Nonlinear Microwave Circuit Design phần 4 pdf

Once the small-signal conductance is evaluated for all bias points, the current curve is computed byintegration with respect to voltage.. The DC curve is now fitted by any fitting function

In the third one, the device is biased at many points along the curve; at eachpoint, the differential (small-signal) conductance is measured by a microwave admit-tance measurement In practice, small-signal S-parameter measurements are performed,

and the reflection coefficients are converted to admittances The small-signal tance corresponds to the tangent of the fast measurement performed from that bias point(Figure 3.28)

conduc-This is easily seen by considering the measurement set-up for large-signal fastmeasurements, and comparing it to a standard vector network analyser Once the small-signal conductance is evaluated for all bias points, the current curve is computed byintegration with respect to voltage

Let us now discuss the three measurements The DC curve must be used for thesimulation of slow or constant phenomena, as the DC bias or rectified voltages andcurrents, the low-frequency second-order intermodulation signal in multi-tone systems orthe down-converted phase-noise in oscillators, when their frequency is very low (belowapproximately 100 KHz) The pulsed curves must be used for the simulation of microwave

large signals; in this case, the curve must be measured from the same quiescent point asthat of the large-signal operation This is however, in general, not predictable, since itincludes rectified terms (see Chapter 1); the model must then include the curves measuredfrom all quiescent points and must be able to adjust to the actual quiescent point obtained

in the analysis

The curve obtained by the integration of the small-signal measurements, finally,

is not physically correct and should never be used Unfortunately, this is a popularmethod to extract nonlinear models, because it is also the most practical and traditionalfrom the point of view of both instrumentation and extraction procedure A completemeasurement set-up for DC, small-signal and pulsedS-parameter measurement has been

demonstrated [63], allowing consistent modelling; however, it is not usually available tomodellers and designers

If the measurements in Figure 3.27 are available, then theI/V characteristics have

the following general form, equivalent to eq (2.2) in Chapter 2:

where the dependence on the instantaneous voltage, on the DC voltage and on temperature

is a consequence of the described phenomena The DC curve is obviously a particularcase of the above formulation, obtained when the instantaneous voltage coincides withthe DC voltage:

A procedure for the correction of small-signal data in order to account for perature and trap effect has been proposed, under certain hypotheses, allowing the use ofsmall-signal parameters for consistent large-signal modelling [64–67] Let us assume that

tem-we are working in isothermal conditions and that the current for a given instantaneousvoltagev(t) when the DC voltage vDCis equal to the current obtained from the DC curvefor a slightly different voltage (Figure 3.29):

The correction can be written as a fraction of the difference between the static andactual voltage value:

iDCi

i (t )

Figure 3.29 The current in large-signal conditions obtained from the DC curve at an offset voltage

If the constant α = α
is the same in both cases, that is, if the correction is the

same fraction of the distance of the actual instantaneous voltage from the bias point, then

it is easily evaluated from a single fast (pulsed) measurement withv(t) = vDC and fromthe DC curve; however, it is also evaluated from a single small-signal measurement at

vDC∼= v(t) and from the DC curve It is easy to see that, for a v(t) very close to vDC,that is, for a small incremental instantaneous signal (Figure 3.30)

in between the two values It is also a reasonable assumption that this effect be linear

Physically different effects can be treated in the same way if they have a lineareffect on the current For instance, mobility is inversely proportional to temperature;therefore, a change in temperature (e.g increase in ambient temperature) for a fixed DCvoltage can be reduced to a change in voltage proportional to the difference in temperature(Figure 3.31):

iDC(vDC, T
) = iDC(uDC, T ) = iDC(vDC + v, T ) (3.38)

Let us allow the device to increase its temperature as a consequence of increaseddissipated power within the device itself, for example, as a consequence of increa-sed applied DC voltage (Figure 3.32) We can write

iDC(vDC, T

) = iDC(uDC, T ) = iDC(v

Figure 3.32 The DCI/V curve for two different temperatures and applied DC voltages

that do not fulfil the condition

iDC(v0) +

 v=v1

for any pair of voltages (v0, v1), because of the problems mentioned The DC curve is

now fitted by any fitting functionf that depends on a number of parameters pi: a set ofvalues for the parameterspDC ,i is determined as a consequence of the fitting procedure:

Then, the difference between the measured small-signal conductance and the vative of the DC curve is computed for all voltages and fitted to the derivative of anotherfitting function:

Alternatively, the small-signal conductance is fitted to the derivative of a fitting function:

So far for the conduction current; when the displacement current contribution isconsidered, things are more complex First of all, the DC measurements are not possible.Second, the fast measurements require that complex data (amplitude and phase) be mea-sured in a short time In fact, this requires pulsed S-parameter measurements: during a

short bias pulse, a microwave small-signal excitation is fed to the device and the scattering

Figure 3.33 The DCI/V curve, the small-signal conductances and the curve derived from

inte-gration of the latter

We now rewrite the above considerations for a two-port device [53] The currents

in a linear device are

Ic1(V1, V2) = Ic1(V10, V20) + dIc1

If the reactive part of the device is reciprocal, the circuit is simplified (C12 = C21)

Figure 3.37 An equivalent circuit with reciprocal capacitances and unilateral transconductances

Often, the conduction currents are unilateral in actual devices (g12 = 0); the

equiv-alent circuit then becomes (Figure 3.37)

For an FET, the quasi-static equivalent circuit is shown in Figure 3.38, where

indepen-E(V1, V2) = Q1(V1, V2 ) · V1 + Q2(V1, V2) · V2 (3.58)

Q1(V1, V2) = ∂E(V1, V2 ∂V1 ) Q2(V1, V2) = ∂E(V1, V2) ∂V2 (3.59)

The second-order partial derivatives of the single-valued energy function E must

fulfil the following relations:

C11(V1, V2) = ∂Q1(V1, V2 ∂V1 )= ∂2E(V1, V2)

∂V2 1

∂2E(V1, V2)

∂V2 2

and the capacitive part of the device then is reciprocal

A nonlinear model of the two-port device can be extracted from the linear signal model by using the small-signal measurements at many bias points, that is, formany values of the bias voltagesVgs ,0andVds ,0 The linearised conductances and capaci-

small-tances are evaluated at each bias point, and their dependence on the applied bias voltages

is found:

gm(Vgs ,0 , Vds ,0 ) gds(Vgs ,0 , Vds ,0 )

Cgs(Vgs ,0 , Vds ,0 ) Cgd(Vgs ,0 , Vds ,0 )

small-the I/V characteristics is easily available; high-frequency data is available from pulsed

measurement or corrected extraction from S-parameters, as described above A suitable

arrangement for the equivalent circuit is shown in Figure 3.41 [72, 75]

If a smoother transition from DC to RF frequency is desired, low-pass and pass filters with transition frequency in the MHz range must be replaced in the DC and

Figure 3.41 An equivalent circuit of an FET including dispersion effects

modelled as the depletion capacitances of the gate–source and gate–drain diodes, andthe drain–source capacitance as a constant parasitic, that is, the substrate capacitance.This circuit automatically fulfils the nonlinear constraints just described Let the diodecapacitances have an expression of the type

3.3.4 Extraction of an Equivalent Circuit from Multi-bias Small-signal Measurements

The behaviour of a real device is distributed by nature; therefore, a good equivalentcircuit can at best be an approximation In general, the higher the number of elements inthe equivalent circuit, the better is the approximation; however, the number of elementsshould be kept as low as possible, both for practical model extraction and for physicalmeaningfulness of the circuit elements On the one hand, the evaluation of the elementvalues should be as easy and straightforward as possible, and this is seriously hampered

by an excessive number of elements in the circuit On the other hand, the behaviour ofthe elements must satisfy the physical constraints (see previous paragraph) that are bestfulfilled by elements with a clear correspondence to actual physical effects inside thedevice Moreover, when physically meaningful, the equivalent circuit gives interestinginformation on the structure of the device, both as a feedback to technology and for aqualitative evaluation of the device performances by the designer

As an example, the correspondence between a simple equivalent circuit of anMESFET and its physical structure is shown in Figure 3.42

Several similar topologies are available for most active devices at microwave andmillimetre-wave frequencies, like MESFETs, HEMTs, MOSFETs, BJTs and HBTs In

Figure 3.42 The physical structure of an MESFET with the equivalent-circuit elements

general, they fit the wide-band small-signal (linear) parameters of the device for a givenbias point; when this changes, the values of the intrinsic elements change too, whileparasitics are unchanged If this is true and the fit to wide-band small-signal parameters

is still good, the equivalent circuit is a valid candidate for nonlinear applications Then,the dependence of the values of the intrinsic elements on the applied voltages is modelled

by some fitting functions, with the limitations described in the previous paragraph; if this

is true, the model is a good nonlinear model

Let us illustrate this procedure with an example The circuit in Figure 3.43 is anequivalent circuit suitable for fitting MESFETs and HEMTs in a wide-frequency band

As an example, the measured S-parameters from 0.1 GHz to 40.1 GHz are shown in

Figure 3.44, together with the S-parameters computed from the equivalent circuit, in a

range of bias points (Vds = 2.5 V, Vgs= −1.8 ÷ 0.5 V).

In Figure 3.45, the values of the intrinsic elements are plotted as a function ofthe gate–source voltage Vgs and of the drain–source voltageVds, together with a fittingfunction, in this case a neural network model [76]

The topology of the equivalent circuit together with the fitting functions identifies

a large-signal equivalent-circuit model

Let us now describe how the values of the elements of the equivalent circuitare extracted from small-signal data for a practical device Two main approaches areavailable: a wide-band fit of the equivalent circuit to the small-signal measured data bymeans of numerical optimisation routines and the selective identification of groups ofparameters by analytical means at special bias points

A wide-band fit of the equivalent circuit to small-signal data is performed bymeans of any commercial CAD package The optimisation variables are the values of

measured ones, in the whole frequency band of interest Alternatively, Y -parameters or

any other linear equivalent parameters can be fitted The optimisation can be performedfor each bias point separately or for all the data from all bias points of interest at once [29]

In the former case, the risk is that the optimised values of the parasitic elements vary frombias point to bias point, contrary to the assumption: this is an indication of bad topology

or bad optimisation In the latter case, the parasitics are forced to have the same values

at all bias points; however, the numerical burden greatly increases In both the cases, theoptimisation algorithm risks to get trapped in local minima, never reaching the absoluteminimum The goal function is usually not very sensitive to some elements, whose valuesare therefore rather uncertain This can be a problem for some applications: for example,the gate resistance in an FET is difficult to extract from normal, operating-point S-

parameters, but its value is meaningful for the evaluation of the noise performances

of the device If this is the case, it is wise to adopt the global fitting procedure, alsoincluding special bias point as in the ‘two-tier’ procedure (see below) On the other hand,this approach has a remarkable advantage: it is very easy to change or adjust the topology

of the equivalent circuit and have a fast feedback on its fitting accuracy In addition, it

is not restricted or dedicated to any topology or device, and there is no need to developdedicated software

The alternative approach, that is, the selective identification of the elements ofthe equivalent circuit, is based on the bias-independence of the parasitics A ‘two-tier’extraction procedure is performed: parasitics are first evaluated at special, suitable biasconditions, and their values are not changed afterwards; then, the intrinsic elements areevaluated at each normal, operating bias point within the region of interest This approachhas a clear advantage: the values of the elements are extracted by means of simple, ana-lytical formulae without any optimisation Usually, a better understanding of the structure

of the equivalent circuit is also gained

A great variety of bias conditions has been proposed for parasitics evaluation [77–87], but almost all require some measurements on a ‘cold’ device, that is, zero drain orcollector voltage This condition greatly simplifies the behaviour of the inner device,and parasitics are better evaluated Basically, the measured small-signal parameters ofthe ‘cold’ device are equated to the corresponding analytic expressions of the small-signal parameters of the model; the equations are then explicitly solved for the values

of the elements Measurements at a single frequency can be used for the evaluation ofthe parameters, but averaging over frequency in a suitable band allows for reduction

of random measurement errors In general, diversity in frequency is a useful tool forimproving the meaningfulness of the extraction

A two-port S-parameter measurement at a ‘cold’ bias condition provides three

complex equations, yielding six real values; if the parasitic elements are more than six,more than one ‘cold’ bias condition must be used Moreover, not all equations yieldreliable results, and it is usually better to have redundant data

Let us illustrate the procedure by an example concerning an FET [83, 87]; theprocedure described hereafter is by no means the only possible one and not necessarilythe best: it is only one of the many proposed so far In fact, it often turns out that a spe-cific device may require modifications of the procedure, because of minor but importantdifferences in the structure of the device; however, the approach is usually similar.Three ‘cold’ bias conditions are used in this example: depleted channel, that is,pinched-off FET (Vds = 0, Vgs< Vpo), open channel (Vds = 0, Vpo< Vgs < Vbi) and gate-

channel junction in weak forward conduction (Vds = 0, Vbi< Vgs) The intrinsic device

behaves very differently in the three conditions in such a way that different parasiticelements are relevant in each bias condition Somehow, this is similar to what is requiredfrom different calibration standards; in fact, in this case also, access elements of a deviceunder investigation (in this case the intrinsic device) must be identified and removed Theequivalent circuits of the device in the three ‘cold’ conditions are shown in Figure 3.46

In the case of pinched-off device, the measured S-parameters are converted to

admittance Y -parameters, which are easier to express analytically for the equivalent

cir-cuit If we limit ourselves to a suitably low-frequency range, the inductances can beneglected, and the simplified equations read as follows:

Im(Y11)po = ω(Cpg+ Cpgs+ Cgs+ Cpgd+ Cgd) (3.69a)

Im(Y22)po = ω(Cin+ Cds+ Cpds+ Cpd+ Cpgd+ Cgd) (3.69c)