Precise Characterization and Multiobjective Optimization of Low Noise Amplifiers

Although practically all function blocks of the satellite navigation receivers are realized using the CMOS digital integrated circuits, it is appropriate to create a separate low noise antenna preamplifier based on a low noise pHEMT. Such an RF front end can be strongly optimized to attain a suitable tradeoff between the noise figure and transducer power gain. Further, as all the four principal navigation systems (GPS,GLONASS,Galileo, andCOMPASS)work in similar frequency bands (roughly from 1.1 to 1.7 GHz), it is reasonable to create the low noise preamplifier for all of them. In the paper, a sophisticated method of the amplifier design is suggested based on multiobjective optimization. A substantial improvement of a standard optimization method is also outlined to satisfy a uniform coverage of Pareto front. Moreover, for enhancing efficiency of many times repeated solutions of large linear systems during the optimization, a new modification of theMarkowitz criterion is suggested compatible with fast modes of the LU factorization. Extraordinary attention was also given to the accuracy of modeling. First, an extraction of pHEMT model parameters was performed including its noise part, and several models were compared. The extraction was carried out by an original identification procedure based on a combination of metaheuristic and direct methods. Second, the equations of the passive elements (including transmission lines and T-splitters) were carefully defined using frequency dispersion of their parameters as Q, ESR, etc. Third, an optimal selection of the operating point and essential passive elements was performed using the improved optimization method. Finally, the s-parameters and noise figure of the amplifier were measured, and stability and third-order intermodulation products were also checked.


Introduction
The optimization algorithms generally require strong convergence behavior and adequate computational efficiency.
In state-of-the-art optimization procedures, these properties are mostly searched as a combination of more reliable, but slower metaheuristic algorithms [1][2][3][4][5][6] with faster, but less stable direct optimization methods [7][8][9][10].We have suggested an original combination of these methods in a robust three-step model identification procedure, which has been used for necessary extraction of pHEMT model parameters.Moreover, for the multiobjective optimization, a uniform coverage of the Pareto front [11] is also very important, especially for multidimensional tasks.We have used our improvement of the goal attainment method [9], which covers the Pareto front at least asymptotically uniformly and enables better automation of the procedure.
The LU factorization is considered the most demanding part of the circuit analysis from the computational point of view, especially in the case of many times repeated solutions during the multiobjective optimization.The elimination process during the LU factorization creates nonzero entries at positions, which correspond to zeros in the original matrix.The fill-in is the set of all the entries, which were originally zeros and took on nonzero values at any step of the factorization.The fill-in is able to highly increase the memory requirement, and the total computational time correspondingly.Generally usable tools [12,13] do not address this problem because they operate with regular matrices only.
For solving this problem, the heuristic Markowitz criterion is still widely used [14][15][16].The Markowitz criterion prevents expressions from swelling in a sparse matrix by selecting a pivot such that the number of entries which remain zero during the elimination is maximal, i.e., the fill-in created by the factorization is minimal.However, the Markowitz criterion is incompatible with a wide class of the fast algorithm modes for the LU factorization because a selection of the pivot from a whole submatrix is not allowed (the fast LU factorization only allows the pivot selection from a row or column).Therefore, the Markowitz criterion must be modified correspondingly for the fast LU factorization algorithm.
At present, there are many pHEMT models available in CAD tools.Their precision is often compared in references in a detailed way [17][18][19][20].We have identified more types of the models used in low noise amplifiers, and we selected the one as a compromise between its accuracy and complexity.

Multiobjective Optimization
Any circuit optimization subject to multiple, mostly mutually contradictory requirements represents so-called multiobjective optimization problem.A comprehensive description of the procedure for solving the multiobjective optimization problem can be found in [21]; in this section, a brief but improved and a bit more precise definition will be given.

The Multiobjective Optimization Problem
The multiobjective optimization problem can be formally written in the standard form of a minimization: Vectors of design variables x belong to a feasible region S, S ⊆ n , which can be specified by several equality constraints, inequality constraints, and/or by bounding allowed intervals for design variables x i .Let us denote f (x) = ( f 1 (x), f 2 (x), . . ., f k (x)) the vector of objective functions and Z = f (S) the image of the feasible region.

Pareto Optimality
Because the applied constraints are opposing one another, it is impossible to minimize them all simultaneously.Some tradeoffs need to be made, which leads to the concepts of Pareto optimality and noninferiority.A noninferior solution is one, whose any component can be further improved only at the cost of deteriorating another one.The set of noninferior solutions is called Pareto front, illustrated in Fig. 1 as the thicker part of the curve between points z A and z B .By solving the problem (1) is then understood the obtaining of a sufficient number of noninferior solutions covering part of the Pareto front with a sufficient density (and uniformity).
Geometrical representation of the (unmodified original) Goal Attainment Method.

(Standard) Goal Attainment Method
The classical (unmodified) Goal Attainment Method (GAM) [7][8][9] solves the multiobjective optimization problem by solving the singleobjective problem minimize in which f i are the individual design goals to be minimized, S is the feasible solution set (feasible region), z * i are some predetermined reference design goal values corresponding to f i , w i , w i ∈ are predetermined weighting coefficients and γ is an artificially introduced variable representing the single objective function.
GAM thus needs 2k parameters z * i and w i provided by the user, although only 2k − 1 degrees of freedom determine the obtained result, as can be seen from the 2D example in Fig. 2. The objective vector z * = (z * 1 , z * 2 ) represents a pivot point in the objective space through which the z S point moves in the direction of w = (w 1 , w 2 ) while varying γ.Therefore only the direction of the vector w is significant.

Suggested Method for an Assymptotically Uniform Coverage of the Pareto Front
Our proposed method further enhances the classical Goal Attainment Method with automatic generation of the predetermined parameters.A normalization is used instead of applying a weighting vector: where zi represent the components of the ideal vector z = ( z1 , z2 , . . ., zk ) found by independent minimization of each objective function z nad i compose the nadir objective vector z nad and they are the maximum (i.e., the worst-case) values of corresponding components ( zi ) j resulting from the independent minimization: A uniform coverage of the Pareto front is secured by a pseudorandom generation of the pivot vector z * in such a way that it uniformly covers a preselected region in the Z space.The random generation of the reference set can be performed as follows: starting with k vertices z 0,1 , z 0,2 , . . ., z 0, k and a (k − 1)-tuple of uniformly distributed and independent random numbers r i ∈ [ 0, 1) for i = 1, . . ., k − 1, this sequence of assignments will be performed to calculate a point z k−1,1 : This procedure calculates a total of N = k−1 i=1 i = k (k − 1)/2 points z i, j in the k-dimensional objective space (i.e., N = 1, 3, and 6 points for k = 2, 3, and 4) from k initially known vertices of the reference set.Much more details about this procedure can be found in [21], and a mathematical proof of the asymptotic uniformity of the coverage is given in [22].

Modifying the Markowitz Criterion
Precise modeling of RF devices generally leads to a number of transmission lines, vias, T-splitters, and parasitic inductors and capacitors to be included to a circuit model.Therefore, the matrix of the system created in the DC, AC, and transient analyses could be quite large, and the LU factorization of the (Jacobian) matrix should be arranged to suppress the fill-in (the set of all the entries, which were originally zeros and took on nonzero values at any step of the LU factorization).For solving this problem, the heuristic Markowitz criterion is still widely used [14][15][16].The Markowitz criterion prevents expressions from swelling in a sparse matrix by selecting a pivot such that the number of entries which remain zero during the elimination is maximal, or phrased the other way around, that the fill-in created by the operation is minimal.However, the Markowitz criterion is incompatible with a wide class of the fast algorithm modes for the LU factorization [23] because a selection of the pivot from a whole submatrix is not allowed (the algorithm modifications for the fast LU factorization only allow the pivot selection from a row or a column).Therefore, the Markowitz criterion must be modified correspondingly for an efficient implementation of the (fast) LU factorization algorithm.

Classical Form of the Markowitz Criterion
The algorithm itself is not too complicated [15].Suppose the LU factorization of A has proceeded through the first k stages.For each i th row in the active i denote the number of entries.Similarly, let c (k )  j denote the number of entries in the j th column.The Markowitz criterion is to select as pivot the entry The usage of this entry as the searched pivot causes r (k )  i − 1 c (k ) j − 1 entry modifications at the k th step.Not all these modifications will result in fill-in -therefore, the Markowitz criterion is actually only an approximation to the choice of the pivot which introduces the least fill-in.

Simplest Fast Mode Compatible Criterion
The entry modifications as a conclusion of applying the Markowitz weights (7) are incompatible with the fast LU factorization algorithm because the matrix structure changes in every stage.Therefore, remembering the structure of the nonzero elements could not have any practical sense under these conditions.For this reason, we should use a criterion that does not change the matrix structure during the LU factorization provided that the pivots remain relatively weighty in comparison with the other matrix elements.The simplest algorithm is natural.The total numbers of the potentially nonzero elements are determined for all the matrix columns.After that, the columns are reordered according to these numbers -the first and last should be those with the minimum and maximum numbers of such elements, respectively.

Modified Form of the Markowitz Criterion
A more sophisticated criterion can be created on the basis of idea of the Markowitz weights (7).However, they must be used in another way compatible with the fast LU factorization algorithm.To each potentially nonzero element located in i th row and j th column, the weight is assigned equal to the product of the numbers of other potentially nonzero elements in the i th row r i − 1 and j th column c j − 1 (r i and c j are the numbers of the nonzero elements in the i th row and j th column, respectively) The weight W i j represents an estimation of the fill-in addition that can arise if the (i, j) element becomes a pivot.The next part of the procedure differs from that in (7) because the column orientation of the LU factorization [23] must be held.
Instead of an immediate reordering in the matrix, the average weights are determined for all columns using the element weights ( 8) Finally, the columns are reordered according to the weights (9) -the first and last should be those with the minimum and maximum weights, respectively.

Approximating the Scattering Parameters
We have selected the Avago ATF-54143 low noise pHEMT [24], and we have identified its model using DC characteristics and s-parameters measured at a number of operating points [24].We have extracted parameters of various nonlinear models (Statz, etc.) and carefully checked their accuracy.A typical comparison of the TriQuint TOM2 and Dobes (called Dobes in [17,18]) models [17][18][19][20]25] is shown in Fig. 3.We have identified the models in the narrower frequency band from 1.1 GHz to 1.7 GHz, because the suggested low noise preamplifier for the satellite navigation systems always works between these two frequencies.For a quantified comparison of the models, we use the criterion where m and n are numbers of the operating points and frequencies at which the s-parameters were measured, respectively.In our case, m = 11 and n = 7, and the values of abs s for the TOM2 and Dobes models were 3.58 % and 3.73 %, respectively.In other words, the two models are comparable using (10a).However, the Dobes model was better using (10b): the values of rms s for the TOM2 and Dobes models were 19.79 % and 17.96 %, respectively.The reason consists in the s 22 accuracy, where the Dobes model was better in nine cases of eleven.(Let us emphasize that the accuracy of the s 22 parameter is well-known problem of the pHEMT models due to frequency dependence of output characteristics. Even contemporary models have this problem as can be seen in [26, Fig. 5], e.g.) Hence, the better s 22 precision was the reason why we selected the Dobes model.Other parameters were comparable, e.g., for i d = 60 mA and v ds = 3 V (it is shown in Fig. 3), the precisions of the s 11 and s 21 parameters were 3.09 % and 7 % for the TOM2 model and (a bit better) 2.94 % and 6.06 % for the Dobes model, respectively.

Modeling the Drain and Gate Noise Sources
For the purpose of the maximal accuracy of the noise modeling, we have used the contemporary Curtice3 model [25,27].(The Curtice3 model incorporates more other ones).The drain noise source is defined by semiclassical equation where k, g m , P, F nc , K f , A f , and F fe are Boltzman constant, transconductance, drain noise coefficient, flicker noise corner frequency, flicker noise coefficient, flicker noise exponent, and flicker noise frequency exponent, respectively.
The relatively novel part of the noise model is represented by gate noise source where C gs and R are zero bias gate-source junction capacitance and gate noise coefficient, respectively.There is also a correlation between the two noise sources (with a modified absolute temperature T j ) where C is gate-drain noise correlation coefficient; and the thermal noises generated by gate, source, and drain ohmic resistances R g , R s , and R d are included, too: The C.I.A. program [23] always contained (11) and ( 14) -however, for the accuracy purposes, the gate noise source model (12) has also been implemented.We have identified this model using datasheet [24], where some of the parameters are defined: F nc = 1 MHz, R = 0.08 Ω, P = 0.2, and C = 0.1, and the exact values have been found by optimization comparing with the set of measured noise parameters.

Description of the Low Noise Preamplifier
Let us mention a design of the low noise antenna preamplifier within the frequency range 1.1 to 1.7 GHz in Fig. 4 (a standard circuit diagram), Fig. 5 (a simplified simulation schematic), and Figs.6, 7, and 8 (a detailed circuit diagram used for the optimization).The terminal impedances were 50 Ω.The multiobjective optimization was used to simultaneously maximize the minimum transducer power gain A pt over the whole frequency range, minimize the maximum noise figure F dB n , while maintaining certain stability criteria and the quiescent operating point within chosen limits.The result is a set of noninferior solutions.The noninferior solution is the one in which an improvement in one objectivein this case, A pt or F dB n -requires a deterioration in another.

Description of the Requested Design Goals
The topology is a natural single stage application of pHEMT in the common source configuration with LC impedance matching networks at the input and the output.There are six design parameters: four component values of L 8 , C 3 , C 2 and L 2 , and two bias voltages, gate E 3 and drain E 4 as shown in Fig. 5. Design goals are also six; they are shown in Tab.1: the above mentioned two objectives, A pt and F dB n , and four inequality constrained goals, the maximum drain current i d , drain-source voltage v ds , Rollett stability factor K Rs , and the determinant of the s-matrix |Δ|.The absolute stability criteria K Rs and |Δ| were still used, even if the transistor's parameters did not allow for achieving a full unconditional stability (K Rs > 1, |Δ| < 1), and had to be relaxed to the region of potential instability (K Rs > 0.5, |Δ| < 0.9).

Description of the Obtained Design Results
Totally 262 solutions have been obtained, spanning the whole part of the Pareto front selected by the requirements A pt 15.5 dB ∧ F dB n 2 dB as shown in Fig. 9.There were no solutions found with A pt above 17.2 dB.Tab. 2 shows a selection of four representative solutions.All have their E 4 (i.e., v DD ) at the chosen upper bound value of 3.75 V. Finally, Fig. 11 shows plots for the four selected solutions as well as for the original unoptimized circuit (marked with 0).It is obvious the multi-objective optimization improved the circuit.For example, the tested solution No. 2 gives better noise figure for all frequencies, and a comparable and more uniform frequency dependence of the transducer power gain.

Measurement of the s-Parameters and Noise Figure
The low noise preamplifier has been carefully measured and the results (as well as the typical data from Avago [24]) confirm a local minimum of the noise figure somewhere in the middle of the band: the unoptimized configuration (marked with 0) has the measured minimum at 1.225 GHz, the solution No. 2 (2) at 1.25 GHz.Moreover, the tested solution No. 2 gives better noise figure in the frequency band from approximately 1.1 to 1.7 GHz as shown in Fig. 11.Moreover, the measured s-parameters of the circuit confirm the good amplifier's properties: both s 11 and s 22 are about −10 dB in the frequency band 1.1-1.7 GHz as shown in Fig. 10.
Fig. 11.Simulated (solid lines and the dotted one) and measured (dashed lines) A pt and F dB n plots for the original (marked with 0) and four selected (marked with 1-4) solutions shown in Tab. 2. (The plot marked with 1' shows the noise figure plot in the case when the gate noise (12) is not included.)Let us emphasize that the (declared) error of the equipment for measuring the noise figure was (at least) 0.1 dB; therefore, the difference between the original solution and the solution No. 2 is more important here than the obtained values themselves.(And this difference clearly shows that the algorithm has improved the noise figure comparing with the original solution.Further, we have detected an interference with an outer signal at some frequencies, and these measurements were excluded.)

Checking the Low Noise Preamplifier Stability
The low noise preamplifier stability has been checked by additional calculations and measurements.It is clear that the solution No. 1 is probably the most stable because its Rollett stability factor K Rs shown in Tab. 2 is greater than one.For the tested solution No. 2, the factors |Δ|, K Rs , and μ have also been computed from measured data a wide band to [28,   Individual input tone frequencies f 1 and f 2 were 1.575 and 1.576 GHz, respectively.The ( f 1 / f 2 ) level ratio was 1. (This method of creating the 3:1 slope line led to a "worst-case" IP 3 estimation.)Let us note the typical shape of the IM 3 curve for the amplifiers with pHEMTs.
First-Order Term

Checking the 3rd-Order Intermodulation Products
We have also measured the third-order intermodulation products for the original unoptimized low noise preamplifier (marked with 0 in Figs. 9 and 11) and for the tested solution No. 2 (see the same figures), and the results are compared in Figs. 12 and 13.The measurements confirm improved properties of the solution No. 2 -let us emphasize better output/input dynamics, and, first of all, the better location of the IP 3 point for the solution No. 2, comparable with [28].Tab. 4. Comparison of the total number of nonzero matrix elements before and after the LU factorization for its tested three versions.

Checking the Efficiency of the Modified Markowitz Criterion
As the Jacobian matrix generated in the DC and frequency & noise analyses of the circuit is quite large, we have also tested the criteria for suppressing the fill-in defined in Sec. 3, and the results are shown in Tab. 4. The worst results are clearly for classical (unmodified) LU factorization: although the matrix has 1570 nonzero elements before the LU factorization, this number increases to the values 4154 and 4340 after the LU factorization for the DC and frequency & noise analyses, respectively.If the simplest criterion for reordering the LU factorization described in Subsec.3.2 is used, the results are better: the total numbers of the nonzero elements are 2826 and 2793 after the (correspondingly modified) LU factorization for the DC and frequency & noise analyses, respectively.However, the best results are obtained using the proposed modified form of the Markowitz criterion defined in Subsec.3.3: the monitored numbers are finally 2604 and 2566 for the DC and frequency & noise analyses.

Conclusions
An improved algorithm for determining the tradeoff between the transducer power gain and noise figure of a low noise preamplifier was suggested.The modified algorithm is based on an improved Goal Attainment Method of the multiobjective optimization, which generates an asymptotically uniform coverage of the Pareto front.The efficiency of the whole procedure of the multiobjective optimization is increased using the implementation of the modified form of the Markowitz criterion.The complete design process is demonstrated using very accurate modeling the circuit active and passive elements followed by a reliable constrained optimization of the whole circuit.The final properties of the circuit including its stability were checked by various types of measurements in linear and nonlinear domains.

Fig. 3 .
Fig. 3. Measured and from-model-generated s-parameters of the ATF-54143 pHEMT in the s-plane for the TOM2 and Dobes models for the frequencies from 1.1 to 1.7 GHz, i d = 60 mA, and v ds = 3 V.At this operating point, the Dobes model gives slightly more accurate s 11 and s 21 .

Fig. 4 .
Fig. 4. Standard circuit diagram of the low noise antenna for the multiconstellation GPS/Galileo/GLONASS/Compass

Fig. 5 .
Fig. 5. Simplified simulation schematic of the low noise antenna preamplifier for the multiconstellation GPS/Galileo/GLONASS/Compass receiver.Actual power supply is brought through the output port and diverted by choke L7.The actual biasing circuitry that transfers it to the places of voltage sources E3 and E4 is not part of the simulation.Nuances like intermediate transmission line segments, T splitters or component models have been omitted here.The controlled sources are used for checking the optimization constraints i d and P diss .The circles mark the six circuit parameters p 1 , . . ., p 6 (L 8 , C 3 , C 2 , L 2 , E 3 , and E 4 ) to be determined by the multiobjective optimization.

Fig. 6 .Fig. 7 .
Fig. 6.Detailed simulation schematic.Transmission line segments and T-splitters are specified with sets of frequency dependent s-parameters, the former also being supplemented with frequency dependent noise sources.Biasing and power supply branches with chokes L1, L3 and L7 are not used for setting the DC operating point as the s-parameter specified blocks do not work appropriately for the DC analysis in the C.I.A. simulator, and also because the actual DC resistances involved are much different from the frequency dependent AC ones.Instead, ideal extremal capacitors C91-C93 and inductors L91-L93 are used to provide transistor biasing in DC analysis with correct resistance values without affecting AC behavior.Matching shunt coils L8 and L2 are modeled by means of controlled voltage sources E78 and E92 (as a function of the current through the inductor and its time derivative) providing inductance, series resistance and its noise contribution, all being nominal value and frequency dependent and based on linearly interpolated catalog data for the equivalent inductance and quality factor.Chokes L1, L3 and L7, which have only a minor effect on AC behavior, are left constant, with constant series resistances R1, R3 and R77 based on an assumed quality factor of 20 at 1.5 GHz.Series capacitances in the signal path C1, C3, C2 and C6 are supplemented with a frequency dependent ESR provided by controlled voltage sources E71, E73, E72 and E76.One-ohm resistances present in the schematic serve as sources of reference noise voltage used by the controlled sources.Transmission line segments marked as 'V' represent PCB vias providing ground connections.Detailed noise models of the pHEMT, transmismission lines and vias are shown in Figs.7 and 8.

Fig. 8 .
Fig. 8.The used noise model of short transmission line segment injects the collective noise of the R and G parts of an equivalent symmetrical T-structure, transferred to the input as noise voltage and noise current.

Fig. 9 .
Fig. 9. Computed Pareto front in the objective space A pt , F dB n .The dot marked with 0 represents the original unoptimized reference design.Solutions marked with 1-4 were selected as the representatives with the results shown in Tab. 2 and Fig. 11 .They were numbered from the lowest A pt of 15.5 dB to the highest A pt of 17.2 dB.The green dots (those for A pt 16.635) mark the part of the Pareto front with K Rs > 1, the red dots (including the original 0) mark the part of the Pareto front with K Rs 1. (Let us also note the uniformity of the Pareto front covering.)

Tab. 2 .
Four selected solutions from the Pareto front with their frequency characteristics shown in Fig.11.

Fig. 5 ]
, and the values inside the satellite navigation band 1.1-1.7 GHz are shown in Tab. 3.

FirstFig. 12 .
Fig. 12. Graphical post-processing of the IP 3 measurement results for the original (unoptimized) amplifier (No. 0).Individual input tone frequencies f 1 and f 2 were 1.575 and 1.576 GHz, respectively.The ( f 1 / f 2 ) level ratio was 1. (This method of creating the 3:1 slope line led to a "worst-case" IP 3 estimation.)Let us note the typical shape of the IM 3 curve for the amplifiers with pHEMTs.

Fig. 13 .
Fig. 13.Graphical post-processing of the IP 3 measurement results for the solution No. 2. Individual input tone frequencies f 1 and f 2 were again 1.575 and 1.576 GHz, respectively.The ( f 1 / f 2 ) level ratio was 1, too.

Fig. 14 .
Fig. 14.Measured spectra of the low noise amplifier.The signal generator has been set to the constant level −20 dBm.The spectra are shown for the case of the signal frequencies 2 and 3 GHz (i.e., it could be clearly seen here how the output voltage decreases with the growing signal frequency -the spectrum was also measured at 1.5 GHz, and the exact result for the frequency 1.507777778 GHz was −10.7 dBm).
Design goals for the antenna low noise preamplifier.