Stability criteria of two-port networks, application to thermo-acoustic systems

Mohammad Kojourimanesh; Viktor Kornilov; Ines Lopez Arteaga; Philip de Goey

doi:10.1177/17568277221088465

Stability criteria of two-port networks, application to thermo-acoustic systems

Kojourimanesh, Mohammad; Kornilov, Viktor; Lopez Arteaga, Ines; de Goey, Philip 2022-03-01 00:00:00 System theory methods are developed and applied to introduce a new analysis methodology based on the stability criteria of active two-ports, to the problem of thermo-acoustic instability in a combustion appliance. The analogy between thermo-acoustics of combustion and small-signal operation of microwave ampliﬁers is utilized. Notions of unconditional and conditional stabilities of an (active) two-port, representing a burner with ﬂame, are introduced and analyzed. Unconditional stability of two-port means the absence of autonomous oscillation at any embedding of the given two- port by any passive network at the system’s upstream (source) and downstream (load) sides. It has been shown that for velocity-sensitive compact burners in the limit of zero Mach number, the criteria of unconditional stability cannot be fulﬁlled. The analysis is performed in the spirit of a known criterion in microwave network theory, the so-called Edwards-Sinsky’s criterion. Therefore, two methods have been applied to elucidate the necessary and sufﬁcient condi- tions of a linear active two-port system to be conditionally stable. The ﬁrst method is a new algebraic technique to prove and derive the conditional and unconditional stability criteria, and the second method is based on certain proper- ties of Mobius (bilinear) transformations for combinations of reﬂection coefﬁcients and scattering matrix of (active) two- ports. The developed framework allows formulating design requirements for the stabilization of operation of a combus- tion appliance via purposeful modiﬁcations of either the burner properties or/and of its acoustic embeddings. The ana- lytical derivations have been examined in a case study to show the power of the methodology in the thermo-acoustics system application. Keywords Burner as an active two-port, Edwards-Sinsky’s criterion, Rollett factor, Conditional stability, Mobius Transformation Date received: 29 November 2021; accepted: 17 February 2022 availability of a purely acoustic characterization of the burner Introduction with ﬂame is the prerequisite of the model. This is achievable Thermo-acoustic combustion instability manifests itself as a within the concept of the transfer matrix (T)orscattering high level of tonal noise, vibration, and may cause the per- 7 matrix (S). Then, a network model of the combustion system formance deterioration or even structural damage of com- is obtained when all two-port components are combined. bustion appliance. The ability to eliminate and/or control The methodological similarity of approaches to and the combustion instability at the appliance design phase is network models equivalence of the electrical circuits and one of the main goals of combustion-acoustics research. The low-order (acoustic network) modeling approach is one of the intensively developing tools which has proven Department of Mechanical Engineering, Eindhoven University its efﬁciency in performing problem analysis, synthesis, of Technology, Eindhoven, The Netherland and eventually the appliances design tasks. Department of Engineering Mechanics, KTH Royal Institute 1–3 of Technology, Stockholm, Sweden Various acoustic network models have been developed that are used to analyze combustion thermo-acoustic instabil- Corresponding author: ities and the design of combustion equipment in recent Mohammad Kojourimanesh, Eindhoven University of Technology, Building 4–7 studies. Thismodelingallowstreatingcombustionappliance 15, Gemini Noord 1.44, PO BOX 513 5600 Eindhoven, the Netherlands. 4,8,9 components as acoustic two-ports. Accordingly, the Email: m.kojourimanesh@tue.nl Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https:// creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as speciﬁed on the SAGE and Open Access page (https://us.sagepub.com/en-us/nam/open-access-at-sage). Kojourimanesh et al. 83 combustion acoustic systems have been shown in various The main focus was on formulating thermo-acoustic papers since 1957. However, the stability analysis and network models, predicting the instability, searching for design methods of the two-port networks have not been unstable frequencies, calculating/measuring the growth developed/applied in the combustion ﬁeld as much as in rate, searching methods for stabilizing systems, etc. microwave theory. The linear two-port network theory References to most of the performed research can be has been an intensively developing research subject and found in the review paper. The conventional methodology the results have been applied ubiquitously in the practice for analyzing the stability of thermo-acoustic systems con- of microwave devices’ design. The closest analogy can be sists of measuring/modeling the Flame Transfer Function established between the combustion thermo-acoustic (FTF). Then, a wave-based 1D linear two-port network instability problem and the problem of stability of operation approach is applied to provide the system matrix. The of microwave ampliﬁers. Here, the burner with ﬂame and eigenfrequencies of the system (zeros of the matrix’ deter- the ampliﬁer (e.g., transistor) both represent a so-called, minant) determine the (in)-stability frequencies and “dependent source” or active element. Furthermore, the growth/decay rates. Therefore, the common practice is to acoustics of the burner upstream and downstream parts in create a system matrix each time when the system is a combustion appliance are an analogy of the “source” altered to check the corresponding effect on the eigenfre- and “load” passive network embeddings of the microwave quencies. This procedure allows resolving the dilemma of ampliﬁer. One of the extremely useful and well-developed the (in-)stability of operation of a particular system and concepts of the microwave ampliﬁer’s design process is gives a reasonably accurate prediction of frequencies of the notion of unconditional stability. oscillation. This approach has been successfully applied 7,25,26 In microwave theory, this means that there is no passive in numerous studies before. However, the conven- source and passive load combination that can cause the tional modeling approach does not provide a good overview circuit with the given ampliﬁer to oscillate. Correspondingly, of conditions and guidelines for designing the upstream and the unconditional stability in a thermo-acoustic context downstream sides of the ﬂame such that the system would means operation stability regardless of the (passive) acoustics be stable. The crux is in the absence of speciﬁc parameters at upstream and downstream sides of the burner/ﬂame. The or criteria to determine the system stability and lack of tools pioneering work on this subject was done by Rollett in (rules) on how to manipulate the system design as it is done 1962. He introduced a quantity (criterion) to characterize in microwave theory. the degree of stability. Later, it was shown that the combin- A new impetus in the development of system-level ana- ation of validity of certain inequality requested from the lysis of thermo-acoustic network models was given by the Rollett factor together with only one other auxiliary condition discovery of the phenomenon of the burner intrinsic 27,28 are necessary and sufﬁcient to provide unconditional stabil- thermo-acoustic mode of instability. This and further 12–15 ity. research on the subject use system theory. Particularly, In 1992, Edwards and Sinsky proposed a single param- the derived system instability conditions are based on the eter, instead of two of Rollett’s conditions, to determine the gain and phase of the TFT for only ITA modes. A necessary and sufﬁcient unconditional stability require- review of literature on this subject can be found in the 16 30 ments. The arguments and analysis were based on a geo- recent publication. metrical approach. Various applications and design tools Another research direction was introduced by Kornilov based on the Edwards-Sinsky criteria were developed and and de Goey who showed the analogy between the thermo- 17–19 31 discussed. Particularly, Balsi et al. extended the geo- acoustic and microwave circuits linear two-port networks metrical approach and derived the necessary and sufﬁcient and use it to investigate two unconditional stability criteria, conditions for a linear active two-port to be conditionally of ‘Rollett’ and ‘Edwards-Sinsky’, for the purpose of 20 32 stable. A recent work of Lombardi and Neri presented evaluation of a burner/ﬂame ﬁgure of merit. In turn, the existence of a duality mapping between the input and this work gave the inspiration to develop a prospective the output of the two-port network; then by using certain method to assess thermo-acoustic instabilities based on properties of Mobius Transformation (MT), they demon- reﬂection coefﬁcients measured only from the upstream strated all possible cases of mapping between the input side of the burner (cold side) by Kojourimanesh et al.. and the output of the system. MT is the bilinear rational In this approach, two reﬂection coefﬁcients, R and up transformation as one of the mathematical concepts R , at the cold side of the ﬂame are measured. As dis- in named after A.F. Mobius. It is well-known that the MT played in Figure 1, R is the reﬂection coefﬁcient of the up maps a line or circle into another line or circle. Çakmak upstream side of the burner and R is the input reﬂection in et al. derived explicit formulas relating the centers and the coefﬁcient of the burner terminated by R .Inother dn radii of the mapped circles. words, if we disconnect the network in Figure 1 from the On the other hand, the beginning of active development R , send in wave f and measure reﬂected wave g then up 1 1, of the acoustic network modeling approach to the problem the corresponding reﬂection coefﬁcient would be: R ≜ in of combustion instability falls in the period after ∼1990. g / f . 1 84 International Journal of Spray and Combustion Dynamics 14(1-2) Figure 1. Thermo-acoustic model of a combustion system. In this method, the stability of the system can be deter- ﬂame for which the purely acoustic representation in the mined by inspection of the Nyquist plot of the measured form of a two-port is known and given, e.g., by the R times R . They showed that the condition applied to burner transfer matrix. However, here we limit the consid- up in the magnitude of R R (iω) being less than 1 for all fre- eration to one particular type of burner, namely, an acoustic up in quencies from 0 to inﬁnity is sufﬁcient (but not neces- velocity-sensitive dependent source of acoustic velocity sary) to result in the thermo-acoustic stability of the (analogy of current sensitive current source in microwave system. Accordingly, in another study, they applied the theory). In this case, we may use some internal symmetries MT properties to provide the necessary conditions of of the transfer and scattering matrices. Physically, this type R to ensure that the magnitude of R becomes less of thermo-acoustic property is appropriate to a wide class of dn in than 1. perfectly premixed gaseous fuel burners operating in the The present paper contributes to the further development limit of low Mach numbers for the mean ﬂow when the of the research on the framework of the system-level ana- heat release zone is compact with respect to the acoustic lysis of thermo-acoustic instability of combustion and uti- wavelength under consideration. In addition, the conditions lizes the close analogy with the theory of microwave presented below should be satisﬁed for all frequencies from networks. The particular goal of the present contribution 0toinﬁnity. For brevity, we may discuss all relations for a is to introduce a new analysis methodology that is based ﬁxed frequency point but ﬁnally, all criteria and conditions on the stability criteria of active two-ports. The criteria should be satisﬁed for the whole frequency range. will be derived using the original approach based on prop- Furthermore, we will work in the frequency domain, and erties of a MT in combination with some algebraic transfor- consider only plane longitudinal waves, 1-D acoustics. The mations. The more general goal is to illustrate the power of network model variables will be represented by the forward the system analysis method in the application to thermo- and backward traveling waves f and g and the convention st acoustic network modeling and introduce an approach for the time dependence is e where s is the complex fre- that allows designing optimal terminations at the quency s = iω + σ, where σ is the growth rate. upstream/downstream sides of the ﬂame. The model of a thermo-acoustic system is ﬁrst written in the form of the network of a scattering matrix for power Stability criteria of thermo-acoustic waves to show the analogy with microwave theory and systems derive system stability conditions. Then, new algebraic For a compact velocity-sensitive ﬂame in the limit of zero proofs of unconditional stability, namely the mean Mach number the transfer matrix takes the form of Edwards-Sinsky criterion, and conditional stability are pro- posed. Besides, an alternative approach using MT is intro- ε + 1 + θ FTF ε − 1 − θ FTF T = 0.5 . (1) duced to determine the stability condition. Next, by ε − 1 − θ FTF ε + 1 + θ FTF combining the outcome of the aforementioned methods, conditional stability criteria for the thermo-acoustic systems are provided. Furthermore, the necessary condition In this notation, θ = − 1 is the temperature jump ratio; is derived which if it is satisﬁed by R ensuring that the T being the temperature at upstream and downstream dn 1,2 ρ c magnitude of R becomes less than 1 in a frequency sides of the ﬂame; ε = is the jump in characteristic in ρ c range. This condition results in passive thermoacoustics sta- acoustic impedance across the ﬂame; and FTF is the bility of the system’s operation. In addition, an optimum ﬂame transfer function which relates the oscillation of value of R is obtained which provides that the value of heat release rate of the ﬂame to the oscillation of acoustic dn R becomes minimum. In its turn, this promises a higher velocity at upstream of the burner and scaled to mean in potential of stability of the system. values of heat release and unburned gas velocity. The results obtained and presented below can be in prin- The corresponding scattering matrix of the thermo- ciple generalized to the case of an arbitrary burner with acoustic two-port can be deﬁned, if one rearranges Kojourimanesh et al. 85 equations of the transfer matrix (T) such that the ingoing In Appendix A.1, it is made clear that for any complex 1−|z| waves appear as inputs to the matrix, and the outgoing number z the function ≤ 1. Accordingly, based on |1−z | waves as outputs. In that case, the scattering matrix (S)of the Rollett criteria, the considered thermo-acoustic system the thermo-acoustic system would be cannot be unconditionally stable because one of the neces- sary conditions of unconditional stability is not satisﬁed, −2T 4 S = , (2) 2 2 namely, K >1. TA T − T 2T 2T 21 11 11 21 Alternately, it is also possible to prove Lemma 1 by ana- lyzing the Edwards and Sinsky parameter, μ. They proved that with the determinant of Δ =−1, where T are transfer ij the necessary and sufﬁcient condition to qualify a two-port as matrix elements (See Appendix A.0). an unconditionally stable element is μ > 1where The scattering matrix representation also includes the ITA mode as a special case when T = 0. However, in 1 −|S | μ = . (6) the present study, we will not focus particularly on pure |S − S Δ|+ |S S | 22 11 12 21 ITA mode analysis. In this notation, the bar symbol is used to denote the con- jugate of a complex number. By substituting parameters, like in Appendix A.2, one Unconditional stability in thermo-acoustic systems 1−|z| can also derive ≤ 1. Hence, for the thermo- |2Im(z)|+|1−z | Unconditional stability of a given burner with ﬂame in a acoustic system which obeys equation (2), μ ≤ 1. TA thermo-acoustic context means that regardless of the acous- Therefore, the thermo-acoustic two-port (a burner with tic reﬂection coefﬁcients of passive upstream and down- ﬂame) for which the transfer (scattering) matrix has sym- stream sides of the burner/ﬂame, the combined system metry properties, as in equation (1), cannot be uncondition- would be always thermo-acoustically stable. To establish ally stable. For brevity of notation, the index of TA is whether the unconditional stability can be ensured for the omitted from K and μ in the rest of the paper. TA TA thermo-acoustic two-port deﬁned in equations (1) and (2), the evaluation of the Rollett stability condition or the Edwards-Sinsky parameter can be performed. Conditional stability in thermo-acoustic systems Lemma 1. The deﬁned thermo-acoustic system in equa- tion (1), cannot be unconditionally stable. When considering the notion of conditional stability of a Proof. The Rollett stability condition says that the com- thermo-acoustic system, the upstream and downstream bination of Rollett stability factor K > 1 where acoustic boundary conditions are playing a role. One needs to search for the range of R (or R ) values in the 2 2 2 up dn 1 −|S | −|S | +|Δ| 11 22 K = , (3) complex domain such that the system would be always 2|S S | 12 21 stable. It is worth mentioning that outside of that range, the system may be stable or unstable. As expressed in together with any one of the following auxiliary condi- paper, the stability of a combustion appliance can be tions given in equation (4) are necessary and sufﬁcient for determined by measuring two reﬂection coefﬁcients at the unconditional stability of an (active) two-port described cold side of the burner, i.e., R and R . In that paper, the up in by the scattering matrix S, expression of R is written in terms of the transfer in ⎧ T R −T 11 dn 21 2 2 2 matrix’s elements, i.e., R = . This expression in B = 1 +|S | −|S | −|Δ| > 0, or ⎪ − T R +T 1 11 22 12 dn 22 2 2 2 can be rewritten with scattering matrix’s entries with help of B = 1 −|S | +|S | −|Δ| > 0, or 2 11 22 equations (1) and (2). The result would be the same expression |Δ|=|S S − S S | < 1, or (4) 11 22 12 21 ⎪ 35 ⎪ 2 used in microwave theory, e.g., equation 7.4.1 in book : 1 −|S | > |S S |, or ⎪ 11 12 21 1 −|S | > |S S |. 22 12 21 −Δ R + S dn 11 R = . (7) in −S R + 1 22 dn Therefore, one can simplify the Rollett stability factor for the Thermo-Acoustic system i.e., K as TA The main idea of the Edwards-Sinsky criterion regarding 2 2 2 the stability of a system expressed by equation (7) is that the T T T 21 21 21 1−− − +|−1| 1 − system is unconditionally stable if the unit disk in the R dn T T T 11 11 11 K = = ≤ 1 . TA plane (note that the interior of the unit circle represents all 2 2 2 2 T − T T 11 21 possible reﬂection coefﬁcients of a passive system/termin- 2 1 − ( ) T 2T T 11 11 11 ation) is mapped by the equation (7) to the interior of the (5) unit disk in the R plane. In other words, the system in 86 International Journal of Spray and Combustion Dynamics 14(1-2) with passive upstream and downstream terminations is Equations (8.b) and (8.c) are almost the same as the con- unconditionally stable if and only if |R | < 1. Besides, ditions suggested by Balsi et al. Appendix C.1 shows how in Kojourimanesh et al. have shown that if |R R | <1then their conditions can be derived using this algebraic method. up in 33,36 the system is conditionally stable. The case of pure ITA Moreover, Appendix C.2 provides a new proof of the modes requires special consideration. For pure ITA modes, Edwards-Sinsky criterion from the mentioned algebraic (i.e., T = 0, R = 0, R = 0) the expression written in technique. 11 up dn equation (2) will be undeﬁned due to division by zero. Also, the conditional stability criterion such as |R R | < 1 up in Mobius transformation between R and R in dn is not applicable because both R and R are already zeros. up dn Accordingly, Lemma 2 which involves requirements to As stated before, the relation between R and R has the in dn the magnitudes of R and R is introduced. It provides form of so-called bilinear (Mobius) transformation. The up dn conditions imposed on the upstream or downstream sides general (so-called, normalized) form of this transformation aZ+b of the ﬂame/burner sufﬁcient for the system to become is H = , where ad − bc = 1. cZ +d thermo-acoustically stable. Among multiple speciﬁc properties of this transform- Lemma 2. For the thermo-acoustic system with the active ation, one which will be used below is that the MT maps two-port deﬁned in equation (2), conditions for the down- a line or a circle in the complex plane of its input Z into stream and upstream terminations which are sufﬁcient to another line or circle in the plane of its output H. qualify the system to be conditionally stable are Particularly, if the mapped contour is the unit circle, then the resulting circle of the unit circle has a speciﬁed 22,23 |R | < , (8.a) center, M, and radius, r , namely, up |R | in b·d − a · c 1 or M = , r = (12) 2 2 2 2 |d| −|c| ||d| −|c| | −|S −|R | S Δ|+|R ||S S | 11 dn 22 dn 12 21 |R | < (8.b) up For a ﬁxed value of the frequency, the entries of the scatter- 2 2 2 (|R | |Δ| −|S | ) dn 11 ing matrix, S, are ﬁxed complex numbers. By looking at the −ΔR +S 2 dn 11 expression for R = , it is clear that it has the form −|S −|R | S Δ|+|R ||S S | in 22 up 11 up 12 21 −S R +1 22 dn |R | < (8.c) dn 2 2 2 of MT. However, it needs to be normalized ﬁrst i.e., divided (|R | |Δ| −|S | ) up 22 by ad − bc = S S . The downstream side of the 12 21 Proof. As mentioned before, the system is conditionally burner/ﬂame is an acoustically passive termination, i.e., stable if |R R | < 1. Therefore, one can say that a sufﬁ- up in | R |≤ 1. Consequently, MT transforms the unit circle dn cient condition for a conditionally stable system could be of R into a circle in the plane of R with a center and dn in |R R | < 1. up in radius as. Considering equation (7) and Δ =−1, Appendix B.1 S + S −2Im(S ) |S S | shows how |R R | < 1 can be extended to get a form of 11 22 22 12 21 up in M = = i, r = . (13) 2 2 2 a quadratic inequality as 1 −|S | 1 −|S | |1 −|S | | 22 22 22 A|R | + B|R |+ C < 0 (9) dn dn Furthermore, the center of the R unit circle transforms dn 2 2 2 where A =|R | |Δ| −|S | , up 22 b into point O = in the R plane which indicates to where in 2 2 2 d B = 2(|S −|R | S Δ|), and C =|R | |S | − 1. 22 up 11 up 11 the inside area of the unit circle of R is mapped to. It can dn As shown in Appendix B.2, the discriminant, be either the inside or outside area of the circle in R plane. in B − 4AC, of equation (9) would be Figure 2 shows four possible cases for this transformation 2 2 2 between R and R planes. This kind of plot provides dn in B − 4AC = 4|R | |S S | (10) up 12 21 new insight into the design strategy, one may follow to Because of B − 4AC ≥ 0, the only way that equation (9) ensure system stability. For instance, considering Case a in becomes always negative is C < 0 & |R | < λ , where λ dn 1 1 Figure 2 one may conclude that for most reﬂection coefﬁ- is the ﬁrst root of the quadratic equation, i.e., cients of the downstream side of the ﬂame, R , the gain of dn R is higher than 1. Accordingly, to stabilize the system for in −B + B − 4AC λ = . (11) this particular case the design of the stabilizing downstream 2A acoustics is more demanding because there is a quite limited range of R which provides |R |≤ 1. Therefore, dn in Appendix B.3 shows that by substituting A, B, C into tuning of the upstream side of the burner becomes a more λ , the right side of equation (8.c) would be the same as 1 appropriate approach. Contrarily, when facing a situation λ . Therefore, if C < 0 & |R | < λ then the considered 1 dn 1 similar to the Cases b,and c in Figure 2, one may conclude thermo-acoustic system is stable. that there is wide freedom to select R such that |R |≤ 1 dn in Kojourimanesh et al. 87 Figure 2. Four possible cases for the Mobius transformation of the unit circle of R into R plane. dn in Figure 3. Mobius transformation of R unit circle into R R . dn up in andevenitispossible todesign a proper R where one can easily show that the location of the center, radius dn |R |= 0. Elaboration of these design guidance ideas is the and point O in the R R plane would be the scaled one in up in subject of the next sections. It should be noted that the in the R plane. Equation (15) and Figure 3 show the loca- in shaded rectangular areas in Cases c,and d in Figure 2 are tion of the center, radius and point O of the circle in the outside domains of the mapped circle. R R plane which is mapped from the R plane. up in dn For the case | R | ≠ 1, one can apply the same strategy up as in equation (7) but for MT of M = R M & O = R & r =|R |r (15) new up new up new up (−R Δ )R + (R S ) up dn up 11 R R = . (14) up in −S R + 1 22 dn Figure 3 depicts the MT of the unit circle in the R plane dn into the R R plane. Comparing Figure 2 with Figure 3, up in Then, the unit circle in R plane maps into a circle in and also equations (13) and (15), one can conclude that dn R R plane. Considering the formula in equation (13), the properties of the disk in the R R plane, i.e., up in up in 88 International Journal of Spray and Combustion Dynamics 14(1-2) Figure 4. MT of R unit circle into R R for 3 different R . dn up in up S +S −B 11 22 M , O , and r , can be deduced from the properties By comparing the expression of with M = and new new new 2 2A 1−|S | of the disk in R plane, i.e., M, O, and r, by scaling them considering S =−S and C < 0, it is obvious that in 11 22 with the factor |R | and rotating it with the phase of R . up up Besides, equation (15) suggests that by decreasing the =|M|. 2A magnitude of R , the center of the corresponding disk in up Therefore, one can relate the ﬁrst root of the quadratic the R R plane and the point O would converge to up in new equation to the MT circle parameters as the origin and the radius of the disk goes to zero as illu- strated in Figure 4. In other words, the whole disk of −B B − 4AC R R will be inside the unit disk, |R R | < 1. up in up in λ = + =−|M|+ r (17) 2A 2A Therefore, decreasing the magnitude of R causes a high up chance of stable system operation (note, that the possibility For the case |R |= 1, |R |= 1, Appendix C.2 provides the up dn of the intrinsic thermoacoustic instability still remains). prove that the Edwards-Sinsky factor is indeed λ .Hence, μ =−|M|+ r (18) The other point is that the values for the center and radius depend on each other for the system deﬁnedinequation(2) Comparing algebraic and MT methods’ results when |R |= 1. Due to the equality A =−C, one can up In this section, we are aiming to search for correspondences write the relation between them as between the results derived from the algebraic and the MT methods. In the aforementioned thermo-acoustic two-port 2 B − 4A(−A) B r = = + 1 = M + 1 (19) system, i.e., obeying equation (2), the symmetry implies 4A 2A that S =−S and therefore Δ =−1, accordingly 11 22 S S = 1 − S . As explained before for any complex z Equations (17) and (18) reveal the relation between 12 21 1−|z| function, ≤ 1. Then one can readily conclude that Edwards-Sinsky factors, μ & μ ,and MT parameters |1−z | the radius of the MT circle in the R plane would be which is shown in Figure 5. As can be seen, the in always bigger than 1, i.e., Edwards-Sinsky factor, μ, is the closest point of the R in disk from the origin i.e., |−|M|+ r|. Moreover, the second |1 − S | root of the quadratic equation (9) is the farthest point of the r = ≥ 1. (16) |1 −|S | | R disk from the origin. Thus, both approaches provide the in same results regarding the elaboration of criteria for the ana- lysis of (in)-stability of the system operation. In addition, for the case |R |= 1, the expression √ up 2|R ||S S | B −4AC up 12 21 = would be the same as r, i.e., 2 2 2 2A 2(|R | |Δ| −|S | ) up 22 B − 4AC. Optimal value of R to obtain minimum value of R dn in r = 2A In this subsection, an optimum value of R is derived dn −|S +S | −B 22 11 The same procedure conﬁrms that = . which provides a minimum value of R at given entries 2 in 2A 1−|S | 22 Kojourimanesh et al. 89 behaves as a passive subsystem and therefore it will be stable in combination with any passive upstream acoustics of the considered appliance. The region of R , which we are looking for, belongs to a dn section of the unit circle in the plane of R which maps dn these R to the double shaded (stable) regions in the dn plane of R , see Figure 2. To ﬁnd the corresponding in area in the R plane, one may use the inverse transform- dn ation of equation (7). For this case, the inverse transform- ation is also of Mobius kind, and it is given by the expression −dR + b in R = . (21) dn cR − a in We are interested where the unit circle of R maps into in Figure 5. Relation of MT, algebraic equation and μ . the R plane. The mapping is similar to the one presented dn in Figure 2 with the only replacement of R and R planes. dn in The described procedure provides the area inside the unit circle of R which is considered as a preferable value of dn of the burner’s scattering matrix, S. As shown in Figure 2, a R to obtain a passive stable system without considering dn minimum value of R is equal to either zero for cases b and in the upstream acoustic’s condition. c or it is equal to k for the cases a and d. It has been shown that |k| is actually the Edwards-Sinsky factor, μ. Therefore, the value of R can be restored which causes the minimum dn Case study value of R by inverse mapping at point k back to R in dn plane. Hence, To show the power of this methodology, a test setup, shown in Figure 6, is prepared to measure the FTF, R , and R in −dk + b dn in R = . (20) dn opt. the frequency domain. A constant temperature anemometer ck − a (CTA) and a photomultiplier tube (PMT) with OH ﬁlter are It is obvious that for the cases b and c the optimum value used to measure the acoustic velocity ﬂuctuation before the of R is R =− . dn dn opt. ﬂame and the varying heat release signal, respectively. Two Bronckhorst mass ﬂow controllers control the methane and airﬂow rates. A water-cooled burner deck holder is used to Possible range of R to obtain a passive stable dn keep the outer perimeter of the burner deck at a ﬁxed tem- system perature to obtain a steady ﬂame and FTF independent from The practical motivation of the considerations presented the time of measurement. A set of National instruments below stems from the fact that many producers of burners DAQ systems connected to a PC running LabVIEW is for industrial or domestic applications tend to combine used to transfer the data with a sampling rate of 10 kHz. the burner with the downstream part of an appliance as a A loudspeaker is installed at the end part of the upstream uniﬁed product. A typical example is the combination of a side to provide a forced excitation supplied by the sequence burner and heat-exchanger as the “power module”. of pure tone signals. Software written in LabVIEW is Accordingly, only the upstream side (for instance, the implemented to control the amplitude of the excitation blower/fan, venturi, snorkel, etc.) is designed by the boiler force, sampling rate, measuring time duration, as well as manufacturer. In this situation, designers of the power setting the desired range, steps of frequencies, data prepro- module would like to optimize their product in terms of cessing, etc. ensuring the thermo-acoustic stability for the module in com- The upstream side has a broadband damper with |R | up bination with any/arbitrary acoustics of the upstream part of less than 0.4 for the frequency range of 15–310 Hz. More an appliance. details of the design of the upstream damper, the setup, In this subsection, the conditions are investigated how to and the measurement procedures can be found in paper. select a passive termination R in such a way that for the The downstream part of the burner is a 110 mm long dn given burner transfer matrix the corresponding magnitude quartz tube with an open end. The reﬂection coefﬁcient of of R becomes less than 1 over a certain frequency range. the downstream side is measured immediately after in The idea to ensure |R | < 1 is motivated by the fact that turning off the ﬂame to obtain a reasonable estimation of in in this case the burner with downstream acoustics R value, i.e., close to the hot condition. Figure 7 shows dn 90 International Journal of Spray and Combustion Dynamics 14(1-2) Figure 6. A test setup to measure the FTF, R , and R . dn in Results and Discussions In this section, the analytical results derived in section 2 are examined for the case study expressed in section 3. Stability analysis using Flame Transfer Function For the case study as deﬁned in the section 3, the Rollett sta- bility factor K, and Edwards-Sinsky parameter μ are calcu- lated based on equations (3) and (6), respectively. The results are presented in Figure 10. As mentioned before, due to symmetry features of the transfer matrix for thermo-acoustic systems, both Rollett factor and Edwards-Sinsky parameter should be less than Figure 7. Measured R , and R of the test setup. up dn 1, K ≤ 1, μ ≤ 1. Therefore, it is not surprising that Th TA this fact is reﬂected in Figure 10 for the case study. By increasing the frequency from 15 to 220 Hz, K and μ the reﬂection coefﬁcients of the upstream and downstream decrease. In the practice of microwave circuits design, the sides of the burner in the frequency domain. value of μ is also used as the qualitative measure/indicator As an example, the ﬂame transfer function for a brass of the active two-port potential of (in)-stability. plate burner with premixed burner-stabilized Bunsen-type Accordingly, it also reﬂects the qualitative measure for the ﬂame is measured. The burner deck is a disk that has a potential of (in)-stability when the considered two-port is thickness of 1 mm, and a diameter of 5 cm. The hexagonal embedded between arbitrary passive up and downstream ter- pattern of round holes with a diameter of holes of 2 mm and minations. The argument which substantiates such the role of the pitch between the holes of 4.5 mm is used. For brevity, μ factor is the following. If to estimate the potential of stabil- the burner is called “D2P4.5”. The total open area of the ity at each frequency as the ratio of the area of the double burner is 399 mm. The measured ﬂame transfer function shaded region and the area of the unit circle in R (shown in is used to calculate coefﬁcients a, b, c, d of the MT. in Figures 2 and 5), then exists the monotonic inverse relation Figure 8 shows the ﬂame transfer function of burner between μ and the area ratio. Therefore, by decreasing μ from D2P4.5, at the mean velocity of the mixture through the 15 to 220 Hz, the ratio of the areas is decreased, also the burner holes of 70 cm/s and equivalence ratio of ϕ = 0.7 potential of stability at that frequency is decreased. For the in the frequency range 15–310 Hz. considered test case the minimum value of μ happens at The impedance tube with 6 microphones shown in Figure 6, the vicinity of the frequency of 220 Hz. It is worth noticing including the low reﬂecting loudspeaker box (damper) at the that it is the same frequency where the phase line of the ﬂame upstream side, is used to measure the reﬂection coefﬁcient transfer function is crossing −π, see Figure 8, which is the R from the combination of a bended supply tube, the burner indication of the burner intrinsic mode. The correlation in with ﬂame, and the downstream subsystem. The measured between the frequency ranges of high potential of instability data of R in the frequency domain is plotted in Figure 9. The of thermo-acoustic system and frequencies of the burner in 33,36 31,32 measurement procedure is explained in detail in papers. intrinsic modes was reported earlier. Kojourimanesh et al. 91 would be the possible upstream reﬂection coefﬁcient to sta- bilize the system. To do that, one can substitute the values of scattering matrix entries and |R | for each frequency dn between 15 and 310 Hz in equation (8.b). If the right-hand side of equation (8.b) becomes negative, then it implies that the system could be unstable at that frequency. Consequently, by changing the upstream part, the system cannot be fully stabilized for all phases of R , i.e., any dn length of the exhaust duct. On the other hand, if the RHS of equation (8.b) is positive then for a ﬁxed value of |R | dn and arbitrary phase of R , the region of |R |where the dn up system remains stable is determined. For the case study under consideration, this region is plotted in Figure 11. Figure 8. Measured ﬂame transfer function of the burner cm As can be seen in Figure 11, for frequencies higher than D2P4.5 at v = 70 and ϕ = 0.7. 120 Hz there is no value of R that guarantees the stability up of the system for the ﬁxed value of |R | with arbitrary dn phase (length) of the downstream side. Note that this fre- quency is exactly the same as the frequency where K and μ start to become negative. However, for a frequency less than 120 Hz, the blue area in Figure 11 gives the possible magnitude of |R | to guarantee the stability of the system up at those frequencies. The reﬂection coefﬁcient of the upstream side can be also designed from the MT strategy. Figure 12 shows the results of mapping the unit circle of R ﬁrst into the dn plane of R i.e., equation (7) (interior area of the dark in blue circle) and second, into the plane of R R plane up in i.e., equation (14) (interior area the light blue circle), at the frequency of 100 Hz. As can be seen, the size of the cm Figure 9. Measured R at v = 70 and ϕ = 0.7. in circle in R R plane is much smaller than the one in the s up in plane of R , and only a section of mapped R (purple in in shading area) is inside the unit circle. However, the whole Upstream design to stabilize the system circle of mapped R R (turquoise shadings area) is inside up in Equation (8.a) reveals that for a speciﬁc R , it is possible to the unit circle. Equation (15) reveals the reason for this in design R such that the system becomes stable, i.e., property of the mapping that applying the low reﬂecting up |R | < . For the particular case under consideration, termination at the upstream side, i.e., |R |= 0.13 at up up |R | in the maximum value of the upstream reﬂection coefﬁcient 100 Hz, the whole disk of R R will endupinsidethe up in is max(|R |) = 0.4 which is two times smaller than unit disk. up Min = 0.81. Therefore, the system is stable at this con- In addition, the system with a high value of the upstream |R | in dition. In general, this system with a deﬁned R value, i.e., reﬂection coefﬁcient, |R | ∼ 1, cannot be fully stabilized/ dn up ﬁxed magnitude and ﬁxed phase, would be stable if passivated. The reason, as can be deduced from equation max(|R |) < 0.8. However, for other values of magnitude (16), is that the radius of the mapped disk in R plane is up in and phase of R ,the value of |R | could be much higher. higher than one. Therefore, some particular values of R dn in dn When this is the case, an upstream termination with a lower can cause |R |>1. In general, it can be argued that by provid- in reﬂection coefﬁcient, for all frequencies, will be needed. ing a low reﬂecting upstream termination, one may attempt to Besides, one can get the advantages of equation (8.b) to increase the potential of stability for any R and/or R . in dn design the upstream reﬂection coefﬁcient. It should be noted that the phase of the reﬂection coefﬁcient at the Downstream design for stability downstream side can be easily changed, like by varying the length of the exhaust duct. However, changing the mag- Equation (8.c) could help to design the downstream part for nitude of the downstream reﬂection coefﬁcient needs a a ﬁxed value of |R | with arbitrary phase (length) of the up dedicated gas path design or even external equipment like upstream side such that the system would be thermo- damper, mufﬂer, etc. Therefore, one of the design strategies acoustically stable. Figure 13 shows the possible values is to infer for the ﬁxed value of |R | with arbitrary phase of of |R | that provide the stability of the system studied as dn dn R , what will happen with the system. It may suggest what the case study, for the ﬁxed value of |R | with arbitrary dn up 92 International Journal of Spray and Combustion Dynamics 14(1-2) Figure 12. MT of R unit circle into R and R R planes at dn in up in 100Hz (interior areas of dark-blue and light-blue circles respectively). Purple and turquoise shadings signify overlapping of the circles’ interior with the unit circle, respectively. Figure 10. Calculated K and μ for the deﬁned FTF. Figure 11. Region of |R | to have a stable system, for the ﬁxed up Figure 13. Region of |R | to have a stable system, for the ﬁxed dn value of |R | and arbitrary phase of R of the case study. value of |R | and arbitrary phase of R of the case study. dn dn up up phase (length) of the upstream side at frequencies of 35– equation allows deﬁning the region of R such that dn 310 Hz. |R | < 1, which in its turn means that the system becomes in Figure 13 demonstrates that even for the ﬁxed low passive and stable for any passive upstream termination. magnitude of the reﬂection coefﬁcient of the upstream To demonstrate this design option for the case study, R dn termination |R |, it is possible that the system circles and the possible ranges of R at which |R |≤ 1 dn in up becomes unstable in the frequency range between 210 are calculated using equation (21). At four arbitrary selected and 225 Hz. It should be noted that this range of frequen- frequencies, the R circles and centers’ locations of the dn cies is close to the intrinsic instability of the burner with circles and those preferable R areas are shown in dn ﬂame. Figure 14 with purple color. As expressed before, some combustion companies prefer By developing these ideas further, one may suggest to sell their product without the upstream side. Accordingly, that the “volume” of possible R values (blue region dn the inverse transformation written in equation (21) could shown in Figure 15) may provide a good indication for provide a method how to ensure the complete system stabil- the kind of “ﬁgure of merit” of the burner. It means ity for any values of magnitude and phase of R . This that a good burner in terms of the thermo-acoustic up Kojourimanesh et al. 93 Figure 14. R circles, centers’ locations, and possible R map to the unit circle of R at 30, 75, 145, 265 Hz. dn dn in stability would have a larger choice of possible (stabi- with a minimum value of R where the system becomes in lizing) R values. passive and therefore stable for any passive upstream ter- dn Figure 15 demonstrates a 3D view of the discussed mination of the system. For the considered case, the area in a frequency range 15–310 Hz. The four slices optimal value of R such that the magnitude of R dn in (joint areas) for particular frequencies as shown in becomes minimum is derived from equation (20) and Figure 14 arealsomarkedinFigure15withthe red results with a corresponding minimum value of R are in color. If the reﬂection coefﬁcient of the downstream demonstrated in Figure 16. side of the burner is lying inside this area (looking as a These graphs also highlight the conclusion that for channel), then the system would be at a passive stability the considered burner/ﬂame the frequencies around condition otherwise, the system will not be passive and 220 Hz may require special measures to ensure system the solution of the dilemma of stability-instability is stability. related to the product of upstream and inlet reﬂection coefﬁcients, i.e., R R . up in Conclusions Figure 15 could also be a productive tool for further analysis. For instance, one may conclude that at low frequen- It is demonstrated that the system-level analysis of a cies, like 15–40 Hz, for almost any downstream termination network of two-ports is a very fruitful tool to perform inves- of the mentioned burner at speciﬁed mean velocity and tigations of many aspects of combustion acoustic instability equivalence ratio condition, the system would be passive phenomena. Particularly, it provides promising approaches and, therefore, stable. The frequency ranges around 220Hz to the task of system design aiming stability of appliance look the most problematic to stabilize because for down- operation. Original algebraic proofs of conditional and stream acoustics the range of values of reﬂection coefﬁcients unconditional stability criteria of linear two-port network needed for stabilizing the system is narrow at these systems are proposed. It has been shown that thermo- frequencies. acoustic systems cannot be unconditionally stable. Hence, Continuing the case study further, by minimizing R , the the conditional stability criteria have been investigated in maximum potential of stability could be achieved. based on the algebraic technique. Also, a complementary Therefore, an optimal R should be found that minimizes framework of analysis is proposed which is based on the dn R . Equation (20) can be applied to ﬁnd the optimal R application of known properties of bilinear MT. The in dn 94 International Journal of Spray and Combustion Dynamics 14(1-2) Figure 15. A 3D view of possible R maps to the unit circle of R in a frequency range 15–310 Hz. The four slices shown in Figure 14 dn in are highlighted in red. applied to analyze the stability of the thermo-acoustic system. However, the technique based on the MT would provide more insightful information and a better visual and intuitive interpretation of results than the two other techniques. The elaborated criteria of system stability can be applied for purposeful design of the upstream and downstream sides of the given burner with ﬂame to provide thermo-acoustic system stability. Furthermore, the elaborated approaches show princi- pal requirements and propose the receipts how to design the acoustics of upstream and downstream sides of the burner such that the system operation would be stable. Also, for each frequency, the area of possible R dn values to obtain a passive stable system is derived. The area depends on the ﬂame transfer function of the burner. Hypothetically, the size (volume or another measure) of the area can be considered as a possible can- didate for an indication for the ﬁgure of merit of the burner. To conﬁrm the validity and usefulness of the proposed ideas related to the combustor design strategies and evalu- ation of the ﬁgure of merit of burners, further theoretical development, and experimental checks are needed. This work is in progress and promises new directions for Figure 16. Optimal R (top plot) to obtain a minimum amount dn of R (bottom plot) at 15–310 Hz. in further research. Particularly, for the experimental veriﬁca- tions of the theoretical results new setup with a wide range of variable reﬂection coefﬁcients at both up- and down- stream terminations is developed. The ﬁnal goal of future comparison of different approaches reveals relations R&D works in this direction would be the elaboration of between the results of the algebraic derivations, the geo- a convenient designer’s toolbox supporting and suggesting metrical approach applied in microwave theory, and the MT technique. Any of these three approaches can be design decisions. Kojourimanesh et al. 95 16. Edwards ML and Sinsky JH. A new criterion for linear Declaration of Conﬂicting Interests 2-port stability using a single geometrically derived par- The author(s) declared no potential conﬂicts of interest with ameter. IEEE Trans Microw Theory Tech 1992; 40: respect to the research, authorship, and/or publication of this 2303–2311. article. 17. Olivieri M, Scotti G, Tommasino P, et al. Necessary and suf- ﬁcient conditions for the stability of microwave ampliﬁers with variable termination impedances. IEEE Trans Microw Funding Theory Tech 2005; 53: 2580–2586. The author(s) disclosed receipt of the following ﬁnancial 18. Marietti P, Scotti G, Triﬁletti A, et al. Stability Criterion for support for the research, authorship, and/or publication of this Two-Port Network With Input and Output Terminations article: This work was supported by the Nederlandse Varying in Elliptic Regions. IEEE Trans Microw Theory Organisatie voor Wetenschappelijk Onderzoek, (grant number Tech 2006; 54: 4049–4055. 16315). 19. Balsi M, Scotti G, Tommasino P, et al. Discussion and new proofs of the conditional stability criteria for multidevice microwave ampliﬁers. IEE Proc - Microwaves, Antennas ORCID iD Propag 2006; 153: 177. 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Combustion thermoacoustics 54: 1617–1618. in context of activity and stability criteria for linear two-ports. 15. Owens PJ and Woods DM. Reappraisal of the unconditional In: Proceedings of the European combustion meeting, stability criteria for active 2–port networks in terms of S para- Dubrovnik, Croatia, 18-21 April, 2017, pp. 18–21. meters (1970). 96 International Journal of Spray and Combustion Dynamics 14(1-2) 33. Kojourimanesh M, Kornilov V, Arteaga IL, et al. A.1) Rollett factor for Unconditional stability of TA Thermo-acoustic ﬂame instability criteria based on 2 2 2 upstream reﬂection coefﬁcients. Combust Flame 2021; Suppose z = x + iy then |z| = zz = x + y , 2 2 2 2 2 225: 435–443. |z|= x + y , and z = (x − y ) + 2xyi. 34. Kojourimanesh M, Kornilov V, Lopez Arteaga I, et al. Rollett factor in thermo-acoustic systems is Mobius transformation between reﬂection coefﬁcients at 1−|z| K = where z = . |1−z | T upstream and downstream sides of ﬂame in thermoacoustics 2 2 2 1−|z| 1−(x +y ) Hence, K = = therefore, systems. In: The 27th International Congress on Sound and 2 2 2 |1−z | |(1−x +y )−2xyi | Vibration. Prague, Czech Republic, 11-16 July 2021. 2 2 2 1 − x + y − 2y 35. Poole C, Darwazeh I and Outcomes IL. Gain and stability of K = . active networks. In: Microwave Active Circuit Analysis and 2 2 2 (1 − x + y ) + (−2xy) Design, 205–244. 2 2 1 − x + y 36. Kojourimanesh M, Kornilov V, Lopez Arteaga I, et al. Also, it is clear that ≤ 1 and Theoretical and experimental investigation on the linear 2 2 2 (1 − x + y ) + (−2xy) growth rate of the thermo- acoustic combustion instability. In: Proceedings of the European Combustion Meeting. −2y ≤ 0. Therefore K ≤ 1. Napoli, Italy, 14-15 April, 2021. 2 2 (1 − x + y ) + (−2xy) Appendix A.2) Edwards-Sinsky factor for Thermo-Acoustic 1 −|S | A.0) Transfer matrix and Scattering matrix of TA μ = |S − S |+|S S | 22 12 21 By considering the burner/ﬂame as an acoustically compact lumped element, the relationship between pressure and vel- 1−− ocity ﬂuctuations at the upstream and downstream parts of 11 = . ∗ 2 the ﬂame (in the limit of zero mean ﬂow Mach number) can T T T 21 21 21 + (−1) + 1 − ( ) be described via the Transfer matrix as T T 21 T 11 11 11 1−− p 10 up dn 11 = . In terms of ′ μ = u 2 u 01 + θ FTF up dn T T 21 21 2Im + 1 − ( ) Riemann invariants (f, g), pressure and velocity can be T T 11 11 written as 2 2 1−|z| 1−|z| ′ ′ As mentioned before ≤ 1 hence, ≤ 1. p = ( f + g )ρ c p = ( f + g )ρ c 2 2 1 1 up 2 2 dn up dn |1−z | |2Im(z)|+|1−z | up dn ′ ′ u = f − g u = f − g 1 1 2 2 up dn B.1) Quadratic equation for Thermo-Acoustic systems ( f + g ) = ε( f + g ) 2 2 1 1 S −ΔR 2 Therefore, . 11 2 We know R = , |R R | < 1, in up in f − g = (1 + θ FTF )( f − g ) 1−S R 2 2 s 1 1 22 2 2Re(z) = z + z, Re(z z ) ≤|z ||z | 1 2 1 2 By calculating g from the second row and put it in the 2 2 2 and |z − z | =|z | +|z | − 2 Re(z z ). Therefore, ﬁrst row then simplify, one can derive 1 2 1 2 1 2 2 2 2 |R R | < 1 |R (S − ΔR )| < |1 − S R | . f ε + 1 + θ FTF ε − 1 − θ FTF f 2 s s 1 up in up 11 dn 22 dn = 0.5 , g ε − 1 − θ FTF ε + 1 + θ FTF g 2 s s 1 Expanding the power results in ρ c 1 1 where ε = . The similar approach should be followed to 2 2 2 ρ c 2 2 |R | (|S | +|ΔR | − 2 Re(S ΔR )) < 1 up 11 dn 11 dn acquire the scattering matrix in terms of TM elements; 2 2 therefore, one should calculate f from the second row +|S | |R | − 2 Re(S R ). 22 dn 22 dn and put it in the ﬁrst row to ﬁnd g ; then substitute the trans- Applying 2Re(z) = z + z, one writes fer matrix elements and simplify the equations. Then the 2 2 2 2 2 2 below expression will be derived. |R | |S | +|R | |Δ| |R | − [|R | S ΔR − up 11 up dn up 11 dn g −2T 4 f 2 2 2 1 21 1 |R | S ΔR ] − 1 −|S | |R | + = . up 11 dn 22 dn 2 2 f T − T 2T g 2T 2 21 2 11 11 21 [S R + S R ] < 0. 22 dn 22 dn Hence, the determinant of the scattering matrix would be 2 2 Factorizing|R | , R , and R concludes dn dn dn (−2T )(2T ) − 4(T − T ) 21 21 11 21 Δ = =−1. 2 2 2 2 2 4T (|R | |Δ| −|S | )|R | + [R (S −|R | S Δ) 11 up 22 dn dn 22 up 11 Kojourimanesh et al. 97 2 2 2 Substituting A, B, C deﬁned in Appendix B.1 leads to +R (S −|R | S Δ)] +|R | |S | − 1 < 0. dn 22 up 11 up 11 −2|S −|R | S Δ|+ 2|R ||S S | 22 up 11 up 12 21 Employing z + z = 2Re(z), one ﬁnds λ = . 2 2 2 2(|R | |Δ| −|S | ) up 22 2 2 2 2 (|R | |Δ| −|S | )|R | up 22 dn Also, the same procedure can be used to prove equation 2 2 2 +2 Re(R [S −|R | S Δ]) +|R | |S | − 1 < 0. dn 22 up 11 up 11 (8.b) from |R R | < 1. dn out C.1) A new proof of Balsi et al. criterion Generally speaking, if A|R | + 2|z ||z |+ C is smaller dn 1 2 than zero then for sure A|R | + 2Re(z z ) + C would be dn 1 2 Balsi et al. showed that necessary and sufﬁcient conditions less than zero because of Re(z z ) ≤|z ||z |. Therefore, 1 2 1 2 for conditional stability can be ascertained by means of a one allows to consider below inequality instead of the last single parameter. Their theorem was, inequality. ‘Provided that the S-parameters deﬁned for at least one 2 2 2 2 2 pair of positive constant reference impedances have no (|R | |Δ| −|S | )|R | + 2|R ||S −|R | S Δ| up 22 dn dn 22 up 11 RHP poles, the necessary and sufﬁcient condition for a 2 2 linear active two-port to be stable is +|R | |S | − 1 < 0. up 11 1 −|S | R Therefore, if 11 up > 1. |S − S ΔR |R +|S S |R R 22 11 dn 12 21 up dn up A|R | + B|R |+ C < 0 dn dn Proof. As shown in Edwards-Sinsky paper, one can readily 2 2 2 where A = (|R | |Δ| −|S | ); up 22 show that B = 2|S −|R | S Δ|; 22 up 11 2 2 2 |S | −|R | |Δ| 22 up 2 2 C =|R | |S | − 1, up 11 2 2 2 2 |S −|R | S Δ| −|R | |S S − Δ| 22 up 11 up 11 22 = . 2 2 then for sure |R R | < 1. up in 1 −|R | |S | up 11 B.2) By substituting the term into the denominator of equation 2 2 2 B − 4AC = 4|S −|R | S Δ| 22 up 11 (8.c) and simplify it, one can derive 2 2 2 2 2 2 2 − 4(|R | |Δ| −|S | )(|R | |S | − 1). up 22 up 11 1 −|R | |S | up 11 |R | < . dn |S −|R | S Δ|+ |R ||S S | Expanding the equation and applying 2Re(z) = z + z, 22 up 11 up 12 21 one writes By moving |R | to the right side of the earlier equation, the dn 2 4 2 conditional stability criterion, provided by Balsi et al., is 4[|S | +|R | |S Δ| − 22 up 11 derived. 2 2 4 2 2 S |R | S Δ − S |R | S Δ −|R | |Δ| |S | 22 up 11 22 up 11 up 11 C.2) A new proof of Edwards-Sinsky criterion 2 2 2 2 2 2 −|R | |Δ| −|S | |R | |S | −|S | ]. up 22 up 11 22 Substituting |R |= 1 and |R |= 1 into the aforemen- up dn tioned equation (last equation in C.1), it is easy to realize Simplifying the equation leads to that the right-hand side of the equation is indeed the 2 2 2 Edwards-Sinsky criterion. B − 4AC = 4|R | |S S − Δ| up 11 22 2 2 2 1 −|S | B − 4AC = 4|R | |S S | ≥ 0. up 12 21 1 < = μ. |S − S Δ|+|S S | 22 11 12 21 The authors disclosed receipt of the following ﬁnancial B.3) support for the research. This work was supported by the −B + B − 4AC Netherlands Organization for Scientiﬁc Research (NWO) λ = 2A [16315] with the project name of STABLE. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png International Journal of Spray and Combustion Dynamics SAGE http://www.deepdyve.com/lp/sage/stability-criteria-of-two-port-networks-application-to-thermo-acoustic-z6o3F4f4pw

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Publisher: SAGE
Copyright: © The Author(s) 2022
ISSN: 1756-8277
eISSN: 1756-8285
DOI: 10.1177/17568277221088465
Publisher site: See Article on Publisher Site

Abstract

System theory methods are developed and applied to introduce a new analysis methodology based on the stability criteria of active two-ports, to the problem of thermo-acoustic instability in a combustion appliance. The analogy between thermo-acoustics of combustion and small-signal operation of microwave ampliﬁers is utilized. Notions of unconditional and conditional stabilities of an (active) two-port, representing a burner with ﬂame, are introduced and analyzed. Unconditional stability of two-port means the absence of autonomous oscillation at any embedding of the given two- port by any passive network at the system’s upstream (source) and downstream (load) sides. It has been shown that for velocity-sensitive compact burners in the limit of zero Mach number, the criteria of unconditional stability cannot be fulﬁlled. The analysis is performed in the spirit of a known criterion in microwave network theory, the so-called Edwards-Sinsky’s criterion. Therefore, two methods have been applied to elucidate the necessary and sufﬁcient condi- tions of a linear active two-port system to be conditionally stable. The ﬁrst method is a new algebraic technique to prove and derive the conditional and unconditional stability criteria, and the second method is based on certain proper- ties of Mobius (bilinear) transformations for combinations of reﬂection coefﬁcients and scattering matrix of (active) two- ports. The developed framework allows formulating design requirements for the stabilization of operation of a combus- tion appliance via purposeful modiﬁcations of either the burner properties or/and of its acoustic embeddings. The ana- lytical derivations have been examined in a case study to show the power of the methodology in the thermo-acoustics system application. Keywords Burner as an active two-port, Edwards-Sinsky’s criterion, Rollett factor, Conditional stability, Mobius Transformation Date received: 29 November 2021; accepted: 17 February 2022 availability of a purely acoustic characterization of the burner Introduction with ﬂame is the prerequisite of the model. This is achievable Thermo-acoustic combustion instability manifests itself as a within the concept of the transfer matrix (T)orscattering high level of tonal noise, vibration, and may cause the per- 7 matrix (S). Then, a network model of the combustion system formance deterioration or even structural damage of com- is obtained when all two-port components are combined. bustion appliance. The ability to eliminate and/or control The methodological similarity of approaches to and the combustion instability at the appliance design phase is network models equivalence of the electrical circuits and one of the main goals of combustion-acoustics research. The low-order (acoustic network) modeling approach is one of the intensively developing tools which has proven Department of Mechanical Engineering, Eindhoven University its efﬁciency in performing problem analysis, synthesis, of Technology, Eindhoven, The Netherland and eventually the appliances design tasks. Department of Engineering Mechanics, KTH Royal Institute 1–3 of Technology, Stockholm, Sweden Various acoustic network models have been developed that are used to analyze combustion thermo-acoustic instabil- Corresponding author: ities and the design of combustion equipment in recent Mohammad Kojourimanesh, Eindhoven University of Technology, Building 4–7 studies. Thismodelingallowstreatingcombustionappliance 15, Gemini Noord 1.44, PO BOX 513 5600 Eindhoven, the Netherlands. 4,8,9 components as acoustic two-ports. Accordingly, the Email: m.kojourimanesh@tue.nl Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https:// creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as speciﬁed on the SAGE and Open Access page (https://us.sagepub.com/en-us/nam/open-access-at-sage). Kojourimanesh et al. 83 combustion acoustic systems have been shown in various The main focus was on formulating thermo-acoustic papers since 1957. However, the stability analysis and network models, predicting the instability, searching for design methods of the two-port networks have not been unstable frequencies, calculating/measuring the growth developed/applied in the combustion ﬁeld as much as in rate, searching methods for stabilizing systems, etc. microwave theory. The linear two-port network theory References to most of the performed research can be has been an intensively developing research subject and found in the review paper. The conventional methodology the results have been applied ubiquitously in the practice for analyzing the stability of thermo-acoustic systems con- of microwave devices’ design. The closest analogy can be sists of measuring/modeling the Flame Transfer Function established between the combustion thermo-acoustic (FTF). Then, a wave-based 1D linear two-port network instability problem and the problem of stability of operation approach is applied to provide the system matrix. The of microwave ampliﬁers. Here, the burner with ﬂame and eigenfrequencies of the system (zeros of the matrix’ deter- the ampliﬁer (e.g., transistor) both represent a so-called, minant) determine the (in)-stability frequencies and “dependent source” or active element. Furthermore, the growth/decay rates. Therefore, the common practice is to acoustics of the burner upstream and downstream parts in create a system matrix each time when the system is a combustion appliance are an analogy of the “source” altered to check the corresponding effect on the eigenfre- and “load” passive network embeddings of the microwave quencies. This procedure allows resolving the dilemma of ampliﬁer. One of the extremely useful and well-developed the (in-)stability of operation of a particular system and concepts of the microwave ampliﬁer’s design process is gives a reasonably accurate prediction of frequencies of the notion of unconditional stability. oscillation. This approach has been successfully applied 7,25,26 In microwave theory, this means that there is no passive in numerous studies before. However, the conven- source and passive load combination that can cause the tional modeling approach does not provide a good overview circuit with the given ampliﬁer to oscillate. Correspondingly, of conditions and guidelines for designing the upstream and the unconditional stability in a thermo-acoustic context downstream sides of the ﬂame such that the system would means operation stability regardless of the (passive) acoustics be stable. The crux is in the absence of speciﬁc parameters at upstream and downstream sides of the burner/ﬂame. The or criteria to determine the system stability and lack of tools pioneering work on this subject was done by Rollett in (rules) on how to manipulate the system design as it is done 1962. He introduced a quantity (criterion) to characterize in microwave theory. the degree of stability. Later, it was shown that the combin- A new impetus in the development of system-level ana- ation of validity of certain inequality requested from the lysis of thermo-acoustic network models was given by the Rollett factor together with only one other auxiliary condition discovery of the phenomenon of the burner intrinsic 27,28 are necessary and sufﬁcient to provide unconditional stabil- thermo-acoustic mode of instability. This and further 12–15 ity. research on the subject use system theory. Particularly, In 1992, Edwards and Sinsky proposed a single param- the derived system instability conditions are based on the eter, instead of two of Rollett’s conditions, to determine the gain and phase of the TFT for only ITA modes. A necessary and sufﬁcient unconditional stability require- review of literature on this subject can be found in the 16 30 ments. The arguments and analysis were based on a geo- recent publication. metrical approach. Various applications and design tools Another research direction was introduced by Kornilov based on the Edwards-Sinsky criteria were developed and and de Goey who showed the analogy between the thermo- 17–19 31 discussed. Particularly, Balsi et al. extended the geo- acoustic and microwave circuits linear two-port networks metrical approach and derived the necessary and sufﬁcient and use it to investigate two unconditional stability criteria, conditions for a linear active two-port to be conditionally of ‘Rollett’ and ‘Edwards-Sinsky’, for the purpose of 20 32 stable. A recent work of Lombardi and Neri presented evaluation of a burner/ﬂame ﬁgure of merit. In turn, the existence of a duality mapping between the input and this work gave the inspiration to develop a prospective the output of the two-port network; then by using certain method to assess thermo-acoustic instabilities based on properties of Mobius Transformation (MT), they demon- reﬂection coefﬁcients measured only from the upstream strated all possible cases of mapping between the input side of the burner (cold side) by Kojourimanesh et al.. and the output of the system. MT is the bilinear rational In this approach, two reﬂection coefﬁcients, R and up transformation as one of the mathematical concepts R , at the cold side of the ﬂame are measured. As dis- in named after A.F. Mobius. It is well-known that the MT played in Figure 1, R is the reﬂection coefﬁcient of the up maps a line or circle into another line or circle. Çakmak upstream side of the burner and R is the input reﬂection in et al. derived explicit formulas relating the centers and the coefﬁcient of the burner terminated by R .Inother dn radii of the mapped circles. words, if we disconnect the network in Figure 1 from the On the other hand, the beginning of active development R , send in wave f and measure reﬂected wave g then up 1 1, of the acoustic network modeling approach to the problem the corresponding reﬂection coefﬁcient would be: R ≜ in of combustion instability falls in the period after ∼1990. g / f . 1 84 International Journal of Spray and Combustion Dynamics 14(1-2) Figure 1. Thermo-acoustic model of a combustion system. In this method, the stability of the system can be deter- ﬂame for which the purely acoustic representation in the mined by inspection of the Nyquist plot of the measured form of a two-port is known and given, e.g., by the R times R . They showed that the condition applied to burner transfer matrix. However, here we limit the consid- up in the magnitude of R R (iω) being less than 1 for all fre- eration to one particular type of burner, namely, an acoustic up in quencies from 0 to inﬁnity is sufﬁcient (but not neces- velocity-sensitive dependent source of acoustic velocity sary) to result in the thermo-acoustic stability of the (analogy of current sensitive current source in microwave system. Accordingly, in another study, they applied the theory). In this case, we may use some internal symmetries MT properties to provide the necessary conditions of of the transfer and scattering matrices. Physically, this type R to ensure that the magnitude of R becomes less of thermo-acoustic property is appropriate to a wide class of dn in than 1. perfectly premixed gaseous fuel burners operating in the The present paper contributes to the further development limit of low Mach numbers for the mean ﬂow when the of the research on the framework of the system-level ana- heat release zone is compact with respect to the acoustic lysis of thermo-acoustic instability of combustion and uti- wavelength under consideration. In addition, the conditions lizes the close analogy with the theory of microwave presented below should be satisﬁed for all frequencies from networks. The particular goal of the present contribution 0toinﬁnity. For brevity, we may discuss all relations for a is to introduce a new analysis methodology that is based ﬁxed frequency point but ﬁnally, all criteria and conditions on the stability criteria of active two-ports. The criteria should be satisﬁed for the whole frequency range. will be derived using the original approach based on prop- Furthermore, we will work in the frequency domain, and erties of a MT in combination with some algebraic transfor- consider only plane longitudinal waves, 1-D acoustics. The mations. The more general goal is to illustrate the power of network model variables will be represented by the forward the system analysis method in the application to thermo- and backward traveling waves f and g and the convention st acoustic network modeling and introduce an approach for the time dependence is e where s is the complex fre- that allows designing optimal terminations at the quency s = iω + σ, where σ is the growth rate. upstream/downstream sides of the ﬂame. The model of a thermo-acoustic system is ﬁrst written in the form of the network of a scattering matrix for power Stability criteria of thermo-acoustic waves to show the analogy with microwave theory and systems derive system stability conditions. Then, new algebraic For a compact velocity-sensitive ﬂame in the limit of zero proofs of unconditional stability, namely the mean Mach number the transfer matrix takes the form of Edwards-Sinsky criterion, and conditional stability are pro- posed. Besides, an alternative approach using MT is intro- ε + 1 + θ FTF ε − 1 − θ FTF T = 0.5 . (1) duced to determine the stability condition. Next, by ε − 1 − θ FTF ε + 1 + θ FTF combining the outcome of the aforementioned methods, conditional stability criteria for the thermo-acoustic systems are provided. Furthermore, the necessary condition In this notation, θ = − 1 is the temperature jump ratio; is derived which if it is satisﬁed by R ensuring that the T being the temperature at upstream and downstream dn 1,2 ρ c magnitude of R becomes less than 1 in a frequency sides of the ﬂame; ε = is the jump in characteristic in ρ c range. This condition results in passive thermoacoustics sta- acoustic impedance across the ﬂame; and FTF is the bility of the system’s operation. In addition, an optimum ﬂame transfer function which relates the oscillation of value of R is obtained which provides that the value of heat release rate of the ﬂame to the oscillation of acoustic dn R becomes minimum. In its turn, this promises a higher velocity at upstream of the burner and scaled to mean in potential of stability of the system. values of heat release and unburned gas velocity. The results obtained and presented below can be in prin- The corresponding scattering matrix of the thermo- ciple generalized to the case of an arbitrary burner with acoustic two-port can be deﬁned, if one rearranges Kojourimanesh et al. 85 equations of the transfer matrix (T) such that the ingoing In Appendix A.1, it is made clear that for any complex 1−|z| waves appear as inputs to the matrix, and the outgoing number z the function ≤ 1. Accordingly, based on |1−z | waves as outputs. In that case, the scattering matrix (S)of the Rollett criteria, the considered thermo-acoustic system the thermo-acoustic system would be cannot be unconditionally stable because one of the neces- sary conditions of unconditional stability is not satisﬁed, −2T 4 S = , (2) 2 2 namely, K >1. TA T − T 2T 2T 21 11 11 21 Alternately, it is also possible to prove Lemma 1 by ana- lyzing the Edwards and Sinsky parameter, μ. They proved that with the determinant of Δ =−1, where T are transfer ij the necessary and sufﬁcient condition to qualify a two-port as matrix elements (See Appendix A.0). an unconditionally stable element is μ > 1where The scattering matrix representation also includes the ITA mode as a special case when T = 0. However, in 1 −|S | μ = . (6) the present study, we will not focus particularly on pure |S − S Δ|+ |S S | 22 11 12 21 ITA mode analysis. In this notation, the bar symbol is used to denote the con- jugate of a complex number. By substituting parameters, like in Appendix A.2, one Unconditional stability in thermo-acoustic systems 1−|z| can also derive ≤ 1. Hence, for the thermo- |2Im(z)|+|1−z | Unconditional stability of a given burner with ﬂame in a acoustic system which obeys equation (2), μ ≤ 1. TA thermo-acoustic context means that regardless of the acous- Therefore, the thermo-acoustic two-port (a burner with tic reﬂection coefﬁcients of passive upstream and down- ﬂame) for which the transfer (scattering) matrix has sym- stream sides of the burner/ﬂame, the combined system metry properties, as in equation (1), cannot be uncondition- would be always thermo-acoustically stable. To establish ally stable. For brevity of notation, the index of TA is whether the unconditional stability can be ensured for the omitted from K and μ in the rest of the paper. TA TA thermo-acoustic two-port deﬁned in equations (1) and (2), the evaluation of the Rollett stability condition or the Edwards-Sinsky parameter can be performed. Conditional stability in thermo-acoustic systems Lemma 1. The deﬁned thermo-acoustic system in equa- tion (1), cannot be unconditionally stable. When considering the notion of conditional stability of a Proof. The Rollett stability condition says that the com- thermo-acoustic system, the upstream and downstream bination of Rollett stability factor K > 1 where acoustic boundary conditions are playing a role. One needs to search for the range of R (or R ) values in the 2 2 2 up dn 1 −|S | −|S | +|Δ| 11 22 K = , (3) complex domain such that the system would be always 2|S S | 12 21 stable. It is worth mentioning that outside of that range, the system may be stable or unstable. As expressed in together with any one of the following auxiliary condi- paper, the stability of a combustion appliance can be tions given in equation (4) are necessary and sufﬁcient for determined by measuring two reﬂection coefﬁcients at the unconditional stability of an (active) two-port described cold side of the burner, i.e., R and R . In that paper, the up in by the scattering matrix S, expression of R is written in terms of the transfer in ⎧ T R −T 11 dn 21 2 2 2 matrix’s elements, i.e., R = . This expression in B = 1 +|S | −|S | −|Δ| > 0, or ⎪ − T R +T 1 11 22 12 dn 22 2 2 2 can be rewritten with scattering matrix’s entries with help of B = 1 −|S | +|S | −|Δ| > 0, or 2 11 22 equations (1) and (2). The result would be the same expression |Δ|=|S S − S S | < 1, or (4) 11 22 12 21 ⎪ 35 ⎪ 2 used in microwave theory, e.g., equation 7.4.1 in book : 1 −|S | > |S S |, or ⎪ 11 12 21 1 −|S | > |S S |. 22 12 21 −Δ R + S dn 11 R = . (7) in −S R + 1 22 dn Therefore, one can simplify the Rollett stability factor for the Thermo-Acoustic system i.e., K as TA The main idea of the Edwards-Sinsky criterion regarding 2 2 2 the stability of a system expressed by equation (7) is that the T T T 21 21 21 1−− − +|−1| 1 − system is unconditionally stable if the unit disk in the R dn T T T 11 11 11 K = = ≤ 1 . TA plane (note that the interior of the unit circle represents all 2 2 2 2 T − T T 11 21 possible reﬂection coefﬁcients of a passive system/termin- 2 1 − ( ) T 2T T 11 11 11 ation) is mapped by the equation (7) to the interior of the (5) unit disk in the R plane. In other words, the system in 86 International Journal of Spray and Combustion Dynamics 14(1-2) with passive upstream and downstream terminations is Equations (8.b) and (8.c) are almost the same as the con- unconditionally stable if and only if |R | < 1. Besides, ditions suggested by Balsi et al. Appendix C.1 shows how in Kojourimanesh et al. have shown that if |R R | <1then their conditions can be derived using this algebraic method. up in 33,36 the system is conditionally stable. The case of pure ITA Moreover, Appendix C.2 provides a new proof of the modes requires special consideration. For pure ITA modes, Edwards-Sinsky criterion from the mentioned algebraic (i.e., T = 0, R = 0, R = 0) the expression written in technique. 11 up dn equation (2) will be undeﬁned due to division by zero. Also, the conditional stability criterion such as |R R | < 1 up in Mobius transformation between R and R in dn is not applicable because both R and R are already zeros. up dn Accordingly, Lemma 2 which involves requirements to As stated before, the relation between R and R has the in dn the magnitudes of R and R is introduced. It provides form of so-called bilinear (Mobius) transformation. The up dn conditions imposed on the upstream or downstream sides general (so-called, normalized) form of this transformation aZ+b of the ﬂame/burner sufﬁcient for the system to become is H = , where ad − bc = 1. cZ +d thermo-acoustically stable. Among multiple speciﬁc properties of this transform- Lemma 2. For the thermo-acoustic system with the active ation, one which will be used below is that the MT maps two-port deﬁned in equation (2), conditions for the down- a line or a circle in the complex plane of its input Z into stream and upstream terminations which are sufﬁcient to another line or circle in the plane of its output H. qualify the system to be conditionally stable are Particularly, if the mapped contour is the unit circle, then the resulting circle of the unit circle has a speciﬁed 22,23 |R | < , (8.a) center, M, and radius, r , namely, up |R | in b·d − a · c 1 or M = , r = (12) 2 2 2 2 |d| −|c| ||d| −|c| | −|S −|R | S Δ|+|R ||S S | 11 dn 22 dn 12 21 |R | < (8.b) up For a ﬁxed value of the frequency, the entries of the scatter- 2 2 2 (|R | |Δ| −|S | ) dn 11 ing matrix, S, are ﬁxed complex numbers. By looking at the −ΔR +S 2 dn 11 expression for R = , it is clear that it has the form −|S −|R | S Δ|+|R ||S S | in 22 up 11 up 12 21 −S R +1 22 dn |R | < (8.c) dn 2 2 2 of MT. However, it needs to be normalized ﬁrst i.e., divided (|R | |Δ| −|S | ) up 22 by ad − bc = S S . The downstream side of the 12 21 Proof. As mentioned before, the system is conditionally burner/ﬂame is an acoustically passive termination, i.e., stable if |R R | < 1. Therefore, one can say that a sufﬁ- up in | R |≤ 1. Consequently, MT transforms the unit circle dn cient condition for a conditionally stable system could be of R into a circle in the plane of R with a center and dn in |R R | < 1. up in radius as. Considering equation (7) and Δ =−1, Appendix B.1 S + S −2Im(S ) |S S | shows how |R R | < 1 can be extended to get a form of 11 22 22 12 21 up in M = = i, r = . (13) 2 2 2 a quadratic inequality as 1 −|S | 1 −|S | |1 −|S | | 22 22 22 A|R | + B|R |+ C < 0 (9) dn dn Furthermore, the center of the R unit circle transforms dn 2 2 2 where A =|R | |Δ| −|S | , up 22 b into point O = in the R plane which indicates to where in 2 2 2 d B = 2(|S −|R | S Δ|), and C =|R | |S | − 1. 22 up 11 up 11 the inside area of the unit circle of R is mapped to. It can dn As shown in Appendix B.2, the discriminant, be either the inside or outside area of the circle in R plane. in B − 4AC, of equation (9) would be Figure 2 shows four possible cases for this transformation 2 2 2 between R and R planes. This kind of plot provides dn in B − 4AC = 4|R | |S S | (10) up 12 21 new insight into the design strategy, one may follow to Because of B − 4AC ≥ 0, the only way that equation (9) ensure system stability. For instance, considering Case a in becomes always negative is C < 0 & |R | < λ , where λ dn 1 1 Figure 2 one may conclude that for most reﬂection coefﬁ- is the ﬁrst root of the quadratic equation, i.e., cients of the downstream side of the ﬂame, R , the gain of dn R is higher than 1. Accordingly, to stabilize the system for in −B + B − 4AC λ = . (11) this particular case the design of the stabilizing downstream 2A acoustics is more demanding because there is a quite limited range of R which provides |R |≤ 1. Therefore, dn in Appendix B.3 shows that by substituting A, B, C into tuning of the upstream side of the burner becomes a more λ , the right side of equation (8.c) would be the same as 1 appropriate approach. Contrarily, when facing a situation λ . Therefore, if C < 0 & |R | < λ then the considered 1 dn 1 similar to the Cases b,and c in Figure 2, one may conclude thermo-acoustic system is stable. that there is wide freedom to select R such that |R |≤ 1 dn in Kojourimanesh et al. 87 Figure 2. Four possible cases for the Mobius transformation of the unit circle of R into R plane. dn in Figure 3. Mobius transformation of R unit circle into R R . dn up in andevenitispossible todesign a proper R where one can easily show that the location of the center, radius dn |R |= 0. Elaboration of these design guidance ideas is the and point O in the R R plane would be the scaled one in up in subject of the next sections. It should be noted that the in the R plane. Equation (15) and Figure 3 show the loca- in shaded rectangular areas in Cases c,and d in Figure 2 are tion of the center, radius and point O of the circle in the outside domains of the mapped circle. R R plane which is mapped from the R plane. up in dn For the case | R | ≠ 1, one can apply the same strategy up as in equation (7) but for MT of M = R M & O = R & r =|R |r (15) new up new up new up (−R Δ )R + (R S ) up dn up 11 R R = . (14) up in −S R + 1 22 dn Figure 3 depicts the MT of the unit circle in the R plane dn into the R R plane. Comparing Figure 2 with Figure 3, up in Then, the unit circle in R plane maps into a circle in and also equations (13) and (15), one can conclude that dn R R plane. Considering the formula in equation (13), the properties of the disk in the R R plane, i.e., up in up in 88 International Journal of Spray and Combustion Dynamics 14(1-2) Figure 4. MT of R unit circle into R R for 3 different R . dn up in up S +S −B 11 22 M , O , and r , can be deduced from the properties By comparing the expression of with M = and new new new 2 2A 1−|S | of the disk in R plane, i.e., M, O, and r, by scaling them considering S =−S and C < 0, it is obvious that in 11 22 with the factor |R | and rotating it with the phase of R . up up Besides, equation (15) suggests that by decreasing the =|M|. 2A magnitude of R , the center of the corresponding disk in up Therefore, one can relate the ﬁrst root of the quadratic the R R plane and the point O would converge to up in new equation to the MT circle parameters as the origin and the radius of the disk goes to zero as illu- strated in Figure 4. In other words, the whole disk of −B B − 4AC R R will be inside the unit disk, |R R | < 1. up in up in λ = + =−|M|+ r (17) 2A 2A Therefore, decreasing the magnitude of R causes a high up chance of stable system operation (note, that the possibility For the case |R |= 1, |R |= 1, Appendix C.2 provides the up dn of the intrinsic thermoacoustic instability still remains). prove that the Edwards-Sinsky factor is indeed λ .Hence, μ =−|M|+ r (18) The other point is that the values for the center and radius depend on each other for the system deﬁnedinequation(2) Comparing algebraic and MT methods’ results when |R |= 1. Due to the equality A =−C, one can up In this section, we are aiming to search for correspondences write the relation between them as between the results derived from the algebraic and the MT methods. In the aforementioned thermo-acoustic two-port 2 B − 4A(−A) B r = = + 1 = M + 1 (19) system, i.e., obeying equation (2), the symmetry implies 4A 2A that S =−S and therefore Δ =−1, accordingly 11 22 S S = 1 − S . As explained before for any complex z Equations (17) and (18) reveal the relation between 12 21 1−|z| function, ≤ 1. Then one can readily conclude that Edwards-Sinsky factors, μ & μ ,and MT parameters |1−z | the radius of the MT circle in the R plane would be which is shown in Figure 5. As can be seen, the in always bigger than 1, i.e., Edwards-Sinsky factor, μ, is the closest point of the R in disk from the origin i.e., |−|M|+ r|. Moreover, the second |1 − S | root of the quadratic equation (9) is the farthest point of the r = ≥ 1. (16) |1 −|S | | R disk from the origin. Thus, both approaches provide the in same results regarding the elaboration of criteria for the ana- lysis of (in)-stability of the system operation. In addition, for the case |R |= 1, the expression √ up 2|R ||S S | B −4AC up 12 21 = would be the same as r, i.e., 2 2 2 2A 2(|R | |Δ| −|S | ) up 22 B − 4AC. Optimal value of R to obtain minimum value of R dn in r = 2A In this subsection, an optimum value of R is derived dn −|S +S | −B 22 11 The same procedure conﬁrms that = . which provides a minimum value of R at given entries 2 in 2A 1−|S | 22 Kojourimanesh et al. 89 behaves as a passive subsystem and therefore it will be stable in combination with any passive upstream acoustics of the considered appliance. The region of R , which we are looking for, belongs to a dn section of the unit circle in the plane of R which maps dn these R to the double shaded (stable) regions in the dn plane of R , see Figure 2. To ﬁnd the corresponding in area in the R plane, one may use the inverse transform- dn ation of equation (7). For this case, the inverse transform- ation is also of Mobius kind, and it is given by the expression −dR + b in R = . (21) dn cR − a in We are interested where the unit circle of R maps into in Figure 5. Relation of MT, algebraic equation and μ . the R plane. The mapping is similar to the one presented dn in Figure 2 with the only replacement of R and R planes. dn in The described procedure provides the area inside the unit circle of R which is considered as a preferable value of dn of the burner’s scattering matrix, S. As shown in Figure 2, a R to obtain a passive stable system without considering dn minimum value of R is equal to either zero for cases b and in the upstream acoustic’s condition. c or it is equal to k for the cases a and d. It has been shown that |k| is actually the Edwards-Sinsky factor, μ. Therefore, the value of R can be restored which causes the minimum dn Case study value of R by inverse mapping at point k back to R in dn plane. Hence, To show the power of this methodology, a test setup, shown in Figure 6, is prepared to measure the FTF, R , and R in −dk + b dn in R = . (20) dn opt. the frequency domain. A constant temperature anemometer ck − a (CTA) and a photomultiplier tube (PMT) with OH ﬁlter are It is obvious that for the cases b and c the optimum value used to measure the acoustic velocity ﬂuctuation before the of R is R =− . dn dn opt. ﬂame and the varying heat release signal, respectively. Two Bronckhorst mass ﬂow controllers control the methane and airﬂow rates. A water-cooled burner deck holder is used to Possible range of R to obtain a passive stable dn keep the outer perimeter of the burner deck at a ﬁxed tem- system perature to obtain a steady ﬂame and FTF independent from The practical motivation of the considerations presented the time of measurement. A set of National instruments below stems from the fact that many producers of burners DAQ systems connected to a PC running LabVIEW is for industrial or domestic applications tend to combine used to transfer the data with a sampling rate of 10 kHz. the burner with the downstream part of an appliance as a A loudspeaker is installed at the end part of the upstream uniﬁed product. A typical example is the combination of a side to provide a forced excitation supplied by the sequence burner and heat-exchanger as the “power module”. of pure tone signals. Software written in LabVIEW is Accordingly, only the upstream side (for instance, the implemented to control the amplitude of the excitation blower/fan, venturi, snorkel, etc.) is designed by the boiler force, sampling rate, measuring time duration, as well as manufacturer. In this situation, designers of the power setting the desired range, steps of frequencies, data prepro- module would like to optimize their product in terms of cessing, etc. ensuring the thermo-acoustic stability for the module in com- The upstream side has a broadband damper with |R | up bination with any/arbitrary acoustics of the upstream part of less than 0.4 for the frequency range of 15–310 Hz. More an appliance. details of the design of the upstream damper, the setup, In this subsection, the conditions are investigated how to and the measurement procedures can be found in paper. select a passive termination R in such a way that for the The downstream part of the burner is a 110 mm long dn given burner transfer matrix the corresponding magnitude quartz tube with an open end. The reﬂection coefﬁcient of of R becomes less than 1 over a certain frequency range. the downstream side is measured immediately after in The idea to ensure |R | < 1 is motivated by the fact that turning off the ﬂame to obtain a reasonable estimation of in in this case the burner with downstream acoustics R value, i.e., close to the hot condition. Figure 7 shows dn 90 International Journal of Spray and Combustion Dynamics 14(1-2) Figure 6. A test setup to measure the FTF, R , and R . dn in Results and Discussions In this section, the analytical results derived in section 2 are examined for the case study expressed in section 3. Stability analysis using Flame Transfer Function For the case study as deﬁned in the section 3, the Rollett sta- bility factor K, and Edwards-Sinsky parameter μ are calcu- lated based on equations (3) and (6), respectively. The results are presented in Figure 10. As mentioned before, due to symmetry features of the transfer matrix for thermo-acoustic systems, both Rollett factor and Edwards-Sinsky parameter should be less than Figure 7. Measured R , and R of the test setup. up dn 1, K ≤ 1, μ ≤ 1. Therefore, it is not surprising that Th TA this fact is reﬂected in Figure 10 for the case study. By increasing the frequency from 15 to 220 Hz, K and μ the reﬂection coefﬁcients of the upstream and downstream decrease. In the practice of microwave circuits design, the sides of the burner in the frequency domain. value of μ is also used as the qualitative measure/indicator As an example, the ﬂame transfer function for a brass of the active two-port potential of (in)-stability. plate burner with premixed burner-stabilized Bunsen-type Accordingly, it also reﬂects the qualitative measure for the ﬂame is measured. The burner deck is a disk that has a potential of (in)-stability when the considered two-port is thickness of 1 mm, and a diameter of 5 cm. The hexagonal embedded between arbitrary passive up and downstream ter- pattern of round holes with a diameter of holes of 2 mm and minations. The argument which substantiates such the role of the pitch between the holes of 4.5 mm is used. For brevity, μ factor is the following. If to estimate the potential of stabil- the burner is called “D2P4.5”. The total open area of the ity at each frequency as the ratio of the area of the double burner is 399 mm. The measured ﬂame transfer function shaded region and the area of the unit circle in R (shown in is used to calculate coefﬁcients a, b, c, d of the MT. in Figures 2 and 5), then exists the monotonic inverse relation Figure 8 shows the ﬂame transfer function of burner between μ and the area ratio. Therefore, by decreasing μ from D2P4.5, at the mean velocity of the mixture through the 15 to 220 Hz, the ratio of the areas is decreased, also the burner holes of 70 cm/s and equivalence ratio of ϕ = 0.7 potential of stability at that frequency is decreased. For the in the frequency range 15–310 Hz. considered test case the minimum value of μ happens at The impedance tube with 6 microphones shown in Figure 6, the vicinity of the frequency of 220 Hz. It is worth noticing including the low reﬂecting loudspeaker box (damper) at the that it is the same frequency where the phase line of the ﬂame upstream side, is used to measure the reﬂection coefﬁcient transfer function is crossing −π, see Figure 8, which is the R from the combination of a bended supply tube, the burner indication of the burner intrinsic mode. The correlation in with ﬂame, and the downstream subsystem. The measured between the frequency ranges of high potential of instability data of R in the frequency domain is plotted in Figure 9. The of thermo-acoustic system and frequencies of the burner in 33,36 31,32 measurement procedure is explained in detail in papers. intrinsic modes was reported earlier. Kojourimanesh et al. 91 would be the possible upstream reﬂection coefﬁcient to sta- bilize the system. To do that, one can substitute the values of scattering matrix entries and |R | for each frequency dn between 15 and 310 Hz in equation (8.b). If the right-hand side of equation (8.b) becomes negative, then it implies that the system could be unstable at that frequency. Consequently, by changing the upstream part, the system cannot be fully stabilized for all phases of R , i.e., any dn length of the exhaust duct. On the other hand, if the RHS of equation (8.b) is positive then for a ﬁxed value of |R | dn and arbitrary phase of R , the region of |R |where the dn up system remains stable is determined. For the case study under consideration, this region is plotted in Figure 11. Figure 8. Measured ﬂame transfer function of the burner cm As can be seen in Figure 11, for frequencies higher than D2P4.5 at v = 70 and ϕ = 0.7. 120 Hz there is no value of R that guarantees the stability up of the system for the ﬁxed value of |R | with arbitrary dn phase (length) of the downstream side. Note that this fre- quency is exactly the same as the frequency where K and μ start to become negative. However, for a frequency less than 120 Hz, the blue area in Figure 11 gives the possible magnitude of |R | to guarantee the stability of the system up at those frequencies. The reﬂection coefﬁcient of the upstream side can be also designed from the MT strategy. Figure 12 shows the results of mapping the unit circle of R ﬁrst into the dn plane of R i.e., equation (7) (interior area of the dark in blue circle) and second, into the plane of R R plane up in i.e., equation (14) (interior area the light blue circle), at the frequency of 100 Hz. As can be seen, the size of the cm Figure 9. Measured R at v = 70 and ϕ = 0.7. in circle in R R plane is much smaller than the one in the s up in plane of R , and only a section of mapped R (purple in in shading area) is inside the unit circle. However, the whole Upstream design to stabilize the system circle of mapped R R (turquoise shadings area) is inside up in Equation (8.a) reveals that for a speciﬁc R , it is possible to the unit circle. Equation (15) reveals the reason for this in design R such that the system becomes stable, i.e., property of the mapping that applying the low reﬂecting up |R | < . For the particular case under consideration, termination at the upstream side, i.e., |R |= 0.13 at up up |R | in the maximum value of the upstream reﬂection coefﬁcient 100 Hz, the whole disk of R R will endupinsidethe up in is max(|R |) = 0.4 which is two times smaller than unit disk. up Min = 0.81. Therefore, the system is stable at this con- In addition, the system with a high value of the upstream |R | in dition. In general, this system with a deﬁned R value, i.e., reﬂection coefﬁcient, |R | ∼ 1, cannot be fully stabilized/ dn up ﬁxed magnitude and ﬁxed phase, would be stable if passivated. The reason, as can be deduced from equation max(|R |) < 0.8. However, for other values of magnitude (16), is that the radius of the mapped disk in R plane is up in and phase of R ,the value of |R | could be much higher. higher than one. Therefore, some particular values of R dn in dn When this is the case, an upstream termination with a lower can cause |R |>1. In general, it can be argued that by provid- in reﬂection coefﬁcient, for all frequencies, will be needed. ing a low reﬂecting upstream termination, one may attempt to Besides, one can get the advantages of equation (8.b) to increase the potential of stability for any R and/or R . in dn design the upstream reﬂection coefﬁcient. It should be noted that the phase of the reﬂection coefﬁcient at the Downstream design for stability downstream side can be easily changed, like by varying the length of the exhaust duct. However, changing the mag- Equation (8.c) could help to design the downstream part for nitude of the downstream reﬂection coefﬁcient needs a a ﬁxed value of |R | with arbitrary phase (length) of the up dedicated gas path design or even external equipment like upstream side such that the system would be thermo- damper, mufﬂer, etc. Therefore, one of the design strategies acoustically stable. Figure 13 shows the possible values is to infer for the ﬁxed value of |R | with arbitrary phase of of |R | that provide the stability of the system studied as dn dn R , what will happen with the system. It may suggest what the case study, for the ﬁxed value of |R | with arbitrary dn up 92 International Journal of Spray and Combustion Dynamics 14(1-2) Figure 12. MT of R unit circle into R and R R planes at dn in up in 100Hz (interior areas of dark-blue and light-blue circles respectively). Purple and turquoise shadings signify overlapping of the circles’ interior with the unit circle, respectively. Figure 10. Calculated K and μ for the deﬁned FTF. Figure 11. Region of |R | to have a stable system, for the ﬁxed up Figure 13. Region of |R | to have a stable system, for the ﬁxed dn value of |R | and arbitrary phase of R of the case study. value of |R | and arbitrary phase of R of the case study. dn dn up up phase (length) of the upstream side at frequencies of 35– equation allows deﬁning the region of R such that dn 310 Hz. |R | < 1, which in its turn means that the system becomes in Figure 13 demonstrates that even for the ﬁxed low passive and stable for any passive upstream termination. magnitude of the reﬂection coefﬁcient of the upstream To demonstrate this design option for the case study, R dn termination |R |, it is possible that the system circles and the possible ranges of R at which |R |≤ 1 dn in up becomes unstable in the frequency range between 210 are calculated using equation (21). At four arbitrary selected and 225 Hz. It should be noted that this range of frequen- frequencies, the R circles and centers’ locations of the dn cies is close to the intrinsic instability of the burner with circles and those preferable R areas are shown in dn ﬂame. Figure 14 with purple color. As expressed before, some combustion companies prefer By developing these ideas further, one may suggest to sell their product without the upstream side. Accordingly, that the “volume” of possible R values (blue region dn the inverse transformation written in equation (21) could shown in Figure 15) may provide a good indication for provide a method how to ensure the complete system stabil- the kind of “ﬁgure of merit” of the burner. It means ity for any values of magnitude and phase of R . This that a good burner in terms of the thermo-acoustic up Kojourimanesh et al. 93 Figure 14. R circles, centers’ locations, and possible R map to the unit circle of R at 30, 75, 145, 265 Hz. dn dn in stability would have a larger choice of possible (stabi- with a minimum value of R where the system becomes in lizing) R values. passive and therefore stable for any passive upstream ter- dn Figure 15 demonstrates a 3D view of the discussed mination of the system. For the considered case, the area in a frequency range 15–310 Hz. The four slices optimal value of R such that the magnitude of R dn in (joint areas) for particular frequencies as shown in becomes minimum is derived from equation (20) and Figure 14 arealsomarkedinFigure15withthe red results with a corresponding minimum value of R are in color. If the reﬂection coefﬁcient of the downstream demonstrated in Figure 16. side of the burner is lying inside this area (looking as a These graphs also highlight the conclusion that for channel), then the system would be at a passive stability the considered burner/ﬂame the frequencies around condition otherwise, the system will not be passive and 220 Hz may require special measures to ensure system the solution of the dilemma of stability-instability is stability. related to the product of upstream and inlet reﬂection coefﬁcients, i.e., R R . up in Conclusions Figure 15 could also be a productive tool for further analysis. For instance, one may conclude that at low frequen- It is demonstrated that the system-level analysis of a cies, like 15–40 Hz, for almost any downstream termination network of two-ports is a very fruitful tool to perform inves- of the mentioned burner at speciﬁed mean velocity and tigations of many aspects of combustion acoustic instability equivalence ratio condition, the system would be passive phenomena. Particularly, it provides promising approaches and, therefore, stable. The frequency ranges around 220Hz to the task of system design aiming stability of appliance look the most problematic to stabilize because for down- operation. Original algebraic proofs of conditional and stream acoustics the range of values of reﬂection coefﬁcients unconditional stability criteria of linear two-port network needed for stabilizing the system is narrow at these systems are proposed. It has been shown that thermo- frequencies. acoustic systems cannot be unconditionally stable. Hence, Continuing the case study further, by minimizing R , the the conditional stability criteria have been investigated in maximum potential of stability could be achieved. based on the algebraic technique. Also, a complementary Therefore, an optimal R should be found that minimizes framework of analysis is proposed which is based on the dn R . Equation (20) can be applied to ﬁnd the optimal R application of known properties of bilinear MT. The in dn 94 International Journal of Spray and Combustion Dynamics 14(1-2) Figure 15. A 3D view of possible R maps to the unit circle of R in a frequency range 15–310 Hz. The four slices shown in Figure 14 dn in are highlighted in red. applied to analyze the stability of the thermo-acoustic system. However, the technique based on the MT would provide more insightful information and a better visual and intuitive interpretation of results than the two other techniques. The elaborated criteria of system stability can be applied for purposeful design of the upstream and downstream sides of the given burner with ﬂame to provide thermo-acoustic system stability. Furthermore, the elaborated approaches show princi- pal requirements and propose the receipts how to design the acoustics of upstream and downstream sides of the burner such that the system operation would be stable. Also, for each frequency, the area of possible R dn values to obtain a passive stable system is derived. The area depends on the ﬂame transfer function of the burner. Hypothetically, the size (volume or another measure) of the area can be considered as a possible can- didate for an indication for the ﬁgure of merit of the burner. To conﬁrm the validity and usefulness of the proposed ideas related to the combustor design strategies and evalu- ation of the ﬁgure of merit of burners, further theoretical development, and experimental checks are needed. This work is in progress and promises new directions for Figure 16. Optimal R (top plot) to obtain a minimum amount dn of R (bottom plot) at 15–310 Hz. in further research. Particularly, for the experimental veriﬁca- tions of the theoretical results new setup with a wide range of variable reﬂection coefﬁcients at both up- and down- stream terminations is developed. The ﬁnal goal of future comparison of different approaches reveals relations R&D works in this direction would be the elaboration of between the results of the algebraic derivations, the geo- a convenient designer’s toolbox supporting and suggesting metrical approach applied in microwave theory, and the MT technique. Any of these three approaches can be design decisions. Kojourimanesh et al. 95 16. Edwards ML and Sinsky JH. A new criterion for linear Declaration of Conﬂicting Interests 2-port stability using a single geometrically derived par- The author(s) declared no potential conﬂicts of interest with ameter. IEEE Trans Microw Theory Tech 1992; 40: respect to the research, authorship, and/or publication of this 2303–2311. article. 17. Olivieri M, Scotti G, Tommasino P, et al. Necessary and suf- ﬁcient conditions for the stability of microwave ampliﬁers with variable termination impedances. IEEE Trans Microw Funding Theory Tech 2005; 53: 2580–2586. The author(s) disclosed receipt of the following ﬁnancial 18. Marietti P, Scotti G, Triﬁletti A, et al. Stability Criterion for support for the research, authorship, and/or publication of this Two-Port Network With Input and Output Terminations article: This work was supported by the Nederlandse Varying in Elliptic Regions. IEEE Trans Microw Theory Organisatie voor Wetenschappelijk Onderzoek, (grant number Tech 2006; 54: 4049–4055. 16315). 19. Balsi M, Scotti G, Tommasino P, et al. Discussion and new proofs of the conditional stability criteria for multidevice microwave ampliﬁers. IEE Proc - Microwaves, Antennas ORCID iD Propag 2006; 153: 177. 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Symp Combust 1957; 6: 500–512. 30. Yong KJ, Silva CF and Polifke W. A categorization of mar- 11. Rollett JM. Stability and Power-Gain Invariants of Linear ginally stable thermoacoustic modes based on phasor dia- Twoports. IRE Trans Circuit Theory 1962; 9: 29–32. grams. Combust Flame 2021; 228: 236–249. 12. Ha TT. Solid-state microwave ampliﬁer design. New York: 31. Kornilov VN and de Goey LPHPH. Approach to evaluate Wiley, 1981. statistical measures for the thermo-acoustic instability proper- 13. Meys RP. Review and discussion of stability criteria for linear ties of premixed burners. In: Proceedings of the European 2-ports. IEEE Trans Circuits Syst 1990; 37: 1450–1452. combustion Meeting, Budapest, Hungary, 30 March - 2 14. Ku WH. Unilateral gain and stability criterion of active two- April, 2015, pp. 1–6. ports in terms of scattering parameters. Proc IEEE 1966; 32. Kornilov V and de Goey LPH. Combustion thermoacoustics 54: 1617–1618. in context of activity and stability criteria for linear two-ports. 15. Owens PJ and Woods DM. Reappraisal of the unconditional In: Proceedings of the European combustion meeting, stability criteria for active 2–port networks in terms of S para- Dubrovnik, Croatia, 18-21 April, 2017, pp. 18–21. meters (1970). 96 International Journal of Spray and Combustion Dynamics 14(1-2) 33. Kojourimanesh M, Kornilov V, Arteaga IL, et al. A.1) Rollett factor for Unconditional stability of TA Thermo-acoustic ﬂame instability criteria based on 2 2 2 upstream reﬂection coefﬁcients. Combust Flame 2021; Suppose z = x + iy then |z| = zz = x + y , 2 2 2 2 2 225: 435–443. |z|= x + y , and z = (x − y ) + 2xyi. 34. Kojourimanesh M, Kornilov V, Lopez Arteaga I, et al. Rollett factor in thermo-acoustic systems is Mobius transformation between reﬂection coefﬁcients at 1−|z| K = where z = . |1−z | T upstream and downstream sides of ﬂame in thermoacoustics 2 2 2 1−|z| 1−(x +y ) Hence, K = = therefore, systems. In: The 27th International Congress on Sound and 2 2 2 |1−z | |(1−x +y )−2xyi | Vibration. Prague, Czech Republic, 11-16 July 2021. 2 2 2 1 − x + y − 2y 35. Poole C, Darwazeh I and Outcomes IL. Gain and stability of K = . active networks. In: Microwave Active Circuit Analysis and 2 2 2 (1 − x + y ) + (−2xy) Design, 205–244. 2 2 1 − x + y 36. Kojourimanesh M, Kornilov V, Lopez Arteaga I, et al. Also, it is clear that ≤ 1 and Theoretical and experimental investigation on the linear 2 2 2 (1 − x + y ) + (−2xy) growth rate of the thermo- acoustic combustion instability. In: Proceedings of the European Combustion Meeting. −2y ≤ 0. Therefore K ≤ 1. Napoli, Italy, 14-15 April, 2021. 2 2 (1 − x + y ) + (−2xy) Appendix A.2) Edwards-Sinsky factor for Thermo-Acoustic 1 −|S | A.0) Transfer matrix and Scattering matrix of TA μ = |S − S |+|S S | 22 12 21 By considering the burner/ﬂame as an acoustically compact lumped element, the relationship between pressure and vel- 1−− ocity ﬂuctuations at the upstream and downstream parts of 11 = . ∗ 2 the ﬂame (in the limit of zero mean ﬂow Mach number) can T T T 21 21 21 + (−1) + 1 − ( ) be described via the Transfer matrix as T T 21 T 11 11 11 1−− p 10 up dn 11 = . In terms of ′ μ = u 2 u 01 + θ FTF up dn T T 21 21 2Im + 1 − ( ) Riemann invariants (f, g), pressure and velocity can be T T 11 11 written as 2 2 1−|z| 1−|z| ′ ′ As mentioned before ≤ 1 hence, ≤ 1. p = ( f + g )ρ c p = ( f + g )ρ c 2 2 1 1 up 2 2 dn up dn |1−z | |2Im(z)|+|1−z | up dn ′ ′ u = f − g u = f − g 1 1 2 2 up dn B.1) Quadratic equation for Thermo-Acoustic systems ( f + g ) = ε( f + g ) 2 2 1 1 S −ΔR 2 Therefore, . 11 2 We know R = , |R R | < 1, in up in f − g = (1 + θ FTF )( f − g ) 1−S R 2 2 s 1 1 22 2 2Re(z) = z + z, Re(z z ) ≤|z ||z | 1 2 1 2 By calculating g from the second row and put it in the 2 2 2 and |z − z | =|z | +|z | − 2 Re(z z ). Therefore, ﬁrst row then simplify, one can derive 1 2 1 2 1 2 2 2 2 |R R | < 1 |R (S − ΔR )| < |1 − S R | . f ε + 1 + θ FTF ε − 1 − θ FTF f 2 s s 1 up in up 11 dn 22 dn = 0.5 , g ε − 1 − θ FTF ε + 1 + θ FTF g 2 s s 1 Expanding the power results in ρ c 1 1 where ε = . The similar approach should be followed to 2 2 2 ρ c 2 2 |R | (|S | +|ΔR | − 2 Re(S ΔR )) < 1 up 11 dn 11 dn acquire the scattering matrix in terms of TM elements; 2 2 therefore, one should calculate f from the second row +|S | |R | − 2 Re(S R ). 22 dn 22 dn and put it in the ﬁrst row to ﬁnd g ; then substitute the trans- Applying 2Re(z) = z + z, one writes fer matrix elements and simplify the equations. Then the 2 2 2 2 2 2 below expression will be derived. |R | |S | +|R | |Δ| |R | − [|R | S ΔR − up 11 up dn up 11 dn g −2T 4 f 2 2 2 1 21 1 |R | S ΔR ] − 1 −|S | |R | + = . up 11 dn 22 dn 2 2 f T − T 2T g 2T 2 21 2 11 11 21 [S R + S R ] < 0. 22 dn 22 dn Hence, the determinant of the scattering matrix would be 2 2 Factorizing|R | , R , and R concludes dn dn dn (−2T )(2T ) − 4(T − T ) 21 21 11 21 Δ = =−1. 2 2 2 2 2 4T (|R | |Δ| −|S | )|R | + [R (S −|R | S Δ) 11 up 22 dn dn 22 up 11 Kojourimanesh et al. 97 2 2 2 Substituting A, B, C deﬁned in Appendix B.1 leads to +R (S −|R | S Δ)] +|R | |S | − 1 < 0. dn 22 up 11 up 11 −2|S −|R | S Δ|+ 2|R ||S S | 22 up 11 up 12 21 Employing z + z = 2Re(z), one ﬁnds λ = . 2 2 2 2(|R | |Δ| −|S | ) up 22 2 2 2 2 (|R | |Δ| −|S | )|R | up 22 dn Also, the same procedure can be used to prove equation 2 2 2 +2 Re(R [S −|R | S Δ]) +|R | |S | − 1 < 0. dn 22 up 11 up 11 (8.b) from |R R | < 1. dn out C.1) A new proof of Balsi et al. criterion Generally speaking, if A|R | + 2|z ||z |+ C is smaller dn 1 2 than zero then for sure A|R | + 2Re(z z ) + C would be dn 1 2 Balsi et al. showed that necessary and sufﬁcient conditions less than zero because of Re(z z ) ≤|z ||z |. Therefore, 1 2 1 2 for conditional stability can be ascertained by means of a one allows to consider below inequality instead of the last single parameter. Their theorem was, inequality. ‘Provided that the S-parameters deﬁned for at least one 2 2 2 2 2 pair of positive constant reference impedances have no (|R | |Δ| −|S | )|R | + 2|R ||S −|R | S Δ| up 22 dn dn 22 up 11 RHP poles, the necessary and sufﬁcient condition for a 2 2 linear active two-port to be stable is +|R | |S | − 1 < 0. up 11 1 −|S | R Therefore, if 11 up > 1. |S − S ΔR |R +|S S |R R 22 11 dn 12 21 up dn up A|R | + B|R |+ C < 0 dn dn Proof. As shown in Edwards-Sinsky paper, one can readily 2 2 2 where A = (|R | |Δ| −|S | ); up 22 show that B = 2|S −|R | S Δ|; 22 up 11 2 2 2 |S | −|R | |Δ| 22 up 2 2 C =|R | |S | − 1, up 11 2 2 2 2 |S −|R | S Δ| −|R | |S S − Δ| 22 up 11 up 11 22 = . 2 2 then for sure |R R | < 1. up in 1 −|R | |S | up 11 B.2) By substituting the term into the denominator of equation 2 2 2 B − 4AC = 4|S −|R | S Δ| 22 up 11 (8.c) and simplify it, one can derive 2 2 2 2 2 2 2 − 4(|R | |Δ| −|S | )(|R | |S | − 1). up 22 up 11 1 −|R | |S | up 11 |R | < . dn |S −|R | S Δ|+ |R ||S S | Expanding the equation and applying 2Re(z) = z + z, 22 up 11 up 12 21 one writes By moving |R | to the right side of the earlier equation, the dn 2 4 2 conditional stability criterion, provided by Balsi et al., is 4[|S | +|R | |S Δ| − 22 up 11 derived. 2 2 4 2 2 S |R | S Δ − S |R | S Δ −|R | |Δ| |S | 22 up 11 22 up 11 up 11 C.2) A new proof of Edwards-Sinsky criterion 2 2 2 2 2 2 −|R | |Δ| −|S | |R | |S | −|S | ]. up 22 up 11 22 Substituting |R |= 1 and |R |= 1 into the aforemen- up dn tioned equation (last equation in C.1), it is easy to realize Simplifying the equation leads to that the right-hand side of the equation is indeed the 2 2 2 Edwards-Sinsky criterion. B − 4AC = 4|R | |S S − Δ| up 11 22 2 2 2 1 −|S | B − 4AC = 4|R | |S S | ≥ 0. up 12 21 1 < = μ. |S − S Δ|+|S S | 22 11 12 21 The authors disclosed receipt of the following ﬁnancial B.3) support for the research. This work was supported by the −B + B − 4AC Netherlands Organization for Scientiﬁc Research (NWO) λ = 2A [16315] with the project name of STABLE.

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International Journal of Spray and Combustion Dynamics – SAGE

Published: Mar 1, 2022

Keywords: Burner as an active two-port; Edwards-Sinsky’s criterion; Rollett factor; Conditional stability; Mobius Transformation

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Stability criteria of two-port networks, application to thermo-acoustic systems

Stability criteria of two-port networks, application to thermo-acoustic systems

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Stability criteria of two-port networks, application to thermo-acoustic systems

Stability criteria of two-port networks, application to thermo-acoustic systems

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