II. DETECTION OF ODD-MODE INSTABILITIES

Let a circuit exhibiting symmetries, such as the one in Fig. 1, be considered. For the analysis of mismatch effects, and assuming sufficient low-pass filtering effects of the output network, the circuit will be terminated in Γ _o at f _in and in 50 Ω at mf _in , where |m| > 1. For instance, in Fig. 1, a 180° shorted stub at 2f _in has been introduced in parallel with the final 50 Ω load (the load that will undergo changes under the mismatch effects), which should help reduce the impact of mismatch at frequencies with |m| > 1. For a given input power P _in and termination Γ _o , the circuit exhibits the steady-state solution $\bar X_o $ , which is the vector of harmonics of the various state variables, obtained with harmonic balance (HB).

Fig. 1. Test-bench power amplifier based on a CLY5 transistor (RO4003C: ε _r = 3.38, h = 0.5 mm). (a) Circuit schematic. The AGs, at the respective phases ϕ _AG1 and ϕ _AG2, are used for the analysis of the modes. The stabilization resistor R _s , connected between the two amplifier branches, is also shown. The small-signal current source is introduced to evaluate the limit-oscillation conditions at an incommensurable frequency f or a subharmonic frequency f _in /2. (b) Photograph.

For a simple analysis of the impact of the circuit topology on the possible oscillation modes, an exemplary outer tier conversion matrix will be calculated. An admittance matrix is obtained by sequentially connecting an AG of voltage type [Reference Quéré, Ngoya, Camiade, Suarez, Hessane and Obregon15, Reference Suárez and Quéré16] at the frequency f, in parallel at the two analysis nodes [1 and 2 in Fig. 1(a)]. The AG should operate in small signal and include a bandpass filter at f _AG  = f. To obtain the first (second) matrix row, the AG is connected to node 1 (2), calculating the ratio between the currents, I ₁ and I ₂, entering the active network from node 1 and node 2, at the particular sideband kf _in  + f frequency, and the AG voltage. In the case of the circuit in Fig. 1, the resulting matrix has the form:

(1) $$\left[ {\matrix{ {Y_{a,11} (f,kf_{in} + f)} & {Y_{a,12} (f,kf_{in} + f)} \cr {Y_{a,12} (f,kf_{in} + f)} & {Y_{a,11} (f,kf_{in} + f)} \cr}} \right],$$

where the subindex “a” stands for “active network”. Due to the circuit symmetry, the form of the outer tier conversion matrix will be the same, no matter the choice of the outer tier sidebands. This matrix topology exhibits the two eigenvectors [1 1] ^T and [1 –1] ^T , as easily derived from (1). On the other hand, the form of the admittance matrix describing the passive network (Fig. 1) is identical to (1) at all frequencies mf _in  + f, and therefore exhibits the same two eigenvectors [1 1] ^T and [1 –1] ^T . The eigenvector [1 1] ^T corresponds to an even mode, with the sidebands mf _in  + f in phase in the two amplifier branches. The eigenvector [1 –1] ^T corresponds to an odd mode, having sidebands with 180° phase shift in the two branches.

The two modes can be excited by simultaneously connecting the two voltage AGs in parallel at the two analysis nodes, at the frequency f, with identical amplitudes V _AG [see Fig. 1(a)]. The mode [1 1] ^T is analyzed by setting equal phase values at the two AGs, ϕ _AG1 = ϕ _AG2 = 0°. Under this simultaneous excitation, the total admittance at f, at each of the two analysis nodes is given by the ratio between each AG current and voltage: Y _AG1,e  = I _AG1(f)/V _AG , Y _AG2,e  = I _AG2(f)/V _AG . Because the excitation agrees with one of the matrix eigenvectors, the two admittances are identical, as demonstrated in [Reference Freitag11]. This admittance will be renamed Y _Te  = Y _AG1,e  = Y _AG2,e . In turn, the mode [1 –1] ^T is analyzed by using two small-signal AGs with equal amplitude and 180° phase shift, ϕ _AG1 = 0°, ϕ _AG2 = 180°. Under this simultaneous excitation, the total admittances Y _AG1,o = I _AG1(f)/V _AG and Y _AG2,e  = I _AG2(f)/V _AG e^{j π} are also equal, and will be denoted as Y _To . This use of multiple AGs to analyze the modes will be most useful in the general case of multi-device PAs, as shown in Section IV.

In the case of odd-mode instability, the two transistor branches will exhibit 180° phase shift at mf _in  + f, so these sidebands will inherently cancel out at the PA output reference plane. This situation in which the stability properties depend on the termination at mf _in but are independent on the terminations at the sideband frequencies at mf _in  + f can be interpreted as a failure of the proviso established in [Reference Suárez, Ramírez and Sancho4] (an extension of Rollet's proviso [Reference Rollett6] to the sideband-impedance problem). This is because the odd-mode instability will be observed even if the sideband frequencies are terminated in open or short circuits at the PA output [Reference Suárez, Ramírez and Sancho4]. The antenna mismatch will only affect the stability properties through the termination impedances at mf _in , which will give rise to a change in the steady-state solution $\bar X_o $ about which the circuit is linearized, and, therefore, to a change in Y _To . However, under the assumption of sufficient low-pass filtering effects, the analysis can be limited to the termination at f _in , expressed as Γ _o . The test bench is illustrated in Fig. 1, where harmonics |m| > 1 are arbitrarily terminated in 50 Ω due to their limited influence.

The prediction of the odd-mode instability will be based on two fundamental properties: (i) when a circuit exhibits a steady-state oscillation, all its nodes exhibit a total admittance function (current-to-voltage ratio) equal to zero, (ii) at the instability boundary, the steady-state oscillation amplitude tends to zero [Reference Suárez18]. When linearizing the circuit with respect to the perturbation at f (as done in the conversion-matrix approach), we implicitly assume amplitude tending to zero at this frequency. Thus, the limit oscillation condition (instability boundary) is defined by the condition: Y _To  = 0. This odd-mode limit oscillation condition can also be evaluated through the connection of a small-signal current source between equivalent device nodes of the two subcircuits (Fig. 1), doing Y _To  = I _test /(V ₁ − V ₂). Note that Y _To has been redefined here.

Pole-zero identification [Reference Ayllon, Collantes, Anakabe, Lizarraga, Soubercaze-Pun and Forestier8–Reference Anakabe, Collantes and Portilla10] would be applicable to detect the odd-mode instabilities under mismatch effects. However, the odd-mode excitation must be preserved as otherwise the impedance at sideband frequencies would be ill represented with a constant Γ _o . Such an analysis would require a sweep in the perturbation frequency f (going from 0 to f _in , in periodic regimes) for each steady-state solution, obtained through a double sweep in the amplitude and phase of Γ _o . Pole-zero identification should be applied to all the transfer functions resulting from this double sweep. The identification interval 0 to f _in should be, in general, divided into smaller intervals, so this analysis will be computationally demanding. Instead, the aim here will be to obtain directly the boundary between stable and unstable values of Γ _o , given by the condition Y _To  = 0, which should be traced in the Smith chart corresponding to Γ _o . Actually, the fulfillment of Y _To  = 0 will depend on the input power P _in , the termination Γ _o , and the oscillation characteristics, since there are two major instability mechanisms from a periodic regime at f _in [Reference Quéré, Ngoya, Camiade, Suarez, Hessane and Obregon15–Reference Rizzoli and Neri20]. One is the onset of an incommensurable oscillation at the frequency f, or Hopf bifurcation, and the other is a division by 2 of the input frequency, leading to a subharmonic regime at f _in /2, which corresponds to a flip bifurcation.

For illustration, the analysis will be applied to the PA in Fig. 1. Introducing a stabilization resistor, R _s  = 170 Ω, in parallel between the two amplifier branches, the circuit is stable for all the P _in values under matched conditions (when terminated in a 50 Ω load), as has been verified with pole-zero identification [Reference Ayllon, Collantes, Anakabe, Lizarraga, Soubercaze-Pun and Forestier8–Reference Anakabe, Collantes and Portilla10]. On the other hand, the amplifier does not exhibit even-mode instabilities under mismatch effects, as verified with the method in [Reference Suárez, Ramírez and Sancho4].

The analysis method relies on the calculation of the Hopf- and flip-bifurcation loci, in terms of the magnitude and phase of the reflection coefficient Γ _o . This must be complemented with a local-stability analysis, such as pole-zero identification [Reference Ayllon, Collantes, Anakabe, Lizarraga, Soubercaze-Pun and Forestier8–Reference Anakabe, Collantes and Portilla10] to distinguish between stable and unstable regions. It will be sufficient to apply this local-stability analysis to representative points in the various regions separated by the loci.

A) Incommensurable oscillation

For each P _in , the boundary of incommensurable oscillations, or Hopf-bifurcation locus [Reference Quéré, Ngoya, Camiade, Suarez, Hessane and Obregon15–Reference Rizzoli and Neri20], will be defined by the two conditions:

(2) $$\eqalign{\bar H(\bar X_o, \rho _o, \phi _o ) = 0, \cr Y_{To} (\bar X_o, \rho _o, \phi _o, f) = 0,} $$

where $\bar H = 0$ represents the whole set of HB equations, acting as an inner tier, ρ _o , ϕ _o are the magnitude and phase of Γ _o and f is incommensurable with f _in . The steady-state solution $\bar X_o $ depends on Γ _o and the limit oscillation condition, Y _To  = 0, is evaluated with the conversion-matrix approach.

The analysis based on (2) should start with a global exploration of the Smith chart, in order to provide a suitable initial value to the optimization/calculation procedure. This is done with a simple graphical technique that takes advantage of the bounded nature of ρ _o and ϕ _o . The perturbation frequency f is swept between 0 and f _in [Reference Ayllon, Collantes, Anakabe, Lizarraga, Soubercaze-Pun and Forestier8–Reference Anakabe, Collantes and Portilla10] and, for each f, a double sweep is performed in ρ _o , ϕ _o , so as to cover the entire Smith chart. For each triplet f, ρ _o , ϕ _o , the total admittance Y _T is calculated as the ratio between the current delivered by the small-signal source and the voltage across its terminals Y _T  = I _test /(V ₁ − V ₂). A closed curve is obtained for each pair of values f, ρ _o . To fulfill Y _To  = 0, there must be changes of sign in both the real and imaginary parts of Y _T under variations of ρ _o , ϕ _o , which is easily evaluated through simple inspection.

Figure 2(a) shows the admittance plots versus f, ρ _o , ϕ _o corresponding to the circuit in Fig. 1, obtained for P _in  = 10 dBm. There are no crossings of the negative real semi-axis, so there is no oscillation boundary within the Smith chart. Because the PA is always stable in matched conditions, the value Γ _o  = 0 is stable and, as a result, the whole Smith chart should be stable too. The same situation is obtained for other P _in values, so one concludes that the PA in Fig. 1 cannot exhibit incommensurable oscillations under mismatch effects.

Fig. 2. Graphical method to obtain initial values, using the total admittance function. (a) Results in the case of an incommensurable frequency f, sweeping in f, ρ _o , and ϕ _o at the constant input power P _in  = 10 dBm. No instability boundary can exist within the unit Smith chart. (b) Results in the case of a subharmonic frequency f _in /2 and P _in  = 10 dBm, sweeping ϕ _in , ρ _o , and ϕ _o . There should be an instability boundary inside the unit Smith chart.

In case the presence of an instability boundary is detected, suitable initial values for the calculation of this boundary should be close to the origin of the admittance plane (Y _T  = 0). This situation may be found in one or several regions of the Smith chart. This initial value (or values) should be introduced in system (2), which will provide an initial point of the instability boundary: $\rho _o ^i, \phi _o ^i, f^i $ . Then the entire Hopf locus will be efficiently traced through continuation [Reference Suárez18], by sweeping ϕ _o from $\phi _o ^i $ and solving (2) to obtain: ρ _o (ϕ _o ), f(ϕ _o ). There will be one Hopf locus for each P _in .

B) Frequency division by 2

One common case of odd-mode instability is the frequency division by 2, associated with flip bifurcations [Reference Paillot, Nallatamby, Hessane, Quéré, Prigent and Rousset13–Reference Rizzoli and Neri20]. This phenomenon occurs when the input signal shifts the circuit natural frequency f to one-half of the input frequency: f → f _in /2, which is often associated with a parametric instability. This evolution involves the splitting of a pair of complex-conjugate poles at f (associated with two dimensions of the differential equation system [Reference Suárez18]) into two independent pairs of complex-conjugate poles at f _in /2 (each associated with one dimension). At the division threshold, the subharmonic-oscillation amplitude will tend to zero, so the flip bifurcations can be detected by setting the frequency of the small-signal current source to f _in /2. Because this perturbation frequency (f _in /2) and the input frequency are commensurable, the phase shift between the input source and the current source is a relevant analysis variable [Reference Wiggins19]. For the bifurcation detection, one can set the phase of the current source to zero and consider the input-source phase ϕ _in as an analysis variable. The mathematical conditions for the flip bifurcation are:

(3) $$\eqalign{\bar H(\bar X_o, \rho _o, \phi _o, \phi _{in} ) = 0, \cr Y_{To} (\bar X_o, \rho _o, \phi _o, \phi _{in} ) = 0,} $$

where $\bar H = 0$ is the whole set of HB equations, acting as an inner tier, and ρ _o , ϕ _o are the magnitude and phase of Γ _o . The steady-state solution $\bar X_o $ depends on Γ _o . Unless a modified conversion-matrix analysis [Reference Di Paolo and Leuzzi21, Reference Pantoli, Suárez, Leuzzi and Di Paolo22] is applied, the above analysis must be carried out with HB at the fundamental frequency f _in /2, due to the frequency commensurability. The initial value(s) is obtained through three nested sweeps, in the input-source phase ϕ _in , varied between 0 and 360°, and in ρ _o , ϕ _o , to cover the full Smith chart. Advantage is taken from the fact that the three sweeps are bounded.

In the case of the PA in Fig. 1, the admittance plots $Y_T (\bar X_o, \rho _o, \phi _o, \phi _{in} )$ provide several crossings of the negative real semi-axis [Fig. 2(b)], indicating the fulfillment of the flip-bifurcation condition (3) for a continuous set of triplets ρ _o , ϕ _o , ϕ _in . Once an initial point $\rho _o ^i, \phi _o ^i, \phi _{in} ^i $ has been obtained, the entire instability boundary is easily traced through continuation, by sweeping the phase ϕ _o from $\phi _o ^i $ , and solving (3) to obtain: ρ _o (ϕ _o ), ϕ _in (ϕ _o ). This provides the flip-bifurcation loci shown in Fig. 3, which constitute the boundary of load-impedance values for which the circuit exhibits a subharmonic oscillation (flip-bifurcation locus). Each locus in Fig. 3 corresponds to a different P _in . The stable and unstable regions of this boundary are easily distinguished since one should know beforehand the stability properties of the matched amplifier, that is, when it is terminated in 50 Ω. Because the amplifier is stable in matched conditions, the stable region corresponds to the outside of the flip loci.

Fig. 3. Instability boundaries, showing also the points fulfilling Re(Y _To ) < 0 and |Im(Y _To )| ≤ 10⁻³ Ω⁻¹. (a) For P _in  = 6 dBm. (b) For P _in  = 10 dBm. (d) For P _in  = 15 dBm.

Processing the data in Fig. 2(b), it has been possible to obtain the Γ _o values giving negative conductance (Re(Y _T ) < 0), with a magnitude of the imaginary part |Im(Y _T )| below 10⁻³ Ω⁻¹ at different P _in values, represented with squares in Figs 3(a)–3(c). The unstable region contains a subset of the points with negative real part of Y _T and low magnitude of the imaginary part. Note that the negative real part and positive-slope resonance of Y _T do not constitute a general instability condition. However, the limit steady-state oscillation condition in (2) and (3) is rigorous and should be fulfilled at any circuit node at the stability boundary.

As shown in Fig. 4, for low P _in , the flip locus does not enter the Smith chart, so there is unconditional stability. From P _in  ≅ 5 dBm, the locus crosses the Smith chart, so the amplifier is potentially unstable under mismatch effects. Due to the natural reduction of the negative resistance from certain signal amplitude, one should expect the loci to escape from the Smith chart from a certain P _in value.

Fig. 4. Evolution of the flip locus obtained with (3) versus P _in . The loci only cross the Smith chart in a certain P _in interval. The two reflection-coefficient points (Γ _t1 and Γ _t2) at both sides of the boundary corresponding to P _in  = 15 dBm (used for the validation of the method) are indicated.

Because the odd-mode instability only depends on the load value Γ _o at f _in , all the possible implementation of this load should give rise to the same kind of behavior, either stable or unstable. This has been validated for two different Γ _o values, one at each side of the flip-bifurcation locus obtained for P _in  = 15 dBm (in a solid red line in Fig. 3), indicated as Γ _t1 and Γ _t2 in Fig. 4. They are relatively close to the stability boundary to evaluate the degree of accuracy. Figure 5(a) presents the results of an independent stability analysis based on pole-zero identification when Γ _t1 and Γ _t2 are implemented with an RL series network. Figure 5(b) presents the results of the parallel-RL implementation. Poles of the Γ _t1 (Γ _t2) load are represented with “+” (“×”). With the two different implementations, the load Γ _t1 is stable and the load Γ _t2 is unstable, in agreement with results from (3).

Fig. 5. Validation of the flip locus corresponding to P _in  = 15 dBm with two different implementations of Γ _t1 and Γ _t2 in Fig. 3. (a) RL-series implementation. Poles of the Γ _t1 (Γ _t2) load are represented with “+” (“×”). (b) RL-parallel implementation. Poles of the Γ _t1 (Γ _t2) load are represented with “+” (“×”).

As stated, to protect the PA against mismatch-induced instabilities, it is convenient to use a low-pass output network, as in the case of the amplifiers considered in this work. In the absence of output filtering effects, the same procedure should be applied for each particular set of reflection-coefficient values at the harmonic frequencies |m| > 1, which would provide a family of Hopf and flip loci. Actually, the flip locus obtained above corresponds to the case Γ _m  = 0 for |m| > 1. For unconditional stability, the PA should be stable in matched conditions and none of the loci should enter the Smith chart. If the goal is just to check whether the PA is unconditionally stable or not, it will be sufficient to inspect the admittance diagrams obtained through consecutive sweeps. To be more precise, for each f (in the case of a Hopf-bifurcation detection) or ϕ _in (in the case of flip-bifurcation detection) a series of nested sweeps are carried out in the magnitude and phase of the reflection coefficients Γ _m , where |m| ≥ 1. To fulfill Y _To  = 0, there must be changes of sign in both the real and imaginary parts of Y _T under the variations of Γ _m , which is easily evaluated through simple inspection.

III. AMPLIFIER STABILIZATION

The amplifier in Fig. 1 will be stabilized under mismatch effects with the aid of the resistor R _s , connected between the two amplifier branches. Because the target is to stabilize the amplifier for all the P _in values, it will be useful to determine the P _in interval with potential instability in an efficient manner. In the particular case of Fig. 4, all the loci cross the boundary of the Smith chart, so one can expect the locus to be tangent to this chart at the limits of the unstable P _in interval. At the boundary of the unit Smith chart, the magnitude of the reflection coefficient is ρ _o  = 1. The locus of P _in and ϕ _o values fulfilling the flip-bifurcation condition under ρ _o  = 1 is expressed as:

(4) $$\eqalign{\bar H(\bar X_o, \rho _o = 1,\phi _o, \phi _{in}, P_{in} ) = 0, \cr Y_{To} (\bar X_o, \rho _o = 1,\phi _o, \phi _{in}, P_{in} ) = 0.} $$

For P _in values such that the flip locus in (3) crosses the unit Smith chart, there will be at least two ϕ _o fulfilling (4) (Fig. 4). This is shown in Fig. 6, where the phase ϕ _o at the intersection points with the Smith chart [calculated with (4)] has been represented versus P _in . At the boundaries of the unstable P _in interval, there will only be one ϕ _o , since the locus is tangent to the Smith chart. To stabilize the circuit under mismatch effects, the resistor R _s , connected between the two amplifier branches, will be reduced from its original value (170 Ω), in order to increase the damping effects in the odd mode. As expected (Fig. 6), the locus (4) decreases in size with R _s and eventually vanishes. For R _s  < 120 Ω, the amplifier should be stable for all the P _in values.

Fig. 6. Calculation of the unstable P _in interval using the locus in (4). The limits of this interval correspond to the edge points of the locus. The calculation has been performed for different values of the stabilization resistor R _s .

The PA has been measured for two R _s values (150 and 100 Ω) and different positions of a triple-stub tuner, used to enable the load variation [Fig. 7(a)]. With R _s  = 150 Ω, the circuit is stable for the measured loads A and B and exhibits a frequency division by 2 for the loads C and D. See the spectra corresponding to B and C in Figs 7(b) and 7(c). The low amplitude of the subharmonic spectral line is due to the near cancellation of this frequency component at the circuit output, due to its odd-mode nature. The region of the unstable loads is in very good correspondence with the analysis in Fig. 4. With R _s  = 100 Ω, the circuit is stable for all the load values [E, F, G, H are shown in Fig. 7(a)] and all the P _in values, in agreement with Fig. 6.

Fig. 7. Measurements for different positions of a triple-stub tuner, connected to the PA output. (a) The loads A, B, C, D correspond to tests under R _s  = 150 Ω. The loads E, F, G, H correspond to tests under R _s  = 100 Ω. (b) Spectrum for R _s  = 150 Ω and load B (stable). (c) Spectrum for R _s  = 150 Ω and load C (unstable).

IV. GENERAL METHODOLOGY FOR THE PREDICTION OF POTENTIAL ODD-MODE INSTABILITIES

The methodology in Section II can be generalized to multi-device PAs, exhibiting symmetries. According to [Reference Freitag11], in general, there are N possible modes when N devices are combined. The method to detect mismatch-induced instabilities should consider all of the possible modes, detected through the eigenvalue/eigenvector analysis [Reference Freitag11] of the outer tier conversion matrix describing the active subcircuit. Each of them should be tested for the two cases of incommensurable and subharmonic oscillations.

For instance, the PA in Fig. 8 is a two-stage extension of the one in Fig. 1, containing four transistor devices. Under matched conditions (50 Ω termination), with the stabilization resistor R _s  = 100 Ω connected as shown in Fig. 9(a), this PA exhibits unstable behavior in a certain P _in interval, as observed when applying pole-zero identification [Reference Ayllon, Collantes, Anakabe, Lizarraga, Soubercaze-Pun and Forestier8–Reference Anakabe, Collantes and Portilla10]. Figure 9 shows the evolution of the real part of the dominant poles versus P _in . For low P _in , there is a pair of dominant complex-conjugate poles at a frequency f, which is incommensurable with the drive frequency f _in . The poles are initially located on the left-hand side (LHS) of the complex plane, but, as P _in increases, they approach the imaginary axis and cross this axis to the RHS at P _in1 ≅ 2.55dBm. At P _in2 ≅ 10.9dBm, the same pair of complex-conjugate poles crosses to the LHS and the amplifier becomes stable. Thus, the matched amplifier is unstable in the input power interval (2.55, 10.9 dBm). To illustrate the possibilities of the new methodology, instead of a two-stage procedure, with a first stage devoted to the stabilization of the matched PA and a second one devoted to its stabilization under mismatch effects, a global stabilization process will be applied, considering all passive terminations Γ _o at f _in .

Fig. 8. PA based on four active devices, operating at f _in  = 1.5 GHz. (a) Schematic. The AGs used for the potential-instability analysis and operating at the frequency f _AG , are also shown. Their phases, ϕ _AG1, ϕ _AG2, ϕ _AG3, and ϕ _AG4 depend on the particular odd mode to be analyzed, as described in the main text. (b) Photograph.

Fig. 9. Stability analysis of the PA terminated in a standard 50 Ω load, with a stabilization resistor R _s  = 100 Ω, connected as shown in Fig. 8. The real part of the dominant poles has been represented versus P _in . The frequency f of these poles is incommensurable with the input frequency f _in . The PA is unstable in the interval comprised between P _in1 = 2.55 dBm and P _in2 = 10.9 dBm.

The method described in Section II to obtain an outer tier conversion-matrix approach can be generalized to PAs containing any number of active devices. A small-signal voltage AG at the perturbation frequency f is sequentially connected in parallel at each of the device nodes. Then, the elements of each row of the admittance-type matrix are given by the ratio between the current entering each device at the particular sideband kf _in +f and the AG voltage. In the particular case of four active devices, considered in Fig. 8, the matrix is expressible as:

(5) $$\left[ {\matrix{ {Y_{11} (f,kf_{in} + f)} & {Y_{12} (f,kf_{in} + f)} & {Y_{23} (f,kf_{in} + f)} & {Y_{23} (f,kf_{in} + f)} \cr {Y_{12} (f,kf_{in} + f)} & {Y_{11} (f,kf_{in} + f)} & {Y_{23} (f,kf_{in} + f)} & {Y_{23} (f,kf_{in} + f)} \cr {Y_{23} (f,kf_{in} + f)} & {Y_{23} (f,kf_{in} + f)} & {Y_{11} (f,kf_{in} + f)} & {Y_{12} (f,kf_{in} + f)} \cr {Y_{23} (f,kf_{in} + f)} & {Y_{23} (f,kf_{in} + f)} & {Y_{12} (f,kf_{in} + f)} & {Y_{11} (f,kf_{in} + f)} \cr}} \right].$$

Note that the matrix structure is independent of the two sidebands chosen for the outer tier analysis. The matrix describing the passive output network at kf _in  + f will have an identical form. Thus, the possible oscillation modes can be derived from the eigenvectors of the active and passive matrices. The matrix (5) (as well as the matrix describing the passive output networks) has the following eigenvectors:

(6) $$\eqalign{\bar e_1 =\; & \left[ {\matrix{ 1 & 1 & 1 & 1 \cr}} \right]^T, \cr \bar e_2 =\; & \left[ {\matrix{ 1 & 1 & { - 1} & { - 1} \cr}} \right]^T, \cr \bar e_3 =\; & \left[ {\matrix{ 1 & { - 1} & a & { - a} \cr}} \right]^T, \cr \bar e_4 =\; & \left[ {\matrix{ { - 1} & 1 & { - b} & b \cr}} \right]^T.} $$

The two last eigenvectors imply different amplitudes in the two subamplifiers. In practice, due to the symmetry of the circuit topology, one can expect identical oscillation amplitudes in all the equivalent circuit nodes. The eigenvectors in (6) indicate four possible situations, in terms of the phase shifts (0 and 180°) between equivalent nodes of the four circuit branches.

Once the possible modes have been determined with the method described above, the potential-instability analysis will be based on an actual excitation of these modes through the simultaneous connection of as many voltage AGs as the number of active devices. These AGs will be connected in parallel between equivalent nodes of these devices and ground, as shown in Fig. 8(a). Note that the study can equally be carried out using current sources in parallel, since the analysis targets the instability boundary, at which the oscillation amplitude tends to zero, so the circuit behaves in linear condition with respect to these auxiliary sources.

The analysis of the possible odd-mode instabilities will consist of two different stages, depending on the desired level of insight into the potential-instability mechanism: (i) a graphical method to detect the potential-instability problem and (ii) a calculation of the stability boundaries in the Smith chart corresponding to Γ _o . The cases of an incommensurable oscillation and a subharmonic oscillation will be distinguished. For each, all the possible odd-mode instabilities, detected with the outer tier conversion-matrix approach, will be checked.

A) Incommensurable oscillation

For the prediction of an odd-mode incommensurable oscillation, the frequency of the AGs (f _AG  = f) must be incommensurable with the input-drive frequency f _in .

1) MODE [1 –1 1 –1] ^T

To analyze the potential instability in the odd mode [1 –1 1 –1] ^T four AGs, at f _AG  = f, with the respective phase distributions: (0, 180, 0, 180°), are connected in parallel at the device output terminals [Fig. 8(a)]. For each P _in value, a triple sweep is performed, in the frequency f, from 0 to f _in , and in the amplitude and phase of Γ _o . Advantage is taken from the fact that the three sweeps are bounded. For each frequency f and magnitude ρ _o , a closed curve is obtained when sweeping ϕ _o . Figure 10 presents the results of the triple sweeps corresponding to the mode [1 –1 1 –1] ^T for different values of the stabilization resistor R _s in Fig. 8(a).

Fig. 10. Graphical prediction of the odd-mode instability [1 –1 1 –1] ^T at an incommensurable frequency f, under mismatch effects. It is based on triple sweeps in the perturbation frequency f and the reflection-coefficient magnitude ρ _o and phase ϕ _o , performed for each P _in value. (a) Stabilization resistor R _s  = 100 Ω. A stability boundary exists within the Smith chart. (b) R _s  = 91 Ω. Tangency situation. (c) R _s  = 82 Ω. There is no instability boundary within the Smith chart. (d) R _s  = 75 Ω. There is no instability boundary within the Smith chart.

Figure 10(a) shows the admittance plots with R _s  = 100 Ω. For each P _in and each f, a closed curve is obtained. For P _in  = −10 dBm, the zero-amplitude steady-state oscillation condition $Y_{To} (\bar X_o, \rho _o, \phi _o, f) = 0$ can never be fulfilled, as derived from a simple inspection of the plots. The entire Smith chart will either correspond to stable or unstable behavior, since there is no instability boundary within this chart. From the pole-zero identification of Fig. 9, for P _in  = −10 dBm, the amplifier terminated in 50 Ω is stable. Therefore, the entire unit Smith chart is stable and the PA is unconditionally stable under mismatch effects. The closed curves become larger for P _in  = 0 dBm, and some of them enclose the center of the complex plane [Fig. 10(a)]. As a result, a continuous set of triplets f, ρ _o , ϕ _o will fulfill the limit oscillation condition Y _To  = 0, so there should be a stability boundary inside the Γ _o Smith chart for this P _in value. The same situation is obtained for other power values comprised between P _in  = 0 dBm and P _in  = 10 dBm.

Using the described graphical method, in combination with a stability analysis of the matched PA, based on pole-zero identification [Reference Ayllon, Collantes, Anakabe, Lizarraga, Soubercaze-Pun and Forestier8–Reference Anakabe, Collantes and Portilla10], it will be straight forward to stabilize the PA under mismatch effects for all the P _in values. Figure 10(b) shows the same admittance plots in the presence of a stabilization resistor of R _s  = 91 Ω, which nearly corresponds to a tangency condition. Figures 10(c) and 10(d) show the admittance plots corresponding to R _s  = 82 Ω and R _s  = 75 Ω, without any possible fulfillment of the limit oscillation condition Y _To  = 0. Provided that the matched PA (under a 50 Ω termination) is stable for all the P _in values with R _s  < 82 Ω, it will also be unconditionally stable under mismatch effects. The results of the pole-zero identification for R _s  = 75 Ω are shown in Fig. 11. For all the P _in values, the poles are located on the LHS, so the amplifier must be unconditionally stable under mismatch effects. This result will be experimentally confirmed in Section IV.C.

Fig. 11. Pole-zero identification of the matched amplifier (terminated in 50 Ω), with a stabilization resistor R _s  = 82 Ω. This analysis complements the one based on the admittance diagrams in Fig. 10. Combining the results of the two analyses, it is predicted that the amplifier will be unconditionally stable under mismatch effects for R _s  < 82 Ω.

Departing from an initial point obtained with the plots in Fig. 10, and by means of a continuation method, it is possible to trace the stability boundaries in the Γ _o Smith chart. For the stabilization resistor R _s  = 100 Ω, this provides the loci shown in Fig. 12, which delimit the termination loads Γ _o that would give rise to unstable behavior. The impact of P _in on the potential-instability boundary can be noticed. There is a significant variation of the shape and size of the boundaries when modifying P _in .

Fig. 12. Evolution of the instability boundary under incommnesurable oscillations in the odd mode [1 –1 1 −1] ^T under variations in the input power P _in . (a) Stability boundaries in the Γ _o Smith chart. Stable and unstable regions are distinguished with a complementary stability analysis of the matched amplifier (terminated in 50 Ω) versus P _in . This analysis is shown in Fig. 9. (b) Validation of the stability predictions obtained with the boundary corresponding to P _in  = 10 dBm, through pole-zero identifiction. Two different loads have been tested, A and B, indicated in (a). The boundary accurately predicts the stability properties.

Stable and unstable regions can be distinguished with the aid of the pole-zero identification in Fig. 9, applied to the matched amplifier. For instance, for P _in  = 1 dBm, the stable region corresponds to the outside of the boundary, since we know that the matched 50 Ω termination is stable and should belong to the stable region. The boundary crosses the origin of the Smith chart at P _in  = 2.55 dBm and P _in  = 10.9 dBm, which agree with the input power values at which the matched PA undergoes Hopf bifurcations in the pole-zero analysis of Fig. 9.

The accuracy of the stability boundaries in Fig. 12 has been validated with pole-zero identification, considering the two fundamental-frequency loads, A and B, one at each side of the boundary corresponding to P _in  = 10 dBm. For A, the poles are on the LHS (Fig. 13), in agreement with the stable behavior predicted by the boundary. For B, the poles are on the LHS, also in agreement with the boundary.

Fig. 13. Prediction of subharmonic instabilities (at f _in /2) under mismatch effects by using a total admittance diagram. Different values of input power P _in have been considered. (a) Odd-mode [1 –1 1 –1] ^T , with AG excitation at the phases (0, 180, 0, 180°). (b) Odd-mode [1 1 −1 –1]^T with AG excitation at the phases (0, 0, 180, 180°). For each of the two AG phase distributions, a triple sweep is performed: in the input-source phase ϕ _in , from 0 to 360°, and in the amplitude and phase of Γ _o .

2) MODE [1 1 −1 – 1] ^T

To predict potential instabilities in the mode [1 1 −1 –1] ^T , four AGs, with the respective phase values (0, 0, 180, 180°), and operating at the incommensurable frequency f, are connected at the device output terminals. Though not shown here, all the resulting admittance plots fulfill Re(Y _T ) > 0 for all the P _in values. Thus, there cannot be any instability boundaries inside the Smith chart. Under this odd-mode excitation, the matched PA is stable for all the P _in values, as verified with pole-zero identification. Taking both results into account, one concludes that the PA does not exhibit this kind of instability.

C) Experimental results

The amplifier has been manufactured and measured. Its output was connected to a triple-stub tuner. Under variations of the triple-stub tuner, subharmonic instabilities were only observed through synchronization mechanisms, once the circuit was in a steady-state oscillatory regime. This is a secondary phenomenon, which takes place when the autonomous frequency of the quasi-periodic regime becomes commensurable with the input-drive frequency [Reference Suárez18]. From a periodic regime at f _in , only transitions to a self-oscillating mixer regime at f _in and f (incommensurable with f _in ) were experimentally observed under variations of the tuner, in agreement with the simulation results. Figure 14 shows the results obtained for the stabilization resistor R _s  = 100 Ω and P _in  = 10 dBm. The theoretical stability boundary is represented in Fig. 14(a), together with the measured load-impedance variations obtained for several positions of the triple-stub tuner. Unstable behavior was obtained inside the instability boundary, as in the case of the loads A and D. As an example, with the spectrum obtained with the load A is shown in Fig. 14(b). Note that the spectrum in Fig. 8(b) corresponds to the fully established autonomous quasi-periodic regime, exhibiting the oscillation frequency f _a . In this regime, only the spectral lines mf _in  + kf _a , with k odd are 180° out of phase. Spectral lines with k even are in phase and are combined by the output network. All the lines comprised between dc and f _in are more than 30 dB below the one at f _in . The spectral line at 1.85 GHz, exhibiting high power, agrees with the second harmonic of one of the autonomously generated line at 0.925 GHz. Stable behavior was obtained outside the boundary, as in the case of the loads B and C. As an example, the spectrum obtained with the load B is shown in Fig. 14(c). When using the resistor value R _s  = 75 Ω, no unstable behavior was obtained for any P _in up to the maximum value (20 dBm), considered in our theoretical analysis.

Fig. 14. Experimental measurement of the PA with four active devices in Fig. 8. (a) Validation of the instability boundary under incommnesurable oscillations in the odd mode [1 –1 1 –1] ^T , corresponding to the input power P _in  = 10 dBm. (b) Unstable behavior for the experimental load A. (c) Stable behavior for the experimental load B.