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2018-07-09
Extraction Of Noise Parameters For Single-Ended Components Inside A Differential Circuit Using Single-Ended Equipment
Huang, Yuxiang
Huang, Y. (2018). Extraction Of Noise Parameters For Single-Ended Components Inside A Differential Circuit Using Single-Ended Equipment (Unpublished master's thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/32349
http://hdl.handle.net/1880/107127 master thesis
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THE UNIVERSITY OF CALGARY
Extraction Of Noise Parameters For Single-Ended Components Inside A Differential Circuit Using Single-Ended Equipment
by
Yuxiang Huang
A THESIS
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF SCIENCE
GRADUATE PROGRAM IN ELECTRICAL AND COMPUTER ENGINEERING
CALGARY, ALBERTA
JULY, 2018
Yuxiang Huang 2018c
Abstract
This thesis proposes an approach of investigating electrical and noise parameters of sub- components inside a fully differential system. To find the single-ended noise parameters, two sets of single-ended noise-parameter measurements and one set of S-parameter mea- surement are performed. A proof-of-concept PCBs is designed, fabricated, and tested after the algorithms are verified using Matlab. The circuit is designed to have a bandwidth from 500 MHz to 1.5 GHz and be unconditionally stable at all frequencies. The designed circuit has the gain of 9.4 dB at 500 MHz and 9.5 dB at 1.5 GHz in schematic simulations, while in measurements, it has the gain of 8.1 dB at 500 MHz and 4.3 dB at 1.5 GHz. The minimum noise figure is 2.4 dB at 500 MHz and 2.4 dB at 1.5 GHz in simulations, while in measure- ments, it is 3.1 dB at 500 MHz and 3.2 dB at 1.5 GHz. This thesis presents schematic and measurement results for the electrical and noise parameters. The measurement results are analyzed in Matlab and compared with the relevant single-ended measurement results to verify the operation of the method.
Acknowledgements
Foremost, I would really like to thank my supervisor Dr. Leonid Belostotski for his guid- ance, patience, and interest throughout my master studies. I appreciate his broad knowledge and experience in the RF circuit and layout design. He is always helpful and spent as much time as I needed to discuss the problems of my research work. I would also like to thank him for his patience and spending days and days helping me write and edit my thesis.
Second, I would like to thank all the MiNT lab students, post-doctoral fellows, my friends throughout the ECE department and department staffs for their help and support.
Many thanks to Donuwan Navaratne, Zhixing Zhao, Nan Zhang, Hao Xie and Vahid Asgari for their helpful discussion and support regarding this thesis work.
Last but not least, I would like to thank my parents for their support and encouragement.
For my parents and my supervisor.
Thank you for your encouragement and support.
Table of Contents
Abstract ii
Acknowledgments iii
Dedication iv
Table of Contents vi
List of Tables vii
List of Figures x
Glossary xi
1 Introduction 1
1.1 Motivation and Objectives . . . 1
1.2 Thesis Outline . . . 2
2 Extraction of Electrical Parameters for Single-Ended Amplifiers Inside a Dif- ferential Amplifier 4 2.1 Introduction and Objectives . . . 4
2.2 Proposed Method of the Circuit Electrical-Parameter Extraction . . . 4
2.3 Verification of the Proposed Method . . . 11
2.4 Schematic Design Process and Simulation Results . . . 12
2.5 Differential Amplifier Design and Simulation Results . . . 16
2.5.1 Schematic Simulation and Extraction Results . . . 17
2.5.2 Layout Design Process and Comparison of Momentum Extraction Results with Momentum Single-Ended Simulation Results . . . 18
2.6 Measurement Results . . . 22
2.7 Summary . . . 25
3 Extraction of Noise Parameters for Single-Ended Amplifiers Inside a Differen- tial Amplifier 26 3.1 Introduction and Objectives . . . 26
3.2 Single-Ended Noise-Parameter Extraction Algorithm . . . 27
3.3 Schematic Extraction and Simulation Results . . . 32
3.3.1 Algorithm Verification Using Ideal Components . . . 32 3.3.2 Comparison Between Schematic Extraction Results and Schematic
Single-Ended Simulation Results . . . 34 3.4 Measurement Noise Extraction Results . . . 35 3.5 Summary . . . 39
4 Conclusions and Future Work 41
4.1 Conclusion . . . 41 4.2 Future Work . . . 44 A Forming Matrix to Solve S-Parameters for Single Amplifier Inside a Differen-
tial Amplifier 55
B Z-Parameters Conversion from 2-Port to 1-Port 58
C Amplifier B with Input and Output Noise Source for Noise Correlation Matrix
in Z Form Calculation 60
D Z-Representation for Combined Common Network and “One Side Terminated”
Differential Amplifier 62
List of Tables
2.1 Circuit Component Values. . . 13 2.2 DC Biasing Conditions. . . 14 2.3 Schematic Design Specifications. . . 16 4.1 Comparison of S-parameters from Schematic Simulations Measurement
Results to the Relevant Single-Ended Simulation and Measurement Results for Single-Ended Amplifiers and the Common Network Impedance . . . 42 4.2 Comparison of S-Parameters from EM-Schematic Momentum to the Rele-
vant Single-Ended Simulation for Single-Ended Amplifiers and the Com- mon Network Impedance . . . 43 4.3 Comparison of Ideal Differential-Circuit Extraction Results with Single-
Ended Simulation Results. . . 45 4.4 Comparison of NFmin, Rn, and Γopt Obtained from Schematic Simulations
to the Relevant Single-Ended Simulation Results . . . 46 4.5 Comparison of S-Parameters from Measurement Results to the Relevant
Single-Ended Measurement Results for Single-Ended Amplifiers and the Common Network Impedance . . . 47 4.6 Comparison of NFmin, Rn, and Γopt Obtained from Measurements to the
Relevant Single-Ended Measurement Results. . . 48
List of Figures
2.1 4-Port Differential Amplifier. . . 5 2.2 4-Port Differential Amplifier with Ports 2 and 4 Terminated with Loads. . . 5 2.3 Overall 2-port Network for Terminated Differential Amplifier. . . 7 2.4 Terminated TransistorB Cascaded with Common Network. . . 9 2.5 System Level Differential Circuit. . . 11 2.6 Comparison of Databox Extracted Results and Single-Ended Results.Top
Graphs: Pre-Stored Data. Middle Graphs: Extracted Pre-stored Data for DataboxA. Bottom Graphs: Extracted Pre-stored Data for DataboxB. (300 MHz to 3 GHz) . . . 12 2.7 Differential Amplifier ADS Schematic. . . 13 2.8 Differential Gain. (300 MHz to 3 GHz) . . . 13 2.9 Differential Circuit Gain, Source and Load Stability Circles, and Stability
Factor. . . 14 2.10 Schematic of the Single-Ended Amplifier. . . 15 2.11 Schematic of a Single-Ended Amplifier S Parameters in dB. (300 MHz to
3 GHz) . . . 15 2.12 Stability Factor, Schematic Single-Ended Source and Load Stability Circles. 16 2.13 Schematic S-parameters Extraction Results vs Schematic Single-Ended Sim-
ulation Results. Red Line: Schematic Simulated Single-Ended S-parameters.
Blue Line: Schematic Extracted Results. (300 MHz to 3 GHz) . . . 17 2.14 Substrate Parameters and Substrate Transverse Plane Picture. . . 18 2.15 3D View of Two-Layer PCB Layout in ADS with Input and Output Trans-
mission Lines. . . 19 2.16 Top View for the Two-Layer PCB. Top Layer: Brown. Bottom Layer: Yellow. 19 2.17 ATF 35143 Footprint. . . 20 2.18 Comparison between Simulated Single-Ended and Momentum Extraction
Results (300 MHz-1.5 GHz). Sky Blue and Red Line: Momentum Extrac- tion Results. Dark Blue Line: Momentum Single-Ended Simulation Re- sults. . . 21 2.19 Top View and Bottom View of Surface-Mounted Two-Layer PCB. . . 22 2.20 ADS De-Embedding Schematic (Left Databox: Measurement Raw Data.
Right Databox: SMA S Parameters from HFSS.). . . 22
2.21 Comparison of Measured S-Parameters Before De-Embedding and After De-Embedding. Blue line: S-Parameter for the Measured 4-port before De-Embedding. Red line: S-Parameter for the Measured 4-port after De-
Embedding. (300 MHz to 3 GHz) . . . 23
2.22 Comparison of De-Embedded Measured Single-Ended Amplifier S-Parameters, and Extracted Single-Ended Amplifier S-Parameters from the Differential Amplifier. (Smith chart) Red Line: Measured Single-Ended S-Parameters. Blue Line: Measured Amplifier A S-Parameters. Purple Line: Measured Amplifier B S-Parameters. (300 MHz to 3 GHz) . . . 24
2.23 Comparison of De-Embedded Measured Single-Ended Amplifier S-Parameters, and Extracted Single-Ended Amplifier S-Parameters from the Differential Amplifier. (Magnitude) Red Line: Measured Single-Ended S-Parameters. Blue Line: Measured Amplifier A S-Parameters. Purple Line: Measured Amplifier B S-Parameters. . . 25
3.1 Divisions of 2-Port Networks. . . 27
3.2 2-Port Upside Down. . . 29
3.3 Cascade AmplifierB with Common Network . . . 29
3.4 Serial Connection of AmplifierA and Common Network . . . 29
3.5 Ideal Differential Amplifier. . . 32
3.6 Comparison of Rnbetween Extracted Results and Prestored Data.. . . 32
3.7 Comparison of NF min between Extracted Results and Prestored Data. . . . 33
3.8 Comparison ofΓopt between Extracted Results and Prestored Data. . . 33
3.9 Schematic Extracted NFmin vs Schematic Single-Ended Simulation Re- sults. . . 34
3.10 Schematic Extracted Rnvs Schematic Single-Ended Simulation Results. . . 35
3.11 Schematic ExtractedΓopt vs Schematic Single-Ended Simulation Results. . 35
3.12 One-Side Terminated NFmin Raw Data from Measurement in Figure 3.4. (300 MHz to 3 GHz) . . . 36
3.13 One-Side Terminated RnRaw Data from Measurement in Figure3.4. (300 MHz to 3 GHz) . . . 36
3.14 One-Side Terminated Γopt Raw Data from Measurement in Figure 3.4. (300 MHz to 3 GHz) . . . 37
3.15 Comparison of NFmin between Measurement Extracted Results and Mea- surement Single-Ended Results (300 MHz to 3 GHz). . . 38
3.16 Comparison of Rnbetween Measurement Extracted Results and Measure- ment Single-Ended Results (300 MHz to 3 GHz). . . 38 3.17 Comparison ofΓoptbetween Measurement Extracted Results and Measure-
ment Single-Ended Results (300 MHz to 3 GHz). . . 39 B.1 Diagram for conversion of a 2 Port Z Parameters into 1 Port. . . 58 C.1 AmplifierB with Input and Output Noise Source for Noise Correlation Ma-
trix in Z form Calculation. . . 60 D.1 AmplifierB with Input and Output Noise Source Cascaded with the Com-
mon Network for Noise Correlation Matrix in Z Form Calculation. . . 62
Glossary
Acronom Definition
ADS Advanced Design Systems
AC Alternating Current
BIT Built-In Test
DC Direct Current
DLNA Differential Low Noise Amplifier
FR4 A NEMA grade designation for glass-reinforced epoxy laminate material
Γ1 Reflection Coefficient at Port1 when Port2 and Port4 are terminated Γ2 Reflection Coefficient at Port2 when Port1 and Port3 are terminated Γ3 Reflection Coefficient at Port3 when Port2 and Port4 are terminated Γ4 Reflection Coefficient at Port4 when Port1 and Port3 are terminated Γopt Reflection coefficient which optimal noise figure can be found
accordingly
HFSS A commercial finite element method solver for electromagnetic structures from Ansys
LNA Low Noise Amplifier
NFmin Minimum Noise Figure
PCB Printed Circuit Board
PNA-X Microwave Network Analyzer
Rn Equivalent Noise Resistance
SDU T31 , SDU T42 S parameters for Device Under Test when measuring one side of the differential amplifier while terminating the other side
SMA Subminiature version A connector
VIA Vertical Interconnect Access
ZDU T31, ZDU T42 Z parameters for Device Under Test when measuring one side of the differential amplifier ad terminatingthe other side
Zin Termination resistor placed at input port of one amplifier in the differential amplifier
Zout Termination resistor placed at output port of one amplifier in the differential amplifier
Chapter 1 Introduction
1.1 Motivation and Objectives
Differential circuits provide designers with significant benefits of supply, ground, and common-mode coupling noise rejection and linearity improvement in integrated analog and RF circuit designs. However, it is hard to examine their operation because they usu- ally have differential inputs and outputs but external measurement equipment is typically single-ended. It is also not possible to measure the behaviour of the subcircuit components inside the differential circuit to verify whether they are operating as expected as they have hidden internal nodes. To determine the faults in a differential amplifier, it is needed to tune the circuit simulation until the measured behaviour is the same as in simulations.
Differential amplifier noise behaviour analysis requires differential noise parameters measurement. And then, a simulation-assisted fault analysis can be carried out. In order to help the fault analysis process, this paper provides a method that can provide an insight into the electrical and noise parameters of the subcomponents inside the differential system, which is designed with following specifications: (a) Passive common network; (b) Only one stage; (c) No internal grounding. This method only needs single-ended measurement equipment, which means there are no transformers, hybrids or baluns included.
Some previous works have been performed to analyze the performance of the amplifiers relying on the knowledge of the internal configurations of the differential amplifiers [1–
9]. Because the differential measurement equipment is usually unavailable in the market, various approaches have been developed to extract the differential parameters using single- ended equipment [2,8–12].
The work in [10] presents a mathematical theory for mixed-mode S-parameters that is developed for characterization of microwave differential circuits. The work in [11] demon-
tion of the correlated input and output noise sources. Procedures of measuring differential LNAs with correlated output noise sources are discussed in [8] and [9]. [1–4] talk about the ways of measuring the differential noise figures without adding baluns. The work in [12]
demonstrated a method of determining the differential noise parameters of differential am- plifiers using hybrids. [13] shows that using hybrids, transformers and baluns is the most general approach currently available as long as the baluns are de-embeded properly. The work in [5] demonstrates a built-in test (BIT) circuit for radio frequency differential low noise amplifiers (DLNAs). However BIT circuits do not apply when the circuit is already fabricated.
The work in [14] shows a theory for combined differential and common mode normal- ized power waves developed in terms of even and odd mode impedances and propagation constants for a microwave coupled line system.
The above works, which have been done previously, allow to examine the operation of the differential amplifier using single-ended measurement equipment. However, they did not provide a way of estimating the operation of components inside the circuit. This topic of
“dissecting” a differential circuit into its subcomponents with the purpose of investigating the behaviour of the subcomponents is discussed in this thesis.
1.2 Thesis Outline
The thesis starts with an introduction of the algorithms that are needed for extracting single- ended noise parameters inside a differential circuit. A differential amplifier is used as the differential circuit to extract the single-ended amplifiers’ electrical parameters as shown in Chapter2and noise parameters as shown in Chapter3.
Chapter2presents the design and the extraction results of a differential amplifier con- stituted by two single amplifiers and a common network. Section2.2presents the algorithm that has been applied to extract the S parameters for single-ended amplifiers and the com-
mon network, which in this example, is an inductor. Section2.3demonstrates the system level extraction results, which are used to verify the algorithms. Section 2.4 shows the process of designing the schematic for the differential circuit, including meeting the design specifications and the selection of the circuit components when taking consideration of the circuit gain and stability. The comparison of the extraction results for the single amplifiers from ADS and single-ended simulation results are also discussed in this section. Section 2.5includes the layout design process and EM momentum simulation results. Section 2.6 shows the measurement simulation and extractions results. A summary is provided in Sec- tion 2.7 to conclude the chapter and briefly summarize the performance of the designed circuit.
Chapter3demonstrates the idea of extracting noise parameters for the single amplifiers inside the differential amplifier by using the electrical parameters extracted in Chapter2and the overall noise parameters, which are taken from terminating one side of the differential amplifier with 50Ωloads. Section 3.2 shows the algorithm that has been used to extract the single-ended noise parameters from the differential circuit. Section 3.3.1provides the extraction results when using ideal databoxs, which have noise parameters data and elec- trical parameters data prestored. Section 3.3.2 gives the comparison between schematic extraction and single-ended simulation results. Section3.4shows the measurement extrac- tion and single-ended results. Summaries are also provided to conclude the chapter and to discuss the overall performance of the algorithms and the design.
Chapter 4 provides a summary of the whole thesis by making tables to compare the extraction results between schematic and measurement. It also talks about the future work that can be done in addition to this project.m
Chapter 2
Extraction of Electrical Parameters for Single-Ended Amplifiers Inside a Differential Amplifier
2.1 Introduction and Objectives
The target of this chapter is to extract electrical parameters for single-ended circuits inside a differential amplifier. The chapter starts with the derivation of the governing algorithm and the associated equations in Section 2.2. To verify the algorithms for the extraction, a system level differential circuit, which contains pre-stored S-parameter data that is selected randomly for each subcircuit components, is built and discussed in Section2.4. After the extraction, a comparison is made between the extracted results and pre-stored S-parameters.
Then, the schematic extraction results as shown in Section2.4, schematic momentum ex- traction results as shown in Section 2.5 and measurement extraction results as shown in Section2.6are made and compared to their relevant single-ended simulation results.
2.2 Proposed Method of the Circuit Electrical-Parameter Extraction
This section demonstrates the derivation of equations that are used in this thesis to extract the electrical-parameters of single-ended circuits forming a differential circuit. A concep- tual diagram of the differential amplifier is shown in Figure 2.1. The assumptions made in this section are that a) the 4-port S-parameters of the differential amplifier are available from either simulations or measurements and b) the single-ended circuits do not have a hidden connection to ground, or in other words, all single-ended circuit ground connec- tions are tied to the common network of the differential circuit. The goal is using the 4-port S-parameters to find 2-port electrical parameters for single-ended amplifiers and the common-network inductor numerically. The key idea explored in this thesis is that by
Figure 2.1: 4-Port Differential Amplifier.
Figure 2.2: 4-Port Differential Amplifier with Ports 2 and 4 Terminated with Loads.
enough linearly independent equations that would allow the extraction of internal param- eters of the subcomponents in the differential circuit. Following this idea, two of the four ports of the differential circuit are terminated by loads turning the 4-port network into a 2-port network. The differential amplifier can be considered as a 2-port network as shown in Figure2.2with two ports (Port 2 and Port 4) terminated for illustration purposes.
The reflection coefficient at the terminated ports can be described by
Γ2= V2+/V2− (2.1)
and
Γ4= V4+/V4− (2.2)
+ − Γ
represents reflection coefficient at Port 2 when looking from Port 2 into some loads, Γ4
represents reflection coefficient at Port 4 when looking from Port 4 into some loads.
If Port 2 and Port 4 are terminated in the differential amplifier in Figure2.7, the newly created 2-port network can be described by its 2-port S-parameters as
S′a=
S′a11 S′a12 S′a21 S′a22
, (2.3)
where S′a represents the 2-port S-parameters for the 4-port differential amplifier with Port 2 and Port 4 terminated.
By repeating this process, several sets of 2-port S-parameter matrices can be formed to find Sai j′ = V
− i
Vj+
Vk6= j+
.
For example, (2.4) is used to solve for Sa11′ .
−S11
−S21
−S41
=
−1 S12Γ2 S12Γ4
0 S22Γ2− 1 S24Γ4
0 S42Γ2 S44Γ4− 1
S′a11
V2− V1+ V4− V1+
, (2.4)
where Si j represents measured S-parameters of the 4-port network. If the loads for Port 2 and Port 4 are known, this matrix can be solved using standard linear algebra approach since there are three equations and three unknowns in this system. S′a12, S′a21, and S′a22can be found in similar ways. Detailed derivation procedures can be found in Appendix A.
When one set of input and output ports of the differential amplifier are terminated with known loads, the resultant 2-port S-parameters, SDUT31 (or SDUT42 when Port 1 and 3 are terminated), of the unterminated single-ended amplifier in series with a cascade com- bination of the terminated amplifier and the common network as shown in Figure2.3 are found.
Once the common-network value is calculated later in (2.19), it is possible to form the
Figure 2.3: Overall 2-port Network for Terminated Differential Amplifier.
de-composition equations of the system so that the electrical parameters (ZA and ZB) for single-ended amplifiers can be found. The following section introduces the way to calculate ZC, and then ZAand ZB.
Because Amplifier A is in series connection with the rest of the system, SDUT31 or SDUT42 are needed to be converted to Z representation. The unknown Z-parameters for Amplifier A and Amplifier B are defined as
Za=
Z11,a Z12,a Z21,a Z22,a
(2.5)
and
Zb=
Z11,b Z12,b Z21,b Z22,b
. (2.6)
Assuming that the Amplifier B is terminated, then 1-port representation of this amplifier is needed for describing the cascade of it with the common network. The 1-port representation Zb,T is expressed as
Zb,T =
Zb,T Zb,T Zb,T Zb,T
(2.7)
where
Zb,T = Zin+ Z11,b
Zout+ Z22,b − Z12,bZ21,b Zin+ Z11,b+ Zout+ Z22,b− Z12,b− Z21,b
. (2.8)
Detailed calculation procedures can be found in AppendixB. If the input and output of the amplifier are terminated with Zinand Zout, the overall Z11becomes Zin+ Z11, the overall Z22 becomes Zout+ Z22.
When cascaded with common network as shown in Figure2.4, the matrix of the cascade can be represented as
Zb,Toverall =
(Zb,T−1+ Z−1c )−1 (Zb,T−1+ Zc−1)−1 (Zb,T−1+ Z−1c )−1 (Zb,T−1+ Zc−1)−1
, (2.9)
where ZCrepresents the impedance for the common network.
And similarly, if Amplifier A is terminated, the 1-port representation Za,T is expressed as
Za,T =
Za,T Za,T Za,T Za,T
, (2.10)
where
Za,T = (Zin+ Z11,a) (Zout+ Z22,a) − Z12,aZ21,a
Zin+ Z11,a+ Zout+ Z22,a− Z12,a− Z21,a. (2.11) When cascaded with the common network, the matrix of the cascade can be represented as
Za,Toverall =
(Za,T−1+ Z−1c )−1 (Za,T−1+ Zc−1)−1 (Za,T−1+ Z−1c )−1 (Za,T−1+ Zc−1)−1
. (2.12)
If Port 2 and Port 4 are terminated by Zinand Zout as shown in Figure2.3, ZDUT31can
Figure 2.4: Terminated TransistorB Cascaded with Common Network.
be represented as
ZDUT31=
Z11,a Z12,a Z21,a Z22,a
+
Zb,Toverall,11 Zb,Toverall,12
Zb,Toverall,21 Zb,Toverall,22
, (2.13)
where Zb,Toverall,11, Zb,Toverall,12, Zb,Toverall,21 and Zb,Toverall,22 are Z parameters for Zb,Toverall.
If, on the other hand, Port 1 and Port 3 were terminated, ZDUT42 is found from
ZDUT42=
Z11,b Z12,b Z21,b Z22,b
+
Za,Toverall,11 Za,Toverall,12
Za,Toverall,21 Za,Toverall,22
, (2.14)
where Za,Toverall,11, Za,Toverall,12, Za,Toverall,21 and Za,Toverall,22 are Z parameters for Za,Toverall.
There are 9 complex unknowns, which are Z parameters for Amplifier A and Amplifier B, as well as ZC. By substituting different Zinand Zoutinto Equation2.8, unknowns can be solved using
ZDUT31(Zin, Zout) = Za+ Zb,Toverall(Zin, Zout) ZDUT42(Zin, Zout) = Zb+ Za,Toverall(Zin, Zout) .
(2.15)
When open, i.e. infinite impedance, and short, i.e. zero ohm impedance, terminations are
substituted into the (2.15), the following two systems of equations are obtained
ZDUT31(∞,0) = Za+ZZ22,bZC
22,b+ZC
1 1 1 1
ZDUT42(∞,0) = Zb+ZZ22,aZC
22,a+ZC
1 1 1 1
(2.16)
and
ZDUT31(∞,∞) = Za+ ZC
1 1 1 1
ZDUT42(∞,∞) = Zb+ ZC
1 1 1 1
.
(2.17)
From the above equations, an observation can be made that
ZDUT31(∞,∞)(1, 1) = Z11,a+ ZC
ZDUT42(∞,∞)(2, 2) = Z22,b+ ZC
ZDUT31(∞,0)(1, 1) = Z11,a+ZZ22,bZC
22,b+ZC.
(2.18)
From this system of equations,
ZC2=ZDUT31(∞,∞)(1, 1) − ZDUT31(∞,0)(1, 1) ZDUT42(∞,∞)(2, 2) (2.19)
and ZC can be determined. In real calculation, there are two roots for ZC. The imagi- nary part of the correct root should be positive and close to the expected impedance of the common-mode network. Once ZC is determined, Z parameters for Amplifier A and Amplifier B can be found from solving (2.20) and (2.17) by substituting ZC into them:
Z11,a= ZDUT31(∞,∞)(1, 1) − ZC
Z12,a= ZDUT31(∞,∞)(1, 2) − ZC
Z21,a= ZDUT31(∞,∞)(2, 1) − ZC
Z22,a= ZDUT31(∞,∞)(2, 2) − ZC
(2.20)
Z11,b= ZDUT42(∞,∞)(1, 1) − ZC Z12,b= ZDUT42(∞,∞)(1, 2) − ZC Z21,b= ZDUT42(∞,∞)(2, 1) − ZC Z22,b= ZDUT42(∞,∞)(2, 2) − ZC
(2.21)
2.3 Verification of the Proposed Method
In order to verify the correctness of the equations derived in Section2.2, an ideal differential amplifier was implemented in Agilent’s Advanced Design System (ADS). Figure2.5shows the simulated schematic. Figure2.6shows that the extraction results of the 4-port network are exactly the same as the original values used to build the differential amplifier. Since the extraction results are perfect, the extraction procedure can be considered correct given the assumptions made in its derivations.
Figure 2.5: System Level Differential Circuit.
Figure 2.6: Comparison of Databox Extracted Results and Single-Ended Results.Top Graphs: Pre-Stored Data. Middle Graphs: Extracted Pre-stored Data for DataboxA. Bot- tom Graphs: Extracted Pre-stored Data for DataboxB. (300 MHz to 3 GHz)
2.4 Schematic Design Process and Simulation Results
The next step is to verify the extraction procedure with simulations of a real amplifier and then experimentally. To verify the extraction algorithm with a real circuit, a differential system is built using two amplifiers and a common network. In order to make a virtual ground at the drain terminals, a big choke inductor is needed to be placed after load resis- tors at the drain terminals. Coil Craft inductors are selected, which have a self-resonant frequency of 1.15 GHz. It is also needed to have a big inductor at the source terminal to create differential ground at the source terminals. In order to reduce the potential instability and the inductor’s self-resonant frequency effect on the extraction results, a 4.7 nH inductor is chosen with a self-resonant frequency of 12.7 GHz. The circuit is constructed in ADS as shown in Figure2.7.
The amplifiers are constructed with Avago ATF-35143 transistors because this type of amplifier has low noise figures and large available gain in the frequency range from 300 MHz to 3 GHz. The biasing network is also formed to make circuit stable, to have wide bandwidth, and to have gain. Table2.1lists all circuit component values. The differential gain for the differential amplifier is shown in Figure 2.8. The circuit has around 9.5 dB
Table 2.1: Circuit Component Values.
Circuit Component Size
Gate resistor (R8 and R9) 50Ω Feedback resistor(R1 and R4) 332Ω Gate biasing resistor (R10 and R11) 10000Ω
Choke inductor (L1, L2 and L3) 220 nH Drain resistor (R12 and R13) 15Ω
Capacitors 56 pF
Figure 2.7: Differential Amplifier ADS Schematic.
Figure 2.8: Differential Gain. (300 MHz to 3 GHz)
differential gain from 300 MHz to 3 GHz. It can also been observed from Figure2.9 that the differential circuit is unconditionally stable at all frequencies.
Because the threshold voltage for ATF 35143 is−0.95 V and the source biasing voltage is 0 V, a 10 kΩresistor is connected between the gate and ground to bias the gate voltage also at 0 V. Therefore, only one power supply is needed to be connected to the drain
(a) Differential Circuit Stability Factor.
(b) Differential Circuit Source and Load Stability Circles.
Figure 2.9: Differential Circuit Gain, Source and Load Stability Circles, and Stability Factor.
Table 2.2: DC Biasing Conditions.
DC biasing conditions Voltages
Gate voltage 0 V
Drain voltage 1.6 V Source voltage 0 V
terminals of the amplifiers. The DC biasing conditions can be found in Table2.2. Since the maximum DC current is 80 mA for ATF 35143 transistor, the biasing conditions are chosen so that the transistor DC current is 62.6 mA, which is less than the maximum.
A single-ended schematic is constructed in Figure 2.10 for comparison purpose. Be- cause ideal 220 nH choke inductors are applied to the schematic, the DC biasing voltage at drain terminal is 1.6 V. When the choke inductors are replace with real inductors (0402AF), there will be a 0.08 V voltage drop across the inductors. The S-parameters for the single- ended circuit are reported in Figure2.11.
Figure 2.10: Schematic of the Single-Ended Amplifier.
Figure 2.11: Schematic of a Single-Ended Amplifier S Parameters in dB. (300 MHz to 3 GHz)
The selection of the gate resistors and feedback resistors is based on the consideration of the stability for the circuit at all frequencies. The stability circles showed in Figure2.12b are all outside the unity circle, which means the circuit is unconditionally stable.
The single-ended simulation results can be concluded in Table2.3. Under current bias- ing condition, the single-ended amplifier has gain of 11.2 dB. It also has a large bandwidth.
In the next section, an electrical parameters extraction typology is introduced.
(a) Stability Factor for Single-Ended Amplifier.
(b) Input and Output Stability Circles for Single-Ended Amplifier.
Figure 2.12: Stability Factor, Schematic Single-Ended Source and Load Stability Circles.
Table 2.3: Schematic Design Specifications.
Design Parameters Specifications Gain at 700 MHz 11.2 dB
Stability unconditionaly stable
Bandwidth 8 GHz
2.5 Differential Amplifier Design and Simulation Results
In this section, the layout design process for the differential amplifier is implemented. The extraction results for the schematic and momentum simulation, as well as their relavant single-ended simulation results are discussed and compared.
2.5.1 Schematic Simulation and Extraction Results
Matlab is used as the tool to simulate the extraction procedures numerically following the equations in Section2.2.
In Figure2.13, extracted S-parameters for single-ended amplifiers inside the differential amplifier as shown in Figure2.7are compared with single-ended amplifier S-parameters as shown in Figure2.10.
Figure 2.13: Schematic S-parameters Extraction Results vs Schematic Single-Ended Sim- ulation Results. Red Line: Schematic Simulated Single-Ended S-parameters. Blue Line:
Schematic Extracted Results. (300 MHz to 3 GHz)
From Figure 2.13, it can be observed that extracted S11, S12 and S21 are very close to the single-ended simulation results both in magnitude plots and on the Smith chart. S22has the largest variation of 1 dB between extracted results and single-ended results. The trend of S22 on the Smith chart is also visible. It is not clear at this time why S22 does not agree exactly with expectations.
2.5.2 Layout Design Process and Comparison of Momentum Extraction Results with Momentum Single-Ended Simulation Results
A two-layer printed circuit board (PCB) was selected to construct the differential amplifier and to verify the extraction algorithms. The PCB substrate is FR4 material, whose substrate parameters and structure are shown in Figure2.14. By using the ADS line calculation tool, it was found that on a 1 mm (i.e. 40 mil) thick substrate, the 50Ωtransmission-line width should be 1.94 mm (i.e. 76.4 mils).
(a) PCB Substrate Parameters.
(b) Substrate Transverse Plane Picture.
Figure 2.14: Substrate Parameters and Substrate Transverse Plane Picture.
Figure 2.15shows a 3D view of the circuit layout with input and output transmission lines. The transmission lines are designed as coplanar waveguides due to low dispersion and the broadband performance. Figure2.16 shows the center area for both top and bot- tom layer of the designed PCB. Vias are placed through out the PCB for good grounding condition.
Figure 2.15: 3D View of Two-Layer PCB Layout in ADS with Input and Output Trans- mission Lines.
Figure 2.16: Top View for the Two-Layer PCB. Top Layer: Brown. Bottom Layer: Yellow.
The assumption made during the derivation of the extraction algorithm is that there are no references to the PCB ground from the single-ended subcomponents inside the differen-
Figure 2.17: ATF 35143 Footprint.
components as the transmission lines may couple to ground planes. To reduce the effects from the internal transmission lines on the extraction results, all circuit components must be placed as close to each other as possible. The feedback loops, which contain feedback resistors and DC block capacitors, are placed on the bottom layer due to the diagonal po- sitions for the drain terminals and gate terminals as shown in Figure2.17. The soldering pads are connected with top layers through vias. These lines are unavoidable and have to be de-embedded during measurements. To make de-embedding process less complicated, all transmission lines are designed to have the same lengths. Since each transistor package has 4 legs as shown in Figure2.17. To make the input line and output lines fully symmetrical, the legs of the transistor on the right-hand side of the PCB are bent over.
Figure 2.18 shows the comparison of extracted results obtained from schematic mo- mentum with a single-ended amplifier also from schematic momentum within a selected bandwidth (500 MHz-1.5 GHz). It can be observed from Figure2.18, S11, S21and S12are in good agreement between expectations and extraction. They all go higher as the frequency increases. Although the momentum single-ended simulation results for those parameters have some variation in magnitude, the shapes for them can still be considered as good since they are similar to each other. S22 is larger in value and shows high variation between what is obtained with direct simulation and what is obtained with extraction. The variation of about 2 dB between the single-ended momentum results and momentum extraction re-
sults is observed. It can be concluded that although the extraction results are not as good as schematic extraction results, the values for S parameters within the bandwidth that is selected are still reasonable. The next section demonstrated experimental results used to verify the extraction process.
Figure 2.18: Comparison between Simulated Single-Ended and Momentum Extraction Results (300 MHz-1.5 GHz). Sky Blue and Red Line: Momentum Extraction Results. Dark Blue Line: Momentum Single-Ended Simulation Results.
2.6 Measurement Results
Figure2.19 shows the fabricated PCB with circuit components and SMA connectors. To obtain accurate S-parameters for the differential network, de-embedding process needs to be performed. HFSS (High Frequency Structure Simulator) is used as the tool to construct the S-parameters for the SMA connector. Since the input and output transmission lines have the same length, they are de-embedded in the same way.
De-embedding process followed in this work is:
1. Measure S-parameters of the transmission line with SMA connectors on each side.
2. In ADS, use the SMA S-parameter data to de-embed one SMA from the transmission line measured in step 1 as shown in Figure2.20.
3. Use the results from step 2 to de-embed transmission lines from each port of the differential circuit.
Figure 2.19: Top View and Bottom View of Surface-Mounted Two-Layer PCB.
Figure 2.20: ADS De-Embedding Schematic (Left Databox: Measurement Raw Data.
Right Databox: SMA S Parameters from HFSS.).
Figure 2.21 shows the 4-port de-embedding structure for S-parameters obtained from the measurement and the comparison of the 4-port S-parameters before and after the de- embedding process.
Figure 2.21: Comparison of Measured S-Parameters Before De-Embedding and After De- Embedding. Blue line: S-Parameter for the Measured 4-port before De-Embedding. Red line: S-Parameter for the Measured 4-port after De-Embedding. (300 MHz to 3 GHz)
From Figure 2.22 and Figure2.23, it can be found that S11 and S12 are good both in
them are both higher than the single-ended simulation results. S21 has 1.017 dB difference at 502.5 MHz and 0.751 dB difference at 1.515 GHz for Amplifier A, while it has 0.633 dB difference at 502.5 MHz and 0.303 dB difference at 1.515 GHz for Amplifier B . S22 has 2.41 dB difference at 502.5 MHz and 5.125 dB difference at 1.515 GHz for Amplifier A, while it has−2.045 dB difference at 502.5 MHz and −5.242 dB difference at 1.515 GHz for Amplifier B. It can still be concluded that the S-parameters extracted from measurement results are good enough to be used as the inputs to the noise calculation in next chapter.
Figure 2.22: Comparison of De-Embedded Measured Single-Ended Amplifier S- Parameters, and Extracted Single-Ended Amplifier S-Parameters from the Differential Amplifier. (Smith chart) Red Line: Measured Single-Ended S-Parameters. Blue Line:
Measured Amplifier A S-Parameters. Purple Line: Measured Amplifier B S-Parameters.
(300 MHz to 3 GHz)
Figure 2.23: Comparison of De-Embedded Measured Single-Ended Amplifier S- Parameters, and Extracted Single-Ended Amplifier S-Parameters from the Differential Am- plifier. (Magnitude) Red Line: Measured Single-Ended S-Parameters. Blue Line: Mea- sured Amplifier A S-Parameters. Purple Line: Measured Amplifier B S-Parameters.
2.7 Summary
To summarize this chapter, the extraction algorithm is presented and experimentally veri- fied. Within the bandwidth (500 MHz to 1.5 GHz), the schematic extraction results are very close to the schematic single-ended simulation results. Momentum extraction results are good in low frequencies. But as frequency goes higher, the agreement gets worse. The measurement results are good in magnitude and phase for S11 and S12, while the biggest variation is the magnitude for S21and S22. Since the target of this project is to extract noise parameters of single amplifiers inside the differential amplifier, it can be examined later to see that whether the variations of S22 have significant influences to the noise parameters extractions. The next chapter will talk about the extraction algorithms and experimental measurement results for the noise parameters of the single-ended amplifiers inside the dif- ferential amplifier.
Chapter 3
Extraction of Noise Parameters for Single-Ended Amplifiers Inside a Differential Amplifier
3.1 Introduction and Objectives
After the electrical-parameter extraction method, which is introduced in Chapter2, the next stage is to determine the noise parameters, which are the minimum noise figure NFmin, the equivalent noise resistance Rn and the optimum source reflection coefficient, which corresponds to minimum noise figure achievement,Γopt, for the subcomponents inside the differential circuit. There are a few approaches to measure the noise parameters for single- ended circuits.
Some of the approaches model noise in terms of power waves [15–21]. There are also other approaches, which perform single noise figure measurements and try to put the results into a DUT noise model determined by using other techniques [22–24]. The most commonly used techniques are performed by using source impedance tuners to generate different signal- source admittances at the DUT input port. And then use the receivers to measure the noise powers at the output port [25–36]. The noise parameters are found by using data fitting techniques as described in [37–41].
The purpose of this chapter is to demonstrate a method of extracting noise parameters of single-ended amplifiers inside a differential amplifier. The same assumptions as in Chapter 2are considered to be applied to the same type of amplifiers here. This chapter starts with the discussion of the noise-parameter extraction algorithm in Section3.2and proceeds with simulation and experimental verification of the algorithm in Section3.3and Section3.4.
3.2 Single-Ended Noise-Parameter Extraction Algorithm
In this noise-parameter extraction algorithm, 2-port single-ended noise-parameter measure- ment equipment is used. Because a differential amplifier has four ports that can interface single-ended equipment, when measuring noise parameters, there are always two unused ports. These unused ports are terminated with 50Ω terminations. In this way, the 4-port network becomes 2-port network as shown in Figure3.1. The following extractions assume that the electrical parameters as derived in Chapter2are available.
Based on Figure3.1, the Z representation of the overall noise correlation matrix for the measured 2-port circuit can be written as
C13,overall = CZ,a+ C′Z,b C24,overall = CZ,b+ C′Z,a
(3.1)
Figure 3.1: Divisions of 2-Port Networks.
where CZ,arepresents the Z-represenntation of the noise correlation matrix of Amplifier A if Amplifier B is terminated as shown in Figure3.1, CZ,brepresents the Z-reprensentation noise correlation matrix of Amplifier B if Amplifier A is terminated, C′Z,b represents the Z-reprentation noise correlation matrix of the terminated Amplifier B cascaded with the
minated Amplifier A cascaded with the common network.
CZ,aand CZ,bcan be represented as
CZ,a=
CZ,a,11 CZ,a,12 CZ,a,21 CZ,a,22
(3.2)
CZ,b=
CZ,b,11 CZ,b,12 CZ,b,21 CZ,b,22
(3.3)
and then
CZa,T = CZ,a+ 2kT
R{Zin} 0 0 R{Zout}
(3.4)
CZb,T = CZ,b+ 2kT
R{Zin} 0 0 R{Zout}
(3.5)
where CZa,T represents the Z-representation noise correlation matrix for terminated Amplifier A, CZb,T represents the Z-representation noise correlation matrix for terminated Amplifier B, T is the absolute temperature, k is Boltzmann’s constant.
Since the ground terminals of the single-ended amplifiers are connected to the common network of the differential circuit, the 2-port network for the terminated amplifiers needs to be “turned upside down” as shown in Figure3.2for further derivations. As can be seen from Figure 3.1, the overall structure of the 2-port network whose noise parameters are measured is an input and output terminated amplifier cascaded with the common network as shown in Figure 3.3, then in series with the amplifier connected to the measurement equipment as shown in Figure 3.4. Seeing from the source terminal of this single-ended amplifier, the terminated single-ended amplifier becomes an 1-port network with identical 4 entries in its noise correlation matrix.
Figure 3.2: 2-Port Upside Down.
Figure 3.3: Cascade AmplifierB with Common Network
Figure 3.4: Serial Connection of AmplifierA and Common Network
If assuming that Amplifier A is “turned upside down” and is terminated, its resultant
input and output referred noise voltages are found as
vn
vn
=
vna,T vnb,T
+ Za,T
1
−1
ia (3.6)
where vna,T refers to the input-referred noise voltage at the terminated input port of Amplifier B, vnb,T refers to the input-referred noise voltage at the terminated output port of Amplifier B, vnrefers to the open circuit noise voltage at Port 1, iarefers to the current flowing into the input port of terminated Amplifier A, Za,T refers to the 2-port Z-parameters of the terminated Amplifier A when turned upside down, derivation can be found in Chapter 2, Section2.2.
In the following analysis, vn,T =
vna,T vnb,T
is used. From (3.6), it can be shown that vn= 1
∆Za,T
Z′a,Tvn,TI2by1, (3.7)
where Z′a,T=
Za,T,22− Za,T,21 Za,T,11− Za,T,12
,∆Za,T=
1 −1
Za,T
1 −1
T
. Detailed calculation procedures can also be found in AppendixC.
The Z-representation of the noise correlation matrix can be found from CZa,T = vnvHnI2by2
=Z
′
a,Tvn,TvHn,TZ′Ha,T
|∆Za,T|2 I2by2
=Z
′
a,T(CZa,T)Z′Ha,T
|∆Za,T|2 I2by2
=Z
′
a,T(CZ,a+CT)Z′Ha,T
|∆Za,T|2 I2by2
(3.8)
where CZa,T represents the noise correlation matrix in Z-representation for the termi- nated 2-port network, CT =2kT
R{Zin} 0 0 R{Zout}
, I2by2is a 2× 2 all-ones matrix.
The noise correlation matrix of the common network in Z-representation is found in [14]
CZC= 2kT
R{ZC} R {ZC} R{ZC} R {ZC}
, (3.9)
where CZC represents the Z-representation of the noise correlation matrix of the com- mon network as shown in Figure3.3and ZC is the common-network impedance.
So the overall Z-representation form of noise correlation matrix can be derived as C′a= CZa,T|ZC|2
|Za,T+ ZC|2+ CZC|Za,T|2
|Za,T + ZC|2, (3.10)
where C′a is the overall noise correlation matrix in Z-representation for Amplifier A cascaded with the common network. Detailed derivation procedures can be found in Ap- pendixD.
C′b, which represents the unknown noise correlation matrix in Z-representation for Am- plifier B cascaded with the common network can also be found in similar way
C′b= CZb,T|ZC|2
|Zb,T+ ZC|2+ CZC|Zb,T|2
|Zb,T + ZC|2. (3.11)
Then the system of matrixes can be formed as below
C13−|Z |ZC|2
b,T+ZC|2
Z′b,TCTZ′Hb,T
|∆Zb,T|2 I2by2−|Z|Zb,T|2CC
b,T+ZC|2 = CZ,a+|Z |ZC|2
b,T+ZC|2
Z′b,TCZ,bZ′Hb,T
|∆Zb,T|2 I2by2 C24−|Z |ZC|2
a,T+ZC|2
Z′a,TCTZ′Ha,T
|∆Za,T|2 I2by2−|Z|Za,T|2CC
a,T+ZC|2 = CZ,b+|Z |ZC|2
a,T+ZC|2
Z′a,TCZ,aZ′Ha,T
|∆Za,T|2 I2by2 (3.12) where C13refers to the measured 2-port overall noise correlation matrix in Z-representation terminating Port 2 and Port 4, C24 refers to the measured 2-port overall noise correla- tion matrix in Z-representation with Port 1 and Port 3 terminated and similar to (3.7), Z′b,T=
Zb,T,22− Zb,T,21 Zb,T,11− Zb,T,12
,∆Zb,T=
1 −1
Zb,T
1 −1
T
.
In 8 equations described by the system in (3.12), there are 8 unknows, which are the terms in Amplifiers A and B noise correlation matrices. Once the matrix system is solved, these noise correlation matrices should be converted to their ABCD-representation in order to calculate their noise parameters [14].
