in the imaginary axis, due to the solution autonomy [16], [31], [35].All the other poles of Osc1 and Osc2 are on the left-hand side of the complex plane (LHS).Thus, these oscillations are stable.In contrast, Osc_c has a pair of complex-conjugate poles about the frequency of Osc1 on the RHS, so it is unstable.The stable behavior of Osc1 and Osc2 has been further verified with a transient analysis.When properly implemented, this analysis only converges to stable solutions.This is because it follows the complete trajectory [32] of the variables from their initial value to the stable steady state.Depending on the initial conditions we will converge to either Osc1 or Osc2, as shown in Fig. 8(d).The stability predictions are also confirmed by the experimental measurements, superimposed in Fig. 7(c).

IV. ANALYSIS IN INJECTION-LOCKED CONDITIONS
The analysis in the presence of an injection source will be based on the recently proposed contour methods [34], here applied for the first time to injection-locked dualmode oscillators.Note that well-known expressions for the injection-locking band in terms of the oscillator quality factor Q [48], [49] and the input amplitude are only approximate.They are based on a linearization of the injection-locked oscillator about its free-running solution, so they are only valid under a small input amplitude.In addition, they neglect the variation of the imaginary part of the oscillator immittance Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.with the oscillation amplitude.As shown in Section II, this variation takes place even when using simple models, due to the impact of the harmonic content.Our objective is not to provide closed-form expressions for the locking bands, but a global prediction of the nonlinear behavior, in the vein of works such as [10], [15], and [44].Note that the commercial HB does not provide (by default) the injectionlocked solutions.The method described below overcomes this limitation and enables the use of commercial HB to obtain the solution curves of injection-locked oscillators through the calculation of specific functions.

A. Locking Bands of the Two-Mode Oscillator
To obtain the injection-locked solutions, we will simultaneously simulate the two circuits shown in the setup of Fig. 9.One is the oscillator in the absence of the input source.It includes the AG of current type [Fig.9(a)], which has one of its terminals connected to the observation Node A. Its aim is to calculate the same impedance function Z T (ε M , I AG , ω), already used in Section III to obtain the steady-state freerunning oscillation.This function will be combined with Thevenin's equivalent voltage of the input network [34], seen between node A and ground [Fig.5(a)].In fact, the goal will be to obtain a (frequency dependent) numerical linear function relating this Thevenin's voltage to the input source.This will be achieved with the aid of the second circuit [Fig.9(b)].We introduce an independent input-current source I g e j0 (of dummy value I g = 1 A) and obtain the output voltage under open-circuit conditions, given by the linear function F Th (ε M , ω).Then, Thevenin's equivalent voltage is related to the input source as follows: Applying Kirchoff's laws at the analysis node (Node A) and using Thevenin's equivalent, we obtain Combining ( 10) and ( 11), the injection-locked oscillator is formulated as follows: where H is a complex function with current dimension and ϕ is the phase shift between the input source and the current at the observation branch (with the amplitude I AG ).From the inspection of ( 12), for any pair of values of I g and ε M , the periodic solution curve(s) versus ω will be given by the constant-value contour of |H (ε M , I AG , ω)| = I g .A significant advantage is the straightforward calculation of the whole family of periodic-solution curves, obtained when varying the input amplitude I g , as constant-value contours of the function |H (ε M , I AG , ω)|, with no need to perform any new HB simulations.As an example, Fig. 10(a) shows the curve families obtained when placing the MUT on the top of Resonator 1 with ε M = 8 and ε M = 11, and increasing I g from 1 to 10 mA.We would like to emphasize that the periodic solution curves agree with the constant amplitude contours of the function H .As the input amplitude I g increases from 0, the system evolution (about each oscillation frequency) is as follows.For I g = 0, we obtain a discrete point with high amplitude (corresponding to the free-running oscillation) and a zero-amplitude line (corresponding to the trivial dc solution).For a small input amplitude I g , the point evolves into an ellipsoidal curve, composed of the locked-solution points, and the zero-amplitude line evolves into an open curve that provides the circuit response when the self-oscillation is not excited.For a further increase of I g , the closed curve, corresponding to the locked operation, is no longer ellipsoidal, and the open curve exhibits a clear maximum.At a certain amplitude I g , the two curves merge in a single open one that exhibits several turning points.This can be seen on the left-hand side of Fig. 10(a), corresponding to solutions about ω 1 .From that I g value, there is only one curve, and the interval of stable locked solutions is delimited by turning points and secondary Hopf bifurcations [35].
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.The results obtained with ( 12) have been validated (circles) through a costly AG optimization and default HB, which provides solutions for which the self-oscillation is not excited.Note that the circuit evolution provided by ( 12) cannot be predicted with explicit models depending on Q and the input amplitude.Instead, ( 12) is based on the complete nonlinear impedance function, calculated with the pure HB system acting as an inner tier.Thus, its accuracy is only limited by the reliability of the active and passive element models, as in any (successful) HB simulation.
As has been verified, the upper section of each closed curve (delimited by the two turning points) is stable, whereas the coexisting section of the lower amplitude curve is unstable.After the merging, the stable locked sections are delimited by both turning points and secondary Hopf bifurcations [10], [15], [44].
When considering I g = 5 mA and varying ε M , we obtain the family of injection-locked solution curves (versus ω), as shown in Fig. 10(b).For each ε M , there are two closed curves, corresponding to the locked oscillations Osc1 and Osc2, plus a low-amplitude open curve.Note that the three disconnected curves are simultaneously obtained from the same contour |H (ε M , I AG , ω)| = I g , which is another advantage of the method (12).The variation of the locking band of Osc2 becomes negligible for the larger ε M values.The analysis has been validated through AG-based HB and default HB, superimposed with circles.The results in Fig. 10 demonstrate the great potential of ( 10) and ( 12) when applied to injectionlocked dual-frequency oscillators.
Fig. 11 shows the comparison between measurements and simulations when placing four substrates, FoamClad, CuClad, Rogers 4003C, and FR4 on the top of the capacitive section of Resonator 1.The experimental locking band is delimited by the abrupt fall in the trace obtained with the Max Hold mode of the spectrum analyzer.Embedded in the figure is an expanded view of the comparison for CuClad and Rogers 4003C, with very similar dielectric constants.Note that in simulation we obtain the amplitude of the gate current, instead of the output power.It would be possible to obtain the output power through interpolation.However, it is not necessary since the detection of the dielectric constant will be based on the central frequency and bandwidth.The results are very repetitive, since numerous measurements have been carried out, spaced by days, weeks, and even months, obtaining errors in the order of tens of kilohertz.
The operation as a dual frequency sensor, by placing the same MUT on the top of the two resonators, has also been analyzed.This provides a significant variation of the two locking bands with ε M , as shown in Fig. 12(a).Measurements and simulations for FR4 are compared in Fig. 12(b) (lower band) and Fig. 12(c) (upper band).The lower band is stable between the turning point T and the secondary Hopf bifurcation, denoted by H Lo .As stated, this bifurcation is typical in injection-locked oscillators and unrelated with the dual operation.In the experiment, the shift of the lower locking band is smaller than in simulation, which can only be due to modeling inaccuracies.To show this, we have performed an additional validation through envelope transient [50], which is a fully independent (and more demanding) analysis method.In the envelope-transient method, the circuit variables are represented in a Fourier series with time-varying harmonic terms [50].When the solution is locked, these harmonic terms take a constant value (after an initial transient).When the solution is unlocked, they oscillates at the beat frequency [51].Fig. 13(a) shows the fundamental-frequency amplitude of the gate current I gate (t) for two different input frequencies.Note that in periodic conditions I gate (t) agrees with I AG .For f = 1.3 GHz, the circuit evolves, after an initial transient, to a locked solution (time-constant amplitude).For f = 1.37 GHz, the circuit evolves to a quasi-periodic (unlocked) steady state, and the amplitude oscillates at the beat frequency.Fig. 13(b) shows the variation of the steady-state amplitude I gate (t) versus the input frequency ω.When I gate (t) is time varying (unlocked solution), we obtain a projection of the time-varying amplitude.
The results are overlapped with those obtained with (12) in the stable sections: between T and H Lo for the lower band and between the two turning points, T 1 and T 2 , for the upper band.Outside these stable intervals, the solution is quasi-periodic.Thus, the fundamental-frequency amplitude is time varying.We conclude that the oscillator can behave as a differential sensor when the expected dielectric constants are relatively high, due to the larger difference between the resonance frequencies.Otherwise, it will operate as a dual sensor.

B. Global Stability Analysis
Assuming that the system is initially locked at Osc1, as the input frequency ω increases it should become unlocked and then locked at Osc2.Thus, a change in ω must be able to switch the system from the original oscillations, Osc1-Osc2.Otherwise, we will obtain a self-oscillation mixing (SOM) solution with two fundamental frequencies: the input frequency ω and the original oscillation frequency ω o1 (slightly affected by the input source).This quasi-periodic solution would coexist with a second stable periodic solution in which the oscillation Osc2 is locked to the input signal (at ω ∼ = ω o2 ).The symbol ∼ = indicates that the input frequency is close to the free-running frequency ω o2 of Osc2.As will be shown, there is a minimum I g enabling the switching from Osc1 to Osc2.To demonstrate this, we will perform a bifurcation analysis setting the input frequency at ω ∼ = ω o2 and increasing I g .We will calculate both the quasi-periodic solution at ω ∼ = ω o2 and ω o1 , and the injection-locked solution at ω ∼ = ω o2 .The quasi-periodic solution will be obtained through an extension of the free-running analysis in Section III-B at this regime, presented here for the first time.We calculate the nonlinear impedance function Z T with the current-type AG (Fig. 9) at the frequency ω o1 , associated with Osc1, in the presence of the input source I g at ω, which requires a two-tone analysis.The quasi-periodic solution must fulfill where H is the inner tier HB system, explicitly shown in (13) to emphasize the two-tone simulation at ω o1 and ω.Note that ω o1 , affected by I g , is an unknown of the problem.
For each I g , we perform a double sweep in ω o1 , I AG and obtain the intersections between the two zero-value contours Z r T (I AG , ω o1 ) = 0 and Z i T (I AG , ω o1 ) = 0, which will provide all the coexisting steady-state quasi-periodic solutions.The results versus I g are shown in Fig. 14(a), where the input frequency is f = 1.937GHz, and the oscillation amplitude (at ω o1 ) is traced in red.Solid and dashed lines indicate the stable and unstable sections, respectively.As expected, the upper section of the curve at ω o1 starts (at I g = 0) from the amplitude of Osc1 in free-running conditions.This oscillation is stable and so it is the quasi-periodic solution for low I g .However, as I g increases, the amplitude at ω o1 starts to vary [Fig.14(a)] and, at T 1 , the curve exhibits a turning-point bifurcation, from which it becomes unstable.The self-oscillation at ω o1 (and, thus, the quasi-periodic regime) is extinguished at the inverse secondary Hopf bifurcation H Lo,2 .The exhaustive quasi-periodic analysis (13) has also detected an unstable quasi-periodic solution having ω and ω oc as fundamental frequencies, traced in gray in Fig. 14(a), which is extinguished at H Lo,1 .
We will now focus on the periodic solution curve at ω ∼ = ω o2 , also as shown in Fig. 14(a).This curve exhibits a nearly degenerate turning T L (at I g ∼ = which corresponds to the transition from a regime at ω and ω o2 (different from the one previously considered at ω and ω o1 ) to the injection-locked regime at ω ∼ = o2 .In Fig. 14(a), T L occurs at I g ∼ = 0, since is very close to the original free-running frequency ω o2 .The periodic curve has been validated with default HB, which provides solutions in the lower section (circles) and then exhibits a discontinuous jump to the upper section.
To understand the system evolution after T 1 , we have carried out a stability analysis along the locked periodic curve at f = 1.937GHz.Because the dc solution was unstable with a pair of complex-conjugate poles (σ 2 ± jω 2 ) on the RHS (as indicated in Section III), the low amplitude section of this periodic curve is initially unstable, with the same pair of poles (σ 2 ± jω 2 ) on the RHS.However, due to the dual nature of the oscillator, there is another pair of critical poles, σ 1 ± jω 1 , with a frequency close to that of the other oscillation: f 1 = 1.71GHz.Fig. 14(b) shows the results of a pole-zero identification about f 1 .Note that due to the difference between the oscillation frequencies, the identification must be carried out separately in each of the two frequency bands.In Fig. 14(b), we have traced the real part of the poles, σ 1 , versus the gate-current amplitude I AG .We should emphasize that there is also a variation of the other critical poles, initially given by σ 2 ± jω 2 , which will be described later.When increasing I AG , the pair of poles σ 1 ± jω 1 crosses to the RHS at H Lo,1 , due to an increase in the negative resistance.This secondary Hopf bifurcation (from a locked regime) is consistent with the extinction of the quasi-periodic solution at ω and ω c , occurring at H Lo,1 [Fig.14(a)].Thus, there is a complete agreement between the two independent analyses.
In the interval comprised between H Lo,1 and T 2P , the locked solution curve has four poles on the RHS: σ 1 ± jω 1 and two other poles, resulting from an evolution of σ 2 ± jω 2 .As I AG increases, ω 2 varies and at certain value, it becomes equal to the input frequency ω 2 = ω.When this happens, the pair complex-conjugate poles transform into two real poles γ 1 and γ 2 [31], [32], initially on the RHS.When following the locked periodic curve from I g = 25.5 mA (corresponding to H Lo,1 ) to I g = 35 mA, passing through the two turning points T 2P and T L , the four critical poles cross to the LHS in the following other: first, γ 1 , then, σ 1 ± jω 1 and finally, γ 2 .The real pole γ 1 crosses to the LHS at the turning point T 2P .When further following the periodic curve, the coexisting pair of complex-conjugate poles σ 1 ± jω 1 crosses to the LHS at the inverse Hopf bifurcation H Lo,2 .At this relevant point, the quasi-periodic solution at ω o1 and ω is extinguished.This can be seen from Fig. 14(a), where the amplitude at ω o1 (of the quasi-periodic solution) tends to 0 at H Lo,2 .Finally, at the turning point T L of the periodic curve, the remaining real pole γ 2 crosses to the LHS, so the upper section of this curve (from T L1 onward) is stable.To summarize, when the quasi-periodic solution reaches the point T 1 , there is a jump to the stable locked solution.When the circuit initially oscillates at ω o2 , we obtain the same qualitative behavior, with T 1 occurring at I g = 5 mA.Thus, it will be convenient to reduce the input frequency (from the upper frequency oscillation to the lower frequency oscillation), since this requires a lower input power.
The above analyses fully explain the dynamics of the injection-locked dual oscillator for the first time to our knowledge.When originally oscillating at ω o1 (ω o2 ) in the presence of an injection signal at ω ∼ = ω o2 (ω ∼ = ω o1 ), the jump to the periodic solution at ω ∼ = ω o2 (ω ∼ = ω o1 ) requires an input power higher than that at the turning point T 1 of the quasi-periodic solution at ω o1 , ω (ω o2 , ω).This turning point occurs because the active core can no longer exhibit negative resistance at ω o1 (ω o2 ) in the presence of a high input amplitude at ω ∼ = ω o2 (ω ∼ = ω o1 ).In the cases analyzed here, when the jump takes place, the poles of the coexisting locked oscillation at ω ∼ = ω o2 (ω ∼ = ω o1 ) are already on the LHS thanks to the inverse Hopf bifurcation H Lo,2 , crucial for the system operation.Thus, under a sufficiently high input power, the system should be able to switch from one oscillation to the other when varying the input frequency.Note that the jump may take place at a different point of the static locked-solution curve, which will not prevent the proper system operation (based on a calibration procedure).For validation, we have measured the circuit when decreasing the input frequency from 2.1 to 1.5 GHz, under the input power P in = −6 dBm.We have obtained the results as shown in Fig. 15, where the static simulations using (12) are also included as a reference.When decreasing the input frequency, we observe the following: 1) the circuit is initially unlocked; 2) the upper oscillation gets locked to the input signal (highpower section of the trace); 3) the circuit becomes unlocked; 4) the lower frequency oscillation gets locked (second highpower section); and 5) the circuit becomes unlocked.

V. DETECTION OF DIELECTRIC CONSTANTS
In Section IV, we have investigated the variation of the injection-locking bands with the dielectric constant of solid substrates placed on the top of the capacitive section of the resonator(s).However, to demonstrate the capability to sense both the dielectric constant ε M and the loss tangent tanδ, it is more convenient to use liquid mixtures, for which both quantities are known and can be used as a reference.Thus, we will consider mixtures of ethanol and water.The liquids are introduced in a deposit made of a UV-sensitive resin manufactured with a PRUSA SL1 3-D printer, located on the top of the capacitive section(s).The thickness of the deposit base (0.2 mm) will necessarily reduce the sensitivity in comparison with the simulated results.The accurate modeling of the complete structure, including the deposit, is beyond the scope of this work, which focuses on the investigation of the nonlinear dynamics.Instead, we will experimentally characterize the passive structure and introduce the results in our nonlinear-analysis setups.Because the aim of the locked operation is to reduce the phase-noise spectral density, we have measured the phase noise at the two frequencies in both free-running and injection-locked conditions.The spectra are shown in Fig. 16.The comparison is carried out with an empty deposit and with a deposit full of water.In the two cases, the injection locking gives rise to a reduction of the spectral density, as theoretically demonstrated in This reduction is a significant advantage of the injection-locked regime.

A. Locking Bands With Liquid Samples
Using the E8364A Vector Network Analyzer, we have initially characterized the passive structure when placing the MUT on the top of one capacitive section only.We have obtained the scattering parameter S 11 for different percentages p of ethanol in water, varied from 10% to 100% in tens.In these conditions, the maximum distance obtained between the two resonance frequencies is 30 MHz.To simulate the locking bands, we have introduced the characterized S 11 parameter in the setup in Fig. 9.To make the system switch from one oscillation to the other when varying ω, we have set the input power to P in = −6 dBm, corresponding to I g = 6.3 mA.Fig. 17(a) and (b) shows, respectively, the lower and upper locking bands obtained for p = 0% and 100 %.Simulations and measurements are compared with good agreement.Because an experimental model has been used for the passive structure and manufacturer models have been used for the lumped components, discrepancies are attributed to additional parasitics and tolerances in the active device.Fig. 17 When two deposits are placed on the top of the two capacitive sections, there are significant changes in the two locking bands.However, the system is bulkier and, thus, less practical.On the other hand, when only one deposit is used (Fig. 17), the nonnegligible variations of the upper band should enable an application as a dual-band sensor, as will be shown next.

B. Determination of the Dielectric Constant and Loss Tangent
As previously stated, multiples of 10 of the concentration p are used for calibration.For each p, we obtain the frequencies that delimit the two locking bands, and calculate the corresponding central frequencies ( f c1 , f c2 ) and widths (BW 1 , BW 2 ).Then, for each p, we obtain the functions f and BW , defined as follows: where 0), and i = 1 and 2. Note that for each percentage p, we know the values of ε M ( p) and tanδ( p) from previous works [51], where they were calculated using the Debye function.Combining these values with the results of ( 14), we will build our sensing function by using piecewise linear interpolation between consecutive pairs of calibrated points Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.where the constant coefficients m i j evolve through the different pairs of calibrated points.Applying the above procedure, we obtain the results in Fig. 18, which provides ε M and tanδ for the pair of values f and BW, corresponding to each p, indicated in the figure.The calibration points are represented with dots.The values obtained with (15), lying between the ones used for the calibration, are represented with squares.These estimated values are successfully compared with those provided in [52], as shown in Table I.

VI. CONCLUSION
An in-depth investigation of the nonlinear dynamics of injection-locked dual-mode oscillators, envisaged for a sensing application, has been presented.It started from an analytical study of a cubic-nonlinearity oscillator, loaded with two distributed LC resonators, one or two affected by the MUT.We analyzed the oscillation modes and their stability properties in free-running regime, and obtained a structural coexistence of two stable periodic modes.Then, the more complex case of a transistor-based oscillator was addressed.We first applied a small-signal resonance analysis, able to detect steady-state oscillations that are generated at turning points of the periodic solution curves.Then, the steady-state free-running and injection-locked oscillations were rigorously calculated through contour analysis methods, which exhaustively provided all the coexisting solutions.They are based on the extraction of a nonlinear impedance function from commercial HB with the aid of an AG.In free-running conditions, the solutions are calculated from the intersections of the zero-value contours of the real and imaginary parts of the impedance function.In injection-locked conditions, the solution curves agree with the contour levels of a nonlinear current and calculate with the same impedance function.Through a global stability analysis, we identified the phenomenon that gives rise to the switching from one oscillation to the other.We where Z o (1, W ) is the characteristic impedance for ε M = 1, W is the linewidth, and the function F is given by where h is the substrate height.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

B. Inductor
For small βl, such that tg(βl) ∼ = βl, and Z L ≪ Z o , (A.1) can be approximated by Z (l) = j Z o βl = j Z o l v g ω.
(A.6) Thus, the line section of length l is equivalent to an inductor of value To ensure condition Z L ≪ Z o , a line with high characteristic impedance is chosen, which implies a narrow width W .The effective dielectric constant and the characteristic impedance are obtained from (A.4) by particularizing ε M = 1.

Fig. 3 .
Fig. 3. Oscillation frequencies versus ε M when the MUT is placed on the top of the capacitive section of Resonator 1.For comparison, we consider a = −49 and a = −10 .The steady-state solutions are calculated with NH = 1, NH = 3, and NH = 31 harmonic components.

Fig. 4 .
Fig. 4.Real parts of the dominant poles of the oscillatory solution at frequency ω 1 versus ε M .They are denoted by λ r m , where m is a counter.

Fig. 5 .
Fig. 5. Sensor based on a dual-mode oscillator.(a) Analysis setup.The schematic shows the two-step-impedance resonators and the active core.(b) Photograph of the microstrip implementation.(c) Photograph with the deposit for the MUT, used in Section V. (d) Experimental characterization.We use the generator R&S SMT06 and connect the sensor output to an FSWP8 Phase noise analyzer with the spectrum analyzer function.

Fig. 7 .
Fig. 7. Periodic oscillations versus ε M .They have been calculated from the intersections of the zero-value contours Z r T (ε M , I AG , ω) = 0 and Z i T (ε M , I AG , ω) = 0. (a) Contours obtained for ε M = 3.The steady-state solutions are given by the intersection points.(b) Oscillation amplitude I AG .Independent HB simulations are superimposed with circles.(c) Oscillation frequency ω.Experimental measurements are superimposed.

Fig. 8 .
Fig. 8. Stability analysis of the coexistent periodic oscillations for ε M = 3.(a) Pole-zero locus for Osc1.The solution is stable.(b) Pole-zero locus for Osc_c.The solution is unstable.(c) Pole-zero locus for Osc2.The solution is stable.(d) Analysis through the time-domain integration.Depending on the initial conditions we obtain convergence to either Osc1 or Osc2.

Fig. 9 .
Fig. 9. Analysis setup to obtain the injection-locked solutions.It is based on the simultaneous simulation of two circuits.(a) First circuit (without the input source) provides the nonlinear impedance function Z T (ε M , I AG , ω).(b) Second circuit (with I g = 1 A) is used to calculate Thevenin's equivalent voltage of the input network.

Fig. 10 .
Fig. 10.Injection-locked behavior when placing the MUT on the top of Resonator 1.(a) Family of solution curves versus the input amplitude I g , obtained as the constant-value contours of the function |H (ε M , I AG , ω)|.The dielectric constants are ε M = 8 and ε M = 11.The results are validated with AG-based and default HB (circles).(b) Variation of the two locking bands (corresponding to Osc1 and Osc2) with ε M .

Fig. 11 .
Fig. 11.Comparison between measurements and simulations for different substrates: FoamClad, CuClad, Rogers 4003C, and FR4.An expanded view of the cases of CuClad and Rogers 4003C is embedded in this figure.

Fig. 12 .
Fig. 12. Oscillator operation as a dual-band sensor when the MUT is placed on the top of the two resonators.(a) Variation of the two locking bands (corresponding to Osc1 and Osc2) versus ε M .(b) Experimental validation for ε M = 4.3.Lower frequency band.(c) Upper frequency band.

Fig. 13 .
Fig. 13.Independent simulation through envelope transient.(a) Fundamental-frequency amplitude of the gate current I gate (t) for the input frequencies f = 1.3 GHz (locked solution) and f = 1.37 GHz (unlocked solution).(b) Variation of the steady-state amplitude I gate (t) versus the input frequency ω.The results are overlapped with those obtained with (12) in the stable sections.

Fig. 14 .
Fig. 14.Bifurcation analysis of the dual-frequency oscillator versus |I g |, under a dielectric constant ε M = 3.(a) Quasi-periodic and periodic solution curves.We inject at the frequency ω ∼ = ω o2 when the circuit originally oscillates at ω o1 .(b) Variation of the real part of the poles σ 1 ± jω 1 of the locked periodic solution at ω ∼ = ω o2 versus the gate-current amplitude (I AG ).

Fig. 15 .
Fig. 15.Experimental behavior when decreasing the input frequency from 2.1 to 1.5 GHz, under the input power P in = −6 dBm.

Fig. 16 .
Fig. 16.Phase noise at the two frequencies in both free-running and injection-locked conditions.(a) Lower oscillation frequency.(b) Higher oscillation frequency.
(c) shows the experimental variation of the lower locking band as p increases.The shift is consistent with the experimental variation (30 MHz) of the resonances of the passive structure.

Fig. 17 .
Fig. 17.Differential operation.Comparison between measurements and simulations.Experimental locking bands for different percentages p of ethanol in water.To calibrate the sensor, the ethanol percentage p has been varied from 10% to 100% in tens.(a) Lower band.With significant variations.(b) Upper band.(c) Experimental variation of the lower locking band as p increases.

Fig. 18 .
Fig. 18.Experimental results.The (x, y, z) coordinates of each graph correspond to (a) ( f, BW, and tanδ) and (b) ( f, BW, and ε M ).Detected and estimated variations of ε M and tanδ.The sensing function (15) is calibrated by varying the percentage p of ethanol from 10 to 100 in tens.For each p, the parameters f and BW in(14) are obtained from measurements.The corresponding ε M and tanδ values (dots) are obtained from published values.For p values lying between the ones used for calibration, ε M and tanδ are estimated (squares) using(15).
presented a piecewise interpolation to obtain the dielectric constant and loss tangent from the central frequencies and bandwidths of the two-locking bands.APPENDIXTo obtain the equivalent inductor L and capacitor C values of the distributed resonators[37], we will make use of the expression of the input impedance of a transmission line of characteristic impedance Z o and length l, terminated with the load impedanceZ L Z (l) = Z o Z L + j Z o tg(βl) Z o + j Z L tg(βl) (A.1)where β = ω/v g , v g is the wave velocity.A. CapacitorFor small βl, such that tg(βl) ∼ = βl, and Z L ≫ Z o , (A.1) can be approximated by Z (l) line section is equivalent to a capacitor of valueC = l Z o v g = l √ ε eff cZ o (A.3)where c is the speed of light and ε eff is the effective dielectric constant in the transmission line.To ensure Z L ≫ Z o , we will use a line with a low characteristic impedance (which implies a relatively large width W ), terminated in an open circuit.When the MUT is placed over the transmission line, ε eff and Z o depend on the MUT dielectric constant ε M

TABLE I COMPARISON
BETWEEN ESTIMATED AND MEASURED VALUES