Nonlinear Microwave Circuit Design phần 8 ppt

Then, the amplitude of the probing signal is stepped up until theamplitude of the oscillation brings the active device into saturation, reducing thus thevalue of the power-generating neg

268 OSCILLATORS

analysis of the oscillator circuit linearised at the bias point, without the probe Then,the probe is introduced, injecting a probing signal with a frequency equal to the lin-ear approximation and with a small amplitude This situation corresponds to the initialoscillation startup, when the oscillation is still growing in time because the power gen-eration at RF by the negative resistance or conductance of the active device prevails onthe power dissipation on the positive resistance or conductance of the passive elements,including the load The control quantity is not zero, as the oscillator delivers power tothe probing generator Then, the amplitude of the probing signal is stepped up until theamplitude of the oscillation brings the active device into saturation, reducing thus thevalue of the power-generating negative resistance or conductance (see Section 5.3) Thevalue of the control quantity will approach zero for a given value of the amplitude ofthe probing signal; it will not probably be exactly zero because the oscillation frequencyusually shifts in large-signal regime with respect to the small-signal linear calculation.However, this is a good starting point for the self-consistent simultaneous solution of

eq (5.113) and (5.121), avoiding the degenerate solution It must be pointed out that aquasi-linear determination of the start-up frequency can alternatively be obtained by per-forming repeated non-autonomous analyses with a small amplitude of the probing signaland fixed frequency; the frequency value is swept within a suitable range, as said above.The frequency at which the control quantity has zero imaginary part and negative realpart is a suitable candidate for oscillation startup and a good first guess of the large-signaloscillation frequency

This method is easily modified to automatically avoid the degenerate solution inthe same way as described in eq (5.118) above Instead of eq (5.121), the Kurokawacondition can be written for the probing port:

Yprobe(Vprobe, ω) = Iprobe(Vprobe, ω)

Vprobe = 0 or Zprobe(Iprobe, ω) = Vprobe(Iprobe, ω)

(5.122)

Convergence may still be problematic, and suitable procedures for accurate guess determination are still needed for improving convergence; however, as said, con-vergence to the degenerate solution is avoided

first-Similar to the above formulation, the probe approach also lends itself to oscillatorsynthesis or tuning The frequency is now fixed to the design value, and the values of one

or more circuit elements are left free to vary in order that the design requirement be met;

if a single element is chosen as a tuning parameter, the number of equations equals thenumber of unknowns The system of equations is formed by eq (5.113) and eq (5.121),and its solution requires care in order to avoid the degenerate solution A tuning curvecan be computed if the analysis is repeated for several frequencies in a suitable range;this approach can also be used for oscillator analysis, as said above

Volterra series formulation is also a viable approach for oscillator analysis andtuning; so far, a ‘probing’ approach has been demonstrated that is analogous to thatimplemented with harmonic or spectral balance [24] The basic arrangement is that shown

in Figure 5.29: the circuit is forced by a probing voltage or current at a single port

Frequency and amplitude of the probing signal are a priori unknown: their correct values,

NOISE 269

corresponding to the fundamental oscillation frequency and voltage or current amplitude

at the probing node or branch of the oscillator, give zero-control current or voltage atthe probing branch or node of the oscillator, indicating that the removal of the probingsignal does not perturb the oscillating circuit In this formulation, the control current

or voltage is computed by standard application of the Volterra series algorithm to theone-port nonlinear circuit being probed Instead of zero-control current or voltage, a zeroadmittance or impedance of the one-port circuit at fundamental frequency can be sought;admittance and impedance are computed by simply dividing the control current or voltage

by the probing voltage or current The correct amplitude and frequency of the probingsignal are the unknowns of a complex equation; their values can be found by very simpleiterative methods with excellent convergence properties A more general formulation hasalso been proposed [25], relying on the Volterra series expression of a mildly nonlinearoscillator formulated as a single-loop nonlinear feedback system

This approach shares the same advantages and drawbacks of the Volterra seriesanalysis described in Chapter 1: it is limited to weak nonlinearities, but it is very fastand reliable Increasing the order of the Volterra series may prove cumbersome, eventhough automated methods have been implemented for high-order nuclei calculationsbased on general procedures [26] An application to feedback amplifiers based on thesame principle has also been demonstrated [27]

Phase noise in an oscillator is so called because it randomly changes the phase ofthe oscillating signal and therefore its instantaneous frequency, causing the widening ofthe spectral line Noise perturbs the noiseless oscillatory state with two different mecha-nisms: for very low frequency noise, that is, for noise components at a small frequencyoffset from the (noiseless) oscillation frequency, the small noise generators modulate theoscillating signal quasi-statically Formally, the noise sources are added to the noiselessKirchhoff’s equation from which the oscillation condition is derived, causing a randomshift of the oscillation frequency, which can be seen as a perturbation of the oscillat-ing signal As said, this mechanism affects the spectrum of the oscillator closest to theoscillation frequency For noise components at larger frequency offset from the (noise-less) oscillation frequency, the noise can be seen as an input signal to a nonlinear circuitunder a large periodic excitation, in this case the self-oscillation; it is therefore frequencyconverted between all harmonics of the oscillating signal, including the DC component.This mechanism is the same as that taking place in mixers; however, it is particularly

Figure 5.30 Widening of the spectral line at the oscillation frequency caused by noise

disturbing because of the strong 1

f noise component in active devices being up-converted

from very low frequency to near the oscillation frequency Both mechanisms can be mented in a nonlinear simulator and are accurately predicted, provided noise sources areaccurately modelled

imple-Frequency conversion from baseband to near the carrier is essentially a order nonlinear effect (see Section 1.3.1) Therefore, its magnitude depends very much

second-on the importance of secsecond-ond-order nsecond-onlinearities A qualitative measure of secsecond-ond-ordernonlinearities is the amplitude of the rectification of the oscillating signal, generating anincrease of the DC bias currents and voltages Therefore, a low-noise oscillator shouldexhibit a fairly constant DC bias current with and without oscillation, to within a fewpercent of the static unperturbed current Generation of current and voltage harmonicshas been treated in detail in Chapter 4, and the reader can refer to that approach for acorrect procedure while designing an oscillator as in Section 5.3 It can be noted that,while third-order (or odd-order in general) nonlinearities are required for the saturation ofthe signal in an amplifier, and therefore for the achievement of the steady-state amplitude

of the oscillation, second-order nonlinearities can be avoided by careful design withoutaffecting the proper performances of the oscillator, especially when emphasis is put onnoise instead of efficiency

Another parameter related to the noise performances of the oscillator is the lation frequency shift with bias voltages A first qualitative indicator is the frequency

oscil-NOISE 271

A

Figure 5.31 Oscillation frequency shift from operating bias to suppression of oscillation

shift from normal operating bias and the reduced bias causing oscillation extinction [28](Figure 5.31)

The larger the shift, the noisier the oscillator This approach, however, can be usedfor accurate, quantitative evaluation of the phase noise in an oscillator The pushing factor

is defined as the shift in oscillation frequency caused by a change in bias voltage:

experi-no low-frequency dispersion is assumed to take place in the active device This is a iting assumption, but it is experimentally verified to an acceptable degree of accuracy, atleast for some types of active devices An example of typical phase noise for a dielectricresonator oscillator (DRO) oscillator is shown in Figure 5.32

lim-Another important point is the correct identification of the control node causingthe shift in oscillation frequency For FET devices, this is usually the gate node; however,

0.5 1 1.5 2 2.5

Figure 5.33 Pushing factor and noise conversion factor as functions of the gate voltage in an FET

experimental evidence suggests that control is not limited to it It can be seen that forsome values of the gate bias voltage the pushing factor is zero; however, the phase noise

at the same gate bias is reduced but does not vanish (Figure 5.33) [29]

As stated above, the spectral characteristics of the low-frequency noise must

be known in order to predict its up-conversion by the pushing factor However, frequency noise is dependent on bias current and voltage, but these are modulated bythe large oscillating signal Therefore, the spectrum of the low-frequency noise changes

The experimental behaviour of low-frequency noise and the single-sideband noiseare sketched in Figure 5.35 for a typical case, where the abscissa is the actual frequencyfor low-frequency noise and the offset frequency from the oscillation frequency for thephase noise Three regions are clearly visible for phase noise: closest to the carrier thenoise has a 30 dB/decade slope; at larger frequency offset, a region with 20 dB/decadeslope is present that becomes a flat noise floor for large offset frequencies.

The 30 dB/decade region is due to modulation of the oscillating signal by the 1

ω law, which becomes

1

ω2 for noise power After the knee voltage, thewhite noise has no frequency dependence, and only the 20 dB/decade of the modulationmechanism remains When the conversion noise is predominant, noise modulates thephase directly and no additional contribution from the conversion mechanism to frequencydependence is introduced; therefore the noise spectrum is flat

First, the noiseless oscillator is analysed, and the unperturbed solution is found.The values of the phasors of the electrical unknown quantities are found at fundamental

The eq (5.125) is perturbed around the noiseless solution:

276 OSCILLATORS

frequency fluctuations with a mean-square value proportional to the available power ofthe noise sources The associated mean-square phase fluctuations are proportional to theavailable noise power divided byω2 Some formulations of the harmonic balance problemrequire a special treatment of the derivatives in eq (5.126) because their value is close tozero, and the solution method requires their inversion (see Chapter 1) [31], while othersare immune from this problem [32]

Other approaches allow the noise performance evaluation, as for instance by means

of direct time-domain numerical integration [33] or Volterra series expansion [34]; ever, so far the harmonic balance approach has proved to be quite successful Lately,

how-an envelope how-analysis harmonic balhow-ance approach has been proposed, with promisingresults: the noise is straightforwardly introduced as an (random) envelope-modulatingsignal (see Chapter 1) [35] Also, a general analytical formulation has been proposed thatincludes both modulation and conversion mechanisms as particular cases [36]; however,its perspectives for implementation do not look very promising because of numericalill-conditioning of some formulae A thorough comparison of the different approachescan be found in [37]

5.7 BIBLIOGRAPHY

[1] K Kurokawa, ‘Some basic characteristics of broadband negative resistance oscillator circuits’,

Bell Syst Tech J., 48, 1937 – 1955, 1969.

[2] G.R Basawapatna, R.B Stancliff, ‘A unified approach to the design of wide-band microwave

solid-state oscillators’, IEEE Trans Microwave Theory Tech., MTT-27(5), 379 – 385, 1979.

[3] G.D Vendelin, Design of Amplifiers and Oscillators by the S-parameter Method, Wiley, New

York (NY), 1982.

[4] R.D Martinez, R.C Compton, ‘A general approach for the S-parameter design of oscillators

with 1 and 2-port active devices’, IEEE Trans Microwave Theory Tech., 40(3), 569 – 574,

1992.

[5] M.Q Lee, S.-J Yi, S Nam, Y Kwon, K.-W Yeom, ‘High-efficiency harmonic loaded

oscil-lator with low bias using a nonlinear design approach’, IEEE Trans Microwave Theory Tech.,

[8] D.J Esdale, M.J Howes, ‘A reflection coefficient approach to the design of one-port negative

impedance oscillators’, IEEE Trans Microwave Theory Tech., 29(8), 770 – 776, 1981.

[9] K Kurokawa, ‘Noise in synchronised oscillators’, IEEE Trans Microwave Theory Tech.,

MTT-16(4), 234 – 240, 1968.

[10] R.J Gilmore, F.J Rosenbaum, ‘An analytic approach to optimum oscillator design using

S-parameters’, IEEE Trans Microwave Theory Tech., 31(8), 633 – 639, 1983.

[11] C Rauscher, ‘Large-signal technique for designing single-frequency and voltage-controlled

GaAs FET oscillators’, IEEE Trans Microwave Theory Tech., MTT-29(4), 293 – 304, 1984.

[12] K.L Kotzebue, ‘A technique for the design of microwave transistor oscillators’, IEEE Trans.

Microwave Theory Tech., MTT-32, 719 – 721, 1984.

[13] T.J Brazil, J.C Scanlon, ‘A nonlinear design and optimisation procedure for GaAs MESFET

oscillators’, IEEE MTT-S Int Symp Dig., 1987, pp 907 – 910.

BIBLIOGRAPHY 277

[14] Y Xuan, C.M Snowden, ‘A generalised approach to the design of microwave oscillators’,

IEEE Trans Microwave Theory Tech., 35(12), 1340 – 1347, 1987.

[15] Y Xuan, C.M Snowden, ‘Optimal computer-aided design of monolothic microwave

inte-grated oscillators’, IEEE Trans Microwave Theory Tech., 37(9), 1481 – 1484, 1989.

[16] Y Mitsui, M Nakatani, S Mitsui, ‘Design of GaAs MESFET oscillator using large-signal

S-parameters’, IEEE Trans Microwave Theory Tech., 25(12), 981 – 984, 1977.

[17] E.W Bryerton, W.A Shiroma, Z.B Popovic, ‘A 5-GHz high-efficiency class-E oscillator’,

IEEE Microwave Guided Wave Lett., 6(12), 441 – 443, 1996.

[18] M Prigent, M Camiade, G Pataut, D Reffet, J.M Nebus, J Obregon, ‘High efficiency free

running class-F oscillator’, IEEE MTT-S Int Symp Dig., 1995, pp 1317 – 1320.

[19] V Rizzoli, A Costanzo, A Neri, ‘Harmonic-balance analysis of microwave oscillators with

automatic suppression of degenerate solution’, Electron Lett., 28(3), 256 – 257, 1992.

[20] C.R Chang, M.B Steer, S Martin, E Reese, ‘Computer-aided analysis of free-running

oscil-lators’, IEEE Trans Microwave Theory Tech., 39(10), 1735 – 1745, 1991.

[21] B.D Bates, P.J Khan, ‘Stability of multifrequency negative-resistance oscillators’, IEEE

Trans Microwave Theory Tech., MTT-32, 1310 – 1318, 1984.

[22] V Rizzoli, A Costanzo, C Cecchetti, ‘Numerical optimisation of microwave oscillators and

VCOs’, IEEE MTT-S Int Microwave Symp Dig., 1993, pp 629 – 632.

[23] V Rizzoli, A Neri, ‘Harmonic-balance analysis of multitone autonomous nonlinear

micro-wave circuits’, IEEE MTT-S Int Micromicro-wave Symp Dig., 1991, pp 107 – 110.

[24] K.K.M Cheng, J.K.A Everard, ‘A new and efficient approach to the analysis and design of

GaAs MESFET microwave oscillators’, IEEE MTT-S Int Microwave Symp Dig., 1990, pp.

1283 – 1286.

[25] L.O Chua, Y.S Tang, ‘Nonlinear oscillation via volterra series’, IEEE Trans Circuits Syst.,

CAS-29(3), 150 – 167, 1982.

[26] L.O Chua, C.Y Ng, ‘Frequency-domain analysis of nonlinear systems: formulation of

trans-fer functions’, Electron Circuits Syst., 3(6), 257 – 269, 1979.

[27] Y Hu, J.J Obregon, J.-C Mollier, ‘Nonlinear analysis of microwave FET oscillators using

volterra series’, IEEE Trans Microwave Theory Tech., MTT-37(11), 1689 – 1693, 1989.

[28] R.G Rogers, ‘Theory and design of low phase noise microwave oscillators’, Proc 42nd Annual Frequency Control Symposium, 1988, pp 301 – 303.

[29] J Verdier, O Llopis, R Plana, J Graffeuil, ‘Analysis of noise up-conversion in microwave

field-effect transistor oscillators’, Trans Microwave Theory Tech., 44(8), 1478 – 1483, 1996.

[30] M Regis, O Llopis, J Graffeuil, ‘Nonlinear modelling and design of bipolar transistors

ultra-low phase-noise dielectric-resonator oscillators’, IEEE Trans Microwave Theory Tech., 46(10),

1589 – 1593, 1998.

[31] W Anzill, P Russer, ‘A general method to simulate noise in oscillators based on frequency

domain techniques’, IEEE Trans Microwave Theory Tech., 41(12), 2256 – 2263, 1993.

[32] V Rizzoli, F Mastri, D Masotti, ‘General noise analysis of nonlinear microwave circuits

by the piecewise harmonic-balance technique’, IEEE Trans Microwave Theory Tech., 42(5),

807 – 819, 1994.

[33] G.R Olbrich, T Felgentreff, W Anzill, G Hersina, P Russer, ‘Calculation of HEMT

oscil-lator phase noise using large signal analysis in time domain’, IEEE MTT-S Symp Dig., 1994,

pp 965 – 968.

[34] C.-L Chen, X.-N Hong, B.X Gao, ‘A new and efficient approach to the accurate simulation

of phase noise in microwave MESFET oscillators’, Proc IEEE MTT-S IMOC ’95 Conf., 1995,

pp 230 – 234.

[35] E Ngoya, J Rousset, D Argollo, ‘Rigorous RF and microwave oscillator phase noise

calcu-lation by envelope transient technique’, IEEE MTT-S Symp Dig., 2000, pp 91 – 94.

278 OSCILLATORS

[36] E Mehrshahi, F Farzaneh, ‘An analytical approach in calculation of noise spectrum in

micro-wave oscillators based on harmonic balance’, IEEE Trans Micromicro-wave Theory Tech., 48(5),

822 – 831, 2000.

[37] A Suarez, S Sancho, S VerHoeye, J Portilla, ‘Analytical comparison between time- and

frequency-domain techniques for phase-noise analysis’, IEEE Trans Microwave Theory Tech.,

50(10), 2353 – 2361, 2002.

be realised both in passive and active configurations, that is, employing passive deviceswith reactive or resistive nonlinearities (diodes under reverse or forward bias respectively)

or active devices (MESFET, HEMT, HBT) biased in a strongly nonlinear operating region.Active multipliers offer the advantage over passive ones of exhibiting conversion gainrather than losses, eliminating or reducing the need for a high-frequency amplifier afterthe multiplier; their bandwidth can also be made to be fairly wide The availability ofactive devices exhibiting a conversion gain well into the millimeter-wave region withnon-negligible bandwidth is actually pushing towards the active solution Moreover, thischoice has the obvious advantage of allowing functional integration in a single technology

if a monolithic implementation is attempted

The intrinsic nonlinear nature of frequency conversion requires the use of nonlineardesign methodologies, both on the side of accurate and efficient nonlinear models andalgorithms, and on the side of clear optimum design conditions and procedures This isespecially true for monolithic solutions, with the aim of reducing unnecessary design timeand increasing the possibility of first-time success of an optimised design

To this moment however, while the availability and accuracy of general nonlinearmodels and analysis algorithms ensures the prediction of the performances of the active

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

280 FREQUENCY MULTIPLIERS AND DIVIDERS

multiplier with sufficient dependability for practical applications, not everything is clearfrom the methodological point of view In recent years, simplified approaches have shedsome light on the basic mechanisms of harmonic generation; however, more complexconsiderations are to be made when full-nonlinear effects are taken into account

It is to be pointed out that the main design goals for a frequency multiplier are

on the one hand high conversion gain from the input signal at fundamental (oscillator)frequency to the converted (multiplied) frequency, and on the other hand a low DCpower consumption The relative importance of these two quantities obviously depends

on the particular application Bandwidth is also important in the case when a frequency signal (and therefore a voltage-controlled fundamental oscillator) is required

variable-by the system Another issue is reliability, which could be impaired variable-by the excitation ofpotentially dangerous nonlinearities of the actual device Stability must also be checked,

as in any high-gain microwave circuit, with the additional complication of strongly linear operations

non-In the following, a short description of passive multipliers is first presented Thisissue has been investigated in detail in the past, and many theoretical and experimentalresults are available Then, active multipliers are presented, which are rapidly gainingpopularity in applications but which still lack a comprehensive treatment and a generallyagreed design methodology

6.2 PASSIVE MULTIPLIERS

Passive multipliers are extensively used for their simplicity The basic principles and some examples are shown.

Passive multipliers are popular for the simplicity of their structure, for the reliability

of the nonlinear frequency-multiplying element, usually a diode, and for the very highmaximum frequency of operations Quite naturally, the frequency multiplication cannotyield any conversion gain, but only losses; this is partly compensated by the low or zero

DC power consumption The cascading of an amplification stage can balance the powerbudget, but requires two circuits for the complete treatment of the signal Both the circuitsare reasonably well established now, and a reliable design can be performed; however,unnecessary complication of the circuitry results, compared to an active implementation.Only the passive multiplier circuit is described in the following, as the amplifier is astandard linear, quasi-linear or high-efficiency amplifier at the multiplied frequency.Passive multipliers can be classified as resistive or capacitive (or reactive, ingeneral) types In the first case, the frequency-multiplying mechanism is the strong nonlin-earity of the conduction current in the diode In the second case, the frequency-multiplyingmechanism is the nonlinear nature of the reactance of the diode, typically the junctioncapacitance In this latter case, the depletion capacitance in reverse bias is used as non-linear reactance in order to avoid the conduction current present when the diffusioncapacitance is not negligible However, especially at high frequency, both mechanismsare found to contribute to frequency multiplication [1] A great variety of diode struc-tures have been developed, especially for very high frequencies, that can reach the THz

PASSIVE MULTIPLIERS 281

range [2–4]; many structures have a back-to-back arrangement and a symmetric C-Vcharacteristic, that allow zero-bias operations and efficient frequency tripling [5].Resistive multipliers [6, 7], in principle, have infinite bandwidth, given the non-frequency-dependent nature of resistive nonlinearities However, the associated junctionreactance and the reactive parasitic elements of the diode imply a frequency-dependentbehaviour of the element Moreover, matching networks will further limit the bandwidth.Nonetheless, a significant bandwidth can be achieved in practice [8] On the other hand,power dissipation is always present in the nonlinear element, which imposes a lowerlimit to frequency conversion losses It can be demonstrated [6, 7, 9, 10] that for theconversion gain the Manley–Rowe relation holds:

GC ≤ 1

n2

where n is the frequency multiplication factor Therefore, resistive multipliers are not

suitable when conversion losses are a critical issue, especially for triplers or plers Unfortunately, losses are important especially at high frequency, where frequencymultipliers are most useful

quadru-Resistive multipliers are efficiently used within balanced configurations Singlybalanced or doubly balanced arrangements have intrinsic fundamental-frequency andodd-harmonic frequency rejection, and, therefore, reduced need for filtering networks.Therefore, bandwidths in excess of one octave can be achieved [8], though conversionloss is quite high (in the order of 10 dB or more)

Reactive multipliers [9–14], in principle, can have zero conversion losses Thisrequires proper reactive loading at all frequencies other that the input (fundamental)frequency and the output (multiplied) frequency; an obvious consequence is that the mul-tiplier has a narrowband, given the strong frequency dependence of reactive impedances.Moreover, when operating at high frequencies, it is difficult to exactly control the values

of reactances because of the high influence of parasitics It is therefore difficult to exactlytune a reactive multiplier

As mentioned above, the general structure of a frequency multiplier requires signal matched terminations at fundamental (input) frequency and multiplied (output)frequency so that most input power reaches the nonlinear device and most output power

large-is transferred to the load All other frequencies are usually referred to as idler frequencies;their loads must be optimised for optimum multiplication and transfer of output power tothe load In principle, reactive loads at idler frequencies are desirable because they do notdissipate active power Their actual value must be determined under large-signal condi-tions, and in general requires detailed optimisation However, short-circuits usually yieldstability of operation, and efficient multiplication, because of the high current circulation

in the diode also at idler frequency

In practice, only the first two or three harmonics will be controlled, resulting in

a conversion loss usually higher than the theoretical minimum Normal conversion lossvalues for a reactive (narrowband) doubler are around 6 to 9 dB, and above 10 dB for atripler [10]

282 FREQUENCY MULTIPLIERS AND DIVIDERS

fundamental-in the output current characteristics [15–21] In particular, current clippfundamental-ing caused bypinching off the channel is most used because of the high distortion introduced, the low

or zero-bias current dissipation and high reliability of operation; this is roughly equivalent

to Class-B or Class-C operations in an amplifier and similarly yields a conversion gain

in the order of 6 to 9 dB less than the fundamental-frequency gain, at least in ple [16] Other strong nonlinearities are less practical: for instance, current clipping bygate-channel junction forward conduction requires Class-A biasing and therefore high DCpower dissipation, and causes possible gate junction damages; however, gain is higherthan that near pinch-off Another very strong nonlinearity, that is gate-drain or channelbreakdown, is never used because of reliability problems, in addition to the necessity forhigh bias voltage The effect of nonlinearities other than the transconductance modula-tion has been investigated [16, 20, 22, 23, 24], leading to the conclusion that a possiblealternative is the modulation of the output conductance (see also Chapter 7) Other non-linearities, for example, those due to nonlinear gate-source and gate-drain capacitances,are less effective for harmonic generation, giving a minor contribution to frequency mul-tiplication However, the presence of reactive feedback elements within the active devicecan not only enhance the harmonic generation but also give rise to instability problemsand must be taken into account for comprehensive and reliable multiplier design.Terminations presented to the active device belong to two basic types: at the signalinput and output ports, that is, gate/base port at fundamental frequency and drain/collectorport at the desired harmonic frequency; and the other ports, usually limited to the second-

princi-or third-harmonic frequency, while pprinci-orts at higher frequencies are shprinci-orted princi-or opened Fprinci-orthe first type, conjugate match allows maximum power transfer, and, therefore, most effi-cient conversion; since the device operates in nonlinear regime, large-signal impedances(or equivalent) must be used For the second type, often called idler ports, reactive termi-nations are a natural choice for minimising power dissipation Terminations at harmonicfrequencies at the input port are sometimes neglected (i.e shorted), assuming that thecontribution of nonlinearities in the input mesh is minor; in the following, it is shownthat this is not usually true Terminations at the output port other than at multiplied fre-quency, on the other hand, play an important role: in principle, they must be as close

as possible to a short circuit for two basic reasons First, it is important that the mosteffective nonlinearity, that is the transconductance, be fully exploited and, therefore, thatthe load line be as vertical as possible in the plane of the output characteristics Second,for high output power the output voltage swing must cover the full range between theknee or saturation voltage and the breakdown voltage, that is, the active region If other

ACTIVE MULTIPLIERS 283

frequency components are present, the multiplied frequency component of the outputvoltage must be reduced consequently in order that the total voltage does not exceed thelimits of the active region These simple considerations are essentially confirmed also by

The analysis and design of active multipliers is presented in the following, first

by simplified piecewise-linear approaches that allow a first insight into the harmonicgeneration principle and give indications on efficient operating regions By these means,

a first qualitative design can be performed: regions of optimum design are found andapproximate values for the terminating networks are derived Then, full-nonlinear analysis

is described for inclusion of detailed nonlinear effects that mandate some care in thenonlinear optimisation of the actual circuit prior to fabrication

6.3.2 Piecewise-linear Analysis

In this paragraph, a piecewise-linear model and analysis is used to derive the main features and behaviour of an active frequency multiplier General guidelines are available that help the designer in the first phase of the circuit design.

Simplified nonlinear methods are useful especially at an early design stage when

a general understanding of the basic principles and mechanisms must be gained in order

to assess the basic structure of the circuit and the expected performances They are also

a useful means to evaluate the performances of the active device and are of help inits selection Such an approach makes use of a simplified device model and is based

on reasonable assumptions on the frequency multiplication mechanisms In general, itsresults qualitatively agree with a full-nonlinear approach, thus providing a valuable toolfor a quick preliminary analysis and a reasonable starting point for full-nonlinear analysis

A simplified model of the active device (in this case a field-effect transistor) isshown in Figure 6.1, where the only nonlinearity is the output current source In general,many elements contribute to the generation of upper harmonics, as for example, gate-source and gate-drain capacitances However, the main harmonic-generating effect isusually provided by output current nonlinearities [16, 18, 19, 21] For an accurate analysis,

a detailed description of the output current dependence on the controlling voltages should

be used; however, the main effects are retained, while at the same time allowing a simpleanalytical treatment when the current is described by a simple piecewise-linear model

An example is shown in Figure 6.2, where the constant transconductance in the regionbetween pinch-off and forward gate-junction conduction is also evidenced

This model allows a very easy design if the assumption is made that the loadline does not reach the ohmic and breakdown regions, that is, if the operating voltage is

284 FREQUENCY MULTIPLIERS AND DIVIDERS

Figure 6.2 A piecewise-linear constant-transconductance model for the output characteristics

always higher than the knee voltage and smaller than the breakdown voltage This can beensured by proper drain voltage biasing and by suitable choice of the output load Then,the output current depends on the gate voltage only Therefore, the current waveform isknown when the gate bias voltage and signal amplitude are known; it has the shape of atruncated sinusoid, as shown in Figure 6.3

The harmonic content of the current waveform of Figure 6.3 is easily and lytically computed, allowing the direct determination of the optimum gate bias and gatesignal amplitude for maximum amplitude of the desired harmonic Then, the optimumoutput bias and load is consequently determined As said above, all output harmonicsmust be shorted except the desired output harmonic; this assumption allows the directdetermination of the optimum output load: it is the resistance that maximises the voltageswing within the hard nonlinear limits imposed by breakdown (upper limit) and kneevoltage (lower limit)

Figure 6.3 Input voltage and output current waveform for a simplified piecewise-linear model

For the sake of clarity, we will first describe multiplier operations when the outputcurrent is clipped only by the pinch-off of the channel (Class C–A); then, the case of

a symmetric clipping by pinch-off and forward gate-junction conduction is described forfrequency tripling (Class-A); and finally, the general case of asymmetric upper and lowerclipping will be described For all the cases, analytic formulae are given

In the former case, and under the mentioned hypotheses, the current has theexplicit expression

wave-286 FREQUENCY MULTIPLIERS AND DIVIDERS

t 2

a

2 a

−

Figure 6.4 Output current waveform in the case of clipping due to pinch-off onlyThe amplitude of the DC component and of the first three current harmonics ofthe output current are explicitly given by the following formulae [25]:

Figure 6.5 Normalised amplitudes of the output current harmonics vs current conduction angleα

The normalised amplitudes of the DC and first three current harmonics as a function

of the conduction angleα are also shown in Figure 6.5.

Clearly, for the best conversion gain, a Class-C bias must be selected For instance,for optimum doubler operations, an operating angle of approximately 126◦ must beselected that yields a second-harmonic current amplitude about one-fourth (0.27) of themaximum output current For a frequency tripler, an operating angle of approximately

75◦ must be selected for a third-harmonic current amplitude equal to about one-sixth(0.185) of the maximum output current

For maximum output power, it is desirable that the peak output current value beequal to the maximum channel currentImax; in this case, the conduction angle is related tothe gate bias voltageVGG and to the sinusoidal input signal amplitude ˆVgsby (Figure 6.6)