Pole-Zero Identification: Unveiling the Critical Dynamics of Microwave Circuits Beyond Stability Analysis

The term pole-zero identification refers to obtaining the poles and zeros of a linear (or linearized) system described by its frequency response. This is usually done using optimization techniques (such as least squares, maximum likelihood estimation, or vector fitting) that fit a given frequency response of the linear system to a transfer function defined as the ratio of two polynomials [1], [2]. This kind of linear system identification in the frequency domain has numerous applications in a wide variety of engineering fields, such as mechanical systems, power systems, and electromagnetic compatibility. In the microwave domain, rational approximation is increasingly used to obtain black-box models of complex passive structures for model order reduction and efficient transient simulation. An extensive bibliography on the matter can be found in [3]-[6]. In this article, we focus on a different application of pole-zero identification. We review the different ways in which pole-zero identification can be applied to nonlinear circuit design, for power-amplifier stability analysis, and more. We provide a comprehensive view of recent approaches through illustrative application examples. Other uses for rational-approximation techniques are beyond the scope of this article.


Overview of Pole-Zero Identification
In the context of microwave amplifier design, pole-zero identification of active microwave circuits was introduced in 2001 as an alternate way to analyze their stability under small-and large-signal excitations [7]. It provided a simple and intuitive way to solve the generalized eigenvalue problem without requiring access to the Jacobian of the system [8] (not available in commercial simulators) or the internal nodes of the nonlinear models of active devices [9], very often not available for industrial property protection. Thus, it is perfectly tailored for use in combination with conventional microwave simulators such as Advanced Design System (ADS) or Microwave Office (MWO).
Pole-zero identification has been increasingly used to analyze the stability of active microwave circuits, mainly amplifiers [10]- [16] but also oscillators [17], [18], frequency dividers [19], [20], or even non-Foster circuits [21]. Perhaps the most attractive features of pole-zero identification for stability are its simplicity (it is actually a probe-based method) and the graphical and intuitive nature of the results. Whenever a pair of complex conjugate poles is found lying on the right-half plane (RHP), i.e., poles with a positive real part, an oscillation will begin to grow from the steady state. The basic steps for the stability analysis are equivalent for both small-and large-signal regimes. Obviously, the simulation engines differ (simple ac analysis in one case and harmonic balance with a conversion matrix in the other), but the processes of probing the circuit, fitting a frequency response, and monitoring the resulting poles in a polezero map are completely analogous.
Obtaining the pole-zero map of a linearized system provides information that goes beyond just a pass/fail stability analysis. In fact, pole-zero identification in the microwave active circuit context has evolved to become a useful tool to gather relevant information regarding the nonlinear dynamics of any kind of active device. In this article, we provide several examples of how pole-zero identification is being used to increase stability margins in power amplifiers, obtain the operation bands in autonomous circuits, and acquire the optimum stabilization solution in multitransistor circuits. We also review recent developments in applying pole-zero identification techniques that can foster their use in active circuit analysis and design.

Fundamentals of Stability Analysis Based on Pole-Zero Identification
Stability analysis based on pole-zero identification is essentially a probe-based technique. In its most basic version, the analysis is carried out in two main steps. The first is to simulate a frequency response in the microwave commercial simulator with the aid of a small-signal current or voltage source (the probe). The second is to fit that frequency response to a transfer function (expressed as the ratio of two polynomials) using a pole-zero identification tool.

Obtaining the Closed-Loop Frequency Response
The first step is obtaining a single-input/single-output (SISO) closed-loop frequency response of the circuit, linearized in terms of its steady-state regime, which can be either a dc bias point or a large-signal periodic regime forced by the input drive. To get the closed-loop frequency response, the circuit is probed, either by a smallsignal current source connected at a circuit node or by a small-signal voltage source inserted into a circuit branch in a series. (See "Closed-Loop Frequency Response.") The frequency of the source is swept along the band of analysis. When a current source is used, the closedloop frequency response is given by the impedance seen by the probe at the connection node [ Figure 1(a)]. In turn, when using a voltage source, the closed-loop frequency response is the total admittance presented to the probe [ Figure 1(b)]. Note that other responses are also valid, such as transimpedances, transadmittances, or voltage and current transfers, as long as they represent a closed-loop response from the linearized system.
Because the system is linear (or linearized), all closed-loop responses provide the same set of poles, except for exact pole-zero cancellations. This means that, in theory, we can probe the circuit at any node/ branch and obtain the same stability information. We know that this is not completely true in practice. Actually, nodes and branches in a circuit can be electrically isolated from part of the circuit's dynamics.
Probing the circuit at those nodes or branches with very low sensitivity (a lack of controllability and observability) can lead to a pole-zero cancellation, and an unstable pole may remain undetected. In control theory, observability refers to the capability of inferring the internal states of a system from external outputs, while controllability relates to the capacity to modify the state of a system acting from an external output. The two concepts are dual aspects of the same problem. This can happen when probing at very low impedance nodes with the current source or at very low admittance branches with the voltage source [22]. A typical example is the case of odd-mode oscillations in powercombining amplifiers. Probing with a current source at a power combination node leads to an exact pole-zero cancellation because the combination node is actually a virtual ground for the odd-mode signal. This is why it is recommended that multistage circuits perform the analysis on at least one node or branch per stage and preferably close to a transistor. We come back to this point later in the article.
If the steady state corresponds to a dc bias point or to a small-signal regime, a common ac analysis carries out the frequency sweep in the simulator. System poles are the eigenvalues of the linearized system, and the frequency sweep should cover the entire band in which the active devices have gain (up to ). fmax When the steady state is a periodic regime driven by a periodic large-signal input at a fundamental frequency f , in a mixer-like analysis based on the conversion matrix algorithm must be used to simulate the frequency response (this is called harmonic balance simulation with small-signal mode in ADS and large-signal small-signal analysis in MWO). In this case, the system poles correspond to the Floquet exponents that appear periodically with the fundamental frequency . fin Consequently, the frequency sweep of the analysis can be constrained to [ / ].

Obtaining the Transfer Function
Once the closed-loop frequency response has been simulated in the commercial CAD tool, we can move on to the second step, which consists of fitting the frequency

Closed-Loop Frequency Response
The first step for stability analysis is obtaining a closedloop frequency response by introducing either a current probe in parallel at a circuit node or a voltage probe in series at a circuit branch. The equivalence between the current or voltage perturbation and the general scheme of a feedback system is shown in Figure S1.
Figure S1. A feedback system including (a) the general feedback system; (b) a basic shunt/shunt feedback topology, in which the input signal is the current is injected at a particular node and the output signal is the voltage vo ; and (c) a basic series/series feedback topology, in which the input signal is the voltage inserted at a particular circuit branch and the output signal is the current flowing through the branch . io

Closed-Loop Transfer Function
Closed-Loop Transfer Function + - response to a transfer function formulated as the ratio of two polynomials ( Figure 2). The fitting mechanisms implicitly approximate the delays introduced by the transmission lines to the polynomial representation. This polynomial approximation of the transmission lines has worked demonstrably well for amplifier designs up to the Ka band and beyond [16]. Once the transfer function is obtained, we can plot the resulting poles and zeros on the complex plane to receive graphical information regarding system stability.
In this context, model stability is not enforced in the identification process because we search for possibly unstable poles. In the case shown in Figure 2, a pair of complex conjugate poles lies on the RHP, indicating that the system is unstable and an oscillation will begin to build up. This oscillation initially starts to grow at the frequency of the poles (≈1.4 GHz). Note that no quantitative values can be given for the final amplitude or frequency of the oscillation from a local stability analysis (in which the system has been linearized about a steady state).
In stability analysis, as in any general linear identification process, the quality assessment of the identification also is a critical step because the order of the transfer function is unknown a priori. The goal is to obtain an approximation that prevents both undermodeling (when a critical pole may be missing in the transfer function) and overmodeling (when the order of the transfer function is unnecessarily high for the system dynamics). In the context of stability analysis, the consequences of undermodeling and overmodeling are illustrated in Figure 3, where a frequency response with a small unstable resonance is identified with different transfer function orders.
Step 1, obtaining a closed-loop frequency response for stability analysis, is shown. (a) With a current source connected at node n, the closed-loop frequency response is given by the impedance seen by the probe at connection node n. (b) With a voltage source connected in a series into branch b, the closed-loop frequency response is given by the total admittance presented to the probe.
With an order n 4 = for both the numerator and denominator, the instability is correctly detected by a pair of RHP complex conjugate poles that are quasicancelled by nearby zeros [ Figure 3(a)]. This quasicancellation indicates a low sensitivity to this dynamic from the observation port to which the probe has been connected. The instability is missed if we use an order as low as n 2 = [ Figure 3(b)]. With an unnecessarily high order, such as , n 6 = we run into an overmodeling problem, with the presence of an additional pole-zero quasicancellation on the RHP [ Figure 3(c)]. This kind of overmodeling quasicancellation has a numerical origin and can be located anywhere on the complex plane, not necessarily on the RHP.
When they appear on the left-half plane (LHP), they are not problematic because they do not modify the stability results and are usually ignored because the goal is not to generate a numerically stable model for transient simulation but to guarantee a reliable detection of critical poles (those that cross to the RHP as some circuit parameter varies). Contrary to the general rational-approximation problem, in the context of stability analysis, the identification process does not need to produce a numerically stable rational model for a later simulation in a time-domain simulator. In this sense, pole-zero identification for stability analysis is less demanding than black-box modeling of passive structures through rational approximations, which must generate numerically stable models in large bands and guarantee stability and passivity.
However, overmodeling quasicancellations that lie on the RHP are a major concern because such quasicancellations can be mistaken for physical poles obtained at a low-sensitivity node or branch, which leads to incorrect conclusions in terms of stability. A prime objective of the identification process is to distinguish critical poles that truly take part in the circuit dynamics from nonphysical poles that may appear numerically due to overmodeling. This can be particularly challenging when analyzing a system with very rich dynamics in a wide frequency band. An obvious approach to tackling this issue is to perform the analysis at different nodes/branches until the RHP physical poles (if any) are detected with high sensitivity. This process entails a clear unstable resonance (as does the one shown in Figure 2) with a pair of complex conjugate poles not quasicancelled by nearby zeros, which cannot be confused with numerical overmodeling.
Another typical solution that does not require analysis at other nodes/branches is based on reducing the complexity of the problem by cutting a broadband frequency response into narrow subbands. In fact, overmodeling numerical quasicancellations depend on the bandwidth of the analyzed frequency range, while physical poles are consistently found at the same location, regardless of the bandwidth. Therefore, a series of reidentifications in narrow subbands centered at the frequency of the critical quasicancellation is commonly carried out to verify/ discard the physical origin of RHP quasicancellations by visual inspection. This concept was used in [23] to provide an automatic procedure to eliminate RHP overmodeling quasicancellations. This kind of strategy is obviously not intended to extract a complete model, but only to reliably detect critical poles. Tools for linear system identification in the frequency domain are available as routines or toolboxes in numerical computing environments, such as tfest.m [24], vectfit.m [25], or FDIDENT [26] in      = , and an overmodeling quasicancellation appears with . n 6 = MATLAB or frep2tf.sci in Scilab [27]. Commercial tools specifically adapted for the pole-zero identification in the context of microwave amplifier stability also exist, such as the STAN tool [28]. Although less frequently employed in nonlinear circuit design, other performing and commercially available tools for rational model generation of passive multiport networks, such as IdEM [29] or Sigrity Broadband SPICE [30], could also be used (provided that the passivity condition is not enforced).
Finally, the reliability of the stability analysis depends more on the accuracy of the simulated frequency response than on the pole-zero identification process that follows. A lack of precise electrical models, incomplete circuit descriptions, or numerical errors in the simulated frequency response such as bad convergence, large numerical noise, or truncation, can lead to inaccurate or even nonphysical frequency responses. Identifying invalid frequency responses can produce false pole-zero plots with incorrect conclusions about the stability of the system.

More Than a Pass/Fail Stability Test
Pole-zero identification is more than a simple pass/ fail stability check. Obtaining the position of the poles and zeros on the complex plane and monitoring their evolution versus a relevant circuit parameter provide useful information that can be used for bifurcation analysis, transient control, optimum circuit stabilization, and stability margin evaluation. We review some examples in different contexts.

Circuit Stabilization
A major value of pole-zero identification lies in its ability to guide designers to an optimal stabilization of the circuit. Pole-zero plots obtained at different nodes and pole trajectories on the complex plane are useful instruments that can help designers select the appropriate stabilization elements that ensure stable behavior with minimal impact on the amplifier's performance. Examples can be found in the literature. In [31], a control design in the frequency domain was applied to stabilize a varactor-based circuit. In [32], a systematic method was proposed to determine the suitable topology, location, and value of the stabilization elements in a multidevice microwave amplifier.
The methodology was based on detecting the sensitive nodes or branches of the circuit, at which the inclusion of a stabilization network (shunt or series resistance, capacitance, inductance, or a combination) is able to eliminate undesired oscillation. This is done by analyzing the position of the zeros relative to the poles, which are obtained at different observation nodes or branches. If the unstable pair of complex conjugate poles is quasicancelled by a pair of complex conjugate zeros that are close by, then stabilization at that location is hardly possible.
A pole-zero quasicancellation denotes low sensitivity to that dynamic from the analysis node or branch. We can illustrate this link between quasicancellations and stabilization with the simple double resonator [see Figure 4(a)]. If the circuit is probed with a current source at node A, the unstable poles are quasicancelled by RHP zeros located nearby [ Figure 4(b)]. Now, consider the connection of a shunt resistor Rstab at node A [green in Figure 4 These results are trivial for this simple circuit, but their conclusions are general. An unstable pole-zero quasicancellation obtained with a current source at a given node indicates that circuit stabilization is very unlikely to be obtained by connecting a shunt resistor at that node. On the contrary, if unstable poles are isolated on the RHP, the amplifier can be stabilized at that analysis node through a shunt resistor. Similar reasoning applies for voltage sources connected at circuit branches and series stabilization resistors.
Developing this concept from a more accurate perspective, the work in [32] proposed pole-placement techniques from control theory (such as root-locus tracing) that can predict the exact trajectory of the critical pair of poles while the stabilization parameter varies (normally, it is shunt or series resistances, capacitances, or inductances) along with the precise value needed to ensure stability. These techniques apply to oscillations that build up from a dc regime.
Power-combining amplifiers include stabilization networks that often are too conservative. In most cases, the presence of these networks causes degraded amplifier electrical performance at the operating bandwidth. The work in [33] presents a design approach to improve the amplifier electrical performance by using pole-zero identification techniques more efficiently. It allows for reducing the number of stabilization networks while maintaining a sufficient stability margin and is based on using a large-signal optimization process that integrates pole-zero identification from the early stages of the design. (See "Stability in Large-Signal Operation.")  The optimization procedure is explained using a two-stage Ku-band monolithic microwave integrated circuit (MMIC) power amplifier for telecommunication space applications that has a risk of oscillation at the divided-by-two input frequency when it is operated in large-signal mode. The original prototype incorporated two kinds of stabilization networks: 1) oddmode parallel resistors between gates and between the drains of the transistors and 2) parallel resistancecapacitance (RC) blocks connected in series to the gates of the transistors (Figure 6). A Monte Carlo large-signal stability analysis accounts for the process variability of the MMIC foundry, which gives the dispersion of the critical pair of poles at / f 2 in for . f 12 7 GHz in = and at 4-dB compression (Figure 7). The poles are stable, with a reasonable stability margin that prevents oscillations due to manufacturing tolerances of the MMIC process. After applying the joint RF and stability-analysis optimization, the stabilization networks can be reduced to only four interbranch resistors, as shown in Figure 8. The same Monte Carlo analysis is performed on the optimized prototype. The dispersion of the critical poles with process variability is shown in Figure 9. Again, poles are stable, and the stability margin is satisfactory. Table 1 gives the gain on circuit performances in the optimized prototype over the original one. This approach results in improved RF performances while preserving sufficient stability margins.

Transient Optimization
Controlling the position of unstable poles on the complex plane can improve the transient characteristics in applications for which switching times are relevant. A method is proposed to improve the startup time in oscillators for switched applications in [17]. Pole-zero identification is used to obtain the pair of complex conjugate poles j ! v~ of the unstable solution from which the oscillation builds up. This pair of poles will dominate the initial transient of the circuit (at least while the amplitude Figure 6. The simplified electric schematics of the original design of the two-stage Kuband MMIC power amplifier with full stabilization networks. The interbranch resistors are between the gates and drains of all of the transistors and RC circuits in series to the gates [33].

Stability in a Large-Signal Operation
An amplifier that was originally stable under dc and small-signal conditions can exhibit spurious oscillations from a certain value of the input power drive. Those oscillations can have two origins. One is the combined effect of feedback with a gain expansion versus power. This typically appears in transistors biased in class B and deep AB, where gain expansion versus input power is common. The other possibility is the negative resistance exhibited by a nonlinear capacitance (such as those included in the models of the active devices) pumped by the input drive. This negative resistance plus a resonance effect facilitates the parametric generation of subharmonic frequencies, as in a pumped varactor diode. A comprehensive explanation of the physical origin of this parametric effect can be found in [S1]. A typical example of this parametric oscillation is the frequency division by two, which is common in power amplifiers with several transistors combined in parallel.

Reference
[S1] A. Grebennikov  of the growing oscillation is still small). Due to the exponential variation of the initial transient, the smaller the real part of the poles , v the longer the transient. The technique in [17] carries out tuning of the oscillator element to increase the positive real part v of the unstable poles, while the required oscillation frequency and first harmonic amplitude are maintained simultaneously.
This approach has been applied to the 2.4-GHz fieldeffect-transistor (FET)-based oscillator in Figure 10. The source capacitance CS is the element selected to control the transient. For each value of , CS the two inductances LG and LD are recalculated to maintain the required oscillation frequency and first harmonic amplitude using an auxiliary generator technique [34]. The variation of the unstable pair of complex conjugate poles versus the set of , CS ( ) L C G S and ( ) L C D S , is shown in Figure 11. The measured transient responses for two values of CS are plotted in Figure 12(a) and are in agreement with the predictions from Figure 11. Although the magnitude of the first harmonic is similar, the two measured waveforms have a very different harmonic content, as can be seen in the detailed plot of Figure 12(b).

Experimental Stability Margin Evaluation
Although stable, LHP complex conjugate poles that are too close to the imaginary axis may have an undesired  . f G Hz 12 7 in = and 4-dB compression [33]. Figure 8. The optimized electric schematic of the amplifier. Only four interbranch resistors remain [33].  . f G Hz 12 7 in = and 4-dB compression. The poles remain stable with a sufficient stability margin [33]. effect on the circuit dynamics. These low-damping poles may increase the risk of oscillation when circuit parameters (such as bias or frequency) or circuit external conditions (such as temperature, mounting, or cabling) change, which affects the robustness of the circuit [35]. In addition, low-damping poles can be responsible for the existence of noise bumps in the output spectrum [14], [36] or long transients in switching-mode amplifiers [37]. Monitoring the path of critical poles that shifts dangerously toward the RHP when a parameter varies, it becomes very important to assess the stability margins, at least qualitatively. This is crucial when critical poles appear at low frequencies in power amplifiers, which involves elements of the bias lines. Even if these critical poles do not become unstable, a location close to the imaginary axis means low damping and a high resonant effect inside the amplifier video bandwidth ( Figure 13). This resonance limits the ability of digital predistortion systems to correct for intermodulation distortion in signals with large instantaneous bandwidths [38], [39]. An experimental method to characterize the lowfrequency stability margins of microwave amplifiers has been proposed in [35]. The experimental approach is especially useful for those situations in which simulation is not fully reliable, either because accurate nonlinear models of the active devices are not available or because there is an incomplete electrical description of the circuit. The low-frequency dominant poles are extracted from reflection coefficient measurements performed at observation ports included specifically in the gate and drain bias paths of the circuit to gain access to the low-frequency dynamics. This method was used in [40] to optimize the design of the bias lines in terms of video bandwidth, relative stability margins, and voltage transfer characteristics in a GaN power amplifier ( Figure 14).
The low-frequency dominant poles have been characterized versus different circuit parameters. In Figure 15(a), we show how a pair of complex conjugate poles scatters when modifying three parameters: the gate and drain bias voltages and input power. The result is a cloud of poles, all stable. However, the pair of poles lies dangerously close to the RHP for some parameter configurations. A redesign of the bias networks (in this case, sets, maintaining a fixed oscillation frequency and first-harmonic amplitude [17].  [17]. The measurements were performed with an HP-83480 digital communications analyzer. As expected from Figure 11, the transient for C F p 2 S = is significantly longer than for . The detailed view of the two waveforms. The harmonic content is very different, although the first harmonic is similar. Figure 10. A schematic of the 2.4-GHz FET-based oscillator [17]. The source capacitance CS is used to control the transient. The drain inductance LD and gate inductance LG are used to maintain the required oscillation frequency and first-harmonic amplitude for each value of . CS increased resistance in series with the low-frequency decoupling capacitors in the gate and drain) manages to shift the critical pair of poles leftward, increasing the stability margin and reducing their resonant effect on the amplifier video bandwidth [ Figure 15(b)].

Bifurcation Analysis
The global stability and bifurcation analysis of nonlinear microwave circuits is another context in which polezero identification is often used as a complementary technique. Bifurcations are qualitative changes in the circuit solution when a parameter is varied continuously [34]. Tracing the bifurcation loci delimits the operation bands of circuits that are autonomous in nature, such as voltage-controlled oscillators, injection-locked oscillators, and frequency dividers. A comprehensive description of the most relevant local bifurcations in microwave circuits and their link to the crossing of the poles through the imaginary axis on the complex plane is given in [41]. Through harmonic-balance simulations, bifurcation-detection techniques (such as those based on the auxiliary generator technique [34]) are often combined with the tracing of pole trajectories on the complex plane to study the stability of solution curves [42]- [47]. This technique provides insight into the dynamic behavior of the autonomous circuit. As an example, a closed-form formulation for the optimized design of coupled oscillators was presented in [43].
The pole-zero identification shows the reduced stability margin when the number of coupled oscillators increases ( Figure 16). As the system grows, it becomes more sensitive to discrepancies between oscillator elements and is more likely to become unlocked. In [45], an in-depth stability and bifurcation analysis of self-oscillating quasiperiodic solutions is presented and applied to the self-oscillating power amplifier (SOPA) in Figure 17. Through a theoretical study, the authors of [45] demonstrated that pole-zero identification can be applied to the stability analysis of quasiperiodic states. For example, Figure 18 shows stable and unstable SOPA solution curves in terms of the oscillation amplitude for different bias conditions. Stability analyses of the quasiperiodic solutions, marked as A and A' in Figure 18, are shown in Figure 19, where the real part of the dominant poles versus input power is plotted.
Examples of bifurcation analysis combined with pole-zero identification can be found in many other articles that carry out detailed investigations of the dynamic behavior of nonlinear autonomous circuits,   [40]. Two observation ports G and D are added in the bias paths, in series with the RC access networks.

Verification of the Rollet Proviso
When investigating the conditional/unconditional stability of a two-port network versus source and load terminations, pole-zero identification is a perfect complement to the Rollet stability criterion [48]; its goal is to determine whether the linear two-port exhibits negative resistance at its ports for some values of passive source and load terminations, which could lead to circuit oscillation. Satisfying the Rollet criterion implies unconditional stability only if the linear two-port contains no internal unstable loops not observable from the external ports. This condition is formulated in a proviso that must be fulfilled for the Rollet stability criterion to be sufficient. Verifying the proviso is important in circuits with complex topologies, such as multistage amplifiers, due to the lack of observability of internal dynamics from the input and output ports. The proviso can be verified by confirming that there are no intrinsic unstable poles when ports are loaded with open and short terminations. This  Figure 16. A pole locus shows the reduction of the stability margin due to discrepancies between oscillator elements, when the n number of oscillator elements increases in a coupled oscillator system [43]. Three cases are considered: , n 4 = 7, and 11. As the system gets larger, the poles approach the imaginary axis, and the system eventually unlocks for . n 11 = Figure 17. A photograph of the SOPA based on an RF class-D power-amplifier topology [45]. The transistor used is an ATF33143 HFET (Avago Technologies), and the switching frequency is in the order of 750 MHz. condition can be easily checked by applying pole-zero identification at internal circuit nodes when the circuit is loaded with open and short terminations.
Things are more complicated if we want to generalize the Rollet stability criterion for analyzing the unconditional stability versus load terminations in large-signal regimes. From a general perspective, this is an incommensurable multidimensional problem because, for each input power , Pin negative resistance at frequency fs depends on load terminations at Kfin and at side bands . Kf fs in ! K is the number of relevant harmonics, and fin is the fundamental frequency of the input signal. A rigorous generalization of the Rollet stability criterion to large-signal periodic regimes under output mismatch effects is given in [49] and [50]. It is applied to practical cases in which the output filter of the amplifier allows consideration of only two (or, at most, three) relevant sidebands at the load-termination port. The fundamental upper and lower sidebands ( ) f fs in ! are normally considered the virtual observation ports for the Rollet analysis. In this case, verifying the Rollet proviso is more demanding because it must be checked for each passive termination at fin, with the two sideband frequencies in all possible combinations of short-and open-circuit terminations. In other words, for each possible combination of the terminations at sideband frequencies, pole-zero identification at internal nodes must be carried out versus variations of load termination at .
fin These variations must cover the entire Smith chart and could be implemented with a double sweep in magnitude and phase. However, in doing so, we receive disconnected circles as we explore the Smith chart, which impedes taking advantage of the continuity properties of the harmonic balance simulation. A smart solution is given in [49]. The analysis follows a single spiral curve on the Smith chart, which depends on a single parameter ( Figure 20). The proviso is checked for all points of the spiral curve.  Figure 18. The SOPA solution curves, represented in terms of the oscillation amplitude versus input power for different gate bias voltages [45].   . N 11 = (b) Checking out the proviso on the poweramplifier prototype in [49] for all points of the h-sweep shows internal unstable poles under short circuit terminations at the relevant sidebands.

Pole-Zero Identification With Vector Fitting for Stability Analysis of Microwave Amplifiers
As explained in the previous section, pole-zero identification carried out at several nodes/branches of the circuit provides valuable information concerning where and how to improve circuit stability. When this is done using a commercial tool, as in [32], the multiple frequency responses are identified as a sequence of independent SISO pole-zero analyses. In theory, all individual SISO transfer functions of a linear (or linearized) system share a common denominator, the characteristic equation. This means that, except for exact pole-zero cancellations, we should get the same poles (only zeros will change) at all analysis ports when performing multiple independent SISO analyses. However, in practice, some disagreement might appear for high damping poles or for poles that are quasicanceled by zeros. Consequently, it seems more convenient to use multiple-input/multiple-output (MIMO) frequency-identification algorithms that maintain a common denominator for frequency responses obtained at different observation ports.
In this direction, an approach based on vector fitting [51], [52] that performs the frequency-domain identification of a vector of transfer functions using a common denominator has recently been proposed in [53] to analyze the stability of multistage amplifiers. An advantage of using a vector fitting-based algorithm for the stability analysis of microwave circuits is that the identified transfer functions H s ( ) where pk are the system poles common to all of the , Hn r , n k is the residue corresponding to pole p k and transfer function , Hn and D is the direct gain. On the contrary, existing commercial pole-zero identification tools for microwave amplifier stability analysis [28] are based on least-squares optimization, and the transfer functions are represented as the ratio of two polynomials: The expansion in partial fractions (1) has a numerical advantage when identifying large bandwidths that involve very high frequencies and high transfer function orders N, compared to the use of the ratio of two polynomials (2). Actually, using (2) results in excessively large numbers due to the terms in powers of s, which leads to an ill-conditioned numerical problem. It is often necessary to divide the original problem into subbands when a ratio of two polynomials is used for the identification [23]. However, with a partial fraction representation as in (1), large numbers are avoided, and the frequency-response identification is numerically better conditioned [51]. This has an important consequence: large bandwidth responses can be identified without dividing the frequency response into narrower subbands, and algorithms for automatic order selection of the transfer function are simplified and can be more effective.

Residue Analysis
Fixing the unstable behavior of a circuit by adding or modifying a stabilization parameter (such as stabilization resistance) at a particular node or branch depends on the sensitivity to the unstable dynamics from that spot. This stabilization capability could be estimated qualitatively by observing the polezero quasicancellations on the complex plane, as explained in the previous section. Isolated pairs of complex conjugate poles reveal an observation port with high sensitivity, while pole-zero quasicancellations indicate low sensitivity to those dynamics from the observation port.
Taking advantage of the representation in partial fractions (1), a procedure based on residue analysis was developed in [53] to quantify the sensitivity of the identification obtained at a particular circuit location. To do so, a normalized factor , n k t is defined to quantify the relative effect of a pair of resonant complex conjugate poles , pk ) pk on the transfer function Hn , where r is the resonant frequency of the poles , pk ) pk and H , n k represents their contribution to the transfer function Hn : Poles identified at high-sensitivity locations will show high values of t , while poles identified at lowsensitivity locations are quasicancelled by zeros and will have small t values. A similar approach to detect spurious poles with a small effect on the frequency response can be found in [54]. The advantages of analyzing t for a set of poles obtained at different nodes or branches are twofold.
We can use the computation of t to minimize the adverse effects of overmodeling in the pole-zero fitting process [23]. If the pole-zero identification is carried out using a very strict optimization goal, we could end up fitting the numerical noise existing in the simulated frequency response. This can lead to the onset of pole-zero quasicancellations that are nonphysical, as shown in Figure 21(a). These nonphysical poles have extremely low values of , t regardless of the circuit location at which we obtain the transfer function. On the contrary, poles representing the actual circuit dynamics will have significantly larger values of t for at least some analysis ports (nodes or branches). Therefore, we are able to identify numerical pole-zero quasicancellations by analyzing their values of t that correspond to the transfer functions obtained at different nodes/branches. This approach has been applied to the example of Figure 21(a) to clean up the pole-zero map from numerical pole-zero quasicancellations, which resulted in the map of Figure 21(b) with only physical poles remaining.
Additionally, by performing the analysis at different nodes and branches and analyzing t for the identified poles, we can classify the different locations of the circuit with respect to their influence on the unstable dynamics and thus deduce the most sensitive places to stabilize the circuit.
A simple example of this approach is given in [53], where the three-stage amplifier based on the GaAs FET transistors (FLU17XM) in Figure 22 is considered. Experimentally, this prototype showed an oscillation of 3.5 GHz for the nominal bias point (Figure 23). A MIMO stability analysis is performed for that bias condition. Eight observation nodes, corresponding to the gate and drain terminals of the four transistors (labeled n n -1 8 in Figure 22), are taken into account. Because the amplifier has a power-combining structure in its third stage, an excitation of the odd and even modes has been considered. (See "Even-Mode/ Odd-Mode Oscillations.") To excite the even mode,  [53]. two in-phase, small-signal current sources are simultaneously applied to nodes n5 and n6 (and the same for nodes n7 and ).
n8 To excite the odd-mode, two 180° out-of-phase, small-signal current sources are simultaneously applied to nodes n5 and n6 (and the same for nodes n7 and ).
n8 Eventually, a total of eight frequency responses is identified with a common denominator. The resulting poles plotted on the complex plane are shown in Figure 24. An unstable pair of complex conjugate poles is obtained at 3.3 GHz. Next, the computation of factor t of this unstable pair is performed for the eight frequency responses. The results are graphically shown in Figure 25.
We can readily infer that the oscillation takes place in the third stage and has an odd-mode nature because its corresponding t values are several orders of magnitude larger than at the rest of the analyzed nodes and modes. To eliminate this oscillation, interbranch resistors between the gates or drains of third-stage transistors can be introduced ( Figure 22). Note that the value of t for the odd-mode frequency response between the gates of the third stage is larger than the value obtained for the odd-mode frequency response between the drains. This suggests that the optimum place for a stabilization resistor is between the gates of the transistors of the third stage, which  n 3 n 2 n 4 n 5, n 6 n 7, n 8 n 5, n 6 n 7, n 8 Figure 25. The residue analysis of unstable poles at 3.3 GHz indicates that the instability is an odd-mode oscillation at the third stage [53].

Even-Mode/Odd-Mode Oscillations
For symmetry, amplifiers with power-combining structures can have different oscillation modes. In the simplest case of two transistors in parallel ( Figure S2), two different oscillation modes are possible: an even mode in which the two transistors oscillate in phase and an odd mode in which the two transistors oscillate 180° out of phase. The type of oscillation mode can be determined with pole-zero analysis. The even and odd modes can be individually excited and observed injecting two current probes with appropriate phases at gates of the parallel transistors, because both probes in phase excite primarily the even mode, while two probes 180° out of phase excite mainly the odd mode ( Figure S2). Note that the odd-mode oscillation is not observable at the power division/combination nodes, which are virtual grounds for the odd mode. Injecting the current probe at those nodes would result in exact pole-zero cancellations. was experimentally confirmed. An interbranch resistor of . 5 1 KX between the gates was enough to eliminate the observed instability at 3.5 GHz, while a much lower value ( ) 910 X was needed between the drains for stabilization. Note that low interbranch resistances can impact amplifier performances when there are appreciable symmetry imbalances due to technological dispersion.