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AN ANALYTICAL APPROACH ON THE DETERMINATION OF GENERATOR REFLECTION COEFFICIENTS USED IN

THE NOISE-PARAMETER MEASUREMENT Robert Hu and Tzu-Hsien Sang

Department of Electronics Engineering National Chiao Tung University

HsinChu, Taiwan R.O.C.

Received 5 September 2005

Abstract

In performing microwave and millimeter-wave noise-parameter measurements, a set of generator reflection coefficients and their corresponding noise temper-atures need to be measured first. Allowing measurement errors, values of these generator reflection coefficients will affect the accuracy of final results. Though simulation alone can be used to find out on the Smith chart the preferred pattern of the generator reflection coefficients, its analytical counterpart has not been explored yet. This paper intends to derive the related mathematical expressions, thus complements and strengthens the simulation method.

Keywords: Generator reflection coefficient, noise parameters, noise

tempera-ture, least-squares fit.

1. Introduction

In the design of microwave and millimeter-wave low-noise amplifiers, both the small-signal model and noise model of the constituting transistors need to be known so they can be imported into the circuit simulators [1]–[6]. Compared with its small-signal model, the transistor’s noise model is more difficult to construct, for it demands accurate measurements of the sensitive, and thus elusive, noise parameters, which in turn impose some stringent requirements on the noise measurement setup [7]–[14]. Of the many expressions for a two-port circuit’s noise parameters, it is the minimum noise temperatureTmin, noise ratioN , and optimum reflection coefficient Γopt (=γoptexp(j θopt)), which is a complex number, that will be used in this paper. The relation between the noise

DOI: 10.1007/s10762-006-9081-z

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RF Probe Post Noise

Source

Tuner

Radius

Amplifier Noise FigureAnalyzer Transistor

Fig. 1. Generator reflection coefficients used for noise-parameter measurement where the Γg pattern contains the zero point and a circle.

temperature and the noise parameters is

Tn= Tmin+ 4 T0N |Γ

g− Γopt|2 (1 − |Γg|2) (1 − |Γopt|2)

(1) where To is 290 Kelvin, Γg (= γgexp(j θg)) is the generator reflection coeffi-cient. Mathematically, the unknown noise parameters can be derived using least-squares fit from a set of generator reflection coefficients and their corresponding noise temperatures [15]–[19].

However, the choice of the Γg set cannot be arbitrary as some may cause the following noise calculation to be divergent, as in the case of a circular Γg on the Smith chart. While it is also known that the mere addition of the zero point, i.e., Γg = 0, into the above circular pattern results in the intended solutions (see Fig. 1). Moreover, when measurement uncertainty, or error, is taken into account, simulation shows that this Γg circle is preferred having a large radius if the uncertainty is embedded in the measuredTn, which is usually the case when measuring an ultra-sensitive transistor. Though it is convenient using the simulation approach to both determine the desired generator reflection coefficients and make an evaluation of different computation algorithms [20]– [23], this empirical approach cannot reveal much about the underlying physics, nor explains why the desired results can be rendered. In this paper, we intend to demonstrate that, through mathematical reasonings, several interesting con-clusions can indeed be obtained without the running of numerous and laborious

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0 0.2 0.4 0.6 0.8 1 10−1 100 101 102 103 0 0.2 0.4 0.6 0.8 1 10−7 10−6 10−5 10−4 10−3 0 0.2 0.4 0.6 0.8 1 10−7 10−6 10−5 10−4 10−3 0 0.2 0.4 0.6 0.8 1 10−4 10−3 10−2 10−1 100 0.4 0.6 0.8 0 0.2 var[ N ]

γ

γ

γ

T

γ

γ

opt

θ

opt var[ ] min var[ ] var[ ] g g g g

Fig. 2. Error analysis on the determination of generator reflection coefficients. The five curves in each subplot correspond to different values of Γopt.

simulations. In developing this analytical approach, better insight into the noise-parameter measurement can also be gained.

2. On the Choice of Γg

In this paper, the transistor, as the device under test, has its measured noise temperature embedded with uncertainty described by a Gaussian distribution with 5% standard deviation; on the Smith chart, the generator reflection coef-ficients contains the zero point and a circle. With different radius for the Γg circle, simulation reveals a preference for large radius, no matterΓopt is zero or not. Fig. 2 shows the error analysis on the noise parameters of the transistor that hasTmin = 30 Kelvin,N = 0.035, and Γopt changing from 0 and 0.8. Subplots (a), (b), (c), (d) are the variance ofTmin,N , γoptandθopt, with the radius of the Γg circle on the X-axis. In each subplot, the five curves correspond to optimum reflection coefficient of 0, 0.2, 0.4, 0.6 and 0.8 respectively, with the direction of increasingΓg indicated by the arrow. This large-radius preference also holds when certain Tn-dependent weighting factor is included in the simulation.

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To begin the formulation, the nonlinear noise temperature expression of Eq. 1 has to be re-arranged as a linear function of four new variables a, b, c and d:

Tn= a + b 1 1 − γ2 g + cγg cos(θg) 1 − γ2 g + dγg sin(θg) 1 − γ2 g (2) with a = −4 T0N 1 − γ1 2 opt + Tmin b = 4 T0N 1 + γ 2 opt 1 − γ2 opt c = −8 T0N γopt1 − γcos(θ2 opt)

opt d = −8 T0N γopt1 − γsin(θ2opt)

opt .

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By setting Δ =√b2− c2− d2, we also have Tmin= a + b + Δ 2 N = Δ 4 T0 γopt=  b − Δ b + Δ θopt= tan−1(d c) . (4)

At each frequency point, the set of different generator reflection coefficientsΓi (=γiexp (j θi)) and their corresponding noise temperatures Ti, withi = 1 ∼ m, can be arranged as ⎡ ⎢ ⎢ ⎢ ⎣ T1 T2 .. . Tm ⎤ ⎥ ⎥ ⎥ ⎦= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 1 1−γ2 1 γ1cos(θ1) 1−γ2 1 γ1sin(θ1) 1−γ2 1 1 1 1−γ2 2 γ2cos(θ2) 1−γ2 2 γ2sin(θ2) 1−γ2 2 .. . ... ... ... 1 1 1−γ2 m γmcos(θm) 1−γ2 m γmsin(θm) 1−γ2 m ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎣ a b c d ⎤ ⎥ ⎥ ⎦ (5)

or in a more concise matrix form

[T ] = [X] [A] . (6)

Applying the least-squares fit, there is

[A] = ([X]t[X])−1[X]t[T ] .

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husa, b, c, d and the four noise parameters at this frequency point can be ob-tained if the generator reflection coefficients and their related noise temperatures are known.

If there is uncertainty embedded in the noise temperature, the resulting noise parameters will show some deviation from their ideal values. By assuming the uncertainty Gaussian, error-analysis simulation can be employed figuring out the variance for each noise parameter. Indeed, the simulated results confirm the large-radius preference for theΓg circle. However, simulation alone cannot explain why it works that way, nor whether the simulated results is valid for other circuits; all this demand a more analytical, and thus comprehensive, approach. First, with the optimum reflection coefficient set to zero, the difference between mismatched and matched noise temperatures is

Tn= b γg2 1 − γ2 g + c γg 1 − γ2 g cos(θg) + d γ g 1 − γ2 g sin(θg) . (8)

So if b, c and d are known, variable a and the original four noise parameters can be calculated from Tn. To simplify the derivation procedure, we let

B = b γ 2 g 1 − γ2 g ; C = c γg 1 − γ2 g ; D = d γg 1 − γ2 g . (9)

Values ofB, C and D can be determined using least-squares fit with error term 2 defined as

2 ≡ 1

mΣ [ B + C · cos(θg) + D · sin(θg) − Tn]

2 . (10)

Here the summation Σ includes all the m sampling points on the Γg circle. Since least-squares fit demands the partial differential of the error term versus variables B, C and D must be zero, i.e.

∂ 2 ∂ B = ∂ 2 ∂ C = ∂ 2 ∂D = 0 , (11) there will be B · Σ1 + C · Σ cos(θg) + D · Σ sin(θg) = ΣTn

B · Σ cos(θg) + C · Σ cos2(θg) + D · Σ sin(θg) cos(θg) = ΣTncos(θg) B · Σ sin(θg) + C · Σ cos(θg) sin(θg) + D · Σ sin2(θg) = ΣTnsin(θg) .

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With a suitable sampling scheme, it is reasonable assuming the following hold: Σ sin(θg) = Σ cos(θg) = Σ sin(θg) cos(θg) ≈ 0 . (13) Eq. 12 can thus be re-arranged as

⎡ ⎣ m0 Σ cos02g) 00 0 0 Σ sin2g) ⎤ ⎦ ⎡ ⎣ BC D ⎤ ⎦ = ⎡ ⎣ ΣTΣTncos(θn g) ΣTnsin(θg) ⎤ ⎦ . (14)

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Variables B, C, D and therefore b, c, d can be easily obtained: b =1 − γ 2 g γg 1 mΣ [ΔTn] c =1 − γ 2 g γg 1 mΣ [ΔTn· cos(θg)] d =1 − γ 2 g γg 1 mΣ [ΔTn· sin(θg)] . (15)

If there is no measurement uncertainty, we will have

b = 4 T0N ; c = 0 ; d = 0 , (16)

which are their original values. If the uncertainty has finite value and is pro-portional to the measured noise temperature, then

Tn= (1 + δ1) Tnmismatched− (1 + δ2) T matched n = (1 + δ1) Tmin+ 4 T0N γg2 1 − γ2 g − (1 + δ2) Tmin (17)

whereδ1andδ2represent the uncertainty values for the mismatched and matched noise temperatures. Neglecting their respective higher-order terms, uncertainty values of the four noise variables will be

Δa = − 1 − γ2g γg2 Tmin 1 m δ1− 4 T0Nm1 δ1+γ12 g Tmin 1 m δ2 Δb = 4 T0Nm1Σ δ1+1 − γ 2 g γg2 Tmin1 mΣ (δ1− δ2) Δc = 8 T0N γg 1 mΣ [δ1· cos(θg)] + 2 Tmin 1 − γ2 g γg 1 mΣ [(δ1− δ2) · cos(θg)] Δd = 8 T0N γg 1 mΣ [δ1· sin(θg)] + 2 Tmin 1 − γ2 g γg 1 mΣ [(δ1− δ2) · sin(θg)] . (18) If δ1 and δ2 are slowly-drifting functions with

0 = Σ [δ1· cos(θg)] = Σ [(δ1− δ2) · cos(θg)] = Σ [δ1· sin(θg)] = Σ [(δ1− δ2) · sin(θg)] ,

(19) then bothΔc and Δd turn to be zero. A large γg in this case reduces the derived Δa and Δb, thus results in more precise solutions for the noise parameters. If both δ1 and δ2 are uncorrelated random variables with

var 1 mΣδ1 = σ12; var 1 mΣδ2 = σ22 (20)

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0 0.2 0.4 0.6 0.8 1 10−1 100 101 102 103 0 0.2 0.4 0.6 0.8 1 10−1 100 101 102 103 0 0.2 0.4 0.6 0.8 1 10−2 10−1 100 101 102 102 0 0.2 0.4 0.6 0.8 1 10−2 10−1 100 101 102 var[ a ] var[ d ] var[ b ] Dashed: theoretical Solid: simulated var[ c ]

γ

γ

γ

γ

g g g g

Fig. 3. Error analysis on the four noise variables. With 100 simulation runs, the variance of the simulated noise variables using least-squares fit agree with their theoretical counterparts.

where var[· · · ] is the statistical variation function, then Eq. 18 leads to

var [ a ] = σ12 1 − γ2 g γg2 Tmin+ 4 T0N 2 + σ2 2  1 γ2g Tmin 2 var [ b ] = σ12 1 − γ2 g γg2 Tmin+ 4 T0N 2 + σ2 2 1 − γ2 g γg2 Tmin 2 var [ c ] = var [ d ] = 2 σ21 1 − γ2 g γ2g Tmin 2 + σ22 1 − γ2 g γg2 Tmin 2 . (21)

All these four expressions show that, in order to minimize the variation, the radius of the Γg circle needs to be large. Accuracy of Eq. 21 can be easily confirmed through simulation, as in Fig. 3. Here the error analysis on the four new noise parameters is carried out assuming 5% Gaussian uncertainty on the measured noise temperatures, and 100 simulation runs are used for averaging.

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Subplots (a), (b), (c), (d) show the variance ofa, b, c and d, with X-axis the ra-dius of theΓg circle. In each subplot, the solid curve is the simulated result with noise parameters derived from Eq. 7, and the optimum reflection coefficient is assumed zero; the dashed curve is its theoretical counterpart calculated directly from Eq. 21. Agreement between the solid and dashed curves in each subplot substantiates our mathematical work. Of course, if we reduce the number of simulations from 100 to 10, then there will be noticeable discrepancy between the simulated variance of the noise variable and its theoretical counterpart.

3. Discussion on the Mathematical Implications

Though the derivation in the previous section is done by assuming zero Γopt to reduce the mathematical complexity, the impact of non-zero Γopt can be un-derstood by applying the sensitivity analysis on the least-squared solutions [24]. In the case of non-zero Γopt, Eq. 14 needs to be revised as

⎛ ⎝ ⎡ ⎣ m0 Σ cos02g) 00 0 0 Σ sin2 g) ⎤ ⎦ + ΔF ⎞ ⎠ ⎡ ⎣ BC D ⎤ ⎦ = ⎛ ⎝ ⎡ ⎣ ΣTΣTncos(θn g) ΣTnsin(θg) ⎤ ⎦ + Δf ⎞ ⎠ . (22)

whereΔF andΔf are the perturbation matrices consisted of terms with magni-tude at the order of [(1 − γg2)/γg] · [γopt/(1 − γopt2 )]. Two observations emerge from this perturbation expression. First, the effect of a non-zero Γopt is quite limited if γopt is small. Second, a large γg helps reducing the impact of non-zero γopt. The latter is because the condition number of a matrix indicates its solution’s sensitivity to perturbation [25], and in our case the diagonal matrix has a small condition number–close to 2, which implies a low sensitivity. It is this property that explains why a non-zero Γopt will not alter the general trend that large generator reflection coefficients are preferred in deriving the noise parameters using least-squares fit.

As in other fields [26], sometimes it is the analytical approach that offers solutions to some peculiar phenomena. For example, in order to explain the seemingly anomalous Tmin variance, i.e., a straight line versus Γg when Γopt is zero (see Fig. 2), it is the knowledge of explicit expressions for the variance ofa, b, c and d that helps. Since there is from Eq. 4

Tmin = a +b + √

b2− c2− d2

2 . (23)

With the mean values of c and d now zero and their variance one order less than that of a and b, the variance of Tmin can be approximated as

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which, apparently, is independent of γg. Thus, the choice of the generator reflection coefficient has negligible impact on the minimum noise temperature when the optimum reflection coefficient is zero.

Finally, we would like to consider other uncertainty modelings. To be sure, there are two possible sources of uncertainty in the noise-parameter measure-ment, one is on the noise temperature itself and the other on the generator reflection coefficient. Our approach is to focus onTnuncertainty while assuming a perfect measurement of Γg. Some authors, however, emphasize the benefits of considering simultaneously both uncertainty sources [27]-[31]. While it is true that by doing so the accuracy of the simulated results can be improved, it will not change the main conclusion of this paper, that is, the radius of the Γg circle should be as large as possible in order to minimize the impact of the measurement fluctuation. This is because if the relation between Tn and γg is a smooth and monotonic function, it is always possible merging the two uncertainty sources into one, for the reason that in this case the Gaussian uncertainty in one quantity can be approximately mapped into the other quantity as Gaussian, and vice versa. As long as it is the small variation involved, leaving out one uncertainty while keeping the other one will not cause problems [32]-[34].

4. Conclusion

In this paper, an analytical approach on the determination of generator reflec-tion coefficients used in the noise-parameter measurement has been developed. The reason why a large generator reflection coefficient is preferred in the noise measurement is now soundly explained. Agreements between the mathematical expressions and their simulation counterparts have also been checked.

Acknowledgments

Author R. Hu would like to thank S. Weinreb, D. Miller, J. Kooi, M. Edgar, F. Rice, G. Chattopadhyay and Prof. J. Zmuidzinas of Caltech, Pasadena, for discussions and encouragement.

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數據

Fig. 1. Generator reflection coefficients used for noise-parameter measurement where the Γ g pattern contains the zero point and a circle.
Fig. 2. Error analysis on the determination of generator reflection coefficients. The five curves in each subplot correspond to different values of Γ opt .
Fig. 3. Error analysis on the four noise variables. With 100 simulation runs, the variance of the simulated noise variables using least-squares fit agree with their theoretical counterparts.

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