Extracted nonlinearity coefficients

The S-parameter characterization method was employed to determine the properties of an Infineon CLY2 GaAs MESFET (Infineon 1996). The S-parameters were measured in pulsed form over a range of bias conditions at temperatures of 0 and 50 °C, and the nonlinearity coefficients extracted are given in the following four figures with respect to the biasing conditions (the x-axis indicates Volts and the y-axis milliamperes).

Fig. 49. Transconductance and output conductance (vertical Id [mA], horizontal Vd [V]).

The first column in Fig. 49 corresponds to the first, second and third order nonlinearities in transconductance (the first row in Equation (28)), while the second column shows the nonlinearity in the output conductance. The shape of the nonlinearity of the transconductance is relatively independent of the drain voltage, and the normalized

1 2 gm 4 5

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nonlinearity in transconductance is in the range 2 - 20% and decreases with increasing bias current. Similarly, the shape of the output conductance is relatively independent of the drain current. The output conductance decreases rapidly with increasing drain voltage at low voltages, and therefore it has a high nonlinearity, but it is almost linear at high drain voltages.

Fig. 50. Cross-terms and electrothermal terms.

Columns 1 and 2 in Fig. 50 give the cross-terms and electrothermal terms. Electrothermal interaction is clearly observable. The DC offset in the I-V curves is of up to 1 mA/K, and if dynamic temperature variations of few Kelvins occur on the surface of the chip and the amplifier is designed to be linear enough, electrothermal distortion mechanisms will have an impact on the electrical distortion in FET amplifiers. Cross-terms are very significant, especially at low drain voltages or currents.

1 2 K2gmgo/go 5

Fig. 51. Nonlinearity of Cgd, including thermal effects and Cds.

The first plot in the first column of Fig. 51 gives the small-signal C_GD, and its nonlinearity is drawn in plots 2 and 3. This nonlinearity of the reverse biased p-n junction is quite weak, but it must be remembered that the nonlinear feedback current is amplified to a significant degree in most cases, as a result of which even weak nonlinearity will cause quite a lot of distortion. For example, the amount of nonlinearity in Cds, given in the last plot of Fig. 51, is similar to that in C_GD. Moreover, the voltages across the two nonlinear capacitors are quite large. Although the nonlinearity of Cds affects only the output of the amplifier, C_GD affects the gate voltage as well. This nonlinear gate voltage caused by the nonlinearity of C_GD is magnified in the output of the amplifier, and it is for this reason that C_GD involves more important nonlinearity mechanisms than C_GS. Plots 1-2 in column 2 present the second and third order electrothermal nonlinearities in C_GD, which might play an important role with regard to distortion in a linear amplifier. In this case they were masked by pronounced electrical distortion, however.

1 2 Cgd 4 5

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Fig. 52. Nonlinearity of Cgs, including thermal effects.

The extracted C_GS changes with gate voltage and temperature. The linear part of C_GS in its second and third order electrical nonlinearities are given in column 1 of Fig. 52, and the thermal second and third order terms in column 2.

4.4 Conclusions

This Chapter has been concerned with characterization of the Volterra model, and presents two techniques, DC and AC characterization. The effects of extrinsic circuit elements coming from lead resistances and inductances cause significant numbers of errors in the extracted coefficients, effects that must be taken into account by de-embedding the extrinsic components. Self-heating is an important factor in nonlinear characterization, because nonlinearities arising from changes in terminal voltages and temperature are very difficult to separate from each other in steady-state measurements.

1 2 Cgs 4 5

Pulsed measurements are therefore employed in order to investigate the transistor under as constant temperature conditions as possible. The effects of optimum pulse length are discussed, and it is noted that the pulse must be wide enough to produce an electrical steady-state, while at the same time it must be as short as possible to avoid self-heating.

Pulse lengths of 1ms and 10 ms were used as a trade-off between the two in the present measurements, but this figure is highly dependent on the transistor type and package, so that the results presented here cannot be generalized. The effects of the optimum fitting range on characterization were also discussed, and the range that corresponds to the actual signal swings in different directions was chosen. Information on the output power and load resistance is needed in order to estimate the voltage and current swings in an I-V plane.

DC characterization is based on the collector and base currents measured over the collector and base voltages. At least 12 measurement points are needed to extract the nonlinearity of gm, go and cross-terms up to order three. This method is very simple, and the measurement equipment is also simple and cheap. Secondly, only resistive extrinsic components have to be taken into account, because capacitances and inductances can be neglected at DC, and this facilitates the extraction procedure. One drawback of DC characterization is dc offset. Since the coefficients of the polynomial must be calculated without the bias voltage, even the smallest dc offset will produce errors in the extracted coefficients, as illustrated in Fig. 53.

Fig. 53. Comparison between AC and DC characterization procedures.

AC characterization, on the other hand, is based on AC measurements performed over ranges of bias and temperature values. Measured S-parameters are converted to Y-parameters, from which the small-signal circuit elements are calculated using well-known small-signal extraction procedures. The nonlinearities of the circuit elements are obtained from circuit elements at surrounding voltage values, and 9 measured small-signal values for gm and go are used here to extract the nonlinearity of gm, go and cross-terms up to order three. This yields 18 Equations and 9 unknowns, and LMSE fitting is employed to solve the matrix equations.

Id

Vg Vg

Id

a) b)

gm ∂Id

∂Vd

---= bias point

problems

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The greatest advantage of the AC method is that first derivatives of I-V curves are measured instead of absolute current values, which means that one circuit element is obtained from just one measurement. This excludes the existence of dc, and consequently avoids dc-related problems. Secondly, the AC method is also more robust in terms of errors in the location of data points in the I-V plane, since only small deviations exist in measured S-parameters with respect to bias point changes. This is not true in a DC method, where small errors in data points can cause significant errors in higher order coefficients (Wambacq & Sansen 1998). Thirdly, capacitive nonlinearities can also be characterized by the AC method, which is not possible in DC characterization. The drawback is that the measurements are quite complicated and a fast NWA is needed.

However, due to the much better accuracy properties of the AC method, the extra work involved in characterization yields more robust distortion simulation results.

The Volterra model can also be characterized by means of a circuit simulator, in which the device model can be simulated in a manner similar to the measurements presented in this Chapter, enabling a full Volterra model to be characterized. This would be a very easy approach compared with measurements, but unfortunately, most of the commercial device simulation models that are available nowadays are far too poor for characterization use as explained in Chapter 3. It does not make any sense to extract nonlinearity coefficients based on model equations that have incorrect higher order derivatives. If better models were available, this approach would become a viable alternative to measurements.

5 Linearization and memory effects

The main effort in previous Chapters has been devoted to analysing the IM3 components of the amplifier as a function of modulation frequency and instantaneous chip temperature by means of the Volterra model. This viewpoint will be extended in three ways in the present Chapter. First, the effects of memory on linearization will be discussed, and second, the effects of signal amplitude will be taken into account in simulations of real PAs, and not only polynomials as in Chapter 2. Thirdly, a technique for measuring both the amplitude and phase of IM3 components will be presented.

Smooth memory effects are not usually harmful for the linearity of the PA itself. A phase rotation of 10-20 degrees or an amplitude change of less than 0.5 dB as a function of modulation frequency has no dramatic effect on the ACPR performance of the device, but the situation is completely different when linearization is employed to cancel out the IM sidebands. Extremely accurate amplitude and phase matching between distortion components and cancelling signals is needed, as pointed out in Section 1.3, and if the IM3 components rotate as a function of modulation frequency, for example, but the cancelling signals do not, the cancellation performance of linearization cannot be good over a wide range of modulation frequencies. Large differences exist in the ability of the linearization techniques presented in Section 5.1 to take memory effects into account, but instead of studying the sensitivity of linearization techniques to memory, Section 5.1 concentrates more on memory effects with respect to predistortion, because predistortion is a very power-efficient choice of linearization method and is therefore important in terms of the power efficiency of the transmitter.

A simulation method for showing the memory effects in IM3 that are related to amplitude and modulation frequency is presented in Section 5.2. This includes a new normalization that is good figure of merit for memory effects in an amplifier. A measurement technique for characterizing memory effects in a real PA is then presented in Section 5.3, and measured results are given for BJT and MESFET amplifiers. The effects of memory on linearization will be summarized in Section 5.4.