# Gain and Noise Boundaries for NFmin or Unity SWRout

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**F****ull ****Characterization ****of ****Gain ****and ****Noise ****Boundaries ****for ****NF****min ****or ****Uni****t****y ****SW****R****out ****O****p****eration**

M. R. M. Rizk1,4, Ehab Abou-Bakr*,2, A. A. A. Nasser3, El-Sayed A. El-Badawy1 and Amr M. Mahros1,5

**Abstract**-For a receiver sub-block, Low Noise Amplifier (LNA) is the first stage after the receiving antenna and as a key device, its amplification and noise figure (NF) affects the whole performance of the receiving part. In this paper we present a full graphical visualization in terms of gain, standing wave ratio (SWR) and noise for a GaAs HJ-FET transistor in two operating cases; i)NFmin, ii) Unity SWRout. The set of curves and contours presented will provide the designer with enough visual information about the transistor operating boundaries and will also visually assist on choosing the appropriate matching points for a wideband operation according to the desired (GT,SWR) for case (i) and (GT,SWRin,NF) for case (ii). Numerical examples are given for each operating case and verified via a microwave circuit design software package to demonstrate the adequacy of the proposed graphical techniques. The results from simulations compare favourably with the visually estimated values.

- INTRODUCTION

Introducing a wide-band LNA with acceptable noise figure and gain depends mainly on choosing a suitable transistor[1, 2, 3, 4]. Several successful techniques have been developed in the literature to develop discrete transistors with super low NF and high associated gain [5, 6, 7, 8, 9, 10, 11]. Different challenging techniques have been used to simultaneously get high gain, low noise figure, good input and output matching and unconditional stability at the lowest possible current draw from the amplifier. In 1982, Yarman et al.[12] introduced a software based non-linear optimization method based on their procedural simplified real frequency technique. This design procedure is applicable to broadband multistage FET amplifiers with no decisions to be made in advance. It was more efficient and accurate than other available CAD methods to fulfill the most optimum gain and SWR over a predefined bandwidth. This method was later extended by Perennec et al.[13] to optimize the noise figure in parallel with the gain and mismatch. Capponi et al. [14], expressed the performance of LNA in input matching condition by analyzing the Combined noise-SWR using the general curve family specified for a given active device. The determination of the required input/output terminations of the active device was explained in [15] when the power gain, noise figure, and input and output mismatch constraints are placed on the amplifier. Bengtsson et al. [16] devised a novel SWR test procedure for GaN-HEMT devices. In [17], the operation conditions of a selected high technology transistor were used along the typical design configurations to find a compromise relations between the gain, noise figure for the output port matching. Recentely graphical methods along with optimization methods for describing the full capacity of the selected transducer under a given set of noise figure and SWR constrains are discussed in [18, 19, 20].

*R**e**c**eiv**e**d **date*

* Corresponding author: Ehab Abou-Bakr ([email protected]).

- Faculty of Engineering, Alexandria University, Alexandria, Egypt.
- The Higher Institute of Engineering and Technology, El-Behera, Egypt.
- Faculty of Engineering, Arab Academy For Science Technology and Maritime Transport, Alexandria, Egypt.
- SmartCI, Alexandria University, Alexandria, Egypt.
- University of Jeddah, Jeddah 21432, Saudi Arabia.

The (noise, gain, SWR) triplets can be expressed on the Smithchart as circles on both the source and load reflection coefficient planes [21, 22]. Choosing matching points on the Smithchart based on the variations of gain circles radii reflects on the noise/SWR performance of the whole amplifier circuit. Pre-Knowledge of the transistors full capacity with respect to gain, SWR and noise could facilitate the choice of the correct part for the targeted design goals.

In this paper, two cases of design restrictions are taken into consideration; i) NFoperation, ii) unity SWRout. For each of these cases, a formed data base is used to create sets of boundaries for the transducer gain GT and NF that will reveal the full operating capacity o the selected transistor. Visual selection of the desired performance is possible and extraction of the appropriate matching points for single frequency or wideband operation is made simple.

The selected active device for our investigation is the GaAs HJ-FET transistor NE3210S01 from Renessa Electronics [24]. The transistor is potentially unstable at *V**D**S*= 2*V*, *I**D*= 10*mA*in the frequency rage below 8.6 GHz [26, 27]. So, by conducting the investigation in a range above this frequency (9-12)GHz, no additional circuit component is required to drive the transistor to its conditional stability region. As a result, the (NFmin) and their corresponding (Î“*op**t*) provided in the manufacturer datasheet are used directly without any modifications. More intermediate dataset that is not provided in the datasheet, is used in our investigation. This was possible by using the interpolation option provided by the Advanced Design Systems (ADS) from Keysight technologies [25].

This manuscript is organized as follows: Numerical example and simulation verification are presented in section.2 for demonstrating the usage of the graphical gain boundaries and the imposing of SWR on them for NFmin operation. In section.3, the idea of correlating noise, gain and SWR on a single graph using NF boundaries are presented and aided by another numerical example. The conclusion is discussed in section.4.

- GAIN BOUNDARIES FOR NFMIN OPERATION

All the basic formulas used in the presented analysis is listed in Table.1.

In [22], three expressions for the gain are provided. These are; the transducer gain (GT), the available gain (GA) and the operating power gain (GP). The design of a microwave amplifier requires utilizing one or more of these gain criteria to reach the required design goals. Graphically, all the previously mentioned types can be represented as circles on the Smithchart. However, choosing which gain type to use in the design, depends on the transistor type and the required design criterion.

+j1.0

+j0.5

GP circles

+j2.0

+j0.2

+j5.0

0.0âˆž

-j0.2

As the radius CP increases, the Value of GP decreases

-j5.0

-j0.5

-j2.0

-j1.0

**Figure****1.**For NFmin operation, Different operating gain circles obtained by changing the GP factor in (15)

**T****able****1.**Basic equations used in the analysis

Î“*in**âˆ’*Î“*âˆ-*

*C*1 = *|S*11 *âˆ’*âˆ†*S**âˆ-**|*(10)

Î“*b*=*S*

*G**A*

1 *âˆ’*Î“*i**n*Î“*S*

*g**A*=

2(11)

Î“*out**âˆ’*Î“*âˆ-*

*|S*21*|*

Î“*b*=*L*

(2)

*g **C**âˆ-*

1 *âˆ’*Î“*ou**t*Î“*L*

*C**P*=*P *2

(12)

*S*12*S*21Î“*S*

1 + *g**P*(*|S*22*|*2 *âˆ’**|*âˆ†*|*2)

Î“*in*= *S*11 + 1 *âˆ’**S *Î“

(3)

I

*P*

__S____1____2____S____2____1____Î“____S__

*r**P*=22

Î“*out*= *S*22 + 1 *âˆ’**S *Î“

(4)

1 + *g**P*(*|S*22*|*

*âˆ’**|*âˆ†*|*)

(13)

*S**W**R*= 1 + *|*Î“*a**|*(5)

*in*1 *âˆ’**|*Î“*a**|*

*C*2 = *|S*22 *âˆ’*âˆ†*S**âˆ-**|*(14)

*G**P*

*S**W**R*= 1 + *|*Î“*b**|*(6)

*out*1 *âˆ’**|*Î“*b**|*

*g**P*=

*|S*21

(15)

*|*2

*G**T*=

1 *âˆ’**|*Î“*S**|*2

*|*1 *âˆ’*Î“*s**S*11*|*2

*|S*21*|*2

1 *âˆ’**|*Î“*L**|*2

*|*1 *âˆ’*Î“*L*Î“*ou**t**|*2

*. *(7)

1 *âˆ’**|S*11*|*2 *âˆ’**|S*22*|*2 + *|*âˆ†*|*2

2*|S*21*S*12*|*

(16)

*C**A*=

*g**A**C**âˆ-*

(8)

*G*= *|S*21*|*I2

1 + *g**A*(*|S*11*|*2 *âˆ’**|*âˆ†*|*2)

I

*P**max*

*|S*12

(*K**âˆ’*

*|*

*K **âˆ’*1)(17)

1 *âˆ’*2*K**|S*21*S*12*|g**A*+ *|S*12*S*21*|*2*g*2

*N**F *= *N**F**min*+

4*r**n**|*Î“*S**âˆ’*Î“*op**t**|*2

(18)

*r**A*=

1 + *g**P*

(*|S*11

(9)

*|*2 *âˆ’**|*âˆ†*|*2)

(1 *âˆ’**|*Î“*S*

*|*2)*|*1 + Î“

*op**t**|*2

- (b)

**Figure****2.**Distribution of SWRout over operating gain circles for NFmin operation at 12 GHz a) A 3D representation with small values of SWRout displayed in lighter colors, b) A plane view of the same figure with actual values of SWRout on the color bar.

2.1. Imposing SWR on GT Boundaries for a Wideband, NFmin Operation

Considering the above choices, the bilateral property of the Device Under Test (DUT) disfavor the usage of GT circles. Also, targeting a NFmin operation forces Î“*S*=Î“*opt*and this prevents the usage of GA circles. As a result, GP circles in the Î“*L *plane of the Smithchart were used.

16

14

1212 GHz11 GHz 10 GHz9 GHz

10

8

6

Maximum attainable G

T

Minimum attainable G

T

2

1212.51313.51414.51515.51616.5

Operating gain (G )

P

**Figure****3.**GT vs. GP, where GTmin *â‰¤*GT*â‰¤*GTmax regions for frequencies 9,10,11,12 GHz are shown in solid and dotted lines respectively.

For a certain frequency of operation, changing the value of the GP factor in (15) will produce different circles for the operating gain as shown in Figure.1. Each point on the circumference of these circles represent a unique value of Î“*L*that can be used for matching according to the desired design goals. For further discovery of the device capabilities, SWR related to these values can be imposed on these circles. For illustration, only the SWRout levels are imposed in Figure.2 where lighter color regions represent lower values of SWRout. Although these are the desired regions to build our design around. However, for a wideband operation, reaching the required GT could prevent choosing matching points from these regions.

Since Î“*S*=Î“*opt*for a NFmin operation, a graphical relation (GT vs. GP) will provide a pre- design information about the limitation of the selected transistor. Figure.3 explains this by specifying GTmin *â‰¤*GT*â‰¤*GTmax over a range of GP for the selected frequency points, the solid lines represent GTmax while the dotted lines correspond to GTmin . In fact a database was constructed for this figure that contain all values of Î“*L*'s that correspond to each GP value. Later on, this database will be very useful in choosing appropriate matching points for wideband operation. A quick look to the figure revels that if targeting a wideband operation the desired GT should not exceed GTmax of the highest frequency. For example, the transistor cannot achieve GT higher than 12.73 dB for a selected frequency of 12 GHz.

However, designing for a suitable SWRin and SWRout requires further correlation between GT and SWR. This is shown in Figure.4 where visual predication of the device operating boundaries are clear. The constructed database is extended by masking the contours of both SWRin and SWRout on the GT boundaries at NFmin operation. Since lighter colors indicate better values of SWR, it is obvious that for this particular transistor, the SWRin and SWRout are worse for lower frequencies. Also, the direction of the color stripes are diagonal for SWRin and horizontal for SWRout, this is an indication that, for this particular transistor, choosing an appropriate GP and its subsequent Î“*L*'s could result in a constant value of SWRin along the entire bandwidth.

As an example to emphasise on using Figure.3 to design a wideband LNA operating at its NFmin, a targeted 12.7 dB is chosen for illustration in the range of 9-12 GHz. From Figure.4, the color contour reveals that the minimum SWRout=1 corresponding to this GT level belongs to a 12GHz operation. Then, the accompanying Î“*L*pairs for frequencies 9,10,11,12 GHz are fetched for matching purpose as shown in Figure.5(a). The displayed Î“*L*pairs on the smith chart of Figure.5(b) were used by ADS to construct matching circuits to verify the expected SWR. the obtained simulation results are listed in Table.2 and compares favourably with those listed in Figure.5(a).

16351650

1430

1225

12 GHz 11 GHz 10 GHz 9 GHz

10

20

8

15

14

12

12 GHz

10

8

11 GHz

10 GHz

45

40

35

9 GHz30

25

20

6615

10

10

44

55

2

12 12.5 13 13.5 14 14.5 15 15.5 16 16.5 **SW****R****in**

Operating gain G

P

(a)

2

12 12.5 13 13.5 14 14.5 15 15.5 16 16.5 **SW****R****out**

Operating gain G

P

(b)

**Figure****4.**Imposing the contours of both SWRin and SWRout on the GT boundaries at NFmin operation,

- for SWRin and b) for SWRout

+j1.0

+j0.5

+j2.0

+j0.2

12 GHz

11 GHz 10 GHz

- GHz

+j5.0

0.0âˆž

11 GHz

(a)

-j0.2

9 GHz

-j0.5

- GHz

-j1.0

(b)

-j2.0

-j5.0

**Figure****5.**a) Extracting the underlying Î“*L*pairs from the constructed database for the shown selected point of operation according to the targeted GT ans SWR, b) Smithchart representation of the extracted Î“*L *pairs

- NF BOUNDARIES FOR A UNITY SWROUT

For the condition of an output conjugate matching (ie. Î“*L*=Î“*âˆ-*

), GA= GT and a unity SWRout is

produced. All values of Î“*S*that corresponds to a particular GA circle gives the same value of SWRin.

This is shown in Figure.6 where a contour of SWRin is imposed on GA= GT circles. The tip of the cone

in Figure.6(a) corresponds to Î“*S*point that will produce a simultaneous conjugate match (ie. Î“*S*=Î“*âˆ- *&

Î“*L*=Î“*âˆ-*

) where (SWRin= SWRout=1).However, this figure alone cannot correlate the (GT,NF,SWR)

triplets to give a full visualization insight of the device capability in this case of operation.

- GT, SWRin and NF Correlation for SWRin=1

Figure.7(a) illustrate the variation of SWRin along a range of GA= GT values where at SWRin=1, a simultaneous conjugate matching occurs. The data in Figure.7(a) alongside GA= GT values and their corresponding NF are used to construct a database to help plotting the NF boundaries shown in Figure.7(b). For a SWRout=1 operation, this figure can be used to visually predict both NF and SWRin for any targeted GT. Since, the marked points on the plot represent SWRin=1 for each selected

**T****able****2.**ADS simulation data results after individually matching the IMN and OMN according to the matching points in Figure.5(a).

Freq |
GT |
NFmin |
NF |
SWRin |
SWRout |

9GHz |
12.742 |
0.31 |
0.31 |
2.472 |
3.073 |

10GHz |
12.710 |
0.32 |
0.32 |
2.438 |
2.319 |

11GHz |
12.751 |
0.33 |
0.33 |
2.434 |
1.869 |

__12GHz ____12.760__ __0.34__ __0.34__ __2.379__ __1.033__

- (b)

**Figure****6.**3D representation of SWRin over a range of GA=GT circles a) Isometric view, b) Plan view

frequency, it is visually clear that a SWRin= SWRout=1 is impossible for a wideband, flat gain design.

For a wideband, flat gain operation with SWRout=1.Figure.7(b) reveals that GT flat max=

GT max 12GHz is the maximum value of GT to attain a flat gain throughout the bandwidth. The

previously constructed database can be used to fetch Î“*S*, Î“*L*that will produce the visually targeted

(GT, SWRout, NF) triplets from Figure.7(b). As an example, a targeted wideband operation (9-12 GHz)

with GT=13.9 dB is chosen for demonstration, Figure.8 present the underlying Î“*S*, Î“*L*for the visually

selected point. this point was selected to give the targeted GT for a simultaneous conjugate matching

at 12 GHz with NF*â‰¤*1.4 dB. the source and load matching points for the selected frequencies are shown

in Figure.9. Again, ADS was used to verify the estimated (GT, NF, SWR) triplets by constructing individual matching networks using Î“*S*and Î“*L*listed in Figure.8. Table.3 present the simulation results which compares favorably with the visually estimated values.

**T****able****3.**ADS simulation data results after individually matching the IMN and OMN according to the matching points in Figure.8.

Freq |
GT |
NF |
SWRin |
SWRout |

9 GHz |
13.96 |
1.34 |
3.06 |
1.02 |

10 GHz |
13.98 |
1.33 |
2.34 |
1.01 |

11 GHz |
13.93 |
1.37 |
1.88 |
1.01 |

__12____GHz ____13.95 ____1.33__ __1.12__ __1.03__

- CONCLUSION

In this paper, rigorous graphical investigation to explore the selected device capabilities in the NFmin and SWRout=1 cases was presented. For the first case; a set of GT boundary curves and contours can be visually used to explore the expected values of SWRin SWRout for a targeted GT at NFminoperation. While for the second case; NF boundary curves were used to visually predict the NF, SWRin levels for

6

Simultaneous conjugate matching point

10Î“ =Î“* , Î“ =Î“* , for 9,10,11,12 GHz

4.5

5

9 GHz

S in L

out

4

410 GHz

8

3.5

- GHz

3

- GHz

2

1

6

4

2

12 GHz11 GHz 10 GHz 9 GHz

3

2.5

2

1.5

0

11.5 12 12.5 13 13.5 14 14.5 15 15.5 16 16.5

Transducer gain G

T

(a)

0

12.5 13 13.5 14 14.5 15 15.5 16 16.5

Transducer gain G

T

(b)

SWRin

**Figure****7.**a) Distribution of SWRin over a range of GT, b) NF boundaries for frequencies 9, 10, 11, 12 GHz

**Figure****8.**Extracted Î“*S*, Î“*L*from the constructed database for the shown selected point of operation. output conjugate matching that will result a SWRout=1. For both cases, a full database was formed to

be used in the extraction of the corresponding matching reflection coefficients for any visually targeted

operating points. The construction and using of this database was found to make termination points extraction easy and accurate. And As described by [19] "*It**c**an**b**e**c**onclud**e**d**that**the**n**e**ar**futu**r**e**mic**r**owave**t**r**ansistor**is**ex**p**e**ct**e**d**to**b**e**identifi**e**d**by**the**Performan**c**e**Data**Bases**whe**r**e**a**l**l**p**ossible**LNA**designs**c**an**b**e**overview**e**d**using**the**fu**l**l**devi**c**e**c**a**p**acity.*"

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+j1.0

+j1.0

+j0.5

+j2.0

+j0.5

+j2.0

+j0.2

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+j5.0

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-j2.0

-j1.0

-j1.0

(a)

(b)

**Figure****9.**Impedance termination for the numerical example of Figure.8 a) Source impedance matching points b) load Impedance matching points

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