Download The Performance of Passive Lumped

ABSTRACT
HEO, SEUNG KYUN. The Performance of Passive Lumped Element. (Under the direction
of Dr. Michael B. Steer).
Lumped elements are the most widely used passive components in Radio Frequency
(RF) circuits. The characteristics vary over frequency, however, limiting their application.
This limitation stems from the physical dimensions of lumped elements, which are, in
general, 0603 (i.e. 3 mm by 1.5 mm), 0402 (i.e. 2 mm by 1 mm), and 0201 (i.e. 1 mm by 0.5
mm) in size in order to avoid the phase shift between the input and the output. By examining
the quality factor and self-resonant frequency, the performance of lumped elements can be
quantified. The quality factor contains the energy storing and dissipation terms, which vary
according to frequency. This quality factor can be obtained using the S-parameters of lumped
elements.
It is difficult to obtain accurate measurements of a non-coaxial packaged devices such
as a lumped inductor, because it cannot be measured using a wide band measurement system.
A Print Circuit Board (PCB) fixture is thus considered to connect a device under test to 50
ohm connectors, and the effects of the fixture are then removed by performing a standards
calibration. The combination of port extension and calibration using a manufacturer’s
calibration kit is employed to collect one-port S-parameters. The best result is achieved using
an SMA fixture that has a short line and good impedance matching.
This thesis presents an extraction method to derive the quality factor of lumped
elements, and the quality factor is obtained using one-port S-parameters.
The Performance of Passive Lumped Element
by
SeungKyun Heo
A thesis submitted to the Graduate Faculty of
North Carolina State University
in partial fulfillment of the
requirements for the degree of
Master of Science
Electrical Engineering
Raleigh, North Carolina
2010
APPROVED BY:
_______________________________
Professor Michael B. Steer
Committee Chair
________________________________
Professor David Schurig
______________________________
Professor Paul D. Franzon
DEDICATION
This dissertation is dedicated to my wife, Mi-Young Kim, and to my parents in Korea.
ii
BIOGRAPHY
Seung-Kyun Heo was born on 4th August, 1977 in Korea. He received the Bachelor
of Science degree in Electrical and Computer Engineering from Chung-Ang University
located in Seoul, Korea, in 2003. He began his master degree in the Electrical and Computer
Engineering Department at North Carolina State University in Raleigh, NC, in 2008. From
2003 to 2007, he worked as an RF engineer in Samsung Electronics Co., Suwon, Korea, and
from June 2009 to January 2010 he worked as an intern engineer in RFMD, Greensboro, NC.
His current interest is focused in the Field of RF circuit design.
iii
TABLE OF CONTENTS
LIST OF TABLES ................................................................................................................v
LIST OF FIGURES..............................................................................................................vi
CHAPTER 1 Introduction ...................................................................................................1
1.1 Motivation ...............................................................................................................1
1.2 Contribution.............................................................................................................2
1.3 Thesis Organization .................................................................................................3
CHAPTER 2 Basic Characteristic of Lumped Elements ......................................................4
2.1 Introduction .............................................................................................................4
2.2 Basic Parameters ......................................................................................................4
2.3 Quality Factor ..........................................................................................................8
2.3.1 Definition of Quality Factor ..........................................................................8
2.3.2 Effect of Quality Factor in an RF Circuit .....................................................10
2.3.3 Extraction of the Quality Factor...................................................................11
2.4 Self Resonant Frequency........................................................................................13
2.5 Summary ...............................................................................................................13
CHAPTER 1 Measurement Method .................................................................................. 17
3.1 Introduction ...........................................................................................................17
3.2 Calibration .............................................................................................................18
3.2.1 Errors in VNA Measurement .......................................................................18
3.2.2 Errors Characteristic by Fixture Measurements ...........................................19
3.2.3 SOLT Calibration ........................................................................................20
3.2.4 TRL Calibration ..........................................................................................22
3.2.5 Port Extension .............................................................................................23
3.2.6 Time Gating ................................................................................................23
3.3 Quality Factor from Two-Port S-parameters ...........................................................24
3.5 Summary ...............................................................................................................28
CHAPTER 4 Measurement Results ..................................................................................... 31
4.1 Introduction ...........................................................................................................31
4.2 One-Port S-Parameter Results ................................................................................32
4.3 Two-Port S-Parameter Results from Manufacturer .................................................34
4.4 Summary ...............................................................................................................36
CHAPTER 5 Conclusions ................................................................................................... 38
5.1 Conclusions ...........................................................................................................38
5.2 Future Work ..........................................................................................................39
APPENDIX......................................................................................................................... 40
Appendix A: Plot of Measurement Result ....................................................................41
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LIST OF TABLES
Table 2.1 Basic parameters of resistor ...................................................................................5
Table 2.2 Critical parameter by material [10][17] ..................................................................6
Table 2.3 Basic parameters of capacitors ...............................................................................7
Table 2.4 Basic parameters of inductors ................................................................................8
Table 3.1 Summary of quality factor calculation.................................................................. 28
Table 4.1 TOKO inductor one-port quality factor and SRF measurement result ................... 33
Table 4.2 Johanson capacitor one-port SRF measurement result .......................................... 33
Table 4.3 Panasonic resistor one-port measurement result ................................................... 34
Table 4.4 Johanson capacitor two-port SRF measurement result .......................................... 35
Table 4.5 TOKO inductor two-port Q and SRF measurement result .................................... 35
Table 4.6 Panasonic resistor two-port measurement result ................................................... 36
v
LIST OF FIGURES
Figure 1.1 RF block of LG300G ............................................................................................1
Figure 2.1 Relationship between I and V in a capacitor .........................................................5
Figure 2.2 Series RLC circuit and parallel RLC circuit ..........................................................8
Figure 2.3 Common source amplifier .................................................................................. 10
Figure 2.4 Resonant circuit method ..................................................................................... 11
Figure 2.5 Reflection coefficient ......................................................................................... 12
Figure 3.1 DUT with fixture ................................................................................................ 19
Figure 3.2 Through standard S11,S22.................................................................................. 20
Figure 3.3 User calibration kit ............................................................................................. 21
Figure 3.4 12-terms error model [2] ..................................................................................... 22
Figure 3.5 Port extension method ........................................................................................ 23
Figure 3.6 Time domain reflection response of a DUT with fixture ..................................... 24
Figure 3.7 ABCD parameter of series impedance ................................................................ 25
Figure 3.8 One-port VNA error model [5] [6] ...................................................................... 26
Figure 3.9 SMA connector fixture ....................................................................................... 28
Figure 4.1 Quality factor of Johanson capacitor [2] ............................................................. 31
Figure 4.2 SRF of Johanson capacitor [2] ............................................................................ 32
Figure A.1 Inductance and phase of impedance ................................................................... 42
Figure A.2 Quality factor of 2.2 nH inductor ....................................................................... 42
Figure A.3 Inductance and phase of impedance ................................................................... 43
Figure A.4 Quality factor of 3.3 nH inductor ....................................................................... 44
Figure A.5 Inductance and phase of impedance ................................................................... 44
Figure A.6 Quality factor of 6.8 nH inductor ....................................................................... 44
Figure A.7 Inductance and phase of impedance ................................................................... 45
Figure A.8 Quality factor of 10 nH inductor ........................................................................ 45
Figure A.9 Inductance and phase of impedance ................................................................... 46
Figure A.10 Quality factor of 33 nH inductor ...................................................................... 46
Figure A.11 Inductance and phase of impedance ................................................................. 47
Figure A.12 Quality factor of 56 nH inductor ...................................................................... 47
Figure A.13 Inductance and phase of impedance ................................................................. 48
Figure A.14 Quality factor of 100 nH inductor .................................................................... 48
Figure A.15 Inductance and phase of impedance ................................................................. 49
Figure A.16 Quality factor of 270 nH inductor .................................................................... 49
Figure A.17 Capacitance and phase of impedance ............................................................... 50
Figure A.18 Quality factor of 2.2 pF capacitor..................................................................... 50
Figure A.19 Capacitance and phase of impedance ............................................................... 51
Figure A.20 Quality factor of 3.3 pF capacitor..................................................................... 51
Figure A.21 Capacitance and phase of impedance ............................................................... 52
Figure A.22 Quality factor of 10 pF capacitor...................................................................... 52
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Figure A.23 Capacitance and phase of impedance ............................................................... 53
Figure A.24 Quality factor of 15 pF capacitor...................................................................... 53
Figure A.25 Capacitance and phase of impedance ............................................................... 54
Figure A.26 Quality factor of 33 pF capacitor...................................................................... 54
Figure A.27 Capacitance and phase of impedance ............................................................... 55
Figure A.28 Quality factor of 47 pF capacitor...................................................................... 55
Figure A.29 Capacitance and phase of impedance ............................................................... 56
Figure A.30 Quality factor of 82 pF capacitor...................................................................... 56
Figure A.31 Capacitance and phase of impedance ............................................................... 57
Figure A.32 Quality factor of 10 Ω resistor.......................................................................... 57
Figure A.33 Resistance, phase of impedance, and reactance ................................................ 58
Figure A.34 Quality factor of 39 Ω resistor.......................................................................... 58
Figure A.35 Resistance, phase of impedance, and reactance ................................................ 59
Figure A.36 Quality factor of 68 Ω resistor.......................................................................... 59
Figure A.37 Resistance, phase of impedance, and reactance ................................................ 60
Figure A.38 Quality factor of 200 Ω resistor........................................................................ 60
Figure A.39 Resistance, phase of impedance, and reactance ................................................ 61
Figure A.40 Quality factor of 300 Ω resistor........................................................................ 61
Figure A.41 Resistance, phase of impedance, and reactance ................................................ 62
Figure A.42 Quality factor of 560 Ω resistor........................................................................ 62
Figure A.43 Resistance, phase of impedance, and reactance ................................................ 63
Figure A.44 Quality factor of 1 kΩ resistor.......................................................................... 63
Figure A.45 Resistance, phase of impedance, and reactance ................................................ 64
Figure A.46 Quality factor of 2 kΩ resistor.......................................................................... 64
Figure A.47 Resistance, phase of impedance, and reactance ................................................ 65
Figure A.48 Quality factor of 3 kΩ resistor.......................................................................... 65
vii
CHAPTER 1 Introduction
1.1 Motivation
Lumped elements, such as inductors, capacitors, and resistors, are the most common
and important components in radio frequency (RF) and microwave circuits. To design RF
circuits correctly and efficiently, knowledge about the lumped elements is essential. Over the
last few decades suppliers have been able to meet this demand by overcoming RF circuit
performance limitations. Lumped elements, however, cannot always be applied in high
frequency applications because of variations in their characteristics. Figure 1.1 shows the RF
section of a mobile handset. Most of the parts, such as a switching module, power amplifier
module, low noise amplifier, mixer, and filters are merged into one package with the
exception of the lumped elements. Lumped elements none the less are still widely used in
biasing and matching networks because of their advantages including small size and low loss.
Figure 1.1 RF block of LG300G
Lumped elements are not the core part of an RF circuit. However, they constitute a
basic block of a big module. A robust system can be built by using high performance
elements that exhibit low variation over temperature, time, frequency, and in the biasing
environment. Characteristics that vary in frequency are often significant factors in RF
1
circuits. Identifying the quality factor and Self-Resonant Frequency (SRF) is useful in
defining and understanding the characteristics of lumped elements over frequency. The
quality factor contains the energy storing and dissipation terms, which are vary by frequency,
and this characteristic affects the performance of the circuit. If a high quality factor inductor
is used as the RF choke in a bias circuit, the gain of the amplifier can be improved. However,
a high quality factor does not always mean high performance. A high performance quality
factor varies according to the purpose of the circuit in which lumped elements are used. By
examining the characteristics of lumped elements over frequency, the lumped elements can
be better understood.
1.2 Contribution
Numerous articles about enhancing the performance of lumped elements have been
published. However, it is difficult to find recent articles that focus on characterizing the
lumped element itself. It is useful, therefore, to investigate the parameters of lumped
elements, especially their frequency-dependent characteristics. The starting point in RF
circuit design is understanding and knowing the limitation of lumped elements. This thesis
focuses on the characterization of lumped element parameters and establishing RF
measurement method. Finally, measurements are taken to obtain the quality factor and SRF
of a number of lumped element components. This thesis lays out the RF measurement
approach. Many data sheets of lumped element do not contain the quality factor over
frequency. By measuring major commercial lumped elements under the same testing
conditions, an understanding of the frequency-dependant characteristics of lumped elements
2
can be acquired. This thesis presents the performance of lumped elements over frequency
extended from S-parameter.
1.3 Thesis Organization
Chapter 2 reviews the basic parameters of lumped elements and the effects of
parasitics, and investigates the quality factor and SRF using S-parameters. Chapter 3 presents
a way to remove the error that is caused in using the vector network analyzer (VNA), which
uses a fixture, and method to obtain the quality factor using the S-parameters of lumped
elements. Chapter 4 summarizes the measurement results.
3
CHAPTER 2 Basic Characteristic of Lumped Elements
2.1 Introduction
The performance of lumped elements has been improved through invention and
applying numerous methods. Many manufacturers have tried to reduce the effects of
parasitics on the ideal inductance, capacitance, and resistance. Lumped elements have a
frequency limitation because of their dimensions and parasitics. Generally, the dimensions of
a lumped element should be smaller than λ/20 to guarantee phase equality at the input and
output ports. In spite of this problem, lumped elements are used widely because of their small
size, wide bandwidth, and low cost. Lumped elements have a large number of applications
such as impedance matching, DC blocking, RF chocking, bypass circuit, and degeneration in
RF circuit. The characteristic parameters of lumped elements must be known in order to use
lumped elements correctly and efficiently. In this chapter the basic parameters of lumped
elements are reviewed including quality factor (Q) and self resonant frequency (SRF).
2.2 Basic Parameters
Resistors are used in lossy impedance matching networks, attenuators, damping
resistors, and termination circuits. The most common resistor is a tick film resistor. When a
resistor is used as an RF component, the thermal noise from the resistor should be
considered. Because of this thermal noise, a resistor is rarely used in an RF matching
network. A resistance value can be calculated using sheet resistance with physical
dimensions. Table 2.1 shows the basic parameters of a resistor.
4
Table 2.1 Basic parameters of resistor.
Parameter
Description
Maximum power that guarantees the basic characteristic and
Power rating
reliability.
Maximum overload voltage
Maximum voltage that can applied for 5 seconds.
Resistance tolerance
Variation of the resistance value.
Temperature coefficient of
resistance (TCR)
Variation of the degree of resistance value over temperature.
Operating temperature range Temperature range that guarantees the performance.
Rated current
Continuous maximum current allowed.
A capacitor stores energy in an electric field. A typical unit range for a lumped
capacitor used in RF and microwave circuits is from picofarads to microfarads. The
impedance of an ideal capacitor can be expressed as
1
jωC
. The impedance of the capacitor
contains ‘-j’ terms, which refers to the current leading the voltage by 90 degrees. This basic
concept is useful when a lumped element is applied to an RF circuit to cancel out the effects
of reactance. By using this concept, the phase relationship between the current and the
voltage in the RF circuit can be predicted easily.
j
1
0.8
I
0.6
-1
1
Amplitude
V
0.4
0.2
0
-0.2
V
I
-0.4
-0.6
-0.8
-1
-j
0
2
4
6
8
Time
Figure 2.1 Relationship between I and V in a capacitor.
5
10
12
14
Parameters such as the dielectric constant, break-down voltage, temperature
coefficient, insulation resistance, and dissipation factor determine the basic performance of a
capacitor [14]. Table 2.2 shows the characteristics of a capacitor for several dielectric
materials, which serve as the dominant performance determining factor of a capacitor. For
example, power loss in a capacitor comes from leakage of current and dissipation of power in
the dielectric. Furthermore, the temperature variation characteristic is also affected by the
dielectric constant. The NP0 material has the lowest dielectric constant has an advantage in
terms of temperature variation and power loss. However, high capacitance values cannot be
obtained by using NP0 material. To obtain a high capacitance value, a high dielectric
constant material is needed, which results in large temperature variations and low efficiency.
The appropriate tolerance to guarantee the performance for capacitance in an RF matching
circuit is ±5%. Table 2.3 shows the basic parameters of a capacitor.
Table 2.2 Critical parameter by material [10] [17].
Material
Dielectric Constant
% Capacitance Change
Dissipation Factor
NP0
15-100
< 0.4% (-55 to 125C)
0.1%
X7R
2000-4000
+/-15% (-55 to 125C)
3.5%
+/-15%(-55 to 85C)
5%
Up to 82% (-30 to 85C)
9%
X5R
Y5V
>16000
6
Table 2.3 Basic parameters of capacitors.
Parameter
Description
Dielectric constant
Constant factor of material used for dielectrics.
Category temperature range
Operating temperature.
Rated voltage
Maximum voltage can be applied.
Rated current
Maximum current can be applied.
Capacitance tolerance
Variation of capacitance value.
Withstand voltage
Voltage that a capacitor can endure for a short time
(above rated voltage).
Dissipation factor
The ratio of loss resistance to capacitance.
Insulation resistance
DC resistor (related to leakage current).
An inductor is used as an RF choke in the bias circuit, and part of the matching
network. The current flowing through a line generates magnetic field, so that a single line has
inductance. For example, a typical bond wire has around 1 nH of inductance. The voltage
across an inductor leads the current by 90 degrees. Table 2.4 shows the basic characteristics
of an inductor.
Table 2.4 Basic parameters of inductors.
Parameter
Description
Maximum current rating
Maximum current can be applied.
Maximum power rating
Maximum power can be applied.
Operating temperature range
Temperature range that guarantees the performance.
Temperature coefficient of inductance Inductance variation over temperature.
Time constant
Voltage charge time in an inductor.
7
2.3 Quality Factor
2.3.1 Definition of Quality Factor
Most literature sources cite the definition of the quality factor as “The definition of
quality factor (Q) is 2π times the ratio of energy stored in a system to the energy dissipated
per cycle” [2] [3] [4].
QQ = 2π
Max Energy Stored
Total Energy Stored
=ω
Average Power Loss
Energy Loss per Cycle
(Equation 2-1)
This definition of a dimensionless quality factor can be applied to all energy stored systems.
AC
L
AC
V
i
C
G
L
C
R
Figure 2.2 Series RLC circuit and parallel RLC circuit.
The definition of quality factor can be applied to a simple series RLC:
1 2
Li
fL ωL XL
Q = 2π 2
= 2π =
=
1 2
R
R
R
Ri T
2
(Equation 2-2)
Similar to Equation 2-2, a parallel RLC circuit quality factor is
1 2
CV
fC ωC
1
R
Q = 2π 2
= 2π =
=
=
1
G
G
ωGL XC
2
2 GV T
8
(Equation 2-3)
The loss of lumped elements can be determined by observing the quality factor. In
reality, the loss of lumped elements varies by frequency, so it is not possible to calculate the
loss using a multimeter. The loss which varies with frequency is due to radiation and skin
effect. In general a high quality factor value means less loss and more energy stored. In a
reactive circuit, the ratio of the reactance in Ωs to the resistance in Ωs is defined as the
quality factor. The quality factor of an inductor can be represented using Equation 2-1 [5].
This general representation of a quality factor is sufficient for practical application.
Y11 Q = ω
� m | − |W
� e |)
(|W
Max Energy Stored
Im|Y11 |
≅ 2ω
= −
Power Loss
Re|Y11 |
Average Power Loss
(Equation 2-4)
This definition can be applied when the operating frequency is much lower than the selfresonant frequency. In a filter, the quality factor is usually taken as the ratio of the center
frequency to the bandwidth [1] and represents the sharpness of the response.
Q=
f0
∆f
(Equation 2-5)
Where f0 is resonant frequency and ∆f is 3dB bandwidth.
The quality factor of a lumped element is referred to as the unloaded quality factor.
The unloaded quality factor is the quality factor of a component or device that does not have
an external source and load. This concept considers only the ratio of the reactive part and
resistive part of the network itself. The loaded quality factor is affected by the choice of load
values and bandwidth. The unloaded quality factor of a lumped element is measured in this
thesis. In a resonant circuit, the quality factor is affected by load conditions rather than the
quality factor of the component itself.
9
2.3.2 Effect of Quality Factor in an RF Circuit
The quality factor is a frequency-defendant parameter and thereby affects RF circuits.
An inductor is used widely as an RF choke in amplifier bias circuits. Figure 2.3 shows a
simple common source Low Noise Amplifier (LNA) structure. Sufficient gain and input
impedance cannot be obtained for a LNA using a single common source amplifier; however,
it is sufficient for observing the effect of an inductor RF choke. The gain of an LNA is
1
1
1
|Av | = gm � 2
+
+ �
Q RD jωLd RL
-1
(Equation 2-6)
Where gm is the transconductance of M1.
The gain of the LNA is proportional to the quality factor of Ld . To maximize the gain of the
LNA, a high quality factor RF choke inductor should be used.
Ld
Rd
Ld
Vout
Vin
M1
D
G
Vout
+
RL
Vgs
gmVgs
-
S
Figure 2.3 Common source amplifier.
10
Ld RL
Rd
2.3.3 Extraction of the Quality Factor
The quality factor can be extracted by applying its definition to the circuit or
component. One of the oldest and most common methods used is a Q-meter measurement. A
Q-meter can measure the quality factor using a resonant circuit. This method is implemented
by applying the specified frequency current source to a DUT and a variable capacitor. Then
the maximum voltage across a variable capacitor, when its value is adjusted, is measured.
The reactance of the DUT is derived from the maximum voltage. Figure 2.5 shows this
resonant circuit method.
DUT
L(XD)
RD
V
C
Milivoltmeter
Figure 2.5 Resonant circuit method.
Determining the quality factor from the critical point method using the equivalent model and
impedance locus is also possible [15]. The quality factor is the ratio of the reactive energy
stored to the dissipated energy. In the case of an inductor, the reactive energy stored is the
difference between the magnetic energy and the electric energy. The quality factor is then
derived using this concept.
11
Lumped Inductor
I1
I2
+
+
V1
V2
-
-
ZL
ΓL
Γin
Figure 2.6 Reflection coefficient.
Im|Z|
This general definition of quality factor,𝑠𝑠Q= Re|Z| , is sufficient for lumped element
measurements since the operating frequency is below the self resonant frequency. From the
general definition of quality factor, the quality factor can be obtained in terms of impedance.
The impedance of a lumped element can be obtained by measuring the input impedance of
the lumped element while placing a perfect short at the output. In this case, 𝑆𝑆11 is the
reflection coefficient and can be converted to the input impedance. By using one port
measurement with a short at the output, the quality factor can be generated. In terms of twoport S-parameter, the input impedance of a network can be derived:
Γin = S11 +
Zin = 50
S12 S21 ΓL
S12 S21
= S11 −
1 − S22 ΓL
1 + S22
1 − Γin
= R + jX
1 + Γin
Z = Zin = R + jX
Q=
(Equation 2-7)
(Equation 2-8)
(Equation 2-9)
|X |
R
(Equation 2-10)
Where Load is a short (ΓL = −1).
If the load is a short, the input impedance is considered to be the impedance of the
network. Equations 2-7 to 2-10 show the derivation of the quality factor using S-parameter.
12
One of the advantages of using S-parameter is that it is easy to obtain data up to high
frequencies. For example, the Anritsu ME7828A VNA can measure up to 100 GHz. Also,
they can be converted to other parameters such as Z-parameter, Y-parameter, and ABCDparameter easily.
2.4 Self Resonant Frequency
A lumped element has parasitics because of the not insignificant physical dimensions.
If a lumped element has large dimensions, it has more parasitic characteristics and a lower
SRF. The effects of the parasitic capacitance and inductance increase as the operating
frequency increasing. The parasitic capacitance and inductance effects are dominant at
frequencies above the SRF. A lumped element must be used below the SRF to guarantee
nominal performance. The SRF can be determined by measuring the S-parameter. The point
at which the sign of the imaginary part of the impedance is changed is the SRF of the lumped
element. By measuring the S21 , the SRF also can be found.
2.5 Summary
The basic parameters of lumped elements and the key parameters of an RF circuit
were reviewed in this chapter. Because of their benefits, lumped elements are used widely.
However, parasitics, which stem from the physical dimensions of lumped elements, limit
their operating frequency. These parasitics introduce loss and affect the characteristics of the
lumped elements over frequency. To characterize their effects the quality factor and self
resonant frequency are used. The definition of quality factor represents the ratio of power
13
storage to power loss in a network, and the SRF indicates the frequency limitations of the
lumped element. The performance of the RF circuit, such as gain and frequency selectivity,
varies according to the quality factor of lumped elements. It was shown that the quality factor
can be generated using measured S-parameters.
14
REFERENCES
[1] I. J. Bahl, Lumped elements for RF and microwave circuits, Artech House, 2003.
[2] M. Sucher, Handbook of microwave and measurements, Polytechnic Press of the
Polytechnic Institute of Brooklyn, 1963.
[3] M. B. Steer, Microwave and RF Design: A Systems Approach, SciTech Publishing, 2010.
[4] D. M. Pozar, Microwave Engineering, 3rd ed. John Wiley & Sons, Inc., 2005.
[5] K. O, “Estimation methods for quality factors of inductors fabricated in silicon integrated
circuit process technologies,” IEEE J. Solid-State Circuits, vol. 33, pp. 1249–1252, Aug.
1998.
[6] H. Jiang, Y. Wang, J. A. Yeh, and N. C. Tien, “On-chip spiral inductors suspended over
deep copper-lined cavities,” IEEE Trans. Microwave Theory Tech., vol. 48, pp. 2415–2423,
Dec. 2000.
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47, pp. 699–713, Aug. 2000.
[8] T. S. Horng, K. C. Peng, J. K. Jau, and Y. S. Tsai, “S-parameter formulation of quality
factor for a spiral inductor in generalized two-port configuration”, IEEE Trans. Microwave
Theory Tech., vol. 51, pp. 2197–2202, Nov. 2003.
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15
[10] Johanson Dielectrics, “Basics of Ceramic Chip Capacitors”,
http://www.johansondielectrics.com/technical-notes/product-training/basics-of-ceramic-chipcapacitors.html.
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16
CHAPTER 1 Measurement Method
3.1 Introduction
The most important part of taking RF circuit measurements is setting up the correct
test environment that can be calibrated to remove errors. This is a time-consuming aspect of
the RF measurement process. A Vector Network Analyzer (VNA) is the most suitable
instrument to use for RF measurements when a passive element is to be measured over
frequency. To measure the S-parameter of a lumped element, the interface between the
lumped element and the VNA must be provided. A suitable fixture must be found to provide
a connection to a non-coaxial device such as a lumped element. An ideal fixture is a material
having no loss, no electrical length, no reflection, and a wide frequency response. A fixture
should guarantee the bandwidth and offer low loss to provide accurate test results. In this
thesis, Printed Circuit Boards (PCB) and SMA connector fixtures are used.
To accurately measure the S-parameters, the effects of the fixture should be removed
by performing calibration fixture standards. The added fixture causes measurement problems
such as insertion loss, phase shift, and mismatch. For example, the transmission lines in a
PCB fixture and in a connector generate insertion loss and phase shift. The mismatch
problem occurs when the connections between the networks have different characteristic
impedance. Fixture error problems can be removed by performing a full calibration. Using a
Through-Reflection-Load (TRL) calibration, better source match and load match corrections
can be achieved than via a Short-Open-Load-through (SOLT) calibration because of the
capacitance variation found in an open standard. However, the SOLT calibration is applied
17
here instead of the TRL calibration because of the absence of TRL calibration standards; the
line standard cannot be implemented in the test fixture.
3.2 Calibration
3.2.1 Errors in VNA Measurement
Three major errors occur in VNA measurements [1], [5], [6]. Drift error occurs when
a test environment is changed. One such environmental variation factor is a change in
temperature. To reduce this effect, most of the laboratory environment contains equipment
that can maintain a stable temperature and humidity. Calibrations should be executed
regularly to minimize the effect of test environment variations. The second major source of
errors is random error. Errors from noise cannot be predicted and removed. A VNA is an
instrument that itself is a noise source, so VNA measurements are affected by such noise.
These errors can be reduced not by calibration but by increasing the number of times
measurements are taken. Third, systematic errors, such as mismatch, insertion loss, and phase
shift, are the dominant types of error in VNA measurement. The root causes of these
systematic errors are signal leakage and unwanted reflection. Such systematic errors could be
removed by using their repeatable characteristics. Systematic errors can be determined during
the calibration procedure. In this thesis, of the three types of common error, only systematic
errors are considered.
18
Figure 3.1 DUT with fixture.
3.2.2 Errors Characteristic by Fixture Measurements
The characteristic impedance of an instrument that is connecting to a lumped element
causes mismatch. If the reflection performance is measured, the reflection should be only
from the Device Under Test (DUT). However, a reflection from connector or trace line could
also occur, which causes measurement error. This error can be removed using a time gating
method. In this thesis, however, the time gating method cannot be applied, because a VNA
does not have a time gating function option. Figure 3.2 shows the input and output reflections
of the through standards used for two-port S-parameter measurements. A mismatch ripple is
evident at around 3.5 GHz. By getting rid of the gap that causes this mismatch reflection
between the SMA connector and PCB board fixture, the mismatch ripple is mitigated
slightly. However, the reduction in the ripple is not enough to obtain an accurate
measurement. This error can be removed using a time gating method that can be applied in a
window of the time domain.
The transmission lines in the fixture and connector generate insertion loss and phase
shift. Insertion loss and phase shift would be small if a short transmission line is used in a
fixture. To avoid these errors a fixture should maintain zero electrical length. However, it is
19
not possible to maintain a zero electrical length in real measurement. The trace line should be
short to minimize the phase shift if an accurate calibration is not possible.
S11
0
1000
2000
3000
4000
5000
-50
-100
Thru
S22
0
0
6000
dB
dB
0
Freq(MHz)
2000
3000
4000
5000
6000
-50
-100
Modified Thru
1000
Freq(MHz)
Thru
Modified Thru
Figure 3.2 Through standard S11 , S22.
3.2.3 SOLT Calibration
The SOLT calibration method uses a 12-term error model, which is calculated by
measuring calibrations standards. To obtain the 12 unknown parameters, 12 measurements
are needed. In order to perform SOLT calibration for an extended reference plane, work for
this thesis required specialized fixture calibration standards unique to this study. Figure 3.3
shows the through board of the RF3025 from RF Micro Devices that is employed as the user
standard for the SOLT calibrations performed in this thesis. However, these user standards
could not predict the correct error terms that cause measurement error. The through should be
zero length. However, because the zero length could not be implemented, the through
standard has added length, and the short standard does not provide an ideal short. The load
must guarantee the broadband response. At the DUT reference point, two 100 Ω resistors
were used to reduce inductance. However, the load standard contains error at high frequency
because of the frequency variations of the resistor. These errors led to inaccurate calibration.
20
Two-port calibration is also possible by using only Open-Short-Load (OSL) standards
calibration [3].
Each standard has its own characteristics. The short standard is a unity reflection with
a 180° phase shift. The short standard contains inductance and should be specified. A short
line interface is needed to reduce the inductance. The open standard is a unity reflection with
no phase shift. However, fringing capacitance generates the phase shift. This fringing
capacitance should be determined in order to insert the calibration standard data. The
characteristics of the fringing capacitance of the open standard can be generated by using a
suitably calibrated instrument. The characteristics of all the standards should be known in
order to execute the correct calibration. However, fringing capacitance varies by frequency,
so an SOLT calibration is useful only below 3 GHz.
Open
Short
Load
DUT
Through
Figure 3.3 User calibration kit.
21
3.2.4 TRL Calibration
The TRL calibration method uses a 12-terms error model. By measuring through,
reflection, and line standard the error terms can be determined. The through standard is a
0
direct connection between Port 1 and Port 2 (S= �
1
1
�). the reflection standard is a short or
0
open response (S11 =S22 =Γshort or Γopen ). The line is a short line inserted between Port 1 and
port 2 ( S= � 0
e-βl
e-βl �).
0
Figure 3.4 12-terms error model, after [2].
If the fixture is symmetric, the through-line (TL) calibration can be applied instead of
the TRL calibration. Short or open standard measurements can be obtained from through
measurements [4]. The advantage of TL calibration is that the random effect, which causes
glitches in measurements, of the arbitrary reflection standard can be removed. The reflection
22
of the short standard is derived from the S-parameter measurement of the through standard. A
detailed derivation is presented in [4].
Γshort = S11 Through − S21Through
(Equation 3-1)
3.2.5 Port Extension
The port extension method is useful when calibration standards are difficult to
implement. This method cannot correct mismatch and losses. So, it is useful when a short
well-matched fixture is available. First, the coaxial reference plane should be a known plane
by performing a full 2-port calibration using manufacturer calibration standards. Phase shift
and added delay are corrected by doing the port extension of a short or open standard.
DUT
2-Port port extension
2-Port calibration
using manufacture Cal
kit
Figure 3.5 Port extension method.
3.2.6 Time Gating
The frequency domain can be converted to the time domain by using the inverseFourier transform. If the through standard is measure using the VNA’s time domain setting,
23
more than one reflection can be observed. One reflection occurs from the DUT and the other
occurs from mismatch. These two reflections are illustrated in Figure 3.6. The first one is
from an SMA-transmission line connection, and the second one is from a DUT. So, only the
second response is the expected measurement. By capturing only the desired reflection, the
reflection from the mismatch error can be removed.
Figure 3.6 Time domain reflection response of a DUT with fixture.
3.3 Quality Factor from Two-Port S-parameters
Network parameters, such as the S-parameter, Z-parameter, Y-parameter, and ABCDparameters represent the characteristics of passive elements. If the two-port S-parameter of a
lumped element is measured, the quality factor of the lumped element can be generated. The
first step of the quality factor calculation is finding the impedance. From [2]
Yij =
Ii
(Vk = 0 for k ≠ j)
Vj
(Equation 3-2)
“Yij can be determined by driving Port j with voltage Vj , short-circuiting all other
ports (so Vk =0 for k≠j), and measuring the short-circuit current at Port i” [2]. From this
definition Y11 is the admittance with an ideal short load at the output port; 𝑌𝑌11 is the
24
admittance of the input port while the output port is shorted to ground. As mentioned in
Chapter 2, S-parameters can be converted to Z-parameters, Y-parameters, and ABCDparameters. By converting the S-parameters to Y-parameters, the quality factor also can be
generated.
Y11 =
(1 − S11 )(1 + S22 ) + S12 S21
(1 + S11 )(1 + S22 ) − S12 S21
𝑄𝑄Q Q = −
Y11 B = Z0
Y11 Q=
(Equation 3-3)
Im|Y11 |
Re|Y11 |
(Equation 3-4)
(1 + S11 )(1 + S22 ) − S12 S21
2S21
Im|B|
Re|B|
(Equation 3-5)
(Equation 3-6)
The impedance of the network can be derived from the ABCD parameter. Series impedance z
is the B of the ABCD parameter according to an ABCD parameter definition [7], see Figure
3.7.
z
1
2
𝐴𝐴 = 1 𝐵𝐵 = 𝑧𝑧
�
�
𝐶𝐶 = 0 𝐷𝐷 = 1
Figure 3.7 ABCD parameters of a series impedance.
25
3.4 Quality Factor from One-Port S-Parameter
A one-port measurement is required if the characteristic of the DUT is determined
using only reflection. The error in one–port measurements can be represented using a threeterm error, see Figure 3.8. These error terms can be determined by examining the SOL
standards.
𝐸𝐸𝑅𝑅𝑅𝑅
𝑆𝑆11𝑀𝑀
𝐸𝐸𝐷𝐷𝐷𝐷
𝐸𝐸𝑆𝑆𝑆𝑆
1
𝑆𝑆11
Reference
Figure 3.8 One-port VNA error model [5] [6].
By using a signal flow graph applied to the network in Figure 3.8:
S11M = EDF +
S11 =
S11 ERF
1-S11 ESF
S11M − EDF
ESF (S11M − EDF ) + ERF
(Equation 3-7)
(Equation 3-8)
The three unknown term, EDF , ESF , and ERF , can be derived by measuring three calibration
standards, and measured value, S11M , convert to the actual value, S11 , using Equation 3-8.
26
In the case of an inductor, the stored reactive energy is the difference between
magnetic energy and electric energy. The quality factor is derived using this concept when
the load is an ideal short. The following calculation is derived from [6]:
Peak Magnetic Energy-Peak Electric Energy
Energy Loss per One Cycle
Q = 2π
= 2π
=
�Wm,avg − We,avg �
�2×Wm,avg − 2×We,avg �
=2ω
Pl,avg ×T
Pl,avg
Im{Zin }
Re{Zin }
where Zin=Z0
1 + S11
1 − S11 )�
=
1+S
Re(50× 1 − S11 )
11
�Im(50×
(1 + Γin )
(1 − Γin )
(Equation 3-9)
So, the quality factor can be derived from a one-port S-parameter using an ideal short. To
obtain the one-port S-parameter of a lumped element, the manufacturer calibration standards
and port extension method are implemented in this thesis. The port extension is not a perfect
calibration method, so the direct SMA connector fixture is applied to minimize measurement
errors. As shown in Figure 3.9, the SMA connector fixture has a very short length and direct
connection to the DUT.
27
Figure 3.9 SMA connector fixture.
3.5 Summary
The test environment is the most important factor in obtaining accurate RF
measurements. Unknown terms and variations in the instruments should be removed. The
appropriate instrument to employ is the VNA to measure the characteristics of lumped
elements over frequency. Possible errors in VNA measurements include drift error, random
error, and systematic error. Several calibration methods, such as SOLT calibration, TRL
calibration, port extension, and time gating, are considered for making the VNA and fixtures
known terms. From the S-parameter, the quality factor can be determined. Table 3.1 presents
a summary of the quality factor calculation methods.
Table 3.1 Summary of quality factor calculation.
One-Port S-parameter
Using term
Q factor
Two-Port S-parameter (with virtual short load)
Zin
Zin
Im|Zin |
Re|Zin |
Im|Zin |
Re|Zin |
28
Y11
−
Im|Y11 |
Re|Y11 |
B
Im|B|
Re|B|
The best approach to obtaining a one-port S-parameter with the limited source
available is the combination of the port extension and manufacturer calibration kit. To
minimize measurement errors, a direct SMA connector fixture is applied. It has a very short
length and direct connection to the DUT.
29
REFERENCES
[1] Agilent, “In-fixture Measurements Using Vector Network Analyzers,” Application Note
AN 1287-9, 2006.
[2] Agilent, “Agilent Network Analysis Applying the 8510 TRL Calibration for NonCoaxial Measurement,” Product Note 8510-8AZ, 2006.
[3] Z. Y. Chen, Y. L. Wang, Y. Liu, and N. H. Zhu, “Two-port Calibration of test fixtures
with OSL method,” in International Conference on Microwave and Millimeter Wave
Technology Proceedings, pp. 138–141, 2002.
[4] M. B. Steer, S. B. Goldberg, G. Rinne, P. D. Franzon, I. Turlik, and J. S. Kasten,
“Introducing Through-Line Deembedding Procedure,” in 1992 IEEE MTT-S International
Microwave Symposium Digest, vol.3 pp. 1455–1458, 1992.
[5] L. F. Chen, Micorwave Electronics: Measurement and Materials Characterization,
Chichester: John Wiley & Sons, Ltd, 2004.
[6] J. M. Golio, The RF and Microwave Handbook, 2nd ed. CRC Press, 2008.
[7] M. B. Steer, Microwave and RF Design: A Systems Approach, SciTech Publishing, 2010.
30
CHAPTER 4 Measurement Results
4.1 Introduction
An Anritsu MS4623 VNA was used in this study to collect S-parameters up to 5 GHz.
To obtain two-port S-parameters, a user fixture, which provides the interface between the
lumped element and the VNA, was employed. Correct measurement result, however, could
not be achieved because of the inaccuracy of the user fixture calibration standards. The
proposed alternative method used was to obtain a one-port measurement, which combines
port extension and calibration using manufacturer’s calibration kit. To improve the
measurement accuracy, a SMA fixture, which has a short line and good impedance matching,
was used. A lumped element was soldered directly to the SMA connector. The quality factor
and SRF could then be obtained using one-port S-parameters and two-port S-parameters from
the manufacturer.
Figure 4.1 Quality factor of Johanson capacitor [2].
31
Figure 4.2 SRF of Johanson capacitor [2].
The part numbers for the lumped elements are TOKO LL1608FSL for the inductor,
Johanson 251R15S for the capacitor, and Panasonic ERJ-3GEYJ100 for the resistor. The size
of each lumped element is 1.6 × 0.8 mm.
4.2 One-Port S-Parameter Results
From one-port parameters, which were collected using a short load, the impedance
values of lumped elements are obtained, and the quality factor and SRF of the lumped
elements are obtained from their impedance value. Table 4.1 shows a summary of quality
factors obtained from one-port S-parameter measurements. Detailed measurement plots are
presented in Appendix A. Each capacitance, inductance, resistance, SRF, and quality factor
of the lumped elements is plotted and presented in Appendix A.
32
Table 4.1 TOKO Inductor one-port quality factor and SRF measurement results.
0603 Toko Inductor
SRF(MHz)
Data sheet
Data sheet
9100±15%
above
5000
5800±15%
above
5000
4700±15%
Measurement
2.2nH
13
300
MHz
26.5
6.469
32.765
29.732
48.698
53.038
42.082
13.8
27.8
36
46.4
52.9
68
7.271
32.494
28.032
43.966
40.799
41.539
15.8
29.4
39
49.3
55.8
67.5
4970
10.634
30.026
37.295
42.685
44.776
47.182
Data sheet
4000±15%
17.3
29.3
38.6
48.1
53.8
56.7
Measurement
4060
13.436
35.263
40.47
43.162
44.069
41.005
Data sheet
2000±15%
19.6
33.2
42.2
48
47.9
15.9
Measurement
2016
18.47
30.42
38.074
33.397
29.951
13.437
Data sheet
1500±15%
21.5
31.7
38.6
38.6
33.5
Measurement
1455
19.532
25.67
32.146
26.901
20.204
Data sheet
900±15%
20.5
35.5
39.1
20.4
Measurement
910
19.6
20.549
20.739
4.454
Data sheet
470±15%
21.9
21.1
Measurement
430
19.476
10.728
Measurement
Data sheet
3.3nH
6.8nH
10nH
33nH
56nH
100nH
270nH
Quality Factor
500
800
MHz
MHz
35.4
44.2
Measurement
100 MHz
1000
MHz
51.9
1800
MHz
63.9
Table 4.2 Johanson Capacitor one-port SRF measurement results.
SRF(MHz)
0603 Johanson Capacitor
2.2pF
3.3pF
10pF
15pF
33pF
47pF
82pF
4500
3500
2000
1600
1100
900
700
The measurement data presented in Table 4.2 are the measurement results obtained
from the Johanson 251R15S capacitors. These results do not include the quality factor results
because of instrument resolution problems. The instrumentation uncertainty of the MS4623 is
±0.15dB [1]. In the case of a high quality factor, that is, above 1000, the results can be
33
inaccurate due to this instrumentation uncertainty. For example, the quality factor of the
Johanson 3.3pF capacitor is about 3000 at 300 MHz, as shown in Figure 4.1. The Equivalent
Series Resistance (ESR) value is very low; a typical value of ESR is from 0.05 to 0.15. So,
the instrumentation uncertainty affects the measurement results, and the measurement results
of capacitor include the instrument error. The ripples caused by instrument error are shown in
the capacitor result plots presented in Appendix A.
Table 4.3 Panasonic resistor one-port measurement results
Quality Factor (at 1 GHz)
0603
Panasonic
Resistor
10 Ω 39 Ω
0.7
0.15
68 Ω
200 Ω 300 Ω
0.075
0.025
0.06
560 Ω
0.15
1000 Ω 2000 Ω 3000 Ω
0.3
0.6
1
Table 4.3 shows the quality factor value of the resistors at 1 GHz. Resistors have very
low quality factor values compare those of capacitors and inductors. Inductance is dominant
from 10 Ω to 68 Ω, and capacitance is dominant from 200 Ω to 3000 Ω as parasitics.
Capacitance parasitics values are 0.1 to 0.3 pF, and inductance parasitics values are around 1
nH. Most of the resistor manufacturers do not provide the quality factor values of their
resistors, so the results cannot be compared with the datasheet.
4.3 Two-Port S-Parameter Results from Manufacturer
To obtain the impedance value of lumped elements, two-port S-parameters are
converted to one-port S-parameters using a virtual short to ground plane. Errors that stem
from the calibration standards, however, affect the results, especially at high frequencies. So,
34
instead of the measurement data, S-parameters provided by the manufacturer are used to
compare the one-port and two-port measurement data. Table 4.4, Table 4.5, and Table 4.6
provide a summary of the inductor results obtained from the calculations.
Table 4.4 Johanson capacitor two-port SRF measurement results.
SRF(MHz)
0603 Johanson Capacitor
2.2pF
3.3pF
10pF
15pF
33pF
47pF
82pF
4000
3600
2000
1600
1200
1100
800
Table 4.5 TOKO inductor two-port Q and SRF measurement results.
Quality Factor
500
800
MHz
MHz
35.4
44.2
0603 Toko Inductor
SRF(MHz)
Data sheet
13
9.493
15.992
22.59
Data sheet
9100±15%
above
6000
5800±15%
300
MHz
26.5
13.8
27.8
Calculation
5290
11.198
Data sheet
4700±15%
Calculation
2.2nH
3.3nH
6.8nH
10nH
33nH
56nH
100nH
270nH
1000
MHz
51.9
1800
MHz
63.9
29.418
31.493
39.069
36
46.4
52.9
68
17.999
24.392
32.699
36.625
43.362
15.8
29.4
39
49.3
55.8
67.5
3850
11.975
21.619
30.031
38.985
42.513
40.708
Data sheet
4000±15%
17.3
29.3
38.6
48.1
53.8
56.7
Calculation
2980
13.821
23.737
32.08
40.254
42.662
29.637
Data sheet
2000±15%
19.6
33.2
42.2
48
47.9
15.9
Calculation
1570
16.903
30.154
36.574
34.386
25.237
Data sheet
1500±15%
21.5
31.7
38.6
38.6
33.5
Calculation
1160
18.023
29.632
32.251
19.843
8.01
Data sheet
900±15%
20.5
35.5
39.1
20.4
Calculation
820
20.875
31.083
26.546
1.825
Data sheet
470±15%
21.9
21.1
Calculation
400
20.363
12.4
Calculation
100 MHz
35
Table 4.6 Panasonic resistor two-port measurement results.
0603 Panasonic
Resistor
10 Ω
39 Ω
0.7
0.15
Quality Factor (at 1 GHz)
68 Ω
560 Ω
1000 Ω
0.075
0.1
0.2
3000 Ω
0.6
4.4 Summary
This chapter presents the quality factors and SRFs of lumped elements using Sparameters. Several methods were applied to obtain the correct S-parameters of the lumped
elements. The two-port measurements, which were taken first, are not appropriate for
deriving the quality factor because of calibration standards errors. To overcome the error of a
fixture, a port extension and calibration using a manufacturer’s calibration kit were applied
for a one-port measurement. In the case of inductors, the measurement results match the
values in the data sheet very well. However, the instrumentation uncertainty affects the
results of capacitors that have low ESR values; the result, ripples are shown in the quality
factor plot.
36
REFERENCES
[1] Anritsu, “MS4622 A/B/C/D, MS4623, MS4624 A/B/C/D Vector Network Measurement
Systems,” 2006. Available at
http://www.us.anritsu.com/Downloads/Datasheet-Configuration%20Guide/MS462xASeriesMS462xCSeries-ME7840A-ME7842B-ME7840/4-MS462xB/DSeriesVariousWirelessBaseStationTransmitFrequencies_downloadmaster.aspx?fileName=1141000288.pdf&fileID=1270&fileType=12
[2] Johanson, “Capacitor Catalog,” Available at
http://www.johansontechnology.com/images/stories/catalog/JTI_MLCC_HighQ_201004.pdf.
[3] Anritsu, “Inductor catalog,” Available at
http://www.toko.co.jp/products/pdf/inductors/ll1608-fsl.pdf.
37
CHAPTER 5 Conclusions
5.1 Conclusions
To observe the characteristics of lumped elements in terms of frequency change, the
quality factor, which represents the ratio of power storage to power loss in a network, is the
appropriate parameter. In this thesis, the quality factors of the lumped elements are derived
from one-port S-parameters. One-port S-parameters, which are useful in RF circuit analysis,
can be obtained over frequency using a VNA. A test fixture should be employed to provide
the interface between the DUT and the VNA, and the effects of the test fixture should be
removed using standard calibration. To minimize the errors caused by a port extension
method, the SMA connector is used which has a very short length and direct connection to
the DUT, is applied. Good measurement results for the inductors are achieved compared to
the resultant data in the data sheet. However, the results for the capacitors, which have low
ESR values show ripples in the quality factor plot. This error comes from instrumentation
uncertainty.
The most challenging part of this research was collecting accurate measurement data,
because the experiments were executed within a limited test environment. If TRL standards
could be implemented, then more accurate results could be achieved at high frequencies. The
quality factor and SRF of commercial lumped RF microwave elements are presented in this
thesis
38
5.2 Future Work
Two-port VNA measurement could not provide accurate data for this research due to
imperfect calibration standards. If correct user fixture calibration standards are employed,
correct two-port S-parameters of lumped elements can be obtained, and a comparison of the
calibration methods can then be exhibited. To apply the correct user fixture calibration
standards, new PCB fixtures or commercial fixture are needed. Experimental results show the
discrepancies among the data sheet data, especially in the case of the capacitor. Another
method should be considered. For instance, a resonant circuit method, which is based on the
resonant system with a variable impedance load, can be employed as the solution to measure
the high quality factors.
39
APPENDIX
40
Appendix A: Plot of Measurement Result
Inductance, capacitance, resistance, phase of impedance, and Q factor was plotted
using S-parameters of lumped elements. Each lumped element has two figures. Reactance
and phase of impedance were plotted in the upper figure and Q factor was plotted in the
lower figure.
41
1. Inductor
10
100
9
90
8
80
7
70
6
60
5
50
4
40
3
30
2
Inductance
Degree
1
0
Degree
Inductance (nH)
0603 Toko 2.2 nH Inductor (LL1608FSL)
20
10
0
0
1000
2000
3000
4000
5000
6000
Freq (MHz)
Figure A.1 Inductance and phase of impedance.
0603 Toko 2.2 nH Inductor (LL1608FSL)
100
90
1-Port S-Parameter
80
2-Port S-Parameter from manufacturer
Q Factor
70
60
50
40
30
20
10
0
10
100
1000
Freq (MHz)
Figure A.2 Quality factor of 2.2 nH inductor.
42
10000
10
100
9
90
8
80
7
70
6
60
5
50
4
40
3
30
2
Inductance
Degree
1
Degree
Inductance (nH)
0603 Toko 3.3 nH Inductor (LL1608FSL)
20
10
0
0
0
1000
2000
3000
4000
5000
6000
Freq (MHz)
Figure A.3 Inductance and phase of impedance.
0603 Toko 3.3 nH Inductor (LL1608FSL)
100
1-Port S-Parameter
2-Port S-Parameter from manufacturer
90
80
Q Factor
70
60
50
40
30
20
10
0
10
100
1000
Freq (MHz)
Figure A.4 Quality factor of 3.3 nH inductor.
43
10000
20
100
18
90
16
80
14
70
12
60
10
50
8
40
6
30
4
20
Inductance
Degree
2
0
0
1000
Degree
Inductance (nH)
0603 Toko 6.8 nH Inductor (LL1608FSL)
10
0
2000
3000
4000
5000
6000
Freq (MHz)
Figure A.5 Inductance and phase of impedance.
0603 Toko 6.8 nH Inductor (LL1608FSL)
100
1-Port S-Parameter
2-Port S-Parameter from manufacturer
90
80
Q Factor
70
60
50
40
30
20
10
0
10
100
1000
Freq (MHz)
Figure A.6 Quality factor of 6.8 nH inductor.
44
10000
0603 Toko 10 nH Inductor (LL1608FSL)
100
30
80
60
40
20
20
Degree
Inductance (nH)
25
0
15
-20
10
-40
-60
Inductance
Degree
5
-80
-100
0
0
1000
2000
3000
4000
5000
6000
Freq (MHz)
Figure A.7 Inductance and phase of impedance.
0603 Toko 10 nH Inductor (LL1608FSL)
100
1-Port S-Parameter
2-Port S-Parameter from manufacturer
90
80
Q Factor
70
60
50
40
30
20
10
0
10
100
1000
Freq (MHz)
Figure A.8 Quality factor of 10 nH inductor.
45
10000
100
100
90
80
80
60
70
40
60
20
50
0
40
-20
30
Inductance
Degree
20
10
Degree
Inductance (nH)
0603 Toko 33 nH Inductor (LL1608FSL)
-40
-60
-80
-100
0
0
1000
2000
3000
4000
5000
6000
Freq (MHz)
Figure A.9 Inductance and phase of impedance
0603 Toko 33 nH Inductor (LL1608FSL)
100
1-Port S-Parameter
2-Port S-Parameter from manufacturer
90
80
Q Factor
70
60
50
40
30
20
10
0
10
100
1000
Freq (MHz)
Figure A.10 Quality factor of 33 nH inductor.
46
10000
0603 Toko 56 nH Inductor (LL1608FSL)
150
100
130
80
60
40
90
20
70
0
50
-20
Inductance
Degree
30
10
Degree
Inductance (nH)
110
-40
-60
-80
-10 0
1000
2000
3000
4000
5000
6000-100
Freq (MHz)
Figure A.11 Inductance and phase of impedance.
0603 Toko 56 nH Inductor (LL1608FSL)
100
1-Port S-Parameter
2-Port S-Parameter from manufacturer
90
80
Q Factor
70
60
50
40
30
20
10
0
10
100
1000
Freq (MHz)
Figure A.12 Quality factor of 56 nH inductor.
47
10000
0603 Toko 100 nH Inductor (LL1608FSL)
100
300
Inductance
Degree
80
60
40
200
20
Degree
Inductance (nH)
250
0
150
-20
100
-40
-60
50
-80
-100
0
0
1000
2000
3000
4000
5000
6000
Freq (MHz)
Figure A.13 Inductance and phase of impedance.
0603 Toko 100 nH Inductor (LL1608FSL)
100
1-Port S-Parameter
2-Port S-Parameter from manufacturer
90
80
Q Factor
70
60
50
40
30
20
10
0
10
100
1000
Freq (MHz)
Figure A.14 Quality factor of 100 nH inductor.
48
10000
0603 Toko 270 nH Inductor (LL1608FSL)
100
500
Inductance
Degree
450
60
350
40
300
20
250
0
200
-20
150
-40
100
-60
50
-80
0
-100
0
1000
2000
3000
4000
5000
Degree
Inductance (nH)
400
80
6000
Freq (MHz)
Figure A.15 Inductance and phase of impedance.
0603 Toko 270 nH Inductor (LL1608FSL)
100
1-Port S-Parameter
2-Port S-Parameter from manufacturer
90
80
Q Factor
70
60
50
40
30
20
10
0
10
100
1000
Freq (MHz)
Figure A.16 Quality factor of 270 nH inductor.
49
10000
2. Capacitor
0603 Johanson 2.2 pF Capacitor (251R15S)
9
8
Capacitance (pF)
100
Capacitance
Degree
80
60
7
40
6
20
5
0
4
-20
3
-40
2
-60
1
-80
0
-100
0
1000
2000
3000
4000
5000
Degree
10
6000
Freq (MHz)
Figure A.17 Capacitance and phase of impedance.
0603 Johanson 2.2 pF Capacitor (251R15S)
10000
1-Port S-Parameter
2-Port S-Parameter from manufacturer
Q Factor
1000
100
10
1
0
1000
2000
3000
4000
Freq (MHz)
Figure A.18 Quality factor of 2.2 pF capacitor.
50
5000
6000
20
100
18
80
16
60
14
40
12
20
10
0
8
-20
6
-40
4
-60
Capacitance
Degree
2
Degree
Capacitance (pF)
0603 Johanson 3.3 pF Capacitor (251R15S)
-80
-100
0
0
1000
2000
3000
4000
5000
6000
Freq (MHz)
Figure A.19 Capacitance and phase of impedance.
0603 Johanson 3.3 pF Capacitor (251R15S)
10000
1-Port S-Parameter
2-Port S-Parameter from manufacturer
Q Factor
1000
100
10
1
0
1000
2000
3000
4000
Freq (MHz)
Figure A.20 Quality factor of 3.3 pF capacitor.
51
5000
6000
50
100
45
80
40
60
35
40
30
20
25
0
20
-20
15
-40
10
-60
Capacitance
Degree
5
0
0
1000
2000
3000
4000
5000
Degree
Capacitance (pF)
0603 Johanson 10 pF Capacitor (251R15S)
-80
-100
6000
Freq (MHz)
Figure A.21 Capacitance and phase of impedance.
0603 Johanson 10 pF Capacitor (251R15S)
10000
1-Port S-Parameter
2-Port S-Parameter from manufacturer
Q Factor
1000
100
10
1
0
1000
2000
3000
4000
Freq (MHz)
Figure A.22 Quality factor of 10 pF capacitor.
52
5000
6000
0603 Johanson 15 pF Capacitor (251R15S)
70
100
80
60
50
40
40
20
0
30
-20
20
-40
-60
Capacitance
Degree
10
0
0
1000
2000
3000
4000
5000
Degree
Capacitance (pF)
60
-80
-100
6000
Freq (MHz)
Figure A.23 Capacitance and phase of impedance.
0603 Johanson 15 pF Capacitor (251R15S)
10000
1-Port S-Parameter
2-Port S-Parameter from manufacturer
Q Factor
1000
100
10
1
0
1000
2000
3000
4000
Freq (MHz)
Figure A.24 Quality factor of 15 pF capacitor.
53
5000
6000
100
100
90
80
80
60
70
40
60
20
50
0
40
-20
30
-40
20
-60
Capacitance
Degree
10
0
0
1000
2000
3000
4000
5000
Degree
Capacitance (pF)
0603 Johanson 33 pF Capacitor (251R15S)
-80
-100
6000
Freq (MHz)
Figure A.25 Capacitance and phase of impedance.
0603 Johanson 33 pF Capacitor (251R15S)
10000
1-Port S-Parameter
2-Port S-Parameter from manufacturer
Q Factor
1000
100
10
1
0
1000
2000
3000
4000
Freq (MHz)
Figure A.26 Quality factor of 33 pF capacitor.
54
5000
6000
0603 Johanson 47 pF Capacitor (251R15S)
100
120
80
60
40
80
20
0
60
-20
40
Degree
Capacitance (pF)
100
-40
-60
Capacitance
Degree
20
0
0
1000
2000
3000
4000
5000
-80
-100
6000
Freq (MHz)
Figure A.27 Capacitance and phase of impedance
0603 Johanson 47 pF Capacitor (251R15S)
10000
1-Port S-Parameter
2-Port S-Parameter from manufacturer
Q Factor
1000
100
10
1
0
1000
2000
3000
4000
Freq (MHz)
Figure A.28 Quality factor of 47 pF capacitor.
55
5000
6000
200
100
180
80
160
60
140
40
120
20
100
0
80
-20
60
-40
40
-60
Capacitance
Degree
20
0
0
1000
2000
3000
4000
5000
Degree
Capacitance (pF)
0603 Johanson 82 pF Capacitor (251R15S)
-80
-100
6000
Freq (MHz)
Figure A.29 Capacitance and phase of impedance.
0603 Johanson 82 pF Capacitor (251R15S)
10000
1-Port S-Parameter
2-Port S-Parameter from manufacturer
Q Factor
1000
100
10
1
0
1000
2000
3000
4000
Freq (MHz)
Figure A.30 Quality factor of 82 pF capacitor.
56
5000
6000
3. Resistor
0603 Panasonic 10 ohm Resistor (ERJ-3GEYJ100)
80
11.5
11
40
10.5
Resistance
10
20
Degree
0
9.5
Inductance (nH)
Degree
60
1.2
3.5
3
2.5
2
1.5
1
0.5
0
1
0.8
Inductance
Capacitance
0.6
0.4
0.2
Capacitance (pF)
Resistance (ohm)
12
0
0
1000
2000
3000
4000
5000
6000
Freq (MHz)
Figure A.31 Resistance, phase of impedance, and reactance.
0603 Panasonic 10 ohm Resistor (ERJ-3GEYJ100)
Q Factor
4
1-Port S-Parameter
3
2
1
0
10
100
1000
Freq (MHz)
Figure A.32 Quality factor of 10 Ω resistor.
57
10000
0603 Panasonic 39 ohm Resistor (ERJ-3GEYJ100)
40
20
40
10
Resistance
39
0
Degree
-10
38
3
Inductance (nH)
Degree
30
41
1.2
Inductance
Capacitance
2.5
2
1
0.8
1.5
0.6
1
0.4
0.5
0.2
0
Capacitance (pF)
Resistance (ohm)
42
0
0
1000
2000
3000
4000
5000
6000
Freq (MHz)
Figure A.33 Resistance, phase of impedance, and reactance.
0603 Panasonic 39 ohm Resistor (ERJ-3GEYJ100)
1
1-Port S-Parameter
2-Port S-Parameter from…
Q Factor
0.8
0.6
0.4
0.2
0
10
100
1000
Freq (MHz)
Figure A.34 Quality factor of 39 Ω resistor.
58
10000
0603 Panasonic 68 ohm Resistor (ERJ-3GEYJ100)
20
71
69
10
68
Resistance
67
5
Degree
66
0
3
Inductance (nH)
Degree
15
70
1.2
Inductance
Capacitance
2.5
2
1
0.8
1.5
0.6
1
0.4
0.5
0.2
0
Capacitance (pF)
Resistance (ohm)
72
0
0
1000
2000
3000
4000
5000
6000
Freq (MHz)
Figure A.35 Resistance, phase of impedance, and reactance.
0603 Panasonic 68 ohm Resistor (ERJ-3GEYJ100)
0.5
1-Port S-Parameter
2-Port S-Parameter from manufacturer
Q Factor
0.4
0.3
0.2
0.1
0
10
100
1000
Freq (MHz)
Figure A.36 Quality factor of 68 Ω resistor.
59
10000
0603 Panasonic 200 ohm Resistor (ERJ-3GEYJ100)
5
Resistance
Degree
195
0
190
-5
185
-10
180
Degree
200
-15
175
170
-20
1.2
60
Inductance
Capacitance
1
0.8
50
40
0.6
30
0.4
20
0.2
10
0
Capacitance (pF)
Inductance (nH)
Resistance (ohm)
205
0
0
1000
2000
3000
4000
5000
6000
Freq (MHz)
Figure A.37 Resistance, phase of impedance, and reactance.
0603 Panasonic 200 ohm Resistor (ERJ-3GEYJ100)
0.3
1-Port S-Parameter
Q Factor
0.25
0.2
0.15
0.1
0.05
0
10
100
1000
Freq (MHz)
Figure A.38 Quality factor of 200 Ω resistor.
60
10000
5
300
0
250
-5
200
-10
150
-15
100
-20
Resistance
Degree
50
0
-25
-30
60
5.0
Inductance (nH)
Degree
350
Inductance
Capacitance
4.0
50
40
3.0
30
2.0
20
1.0
10
0.0
0
0
1000
2000
3000
4000
5000
Capacitance (pF)
Resistance (ohm)
0603 Panasonic 300 ohm Resistor (ERJ-3GEYJ100)
6000
Freq (MHz)
Figure A.39 Resistance, phase of impedance, and reactance.
0603 Panasonic 300 ohm Resistor (ERJ-3GEYJ100)
0.6
1-Port S-Parameter
Q Factor
0.5
0.4
0.3
0.2
0.1
0
10
100
1000
Freq (MHz)
Figure A.40 Quality factor of 300 Ω resistor.
61
10000
0603 Panasonic 560 ohm Resistor (ERJ-3GEYJ100)
500
0
-10
400
-20
300
-30
200
100
-40
0
-50
70
60
50
40
30
20
10
0
Inductance
Capacitance
4.0
3.0
2.0
1.0
0.0
0
1000
2000
3000
4000
5000
Degree
600
5.0
Inductance (nH)
10
Resistance
Degree
Capacitance (pF)
Resistance (ohm)
700
6000
Freq (MHz)
Figure A.41 Resistance, phase of impedance, and reactance.
0603 Panasonic 560 ohm Resistor (ERJ-3GEYJ100)
1.2
1-Port S-Parameter
2-Port S-Parameter from…
Q Factor
1
0.8
0.6
0.4
0.2
0
10
100
1000
Freq (MHz)
Figure A.42 Quality factor of 560 Ω resistor.
62
10000
Resistance
Degree
1000
800
600
400
200
0
Inductance (nH)
5.00
Inductance
Capacitance
4.00
3.00
2.00
1.00
0.00
0
1000
2000
3000
4000
5000
10
0
-10
-20
-30
-40
-50
-60
-70
Degree
Resistance (ohm)
1200
35
30
25
20
15
10
5
0
Capacitance (pF)
0603 Panasonic 1000 ohm Resistor (ERJ-3GEYJ100)
6000
Freq (MHz)
Figure A.43 Resistance, phase of impedance, and reactance.
0603 Panasonic 1000 ohm Resistor (ERJ-3GEYJ100)
Q Factor
2
1-Port S-Parameter
2-Port S-Parameter from…
1.5
1
0.5
0
10
100
1000
Freq (MHz)
Figure A.44 Quality factor of 1 kΩ resistor.
63
10000
0603 Panasonic 2000 ohm Resistor (ERJ-3GEYJ100)
20
Resistance
Degree
0
1500
-20
1000
-40
500
-60
0
-80
5.00
8
7
6
5
4
3
2
1
0
Inductance
Capacitance
4.00
3.00
2.00
1.00
0.00
0
1000
2000
3000
4000
5000
Degree
2000
Capacitance (pF)
Inductance (nH)
Resistance (ohm)
2500
6000
Freq (MHz)
Figure A.45 Resistance, phase of impedance, and reactance.
0603 Panasonic 2000 ohm Resistor (ERJ-3GEYJ100)
3
1-Port S-Parameter
Q Factor
2.5
2
1.5
1
0.5
0
10
100
1000
Freq (MHz)
Figure A.46 Quality factor of 2 kΩ resistor.
64
10000
0603 Panasonic 3000 ohm Resistor (ERJ-3GEYJ100)
Resistance
Degree
3000
2500
0
Degree
-20
2000
-40
1500
-60
1000
-80
500
0
-100
5.00
Inductance (nH)
20
14
12
10
8
6
4
2
0
Inductance
Capacitance
4.00
3.00
2.00
1.00
0.00
0
1000
2000
3000
4000
5000
Capacitance (pF)
Resistance (ohm)
3500
6000
Freq (MHz)
Figure A.47 Resistance, phase of impedance, and reactance.
0603 Panasonic 3000 ohm Resistor (ERJ-3GEYJ100)
5
1-Port S-Parameter
2-Port S-Parameter from…
Q Factor
4
3
2
1
0
10
100
1000
Freq (MHz)
Figure A.48 Quality factor of 3 kΩ resistor.
65
10000