Nonlinear Microwave Circuit Design phần 7 ppsx

Stability in nonlinear regime for the design of generalmicrowave circuits free of spurious oscillations of nonlinear origin or for the design ofintentionally unstable nonlinear circuits,

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non-linear microwave circuits’, IEEE Trans Microwave Theory Tech., MTT-43(12), 2851 – 2855,

1995.

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series representation’, IEEE Trans Microwave Theory Tech., MTT-28(1), 1 – 8, 1980.

[45] R.S Tucher, C Rauscher, ‘Modelling the third-order intermodulation-distortion properties of

GaAs FET’, Electron Lett., 3, 508 – 510, 1977.

[46] M.R Moazzam, C.S Aitchison, ‘A low third order intermodulation amplifier with harmonic

feedback circuitry’, IEEE MTT-S Int Microwave Symp Dig., 1996, pp 827 – 830.

[47] P Colantonio, F Giannini, G Leuzzi, E Limiti, ‘IMD performances of harmonic-tuned

microwave power amplifiers’, Proc of the European Gallium Arsenide Applications Symposium, Paris (France), Oct 2000, pp 132 – 135.

[48] P Colantonio, F Giannini, G Leuzzi, E Limiti, ‘High-efficiency low-IM microwave PA

design’, IEEE MTT-S Int Microwave Symp Dig., Vol 1, Phoenix (AZ), May 2001, pp.

511 – 514.

[49] A Betti-Berutto, T Satoh, C Khandavalli, F Giannini, E Limiti, ‘Power amplifier second

harmonic manipulation: mmWave application and test results’, Proc of the European Gallium Arsenide Applications Symposium, Munich (Germany), Oct 1999, pp 281 – 285.

‘unstable’ circuit design is provided.

Oscillators can, in principle, be considered as linear circuits, since an instabilitygiving rise to an oscillatory behaviour, for instance sinusoidal, is a linear phenomenon

In fact, most oscillators are designed by means of linear concepts and tools, and theirperformances are satisfactory, at least for basic applications However, many oscillatoryperformances have an intrinsically nonlinear nature, and they are becoming increasinglyimportant in microwave applications First, the amplitude of the oscillation cannot bepredicted by linear considerations only, and also the frequency of oscillation is often notaccurately predicted; however, simple empirical considerations can yield a reasonableestimation of the power being produced by the oscillator, and the use of high-Q resonatorscan force the frequency to be very close to the desired value Nonetheless, a fully nonlinearmethod can give a better and more accurate evaluation of the actual performances of theoscillator, ensuring a first-pass design Still more important, there are phenomena that can

be described only by means of purely nonlinear considerations The circuits that exploitsuch features are becoming increasingly important in microwave systems: for instance,injection locking of an oscillator, which is fundamental in an oscillator array; phase-noise reduction for accurate phase modulation/demodulation; subharmonic generation forphase-locked loops; chaos prediction for chaotic communication or for chaos avoidance

In this chapter, the linear conditions for stability and oscillation are first recalled; then,methods for large-signal behaviour prediction are briefly summarised The most commonand practical fully nonlinear analysis methods that are becoming increasingly importantfor accurate oscillator design are then reviewed Methods for noise evaluation, mostly ofnonlinear nature in oscillators, are also briefly discussed together with the guidelines forlow phase-noise oscillator design Stability in nonlinear regime for the design of generalmicrowave circuits free of spurious oscillations of nonlinear origin or for the design ofintentionally unstable nonlinear circuits, as frequency dividers and chaotic oscillators, and

an overview of frequency locking in microwave oscillators are treated in Chapter 8

Nonlinear Microwave Circuit Design F Giannini and G Leuzzi

 2004 John Wiley & Sons, Ltd ISBN: 0-470-84701-8

5.2 LINEAR STABILITY AND OSCILLATION CONDITIONS

In this paragraph, the stability or instability of linear circuits are described as a nary step for both nonlinear oscillation design and for nonlinear stability determination.

prelimi-The behaviour of a linear autonomous network, that is, a network without externalsignals, is represented by a homogeneous system of linear equations The standard case inelectronic circuits, however, involves a nonlinear network including solid-state nonlinearcomponents (diodes, transistors) biased by one or more power supplies establishing anoperating DC point The operating or quiescent point is usually found by approximategraphical methods (load line), by approximate nonlinear analysis making use of simplemodels for the nonlinear device(s) or by accurate numerical nonlinear network analysis,usually by means of a CAD program Once the quiescent point is determined, the cir-cuit is linearised and linear parameters are evaluated, as for instance the hybrid model,the Giacoletto model, or whatever equivalent (see Chapter 3), or by black-box data asscattering parameters, usually found by direct measurement As long as any RF signalestablishing itself in the circuit remains small, that is, as long as its amplitude does notexceed the range for which the linearisation holds, this is accurate enough for the anal-ysis and design of the RF behaviour of the circuit In this paragraph, this hypothesis isassumed to hold and purely linear considerations are made

The unknowns of the homogeneous (Kirchhoff’s) system of equations are voltages,currents, waves or a mixture of these, depending on the type of equations selected In allcases, a trivial or degenerate solution is always possible when all voltages and currents arezero This is the solution at all frequencies when the circuit is stable or at all frequenciesexcept one or more than one when the circuit oscillates At each of these frequencies, thedeterminant of the system of equations is zero and a non-trivial or non-degenerate solutionexists This or these solutions represent the oscillation(s) in the circuit Let us assume anodal analysis of the circuit (KCL), and therefore an admittance-matrix representation ofthe circuit:

I = ↔

where Y is an n × n complex matrix and
↔ V ,
I and
0 are n × 1 complex vectors of the

unknown voltages, of the node currents and the zero vector respectively, for a circuitwith n nodes The admittance matrix Y is a function of the values of the elements of↔

the circuit, both linear (passive elements and parasitic elements of the active device) andlinearised (intrinsic elements of the active device); it is also a function of the (angular)frequencyω The condition for the existence of an oscillation at a generic frequency ω0therefore is the scalar equation:

The left-hand side of the equation can be seen as a function of the frequency,which is the unknown of eq (5.2), for fixed values of the elements of the circuit: this isthe case in which an existing circuit is analysed for determination of its stability:

If the equation has no solution for any frequency ω, then the circuit is stable;

otherwise, if one or more solutions exist, the circuit will oscillate at all the frequenciessolution of the equation Nothing can be said on the amplitudes of the oscillations or onthe existence of other spurious frequencies generated by their interactions

Otherwise, the left-hand side of eq (5.2) can be a function of the value(s) of one

or more circuit elements, while the frequency has a fixed valueω0:

The set of values of the circuit elements that satisfy the equation, if any exists,

is the solution to the problem of the design of an oscillator at a given frequencyω0 Inorder for the solution to be satisfactory from a practical point of view, it must be verifiedthat eq (5.3) with the designed values of the circuit elements has no solution for anyother frequencyω.

This is complete from a mathematical point of view, but is not practical from thedesigner’s point of view Therefore, simpler approaches are developed First of all, thenetwork can be divided into two subnetworks at an arbitrary port (Figure 5.1) The twosubnetworks are represented in a nodal approach by two scalar complex admittances.Systems (5.1) and (5.2) become scalar equations:

Equation (5.6) is also known as the Kurokawa oscillation condition [1] It can

be shown (e.g [2–4]) that if eq (5.6) is satisfied, the whole network oscillates, unlessthere are unconnected parts of the network A simple illustration is given here for acascaded network with two nodes; the two-port network in the middle typically standsfor a biased active device (a transistor), while the two one-port networks are the input-and output-matching networks For the circuit shown in Figure 5.2, eq (5.1) reads as



Y11 Y12

Y21 Y22

+

+

−

Figure 5.1 A port connecting the two subnetworks of an autonomous linear circuit

V1 Y2p V2 Yload

Ysource

Figure 5.2 A cascaded two-node network

Equation (5.8) can be rearranged in two different ways:

Let us now come back to eq (5.6): this complex equation can be split into tworeal ones:

whereYrandYjare the real and imaginary parts respectively of the complex admittanceparameter Y = Yr+ jYj System (5.11) can be interpreted in the following way: one ofthe two subnetworks must exhibit a negative conductance, whose absolute value mustequal the positive conductance of the other subnetwork; moreover, the two susceptancesmust resonate In practice, the subnetwork including the active device provides the neg-ative conductance, while the other subnetwork must be designed in order that eq (5.11)

In microwave circuits, waves and scattering parameters are normally used instead

of voltages, currents and impedance parameters Equivalently, the network shown inFigure 5.3 is modified to the network as shown in Figure 5.5, and eqs (5.1), (5.2), (5.9),(5.10) and (5.11) become

−

VR+

−

Figure 5.4 A series connection of the two subnetworks of an autonomous linear circuit

way that a growing instability is present in the circuit, so that the noise always present

in the circuit can increase to a fairly large amplitude at the oscillation frequency only.Therefore, all signals have a time dependence of the form

Admittance, impedance or scattering parameters in eqs (5.2), (5.3) and (5.14) pectively must be computed as functions of the complex Laplace parameters = α + jω

res-instead of the standard (angular) frequency ω In the analysis case, eq (5.3) and its

equivalent condition for an impedance or wave representation are

The circuit is an oscillator if one or more solutions exist for one or more values

ofω0 with alsoα0> 0 Conversely, an oscillator must be designed from eq (5.4), or its

equivalent condition for an impedance or wave representation, so that

are satisfied for the desired value of the Laplace parameterω = ω0 andα = α0 > 0.

A practical problem when using eq (5.22) or eq (5.23) instead of eq (5.3) and

eq (5.4) arises from the fact that CAD programs do not usually compute network eters in the Laplace domain Equivalent conditions must therefore be available requiringonly standard frequency-domain expressions Typically, an oscillator includes a resonatorthat forces the circuit to oscillate near its resonant frequency, more or less independently ofthe amplitude of voltages and currents in the circuits Therefore, eq (5.11b) or eq (5.18b)mainly involving the frequency-dependent elements can typically be computed in the fre-quency domain, that is with α = 0, to a good degree of approximation Contrariwise,

param-eq (5.11a) or param-eq (5.18a) mainly involving the negative- and positive-resistance termstypically are sensitive to voltage and current amplitudes in the circuit From what hasbeen said above, it seems to be a reasonable assumption that the circuit be designed insuch a way that the total conductance or resistance be negative or that the total reflection

be greater than one:

Gs Cs Ls

Gd

Figure 5.6 A parallel resonant circuit

Kirchhoff’s current law at the only node, written in the form of eq (5.22a), is

This condition gives a growing instability, thus confirming the validity of eq (5.24a)

In particular, if the quality factor (Q) of the circuit is high, that is, if

and the oscillation frequency is

ω0= √1

independent of the resistive elements in the circuit

Let us now check whether eq (5.24c) is also valid If condition (5.31) holds,the reflection coefficients of the left and right subcircuits in Figure 5.6 can be approxi-mated by

s∼ G0− Gs

G0+ Gs d∼ G0− Gd

whereG0 = Z1

0 = 20 mS It is by no means true that if eq (5.30) holds, then eq (5.24c)

is satisfied Let us show this by assigning actual values to the resistive elements of thecircuit For instance, we can take

s0 = −Rtot

2L ±


Rtot

in the frequency domain with a design criterion similar to eq (5.24a) or eq (5.24b) is thefollowing The instability criterion is computed at the port where the external resistiveload is connected to the oscillator (Figure 5.8).

Oscillator ZL= Z 0

Γ osc Γ load

Figure 5.8 An oscillator partitioned at the output port

In this case, no ambiguity is present Since the load is real, for stable oscillationsthe phase ofosc must be zero and its amplitude must be infinite so that

This situation corresponds to Zosc = −Z0= −50  For growing instability, the

oscillator can be approximated as a series or parallel resonator in the vicinity of theresonant frequency ω0 where  osc(ω0) = 0 In the case that the resonance is a series

A more rigorous and general formulation is as follows [5, 6] Let us come back tothe general equation system in eq (5.1); its determinant in the Laplace domain has beenintroduced in eq (5.22) For typical oscillators, a solutions0= α0+ jω0 of eq (5.22) inthe complex Laplace plane is located in the vicinity of the frequencyω1 where the phase

of the function F (jω) becomes zero (Figures 5.10, 5.11 and 5.12); from an electrical

point of view, this corresponds to resonating the circuit reactances computed in periodicregime s = jω.

We can therefore write the solution of eq (5.22) as

s0= α0+ jω0 = α0+ j (ω1+ δω) (5.51)

where bothα0 andδω are small compared to ω1 Expanding in Taylor seriesF (s + jω)

arounds = jω1 we get (see Appendix A.10)

Figure 5.9 The four resonance types of an oscillator at its output port: parallel unstable, parallel stable, series unstable and series stable

Equation (5.52) is solved forα0, yielding

Figure 5.11 A qualitative behaviour of the complex functionF (s) = F (α + jω) along the

imag-inary axis of the complex Laplace plane

From an electrical point of view, the four cases correspond to the series and parallelresonances described above.

Let us illustrate this result with our example parallel resonant oscillating circuit

in Figure 5.6 The determinant of Kirchhoff’s equation system in our case becomes(eq (5.25)):

Its imaginary part becomes zero for

always; therefore, ifGtot> 0, then F (jω1) > 0, and the circuit is stable Otherwise, when

Gtot< 0, then F (jω1) < 0, and the circuit is unstable, as found earlier The real part of

the Laplace constant is also evaluated from eq (5.53), with the same result as above

5.3 FROM LINEAR TO NONLINEAR: QUASI-LARGE-SIGNAL OSCILLATION AND STABILITY CONDITIONS

In this paragraph, the linear stability and oscillation conditions so far described are modified to take into account the nonlinearity, that is, the dependence of the oscillator parameters on the amplitude of the signal.

The stability and oscillation conditions given so far are valid only in linear regime.However, the behaviour of actual oscillating circuits always involves the nonlinear char-acteristics of the active device A rigorous study requires the use of full-nonlinear analysismethods that will be described in Section 5.4 However, many conclusions of the previ-ous paragraph are extended to the nonlinear regime by means of simple considerationsrequiring a general knowledge of the dependence of circuit parameters on the amplitude

of the signal

First of all, let us extend the stability and oscillation considerations described inthe previous paragraph to a circuit with parameters varying with the amplitude of thesignal Let us first consider the reflection coefficients (e.g those shown in Figure 5.5)dependent on the amplitude of the signal within the oscillator Intuition and experi-ence suggest that a growing instability will not grow forever but will saturate at acertain amplitude, because of the limitations of the active device Therefore, let us

assume that the steady-state periodic regime has been attained when the signal withinthe oscillator has reached the equilibrium amplitude A0 with oscillation frequency ω0.Equations (5.11), (5.13) and (5.18) are now rewritten, in complex form, and explicitlyindicate the network parameter dependence on signal amplitude:

It is now interesting to derive a formal stability criterion [1, 4, 7] Let us apply

a small perturbation δA to the amplitude of the oscillating signal; if the oscillation is

stable, the amplitude will come back to the same value as it was before the perturbation.Since the perturbation is small, the perturbed admittance in eq (5.61a) can be expanded

in Taylor series to the first order, and eq (5.61a) becomes

Ytot(A0, jω0) + ∂Ytot(A, s)

SinceYtot(A0, ω0) = 0, we have

∂Ytot(A, s)

∂Ytot(A, s)

where the partial derivatives are computed in the unperturbed oscillation state, that is for

s = jω0, and the star denotes complex conjugation By dividing eq (5.67) into real andimaginary parts we get

If the eq (5.71) is satisfied, the frequency of oscillation is stable and will not change for

a small perturbation of the amplitude of the oscillating signal The smaller the expression

at the left-hand side of the first of eq (5.71), the smaller the sensitivity of the oscillationfrequency to an amplitude perturbation

Equations (5.70) and (5.71) can be equivalently rewritten in terms of the impedanceand reflection coefficient representation of the network, with equivalent results

We can also rewrite eq (5.70) in the following form:

where the function T is any of the network functions in eq (5.61) or eq (5.62)

Geo-metrically, eq (5.72) can be interpreted as follows: the oscillatory state is stable if theangle between the derivative of the function T (A, ω) with respect to frequency and the

derivative with respect to amplitude is greater that 0◦ and less than 180◦ in the complexplane of the functionT (A, ω), when taken counterclockwise.

Gs(A) j Bs(A) Gd( w) j Bd( w)

Active device Passive embedding network

Figure 5.13 A parallel resonant circuit partitioned into an active and a passive subcircuit

Let us illustrate the rule for a parallel resonant circuit, as partitioned in Figure 5.1,and repeated in Figure 5.13, where one subcircuit includes the nonlinear amplitude-dependent active device and the other the linear amplitude-independent passive embeddingnetwork We can assume that the reactive part of the active device is frequency-independent when the embedding network has a very strong frequency dependence, sothat the former can be neglected in the narrowband near the resonant frequency; this isusually the case for single-frequency oscillators that include a high-Q resonator in thepassive network for frequency stabilisation

The admittances are

Ys(A) = Gs(A) + jBs(A) Yd(ω) = Gd(ω) + jBd(ω) Ytot(A, ω) = Ys(A) + Yd(A)

∂Y s (A)

∂A

Figure 5.14 Loci of the device and embedding network admittances

and it can be interpreted graphically that the angleα between the vector of the derivative

of the two curves with respect to the curve variable have a value between 0◦ and 180◦,with maximum stability when α = 90◦

The same considerations can be repeated if the two subnetworks are represented

by impedance parameters or reflection coefficients, leading to similar results The resultsfor the impedance parameter representation are easily deduced by the use of duality Forthe reflection coefficient, the oscillation condition reads

d(ω0) = 1

For the derivation of the stability condition, we can remark that the reflection ficient is related to admittance and impedance parameters by conformal transformations;therefore, angles are preserved, and so the requirement that (Figure 5.15)

coef-0◦ < β < 180◦ and β = 90◦ for maximum stability

This can be obtained by analytical calculations also [8] We first write the reflectioncoefficients as