Nonlinear Microwave Circuit Design phần 10 potx

NONLINEAR ANALYSIS, STABILITY AND BIFURCATIONS 349 loads at the relevant frequencies can be chosen so to satisfy Barkhausen’s criterion or the instability criterion as defined in Section 5.2. If the stability factor is greater than 1, then the large-signal state must be modified, for example, by increasing the amplitude of the large-signal source, or the bias point, or the loads at harmonic frequencies, in order to enhance the mixing properties of the active device in large-signal regime. If the design is performed at two ports at different frequencies, the potential stability of the reduced network may be caused by the frequency-converting (out of diagonal) terms in the conversion matrix, which connect the input and the output of the network; if this is the case, the potential instability vanishes for decreasing amplitude of the large periodic signal at ω 0 , since no frequency conversion takes place for a small amplitude of the local oscillator. The instability appears only for a sufficiently large amplitude of the periodic large signal. In the opposite case that instability must be avoided, the loads of the two-port network must be chosen so that the circuit is stable for all perturbation frequencies 0 <ω osc < ∞, very much as in linear amplifiers. This approach is quite general, but requires a modified formulation in the case that ω osc = ω 0 2 , because of the coincidence of upper and lower sidebands [6]. 8.3 NONLINEAR ANALYSIS, STABILITY AND BIFURCATIONS 8.3.1 Stability and Bifurcations In this paragraph, the parametric analysis of nonlinear circuits is introduced, which allows the detection of bifurcations and the determination of the stable and unstable regions of operations of the circuit. In the previous paragraph, a criterion for the determination of the stability of a nonlinear regime has been described. The reason why this criterion has been introduced lies in the capability of the harmonic or spectral balance algorithm to yield a solution, even if the solution itself is not stable. A basic assumption of harmonic balance algorithms (see Section 1.3.2) is that the signal must be expanded in Fourier series, with one or more fundamental frequencies. If the actual solution includes any additional real or complex basic frequency, but this is not included in the expansion, the algorithm does not detect it. The solution, if any, is only mathematical, being physically unstable. A time-domain analysis on the other hand will not yield any unstable solution, always preferring the stable one. Therefore, harmonic balance analysis always requires a stability verification, typically of the Nykvist type. However, the possibility to find unstable solutions allows the designer to get a complete picture of the behaviour of the circuit. A particularly illuminating approach requires the tracking of a solution as a function of a parameter of the circuit. In many cases, the value of a bias voltage or of an element of the circuit may determine the behaviour of the circuit, whether stable or not. By changing the value and checking the stability properties of the solution, the operating regions of the circuit are found. It is particularly important to detect the values of the parameter at which a stable solution becomes unstable, or vice versa. These particular 350 STABILITY AND INJECTION-LOCKED CIRCUITS values are called bifurcations because two solutions are found after the bifurcation, one of which is usually unstable. As an example, let us consider again the shunt oscillator in Figure 5.6 in Chapter 5, which is repeated here in Figure 8.5. Let us consider the behaviour of the solution as a function of the parameter G tot = G s + G d , that is, of the total conductance; we will first consider the linear solution. Kirchhoff’s equation for the circuit is 1 L ·  v(t) ·t + G · v(t) +C · dv(t) dt = 0 (8.24) The equation always has the trivial solution v DC (t) = 0; it also has the non-trivial solution: v osc (t) = v 1 · e (α+jω)·t + v 2 · e (α−jω)·t α =− G tot 2C ω =   G tot 2C  2 − 1 LC if G tot < 2 ·  C L (8.25a) v osc (t) = v 1 · e −α 1 ·t + v 2 · e −α 2 ·t α 1 =− G tot 2C +   G tot 2C  2 − 1 LC α 2 =− G tot 2C −   G tot 2C  2 − 1 LC if G tot > 2 ·  C L (8.25b) It easy to see that for G tot < 0 an oscillatory solution growing in time is present; in this case, the DC solution is not stable, which is only mathematically possible. For positive values of the total conductance G tot , a damped oscillatory solution or two exponentially decaying solutions are present in the circuit, indicating the stability of the DC solution. The values of the total conductance for which the nature of the solutions changes is the origin (G tot = 0); an additional change takes place for G tot = 2 ·  C L , but no stable solution is involved in the change, which is not observable in steady-state conditions. The root locus can be plotted as in Figure 8.6. G s G d C s L s v ( t ) + − Figure 8.5 A parallel resonant circuit NONLINEAR ANALYSIS, STABILITY AND BIFURCATIONS 351 j w G tot = 0 G tot = 0 G tot < 0 G tot < 0 G tot = 2 C L a Figure 8.6 Root locus for the parallel resonant circuit in Figure 8.5 Correspondingly, the value of an electrical quantity in the circuit may be plotted as a function of the parameter; for instance, let us a ssume that the amplitude of the oscillation saturates to a finite value because of the nonlinearities of the circuit, so far neglected for stability analysis. Therefore, a plot of the amplitude of the oscillating voltage in the parallel resonant circuit after the oscillations have reached the equilibrium amplitude (see Section 5.3) looks as shown in Figure 8.7. For G tot > 0, eq. (8.24) has only one solution, which is the stable DC solution with v osc (t) = 0. For G tot < 0, eq. (8.24) has two solutions, of which the DC one with v osc (t) = 0 is only mathematical because it is unstable, while the other with v osc (t) > 0 is stable. The point G tot = 0 is a bifurcation point. In this case, the generation of a new branch is caused by the sign change of the real part of the complex roots of the perturbation equation (8.24), or in other words by the onset of an autonomous oscillation; this is called a Hopf bifurcation. Other types of bifurcation are described below. The circuit in Figure 8.5 becomes closer to real life if we consider it as the linearisation of a circuit including an active device, exhibiting a negative differential (small-signal) conductance in a given bias point range. By changing the bias, the total conductance may change sign and quench the oscillations. Therefore, the parameter that controls the bifurcation may be the bias voltage of the active device, for example, a tunnel diode. Another example of control parameter is the gate bias voltage of an FET oscillator as described in Section 5.4, and shown in Figure 8.8. The condition for the onset of the oscillation is A(P in = 0) · β>1 (8.26) 352 STABILITY AND INJECTION-LOCKED CIRCUITS v osc > 0 v osc = 0 v osc = 0 v osc G tot Figure 8.7 Amplitude of the oscillating voltage in the parallel resonant circuit in Figure 8.5 P f = b − P out P out = A − P in P in = P f b A Figure 8.8 An oscillator based on an amplifier and a feedback network If condition (8.26) is satisfied, the amplitude of the oscillation grows until the oscillation condition at equilibrium is fulfilled. A(P 0 ) · β = 1 (8.27) The equilibrium is reached because the transistor has a gain compression for increas- ing operating power (see Section 5.3). The small-signal gain of the amplifier is usually NONLINEAR ANALYSIS, STABILITY AND BIFURCATIONS 353 controlled by the gate bias voltage, which therefore determines whether the condition for the onset of the oscillations is fulfilled or not, and the equilibrium operating power changes accordingly (Figure 8.9). The bifurcation diagram for the gate bias voltage is shown in Figure 8.10. The gate bias voltage has the same qualitative behaviour as shown in Figure 8.7. Let us now treat the case in which the amplifier is biased near Class-B and the transistor has a transconductance increase for increasing gate bias voltage; the amplifier therefore exhibits a gain expansion at low input power, then a gain compression for higher input power when the limiting nonlinearities (forward gate junction conduction, breakdown, etc.) come into play (Figure 8.11). This case lends itself to illustrating a different type of bifurcation; in the following, reference is made to Figures 8.11 and 8.12. Let us start the analysis with a gate bias A A 1 b V gs =−3.5 V V gs =−3 V V gs =−2.5 V V gs =−2 V V gs =−1.5 V B C D P 1 P 2 P 3 P in Figure 8.9 Power gain of an amplifier as a function of input power for different bias gate voltages, compared to the inverse of the attenuation of the feedback network A B C D P in,0 −3.5 V −3 V −2.5 V −2 V −1.5 V V gs P 3 P 2 P 1 Figure 8.10 Bifurcation diagram for a feedback oscillator with the gate bias voltage as a control parameter; letters refer to points in Figure 8.9 354 STABILITY AND INJECTION-LOCKED CIRCUITS 1 b BC D E F A A V gs = −2V V gs = −1.5 V V gs = −2.5 V V gs = −3V V gs = −3.5 V P 1 P 2 P 3 P 4 P 5 P in Figure 8.11 Power gain of an amplifier with gain expansion as a function of input power for different bias gate voltages, compared to the inverse of the attenuation of the feedback network C −3.5 V −3 V −2.5 V −2 V −1.5 V V gs D B A E F P in,0 P 5 P 4 P 3 P 2 P 1 Figure 8.12 Bifurcation diagram for a feedback oscillator with the gate bias voltage as a control parameter when the amplifier has gain expansion; letters refer to points in Figure 8.11 voltage V gs =−3.5 V; no oscillations are present in the circuit, which remains in the DC state. Then, let us increase the gate bias voltage; the small-signal condition for the onset of oscillation (8.26) is fulfilled for V gs =−2 V, corresponding to point A, where also the equilibrium condition (8.27) is fulfilled. However, this equilibrium point is not stable, as described in Section 5.3. After a transient, the other equilibrium point E, which is reached is stable. If the gate bias voltage is further increased, the oscillation amplitude increases reaching point F and beyond. Let us now come back towards decreasing gate bias voltages while the circuit is still oscillating. From point F, point E is first reached when V gs =−2 V. Then, a further decrease of the gate bias voltage does not quench the oscillation, since a stable NONLINEAR ANALYSIS, STABILITY AND BIFURCATIONS 355 equilibrium exists, that is, point D for V gs =−2.5 V. The circuit oscillates even though the small-signal start-up condition is not fulfilled, since the oscillation had started at higher gate bias voltages. Continuing along the path for decreasing leads the circuit to point C for V gs =−3 V, where the oscillation is stopped. The path from point C to point A through point B is only mathematical, since point B and all other points along the path are unstable equilibrium points. The oscillator then exhibits hysteresis in its behaviour when the gate bias voltage is swept from pinch-off towards open channel and vice versa. The bifurcation point C is called a turning-point or direct-type bifurcation; the bifurcation point A at which an unstable branch departs is a subcritical Hopf bifurcation, while the Hopf bifurcation described above (see point A in Figure 8.10) that gives rise to a stable oscillation is called supercritical. At the turning point C, coming from the point D and continuing towards point B, a real Laplace parameter, solution of the stability equation (8.21) becomes positive, causing the amplitude of the oscillating solution to grow in time, reaching the stable branch with larger oscillation amplitude near point Dagain. Bifurcations may be encountered also when a parameter is changed in a periodic large-signal steady state. A method for the determination of the stability of a periodic steady state has been described in the previous paragraph, and a procedure for the design of an instability has also been introduced; both are based on the conversion matrix. When a parameter of the circuit changes in such a way that the circuit starts oscillating at a frequency different from that of the large signal, a secondary Hopf bifurcation takes place, either supercritical or subcritical. If the frequency of the instability is one half that of the periodic steady state, the bifurcation is called a flip or indirect or period- doubling bifurcation; this is normally encountered in regenerative frequency dividers (see Section 6.4). A typical bifurcation diagram is shown in Figure 8.13, where the amplitude of both the fundamental frequency and subharmonic frequency of order two are plotted versus the input power of the frequency divider at fundamental frequency ω 0 .When the input power is P I , the conditions for a subharmonic of order two with ω = ω 0 2 to exist are fulfilled, and the two signals coexist within the circuit; the amplitude of the output signal at fundamental frequency decreases because of the simultaneous presence of another signal at one-half this frequency. If the subharmonic is not detected, the unstable solution with only the fundamental-frequency signal is found; this is only a mathematical solution, as said before. Similar plots are found for secondary Hopf bifurcations, where the second branch represents a signal with a frequency different from the period-doubling subharmonic. Turning points or direct-type bifurcations are also found in periodic regimes, with similar characteristics as seen above for the stability of DC regimes. Other types of bifurcations can be encountered in a nonlinear system [7], which are not described here, as they are less common to be found in practical circuits. It is not unusual that successive bifurcations are encountered along the branches of a bifurcation diagram. This mechanism usually leads to chaotic behaviour of the circuit for higher values of the controlling parameter. A chaotic system is not a system with random solutions, strictly speaking: it is a system where two solutions starting from two initial points lying very close to one another diverge from one another. It is possible to identify the characteristics of a nonlinear system that leads to chaotic behaviour [8, 9]; 356 STABILITY AND INJECTION-LOCKED CIRCUITS P out P out (w 0 ) P out P l P in (w 0 ) w 0 2 Figure 8.13 Bifurcation diagram vs input power for a regenerative frequency divider; the unstable branch is the dotted line, while the stable branches are the dashed lines however, this is not the object of this description, as chaotic systems have not so far found any established application in electronic systems. Other examples of bifurcations are described below, in Section 8.4. 8.3.2 Nonlinear Algorithms for Stability Analysis In this paragraph, the nonlinear algorithms for the analysis of nonlinear circuits with autonomous oscillations and external excitation are described. Circuits where an oscillation coexists with a forced periodic state are, in principle, quite naturally analysed by means of time-domain algorithms. No special modification is required in the formulation of the algorithm with respect to the analysis of autonomous circuits; however, the considerations made both for the case of two-tone analysis and for the case of oscillator analysis hold. It is worth repeating, however, that the time-domain analysis always yields a sta- ble solution, that is, the solution actually present in the circuit; moreover, the transient behaviour is correctly found. This is important, especially in the cases in which the state actually reached by the circuit is uncertain because of the presence of possible instabilities. Harmonic or spectral balance is another viable method for this case. Formally, the electrical quantities are expressed as in the case of a two-tone analysis (see Chapter 1), with the first frequency being that of the external signal source ω ext and the second being that of the oscillating signal ω osc ; in this case, the second basis frequency is unknown and must be added to the vector of the unknowns of the problem; however, the phase of the oscillation is undetermined and can be set to zero (see Chapter 5). Therefore, the number of unknowns is again equal to the number of equations, and the system can be solved by a numerical procedure. The problem always has a trivial solution, where the amplitudes of all phasors relative to the second basis frequency ω osc and of the intermodulation NONLINEAR ANALYSIS, STABILITY AND BIFURCATIONS 357 frequencies n ext ω ext + n osc ω osc are zero, corresponding to absence of oscillation. This solution can be real or can be unstable and only mathematical. With the aid of the same procedures as those described for oscillator analysis, the trivial solution can be avoided. For example, a port of the circuit where non-zero amplitude of the signal is expected to appear at the oscillation frequency is selected. Then, Kirchhoff’s equation at that port and frequency can be replaced by the Kurokawa condition: Y(ω osc ) = I(ω osc ) V(ω osc ) = 0 (8.28) Alternatively, Kirchhoff’s equations at all nodes and frequencies can be rewritten as I L (V n osc ) + I NL (V n osc )   n |V n osc | 2 = 0 (8.29) Both the approaches remove the trivial solution, being in fact extensions of the previously described methods. An alternative point of view that is an extension of that illustrated in Figure 5.23 in Section 5.5 is now described [10]. A probing voltage or current at frequency ω is intro- duced at a single port of the nonlinear circuit driven by the large signal at frequency ω 0 (Figure 8.14). The nonlinear circuit is analysed by means of a non-autonomous, two-tone harmonic or spectral balance algorithm. Frequency and amplitude of the probing signal are swept within a suitable range; an oscillation is detected when the control quantity (the probing current or voltage respectively) is zero, indicating that an autonomous oscillating signal is present in the circuit and that the removal of the probing signal does not perturb the circuit. + − V LO (w 0 ) + − V probe (w) Z (w) = 0 Z ( n w) =∞ Z ( k w 0 ) =∞ I control (w) Z L Nonlinear circuit + − V LO (w 0 ) + + − − I probe (w) V control (w) Y (w) = 0 Y ( n w) =∞ Y ( k w 0 ) =∞ Z L Nonlinear circuit Figure 8.14 Voltage and current probes for instability detection 358 STABILITY AND INJECTION-LOCKED CIRCUITS A filter ‘masks’ the presence of the probe at all other frequency components; we remark that harmonics of the probing frequency ω canbepresent,sinceatwo- tone harmonic or spectral balance analysis is performed, and arbitrary amplitude of the probing tone is accounted for. As seen in the case of oscillators, both probing amplitude and frequency are apriori unknown; they are found from the real and imaginary parts of either of the following complex equations: I probe (V probe ,ω)= 0orV probe (I probe ,ω) = 0 (8.30) Quite naturally, the same problems in identifying a suitable starting point for easing the convergence of the analysis are present in this case also. Volterra analysis could be used for this type of analysis; however, the authors are not aware of such an algorithm being proposed so far. The methods described have been extensively used for the analysis of two-tone mixed autonomous/non-autonomous circuits, and in particular, they have been used for the determination of the bifurcation diagram of the circuits. To achieve this goal, continuation methods are applied, in order to ‘follow’ the solution of the circuit as a parameter is varied and to detect the qualitative changes in its behaviour at bifurcation points [11–13]. As stated above in Section 8.3.2, the harmonic balance method can also find unstable solutions and is therefore ideal for a complete study of the behaviour of a circuit; however, the stability of branches or solutions must be verified. In general, this is straightforwardly done by application of Nykvist’s stability criterion as described above. The analysis along a branch of the bifurcation diagram requires, in principle, simply the repeated application of the methods described above, as the value of the parameter is varied. However, problems arise both at turning points and at Hopf bifurcations. Referring to Figure 8.12, the diagram is, in principle, computed by selecting the gate bias voltage V gs as a parameter, and the input power to the FET as one of the problem unknowns that identifies the branch. However, near the turning point C, the curve becomes multi- valued, and numerical problems arise. When the turning point is approached, therefore, it is advantageous to switch the role of the two axes in the plot, use the amplitude of a given frequency component at a given port (e.g. the fundamental-frequency component at the gate port, related to the input power) as a parameter and set the gate bias voltage as a problem unknown. This requires a modification of the analysis algorithm, as sketched in Section 5.5. The analysis then gets through the turning point, and the whole branch can be followed. The approaching of the turning-point bifurcation can be detected by inspection or automatically by monitoring the quantity dP out dV gs , and setting a maximum value for it. At the turning point, the derivative becomes infinite. The two quantities can be switched back to the original role when the derivative becomes reasonably small again. Another problem arises when a Hopf of a flip bifurcation is encountered along a branch. Referring to Figure 8.13 for a frequency divider-by-two, the diagram is plotted by starting from a low input power, where the solution is quasi-linear and no subharmonic is present; the solution is a Fourier expansion on the basis frequency ω 0 . When the input power P I is reached by stepping the input power, a second branch appears on the [...]... Stability Analysis of Nonlinear Circuits, Artech House, Norwood (MA), a ee 2003 [8] G Chen, T Ueta (Ed.), Chaos in Circuits and Systems, World Scientific, Singapore, 2002 [9] L.O Chua, ‘Dynamic nonlinear networks: state-of-the-art’, IEEE Trans Circuits Syst., CAS27, pp 105 9 108 7, 1980 [10] A Su´ rez, J Morales, R Qu´ r´ , ‘Synchronisation analysis of autonomous microwave circuits a ee using new global... Trans Microwave Theory Tech., 46, 494–504, 1998 [11] V Rizzoli, A Neri, ‘Automatic detection of Hopf bifurcations on the solution path of a parameterised nonlinear circuit , IEEE Microwave Guided Wave Lett., 3(7), 219–221, 1993 [12] V Iglesias, A Su´ rez, J.L Garcia, ‘New technique for the determination through commercial a software of the stable-operation parameter ranges in nonlinear microwave circuits’,... frequency The nonlinear current Ig,NL is obtained by means of a discrete Fourier transform from 1 ↔ −1 Ig,NL = R · i (A.22a) T with  1 1 0 1 0 1 0.6235 −0.7818 −0.2225 −0.9749 −0.9 010 1  0.4339 0.6235  1 −0.2225 −0.9749 −0.9 010 ↔  0.6235 0.7818 −0.2225 R =  1 −0.9 010 −0.4339  0.4339 0.6235 −0.7818 −0.2225  1 −0.9 010  1 −0.2225 0.9749 −0.9 010 −0.4339 0.6235 1 0.6235 0.7818 −0.2225 0.9749 −0. 9101 ... IEEE Microwave Guided Wave Lett., 8(12), 424–426, 1998 [13] S Mons, J.-Ch Nallatamby, R Qu´ r´ , P Savary, J Obregon, ‘A unified approach for the ee linear and nonlinear stability analysis of microwave circuits using commercially available tools’, IEEE Trans Microwave Theory Tech., MTT-47(12), 2403–2409, 1999 [14] K Kurokawa, ‘Some basic characteristics of broadband negative resistance oscillator circuits’,... of microwave solid-state oscillators’, Proc IEEE, 61, 1386–1 410, 1973 [16] H.C Chang, A.P Yeh, R.A York, ‘Analysis of oscillators with external feedback loop for improved locking range and noise reduction’, IEEE Trans Microwave Theory Tech., MTT-47, 1535–1543, 1999 [17] F Ram´rez, E de Cos, A Su´ rez, Nonlinear analysis tools for the optimised design of ı a harmonic-injection dividers’, IEEE Trans Microwave. .. independent and coexist in the same circuit without interaction In a nonlinear circuit, that is, in all practical active circuits, an injected signal interacts with the nonlinear active device, locking the free oscillation frequency to that of the external signal, provided that some conditions are fulfilled In case the conditions are not fulfilled, either the two signals coexist in the circuit or the injected one... ‘A design approach for sub-harmonic generation or suppression in nonlinear circuits’, Proc GAAS 2002 , Milan (Italy), 2002 [5] S.A Maas, Microwave Mixers, Artech House, Norwood (MA), 1993 [6] F Di Paolo, G Leuzzi, ‘Bifurcation synthesis by means of harmonic balance and conversion matrix’, Proc GAAS 2003 , M¨ nchen (Germany), 2003 u BIBLIOGRAPHY 369 [7] A Su´ rez, R Qu´ r´ , Stability Analysis of Nonlinear. .. the critical behaviour of the circuits 8.4 INJECTION LOCKING In this paragraph, some circuits using injection locking of self-oscillating signals for oscillation synchronisation or for frequency multiplication or division are described Injection locking is a nonlinear mechanism that synchronises a free oscillation in a circuit to an externally injected signal In a linear circuit, signals at different... the circuit, as shown in Chapters 1 and 3 A.5 THE HARMONIC BALANCE SYSTEM OF EQUATIONS FOR THE EXAMPLE CIRCUIT WITH N = 3 For our example circuit, the harmonic balance (Kirchhoff’s) system of equations in the nodal formulation is described in detail for the sake of clarity First, it can be seen that 376 APPENDIX the nonlinear conductance ig (v) has a linear behaviour for small voltages with a nonlinear. .. 8.27 8.5 BIBLIOGRAPHY [1] R Grimshaw, Nonlinear Ordinary Differential Equations, CRC Press, Boca Raton (FL), USA, 1993 [2] T.S Parker, L.O Chua, Practical Numerical Algorithms for Chaotic Systems, Springer-Verlag, New York (NY), 1989 [3] V Rizzoli, A Lipparini, ‘General stability analysis of periodic steady-state regimes in nonlinear microwave circuits’, IEEE Trans Microwave Theory Tech., MTT-33(1), 30–37, . non- linear microwave circuits’, IEEE Trans. Microwave Theory Tech., MTT-33(1), 30–37, 1985. [4] F. Di Paolo, G. Leuzzi, ‘A design approach for sub-harmonic generation or suppression in nonlinear circuits’,. independent and coexist in the same circuit without interaction. In a nonlinear circuit, that is, in all practical active circuits, an injected signal interacts with the nonlinear active device, locking. described [10] . A probing voltage or current at frequency ω is intro- duced at a single port of the nonlinear circuit driven by the large signal at frequency ω 0 (Figure 8.14). The nonlinear circuit