Modelling of passive intermodulation in RF systems

IN DEGREE PROJECT ELECTRICAL ENGINEERING,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2020

Modelling of passive intermodulation in RF systems

MARTIN PETEK

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE

Modelling of passiveintermodulation in RFsystems

MARTIN PETEK

Master in ElectrophysicsDate: June 14, 2020Supervisor: Christos Kolitsidas (Ericsson), Qingbi Liao (KTH)Examiner: Oscar Quevedo-TeruelSchool of Electrical Engineering and Computer ScienceHost company: Ericsson ABSwedish title: Modellering av passiv intermodulation i RF-system

iii

AbstractModern communication systems are increasing in complexity, power and dy-namic range. As a result, weaker signals can be detected which improves rangeand quality of service. However, this also results in greater sensitivity to spu-rious signals which can impact system performance. One of the mechanismsthat spurious signals can be created is by mixing that occurs by the nonlin-ear behavior of components. These components are typically active, such astransistors and diodes. Recently, there has been increased focus on studyingnonlinearities of passive components. These are usually considered linear andoperate under low power, such as connectors and waveguides. Such elementsexhibit weak nonlinearity and can produce measurable levels of unwanted sig-nals when enough power is injected into the system. Since the source is pas-sive, these signals are referred to as passive intermodulation products.

In this thesis, the fundamental behavior of passive intermodulation is ex-plored and modelled. A literature review of physical sources is conducted andkey contributors to nonlinearities are identified. A general model is developedfor both discrete and distributed structures and can be used to characterize thecurrent-voltage relation, regardless of the physical origin of nonlinearity. Ad-ditionally, different numerical methods are presented for calculating the cur-rent amplitudes of individual spectral components, allowing for extraction ofpower of intermodulation products. The models are first verified through com-parison with commercial circuit simulation software. The effects of losses anddispersion in transmission lines on passive intermodulation generation are alsostudied.

Finally, the model is used for characterizing a transition from stripline tomicrostrip line. The presence of other nonlinear components is separatelycharacterized and accounted for. The developed model is shown to be ableto closely fit the third order intermodulation products. Additionally, it canpredict higher order intermodulation products with a good degree of accuracyfor two out of three components measured.

iv

SammanfattningModerna kommunikationssystem blir alltmer komplexa, får allt högre effektoch allt större dynamiska intervall. Som ett resultat av detta kan allt svagaresignaler upptäckas, vilket förbättrar signalens räckvidd och kvaliteten på upp-kopplingen. Det resulterar dock även i större känslighet för oönskade signaler,vilket kan påverka systemets prestanda. En av mekanismerna som kan ska-pa oönskade signaler är blandandet av signaler som uppstår av det ickelinjärabeteendet hos komponenter som ingår i systemet. Dessa komponenter är oftaaktiva, så som transistorer och dioder. På senare tid har det funnits ett ökandefokus på att studera ickelinjäriteter i passiva komponenter. Dessa komponen-ter antags ofta vara linjära och opererar vid låga effekter. Exempel på sådanakomponenter är kontakter och vågledare. Sådana komponenter uppvisar sva-ga ickelinjäriteter och kan producera mätbara nivåer av oönskade signaler närtillräckligt mycket effekt finns i systemet. Eftersom källan till dessa signalerär passiv, benämns signalerna passiva intermodulationsprodukter.

I detta arbete modelleras och utforskas det grundläggande beteendet hospassiv intermodulation. En litteraturstudie över de fysikaliska källorna till ic-kelinjäriteterna genomförs och de huvudsakliga källorna identifieras. En ge-nerell modell för både diskreta och distribuerade strukturer tas fram och kananvändas för att bestämma sambandetmellan ström och spänning, oavsett icke-linjäritetens fysikaliska ursprung. Dessutom presenteras olika numeriska me-toder för beräkning av strömamplituden för enskilda spektrala komponenter,vilket medger beräkning av effekten hos intermodulationsprodukter. Model-lerna verifieras genom jämförelser med kommersiella mjukvaror för kretssi-mulering. Effekten av förluster och dispersion i transmissionsledning på passivintermodulation studeras också.

Slutligen används modellen för att bestämma övergången från stripline tillmicrostrip. Även påverkan av närvaron av andra ickelinjära komponenter be-stäms och tas hänsyn till. Den framtagna modellen visas ge bra överensstäm-melse med tredje ordningens intermodulationsprodukter. Dessutom kan denmed god träffsäkerhet bestämma högre ordningens intermodulationsproduk-ter för två av tre studerade komponenter.

Contents

1 Introduction 31.1 What is passive intermodulation? . . . . . . . . . . . . . . . . 3

1.1.1 From nonlinearity to PIM . . . . . . . . . . . . . . . 51.2 Sources of PIM . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Contact nonlinearity . . . . . . . . . . . . . . . . . . 81.2.2 Electrothermal . . . . . . . . . . . . . . . . . . . . . 91.2.3 Nonlinear materials . . . . . . . . . . . . . . . . . . . 9

1.3 Measuring PIM . . . . . . . . . . . . . . . . . . . . . . . . . 101.3.1 Forward and reverse PIM . . . . . . . . . . . . . . . . 101.3.2 Distance to PIM . . . . . . . . . . . . . . . . . . . . 111.3.3 Near field scanner . . . . . . . . . . . . . . . . . . . 111.3.4 Phase measurements . . . . . . . . . . . . . . . . . . 12

1.4 Aim of the thesis . . . . . . . . . . . . . . . . . . . . . . . . 121.5 Overview of the literature . . . . . . . . . . . . . . . . . . . . 13

2 Modelling 142.1 Discrete source . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.1 Taylor series calculation . . . . . . . . . . . . . . . . 172.1.2 FFT calculation . . . . . . . . . . . . . . . . . . . . . 182.1.3 Orthogonality calculation . . . . . . . . . . . . . . . 192.1.4 Calculating power from PIM current . . . . . . . . . . 202.1.5 Verification . . . . . . . . . . . . . . . . . . . . . . . 212.1.6 Loading effects . . . . . . . . . . . . . . . . . . . . . 22

2.2 Distributed nonlinearity . . . . . . . . . . . . . . . . . . . . . 232.2.1 Verification and examples . . . . . . . . . . . . . . . 26

3 Measurements and results 323.1 Connector characterization . . . . . . . . . . . . . . . . . . . 34

3.1.1 Right angle connectors . . . . . . . . . . . . . . . . . 35

v

vi CONTENTS

3.1.2 Straight connectors . . . . . . . . . . . . . . . . . . . 413.2 Microstrip to stripline transition . . . . . . . . . . . . . . . . 453.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4 Future work and conclusion 514.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Bibliography 53

Acknowledgements

As my time at KTH reaches its conclusion, I feel that it is important to ac-knowledge everyone who helped me reach this point.

Firstly, I am sincerely thankful to Dr. Christos Kolitsidas for his mentor-ship during the thesis project. He went above and beyond of what is expectedfrom a mentor and helped me grow both personally and professionally. Ad-ditionally, I thank Qingbi Liao for reviewing my thesis. I would also like tothank Björn Ljungberg for his help in translating the thesis abstract.

Secondly, I would like to thank all teachers and professors whowere instru-mental in my personal and professional development. Particularly, my thanksgoes to Prof. Nickolay Ivchenko for supervising the REXUS project, whichwas a good introduction to the world of rocketry and space engineering. Fur-thermore, I am very thankful to Prof. Oscar Quevedo-Teruel and Oskar Zetter-ström for their mentorship and guidance in exploring the wonderful world oflenses and metamaterials.

Thirdly, I would like to thank my friends, family and my girlfriend Ana. Ifeel very privileged to have such excellent people surrounding me. I couldn’thave gotten this far without you.

Finally, I am thankful to Republic of Slovenia for the Ad Futura scholarshipwhich helped me finance my stay in Sweden.

1

Acronyms

PIM - Passive intermodulationET-PIM - Electrothermal passive intermodulationTCR - Temperature coefficient of resistanceMM contact - Metal-to-metal contactMIM contact - Metal-insulator-metal contactDUT - Device under testRF - Radio frequencyFFT - Fast Fourier TransformAWR - Simulation tool for circuit analysis

2

Chapter 1

Introduction

The rising need to always be connected in the world has resulted in an in-creased strain on the telecommunication systems. As a result, there has beena rapid development of channel capacity in terms of speed, latency and band-width. Thus newer systems have increased in bandwidth, complexity, powerand dynamic range. As a result, weaker signals can be detected which al-lows for greater stability and longer range of communication. In turn, it alsoresults in greater susceptibility to spurious signals. These signals can be re-ceived by the antenna or created inside the system through mixing of carriersignals, caused by nonlinear components. In this thesis, we first explain howtypically linear components such as connectors, terminations and waveguidesmight become weakly nonlinear at high powers and thus create intermodu-lation products. Then, a general model for both electrically small and largeelements is developed to calculate the power level of these products. Finally,measurements of typical microwave components are presented and the modelis used to extract the nonlinear current-voltage relation.

1.1 What is passive intermodulation?Passive intermodulation (PIM) is a nonlinear phenomena occuring in passivecomponents of electrical systems, which has recently attracted some attentionin microwave community due to its detrimental effect in radio frequency (RF)systems. In systems excited with two tones, see Fig. 1.1a, the non-linearitycauses mixing which creates new frequencies at

fPIMN = mf1 + nf2, m, n ∈ (−∞,∞), N = |m|+ |n|, (1.1)

3

4 CHAPTER 1. INTRODUCTION

where f1 and f2, f1 < f2, are the frequencies of first and second tone, respec-tively. These combinations are referred as N-th order products and are furtherdenoted as "upper" or "lower", depending onwhether they have lower or higherfrequency in comparison to carrier signals. For example, the intermodulationfrequency 2f1 − f2 is referred as the lower PIM3 (PIM3l) and the frequency2f2 − f1 as upper PIM3 (PIM3u). They are of particular interest as they arestrongest and closest to the excitation tones, which often makes them fall inthe receive band. A diagram of spectrum can be seen on Fig. 1.1b, where thelower components fall within the receive band and can thus interfere with thesystem.

PIM is often divided into its relation to the system it interferes with. Con-sider the illustration on Fig. 1.2. The system contains transmission lines, con-nectors and antenna. Based on the direction of propagation, forward (travellingtowards the antenna) and reverse (travelling towards the generator) PIM arenoted. Additionally, based on the location of the PIM source, it can be dividedinto internal (created in the system) and external (created outside of the systemand being received by the antenna) PIM. Internal PIM can further be subdi-vided into distributed and discrete PIM sources. Distributed PIM is createdin structures spanning multiple wavelengths and and has different propertiesfrom discrete PIM, which is created in lumped elements like connectors andresistors.

0 0.2 0.4 0.6 0.8 1

Time/s 10-8

-1

0

1

Voltage/V

f1 = 1 GHz

f2 = 1.1 GHz

0 0.2 0.4 0.6 0.8 1

Time/s 10-8

-2

0

2

Voltage/V

(a)

f1 f22f1-f23f1-2f2 2f2-f1 3f2-2f1

PIM5l PIM3l PIM5uPIM3u

TXRX

(b)

Figure 1.1: a): The two tone excitation. The upper plot shows individual tonesand the bottom shows their sum. b): An illustration depicting the spectrum ofthe signal in a weakly nonlinear system excited by two tones. The transmit(TX) and receive (RX) band are indicated with dashed lines.

CHAPTER 1. INTRODUCTION 5

Connector

Transmission line

Reverse PIM Forward PIM

AntennaRadiated signal/PIM

External PIM

Excitation signal

Figure 1.2: An example of antenna system. The connector acts as a PIMsource, launching forward and reverse PIM waves. The forward waves areradiated through the antenna together with the signal. The radiated signal mayencounter nonlinearities, causing PIM to be created and received by the an-tenna. This is called external PIM.

1.1.1 From nonlinearity to PIMTo further illustrate how nonlinearity creates intermodulation products andexplore some properties of the phenomenon, we will now consider a nonlinearresistor, directly excited by a two tone source, as presented in Fig. 1.3. Theresistor has the following current-voltage relation

I(V ) = a1V + a2V2 + a3V

3, (1.2)

where I is current, V is voltage and ak are arbitrary coefficients. If the elementis connected to a two tone source, as presented in Fig. 1.3 with frequencies ωand phases ϕ, the voltage accross it is:

V = V1 sin (ω1t+ ϕ1) + V2 sin (ω2t+ ϕ2). (1.3)

Inserting the voltage into resistor’s I(V ) curve, expanding and using trigono-metric formulas we obtain

V

I

Figure 1.3: A simple circuit consisting of a nonlinear resistor and a voltagesource.


I(V ) =a2V

21

2+a2V

22

2+

a2V1V2 cos((ω1 − ω2)t+ ϕ1 − ϕ2

)+

3a3V21 V2

4sin((2ω1 − ω2)t+ 2ϕ1 − ϕ2

)+(

a1V1 +3a3V

31 + 6a3V1V

22

4

)sin (ω1t+ ϕ1)+(

a1V2 +3a3V

32 + 6a3V

21 V2

4

)sin (ω2t+ ϕ2)+

3a3V1V22

4sin((2ω2 − ω1)t+ 2ϕ2 − ϕ1

)−

a2V21

2cos (2ω1t+ 2ϕ1)−

a2V1V2 cos((ω1 + ω2)t+ ϕ1 + ϕ2

)−

a2V22

2cos (2ω2t+ 2ϕ2)−

3a3V21 V2

4sin((2ω1 + ω2)t+ 2ϕ1 + ϕ2

)−

3a3V1V22

4sin((2ω2 + ω1)t+ 2ϕ2 + ϕ1

)−

a3V31

4sin (3ω1t+ 3ϕ)−

a3V32

4sin (3ω2t+ 3ϕ2). (1.4)

By looking at this example, there are three interesting conclusions we candraw. First, it is clear that the order of the intermodulation product does notexceed the order of polynomial creating it. Second, by looking at one of thespectral components,

IPIM3l =3a3V

21 V2

4sin((2ω1 − ω2)t+ 2ϕ1 − ϕ2

), (1.5)

which is the lower PIM3 product denoted by IPIM3l, we see that the only con-tributing term comes from cubic nonlinearity, as evident by the presence ofonly a3, and the linear and quadratic parts have no effect on its magnitude.The statement can be generalized: if we imagine the I(V ) relation to be ofarbitrary order, only odd terms larger or equal to the order of intermodulationcontribute to odd ordered intermodulation products. Additionally, the phaseis shown to follow the same structure as the frequency.

The current, expressed in equation (1.5) creates a voltage drop in the inter-nal resistance of the generator Rg, taken to be 50 Ω, and dissipates entirety of


20 25 30 35 40

Input power/dBm

-120

-110

-100

-90

-80

-70

-60

PIM

3 p

ow

er

/dB

m

0

1

2

3

4

Re

gro

wth

ra

te d

B/d

B

PIM3 power

PIM3 regrowth rate

a3 = 10-8

P1 = P

2 = P

input

(a)

10-4 10-2 100 102 104

Ratio of power of two tones, k

-120

-110

-100

-90

-80

-70

-60

PIM

3 p

ow

er/

dB

m

PIM3 lower

PIM3 higher

P1 + P

2 = 10 W

P2 = k P

1

(b)

Figure 1.4: Behavior of PIM3 caused by third order nonlinearity. a): ThePIM3 power vs input power of individual tones rises 3 dBm/dBm. b): PIM3power versus ratio of power of two tones where the sum of the power of twotones is constant at 10 W.

its power. This power can be calculated with the following equations (the hatdenotes amplitude of current)

PPIM3 = RgI2PIM3

2, PPIM3,dBm = 10 log10

(PPIM3

0.001

), (1.6)

in linear or dBm (note the reference in the logarithm is 0.001 W). A commonway to characterize PIM is to plot individual input power of the two tonesversus PIM power. For our example, this has been done for PIM3 with valueof a3 = 10−8 and can be seen on Fig. 1.4a. The figure also depicts derivativeof this curve, which is sometimes called regrowth rate [1]. Thus, for a thirdorder polynomial nonlinearity, the regrowth rate is exactly 3 dBm per dBm ofinput power.

Another interesting remark comes from analyzing the amplitude of equa-tion (1.5) and its unequal scaling with two tones. Thus, the maximum of PIM3power for a given total (P1 + P2) power is not when two tones are equal butrather when the tone closer in frequency has two times more power. This canbe seen on Fig. 1.4b, where the total power is kept constant at 10 W (40 dBm)and the ratio of powers of two tones k = P2

P1is swept from 10−3 to 103.

1.2 Sources of PIMIn the previous section, the nature and mathematical origin of PIM was de-scribed, but no physical mechanisms were mentioned to explain why nonlin-


earities exist in the first place. Here, a brief overview of the most commonlyexplored is provided.

1.2.1 Contact nonlinearityContact nonlinearity occurs in components where two metals are joined to-gether by mechanical load (usually using screws), for example as in waveg-uide flanges [2, 3, 4] or connectors [5, 6], and is caused by microscopic im-perfections of the two surfaces. Real metallic surfaces are uneven with valleysand peaks called asperities and are usually covered in a thin film of oxide.This results in a reduction of effective contact area, decreases the amount ofdirect metal-metal (MM) contacts and introduces nonlinear metal-insulator-metal (MIM) connections.

(a)

Cc

Rconst

Rnl,c

Rc

Rnl,nc

Cnc

(b)

Figure 1.5: a): Schematic of contact between two metals. b): Equivalent cir-cuit of the contact nonlinearity.

A real contact between two metals can be seen on Fig. 1.5a and can bemodeled with a circuit on Fig. 1.5b. Here, Rc is the envelope constrictionresistance, Cc and Cnc are the contact and noncontact capacitances, Rconst isthe constriction resistance,Rnl,c is the contact nonlinear resistance andRnl,nc isthe noncontact nonlinear resistance. Their possible sources of its nonlinearityare thermionic emission, tunnelling, or Poole-Frenkel in the MIM contacts(Rnl,c) and breakdown or field emission in empty region in the non contactregions (Rnl,c) [2]. Of these, tunnelling has been given most attention andmodels produce good agreement with data [2, 5].

The properties of contact nonlinearity and its dependance on various ma-terial types have been explored in [7]. A prominent feature of contact non-


linearity is its strong dependence on mechanical load applied [2, 3, 4, 5]. Byincreasing the pressure, the oxide layer starts to crack, increasing the amountof MM connections and thus reducing the strength of nonlinearity causing adecrease in PIM. Additionally, dirty or degraded contacts [6] and higher sur-face roughness [2, 5] at the interface were shown to increase PIM caused bycontact nonlinearity.

1.2.2 ElectrothermalThe main principle of electrothermal PIM (ET-PIM) is the coupling betweenelectrical currents and heating. For two tones excitation the envelope of thesummed signal becomes modulated with a much lower frequency ∆f = f2−f1, see Fig. 1.1a. This allows currents to periodically heat up the componentand its surroundings, causing variation of material properties with respect totime. This time variation affects the current and introduces a nonlinear behav-ior.

Electrothermal PIM has been subject of various papers and is one of thefew models which is (or can be) fully analytical. Studies have been made toanalyze the effect changing of resistivity in conductors due to their temper-ature coefficient of resistance (TCR) [8, 9, 10]. This mechanism appears tobe most prominent when two tones are close in frequency and has a charac-teristic 10 dBm per decade fall as ∆f increases [8, 9, 10]. As the TCR de-fines the strength of change in resistance when the element is heated, largerTCR implies higher PIM levels. Models were developed for both distributedand lumped elements, with good agreements for frequency spacing of up to10 kHz for microstrip lines [9], attenuators [11], terminations [8] and patchantennas [10]. Additionally, a model of coplanar waveguide was developedin [12], showing good agreement with measurements up to 10 MHz. How-ever, it was discarded as a dominant source of nonlinearity in connectors [1]and microstrip lines [13], as the measurements show no presence of its char-acteristic behavior.

1.2.3 Nonlinear materialsThe presence of nonlinear materials in an RF system can drastically affect itsPIM level. Some of these are:

Ferrites are a class of ferrimagnetic materials which are used in manynon-reciprocal microwave components, like circulators and isolators. Unfor-tunately however, they exhibit a strong third order nonlinearity [14], character-


ized by a stark difference between the PIM3 level and higher order products.As such, use of ferrites is generally avoided in high power circuits where PIMlevels are of importance.

Ferromagnetic materials, like nickel and iron, have been found to createappreciable levels of PIM [15]. The main mechanism of nonlinearity is thenonlinear hysteresis of the B-H curve. Particularly, nickel is used in certainconnectors which has been found to have a strong effect on PIM power [16,17].

Superconductors have recently become interesting for application in RFcircuits, particularly with the development of high temperature superconduc-tors. Due to their extremely low resistance and loss they allow for filters withhigh quality factors [18] and highly efficient transmission lines. Unfortunately,the relation between superfluid density and current density is intrinsically non-linear [19] and components which use superconductivematerials exhibit a highlevel of PIM response [19, 20, 21] when compared to non-superconductive de-signs [22, 23, 24]. Modelling andmitigation of these nonlinearities is an activeresearch topic.

1.3 Measuring PIMMeasuring PIM is a difficult task as it involves measuring weak signals whichare created when high power carrier signals pass through the structure. Thus,the measurement setup must be able to not only measure low power signals,but do so in the presence of carrier tones which can be stronger by more than100 dBm. Furthermore, the PIM produced by the measurement setup mustbe very low so that it does not interfere with the measurement. To overcomethese challenges, several measurement techniques have been developed whichcan not only measure level, but also help localize sources of PIM. Here, a briefoverview of general techniques used is provided.

1.3.1 Forward and reverse PIMAs PIM power level is the defining factor in its interference with the system,various measurement techniques and systems have been developed to measureits value. Reverse PIM measurements are very common and a general testsetup can be seen on Fig. 1.6a. The two signals are first combined and sentthrough duplexer to device under test (DUT). If the DUT has more than oneport, other ports are terminated with a low PIM load. The forward PIM createdin the device is then dissipated in the terminations, while the reverse travels


through the duplexer and is received by the receiver. For forward PIM mea-surements, the locations of low PIM termination and receiver are switched.The circuit is depicted on Fig. 1.6b.

DUT

Receiver

Duplexer

(a)

DUT

Duplexer

Receiver

(b)

Figure 1.6: a): Reverse PIM measurement setup. b): Forward PIM measure-ment setup.

Before any of the two measurements can be done, however, it is importantto measure the PIM level produced in the measurement setup itself, knownas residual PIM. This is done by removing the DUT and measuring the PIMlevel. The value obtained is the lower bound of PIM which can be detected bythis system.

1.3.2 Distance to PIMIf the system is assembled and PIM level found unsatisfactory it is importantto localize its source. While the methods described in previous section canbe used to assess the PIM level of individual components, in practice theircontribution to overall PIM level can be higher or lower due to mismatches ordifferent speeds of degradation with time. Moreover, if the system is assem-bled in the field, it is possible that the main source of elevated PIM levels isexternal. The location of PIM source can in principle be determined by vary-ing the frequency of the carrier tones and doing an inverse Fourier transform toconvert the signal to time domain [25]. If the propagation velocity of electro-magnetic waves in the system is known, the time can be converted to distanceand the major contributors determined.

1.3.3 Near field scannerIf the device under test is open and the location of the fields is readily ac-cessible, near field scanning can be used for accurate localization and mea-surements of amplitude and phase. The principle behind this measurement


is to couple a portion of the PIM signal to a probe which is placed at a dis-tance smaller than one tenth of a wavelength [26]. Then, a small signal at PIMfrequency is sent through the structure to calibrate the system. After the cal-ibration is complete, the two tones are sent to the structure and PIM sourcescan be localized by moving the probe [27].

1.3.4 Phase measurementsIn order to fully characterize a system, measurements of both phase and ampli-tude are required. Unfortunately, there are very few studies of phase behaviorof passive intermodulation. To the authors’ knowledge, the only study so faron PIM phase is [28], where the relative behavior of PIMs’ phase of an iso-lator is studied. By changing phase of each signal, it is shown that the phaseof PIM3 follows the behavior of 2φ1 − φ2 as seen in eq. (1.5), but a discrep-ancy is observed when the same treatment is given to PIM5. A measurementsetup which can measure absolute value of phase of intermodulation productsis presented in [29], where it is used to measure the phase of third order inter-modulation product from an amplifier.

1.4 Aim of the thesisThe thesis will:

1. Conduct a literature review and identify key sources of PIM.

2. Develop a model which can be used in predicting or evaluating effectsof weak nonlinearities in RF systems.

3. Validate the model through simulations and measurements.

4. Evaluate the PIM performance of some key microwave components.


1.5 Overview of the literatureIn this section, we provide overview of the references.

Element of study PaperConnectors [1, 5, 6, 14, 16, 30]Terminations [8]Microstrip line [9, 22, 23, 31, 32, 33, 34]

Coplanar waveguide [12, 35]Microstrip filters [22]Cavity filters [24, 36]Antennas [10]Circulators [14]Metals [7]

Table 1.1: Overview of literature by components studied.

Physical source PaperPhenomenological [1, 5, 6, 16, 30]Electrothermal [8, 9, 10, 11, 32]

Contact nonlinearity [2, 3, 4, 5, 6, 7]

Table 1.2: Overview of literature by physical source studied.

Chapter 2

Modelling

The presence of internal PIM in a system can be a result of various physi-cal sources and physics-based models have been developed, for example forET-PIM [8], contact nonlinearity [5] and nickel coating in connectors [37].However the usage of these models requires the underlying physical mecha-nisms to be dominant and present in PIM generation. This is difficult to predictin practical systems where there are multiple sources present. Furthermore, insome cases the physical source is still actively discussed [13].

In this work, we wish to obtain a general model which could be appliedto an arbitrary component. To achieve this goal we omit modelling physicalsources directly. This approach follows and builds on the work in [1], wherethe nonlinearity is represented with a polynomial series current-voltage rela-tion. Building up from this work the models presented in this thesis and theirmathematical solutions present a new insight on PIM characterization. In thismodel, the I(V ) relation can be presented as a truncated power series

I(V ) =N∑n=1

anVn, (2.1)

where an are coefficients obtained with a fit to data and N is the order of thepolynomial. The goal is to create a model which would accurately predict PIMlevels even in a changed environment. This is achieved by extracting the coef-ficients an through a model which accounts for the effect of the measurementsystem on production of PIM products. This approach has been used to modela connector in [1, 30] with an excellent degree of success. Furthermore, themodel does not address a specific source of nonlinearity and captures all dif-ferent effects in a single set of parameters. This means that the same procedurecan be applied to any type of nonlinearity, regardless of its physical source.

14

CHAPTER 2. MODELLING 15

Vg

Vnl

RL

I

(a)

IPIMRL

IPIMr

Rg

IPIMf

(b)

Figure 2.1: a): Circuit with a series nonlinearity b): Equivalent currentsource representing scattered PIM current waves in forward (IPIMf) and re-verse (IPIMr) direction.

Nevertheless, the model accounts for the presence of the linear resistor,used in the measurements to terminate the circuit. A simplified model of themeasurement is presented in Fig. 2.1a. The voltage, produced by the generatorsplits between the nonlinear resistor and linear load RL. The excitation of thenonlinear resistor creates PIM products, which are launched as waves in bothforward and reverse directions. This behavior can be illustrated by replacingthe nonlinear resistor with a parallel current source, as seen on Fig. 2.1b.

2.1 Discrete sourceConsider the schematic of the circuit on Fig. 2.1a. The nonlinear resistor isrepresented with a third order polynomial I(Vnl) = a1Vnl + a3V

3nl and is con-

nected in series with a linear resistor RL (usually 50 Ohm). The resultingequation for I(Vg) is then:

I(Vg) = a1(Vg − IRL) + a3(Vg − IRL)3 (2.2)

which is a cubic equation with three solutions:

I(Vg) =VgRL

−

21/3(3a3R3L + 3a1a3R

4L)

3a3R3L

(− 27a23R

5LVg +

√4(3a3R3

L + 3a1a3R4L)3 + 729a43R

10L V

2g

)1/3+

(− 27a23R

5LVg +

√4(3a3R3

L + 3a1a3R4L)3 + 729a43R

10L V

2g

)1/321/3 3a3R3

L

, (2.3)

16 CHAPTER 2. MODELLING

I(Vg) =VgRL

+

(1 + i√

3)(3a3R3L + 3a1a3R

4L)

22/3 3a3R3L

(− 27a23R

5LVg +

√4(3a3R3

L + 3a1a3R4L)3 + 729a43R

10L V

2g

)1/3−(1− i

√3)(− 27a23R

5LVg +

√4(3a3R3

L + 3a1a3R4L)3 + 729a43R

10L V

2g

)1/321/3 6a3R3

L

,

(2.4)

I(Vg) =VgRL

+

(1− i√

3)(3a3R3L + 3a1a3R

4L)

22/3 3a3R3L

(− 27a23R

5LVg +

√4(3a3R3

L + 3a1a3R4L)3 + 729a43R

10L V

2g

)1/3−(1 + i

√3)(− 27a23R

5LVg +

√4(3a3R3

L + 3a1a3R4L)3 + 729a43R

10L V

2g

)1/321/3 6a3R3

L

,

(2.5)

some of which are not physically correct. The value of a1 can only be posi-tive as it represents the linear ohmic resistance of the component, whereas a3can be either positive and negative. For the case of positive a3, it is clear thateq. (2.3) is the only possible solution, as the values of eq. (2.4) and eq. (2.5)are always complex. However, the for the case of negative a3, there are caseswhere all solutions yield a real number. The correct solution can be deter-mined by plotting the I(V ) curve together with a load line, as presented onFig. 2.2. By looking at the figure we can quickly determine equation (2.4) tobe the physical solution, as eqs. (2.3) and (2.5) would result in nonzero cur-rents flowing through the circuit when the Vg = 0 V. Finally, it can be seen thatat a certain value of Vg the model breaks down and the only solution becomeseq. (2.5).

The significance of choosing a model of positive a3, or negative a3 is thetype of physical behavior causes the phenomenon. A positive a3 implies in-creasing resistance nonlinearity, while a negative a3 implies decreasing resis-tance nonlinearity. For example, the former could be caused by electrothermaleffects, whereas the latter could be caused by tunneling [1]. Regardless of themodel we choose, a calculation procedure is needed for the desired PIM cur-rents. In the next few sections, we will describe three possible methods.


-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

Current/A

-1.5

-1

-0.5

0

0.5

1

1.5

Vo

lta

ge

/V

Load line

Nonlinear resistor

Nonlinear res, +a3

Equation 1

Equation 2

Equation 3

Equation 1, +a3

Figure 2.2: Load line analysis of a resistive nonlinearity I(Vg) = a1Vg +a3Vg.The light blue line has values of parameters a1 = 10 and a3 = −100 and thegreen line has a1 = 10 and a3 = 100. The markers show calculated values ofsolutions from eqs. (2.3), (2.4) and (2.5).

2.1.1 Taylor series calculationThe Taylor series approach is starts by expanding the appropriate equation forI(Vg) in a Taylor series

I(Vg) =N∑n=1

cnVng , cn =

I(n)(0)

n!(2.6)

where N is the truncation order of the series and cn are Taylor coefficients.Then, we insert the equation representing two tone excitation for the generatorvoltage,

Vg = V1 sin (ω1t) + V2 sin (ω2t) (2.7)


and expand the terms, exctracting only the sin((2ω1 − ω2)t

)component. We

end up with a truncated series for the lower PIM3 current:

IIM3 =

(3

4V 21 V2c3+

5

8V 21 V2(2V

21 + 3V 2

2 )c5+

15

64V 21 V2(V

41 + 4V 2

1 V22 + 2V 4

2 )c7+

(63

64V12V2(2V16 + 15V14V22 + 20V12V24 + 5V26))c9

+ ....

)sin((2ω1 − ω2)t

). (2.8)

The expression in the brackets in eq (2.8) thus represents the amplitude ofPIM3 current, IIM3.

While the procedure described calculates the PIM level in theory, it suffersfrom three major problems. First, the coefficients cn are dependent on thechoice of I(Vg) and need to be developed for each current-voltage relationseparately. Second, they quickly become large and difficult to calculate evenwith programs such as Wolfram Mathematica [38]. Finally, too many termsare needed to calculate PIM levels precisely. Even with up to 27 terms, thePIM versus power curve is not adequately described for certain values of a1and a3, making the model unreliable.

2.1.2 FFT calculationAnother approach to calculating the currents of spectral components createdby the nonlinearity is to take the equation for I(Vg), inserting the two toneexcitation with appropriate amplitude and taking the Fast Fourier Transform(FFT) of the expression. The benefit of this method is that it calculates allharmonic components, given that we choose the maximum time and time stepappropriately. Furthermore, the method is usually readily available as a func-tion in most numerical libraries and is easy to implement. The drawback ofthis method is that it requires a high amount of sampling points to achievegood accuracy, as presented in Fig 2.3b. This prohibits the use of the methodas a calculation procedure.


-100

-50

0

50

Pow

er

spectr

um

/dB

m

0 0.5 1 1.5 2

Frequency/GHz

AWR

FFT

(a)

102

104

106

108

1010

Number of samples of FFT

10-4

10-3

10-2

IM3 c

urr

ent/A

FFT calculation

Correct value

(b)

Figure 2.3: a): example of an calculated spectrum of a series nonlinearitywith FFT and AWR. b): Value of calculated current for IM3 versus number ofsamples used in FFT calculation.

2.1.3 Orthogonality calculationAs calculating the intermodulation products using the Taylor series is numer-ically unstable and the FFT method is too slow, a new method of calculationis developed and presented in this thesis. The underlying intuition in this ap-proach is to note that while the Taylor series expansion of eq. (2.3) is infinite,the resulting spectrum contains components which are evenly spaced, as cor-roborated by FFT calculation on Fig. 2.3a. Since the model is a resistor andthus frequency independent, we can choose the frequency freely to simplifythe calculation. If we choose

f1 = k(f2 − f1), (2.9)

where k is a positive integer, we see that the lowest frequency componentproduced in the system is f1 − f2 and all other frequencies are a multipleof this frequency. This means that we can represent the function I(Vg) as aFourier series

I(Vg) =∞∑k=0

Ik sin(k(ω1 − ω2)t

), (2.10)

where the Ik denotes amplitude of the spectral component. Therefore, if we in-sert the appropriate (two tone) excitation into our chosen I(Vg), we can extractthe amplitude of current at a specific frequency by multiplying the current-voltage relation with a sine function of desired frequency and integrating over


the period of the longest spectral component, T = 1f2−f1 .∫ T

0

I(Vg) dx =∞∑k=0

∫ T

0

Ik sin(k(ω1 − ω2

)t) sin

((mω1 − nω2

)t) dx.

(2.11)Due to the orthogonality of the trigonometric functions, the result of the com-putation is proportional to the current amplitude of a single component atf = mf1 − nf2. The only remaining step is to choose a proper scaling factorso that we obtain the amplitude. We calculate the scaling factor by looking atthe solution of the integral for the desired component,

IPIM = C

∫ T

0

IPIM sin2((mω1 − nω2)t

)dt, (2.12)

where we solve for the scaling factor C. For our choice of T and f1, the wecalculate the scaling factor to beC = 2/T = 2(f2−f1) for all intermodulationproducts.

The result of the derivation are equations (2.13), (2.14) and (2.15) for thethird, fifth and seventh intermodulation product respectively. These equationsgive us a direct relation between current-voltage relation, excitation and indi-vidual PIM products. The procedure is a fast, numerically stable and flexiblesolution to the problem of calculating individual current amplitudes. Further-more, it is easy to adapt the equations listed below for different choices ofcurrent-voltage relations I(Vg).

IPIM3 = 2(f2 − f1)∫ T

0

I(Vg) sin((2ω1 − ω2)t

)dt (2.13)

IPIM5 = 2(f2 − f1)∫ T

0

I(Vg) sin((3ω1 − 2ω2)t

)dt (2.14)

IPIM7 = 2(f2 − f1)∫ T

0

I(Vg) sin((4ω1 − 3ω2

)t) dt (2.15)

IPIM = 2(f2 − f1)∫ T

0

I(Vg) sin((nω1 −mω2

)t) dt (2.16)

2.1.4 Calculating power from PIM currentAfter the total current is calculated, we can compute the PIM in forward andreverse direction by transforming the nonlinear resistor as a parallel currentsource, as presented in Fig. 2.4. Here, it is important to account for the fact


IPIMRL

IPIMr

Rg

IPIMf

Figure 2.4: Equivalent model of PIM generation. The total PIM current splitsbetween the generator and load branch.

that the current branches into two different paths, with the load resistor RL

and generator resistance Rg. The power of reverse and forward PIM currentsis then calculated with

PPIM,r = Rg

( RL

RL +Rg

IPIM√2

)2, PPIM,f = RL

( Rg

RL +Rg

IPIM√2

)2. (2.17)

We can further simplify the equations by noting that the load and the generatorare typically well matched, RL = Rg = 50 Ω. Under this asumption, the wecan write the equations as

PPIM,r = RcI2PIM

4, PPIM,f = Rc

I2PIM4

Rc = 50 Ω. (2.18)

Finally, the power is converted to dBm with the following equation

PPIM,dbm = 10 log10

(PPIM

0.001

). (2.19)

The procedure of calculating PIM power is then as follows:

1. Choose the appropriate I(Vg): use equation (2.3) for positive a3 andequation (2.4) for negative a3

2. Use equations (2.13)-(2.16) to calculate current of desired PIM product.

3. Calculate power of forward and reverse component with equation (2.17)and convert them to dBm with eq. (2.19).

2.1.5 VerificationThemodel was first verified through comparisonwithAWRandmeasurementsobtained from the literature [1, 6]. First, the PIM3 power curve was fitted and


coefficients a1 and a3 were extracted. Then, the model was simulated withharmonic balance simulator in AWR. The PIM power, obtained with the or-thogonality calculation and the HB simulations match closely, which can beseen from Fig. 2.5. Finally, the validity of the model can be verified throughhow well it predicts the higher order products. Since only data from PIM3measurements is used with the fit, comparing the PIM5 power, predicted bythe model, and measurements can be used to independently assess the modelchoice. When doing so in Fig. 2.5b, we observe good agreement, indicatingthat the model of series nonlinearity with the chosen I(V ) curve is appropri-ately selected.

20 25 30 35 40 45

Input power/dBm

-130

-120

-110

-100

-90

-80

-70

-60

PIM

po

we

r/d

Bm

PIM3 measurements

PIM3 model fit

PIM5 model result

PIM3 AWR simulation

PIM5 AWR simulation

(a)

30 35 40 45

Input power/dBm

-130

-120

-110

-100

-90

-80

-70

-60

PIM

po

we

r/d

Bm

PIM3 measurements

PIM5 measurements

PIM3 model fit

PIM5 model prediction

PIM3 AWR simulation

PIM5 AWR simulation

(b)

Figure 2.5: Verification of the model and comparison with measurements andsimulation in AWR. a): Measurements taken from [1]. The obtained coef-ficients are a1 = 141.81 and a3 = 8.0267 · 106. b): Measurements takenfrom [6]. The obtained coefficients are a1 = 335.943 and a3 = 3.2445 · 106.

2.1.6 Loading effectsWe can now explore how the values of parameters a1 and a3 affect the PIM3 vspower curve. On Fig. 2.6, a parameter sweep of a1 and a3 is presented. FromFig. 2.6a, we see that a higher value of a1 decreases the PIM power, as the I(V )

curve is more linear. On the other hand, a higher value of a3 tends to increasePIM power, as depicted on Fig. 2.6b. Furthermore, as a3 increases, the curve’sslope reduces and for higher values of a3, resulting in a lower PIM power athigh input powers. This can be explained by understanding the behavior oflinear-nonlinear interaction. As the power and therefore voltage, is increased,the resistance of nonlinear resistor falls due to the nonlinear component. Thus,


as the overall voltage over the linear and nonlinear resistor increases, the in-crement of voltage over the nonlinear resistor is decreasing, causing the PIMcurve to have a reduction in slope.

20 25 30 35 40 45

Input power/dBm

-160

-140

-120

-100

-80

-60

-40

PIM

po

we

r/d

Bm

a1 =10

a1 =100

a1 =200

a1 =300

a1 =400

(a)

20 25 30 35 40 45

Input power/dBm

-160

-140

-120

-100

-80

-60

-40

PIM

po

we

r/d

Bm

a3 = 104

a3 = 105

a3 = 106

a3 = 107

a3 = 108

(b)

Figure 2.6: Parameter sweep of PIM3 power vs input carrier power. a): Effectof changing a1 when a3 = 106. b): Effect of changing a3 when a1 = 100.

2.2 Distributed nonlinearityThe model for a discrete nonlinearity, presented in previous section can beexpanded in order to deal with multiple nonlinear sources distributed alongthe signal path. These can be multiple discrete sources, such as a cascade ofconnectors or large electrical structures, such as waveguides and filters. Here,it is important to account for the phase shifts the signal experiences as it travelsthrough the structure. This can be done by cascadingmultiple PIM sources andphase shifts in the model, similarly to [16, 39], as can be seen on Fig. 2.7. Thestructure consists ofK nonlinear resistors andK + 1 phase shifters. A singlephase shifter represents physical length of Lsec, which is chosen such that thephase shift is sufficiently small.

An important observation when looking at Fig. 2.7 is that while the struc-ture is symmetric, the phase shifts of individual elements’ forward and reversePIM products are different due to different path lengths. This has a profoundimpact on properties of forward and reverse PIM due to a process called four-wave mixing. Consider an single cell at the kth section. Two waves of differentfrequencies start with a zero phase shift at the generator, as depicted in Fig. 2.8.


Vg

Vnl

RL

IVnl

VPIMr VPIMfK sections

Figure 2.7: Model of distributed series nonlinearity. VPIMf and VPIMr are thevoltages of the PIM signal for forward and reverse PIM, respectively.

Vg

Vnl

kLsec

RL

(K+1-k)Lseck = 1:K

2=0

1=0

Figure 2.8: Step one of four wave mixing.

Then, the wave obtains a phase shift ϕ1,2 = βf1,2kLsec according to thepropagation constant β as depicted in Fig. 2.9. Since the two tones are atdifferent frequencies, their phase shifts are different.

Vg

Vnl

kLsec

RL

(K+1-k)Lseck = 1:K

2

1

Figure 2.9: Step two of four wave mixing.

The two tones then mix at the nonlinear resistor, creating PIM products inboth forward and reverse directions as seen in Fig. 2.10. The phase is alsomixed accordingly to the discussion in section 1.1.1. The forward and reverseproducts then travel to the generator and the load, see Fig. 2.11, resulting inadditional phase shifts ϕm,n and ϕ3. This results in a final phase shift for the


Vg

Vnl

kLsec

RL

(K+1-k)Lseck = 1:K

PIM = n1-m2

Figure 2.10: Step three of four wave mixing.

reverse wave arriving at the generator

ϕPIMr,k = (mβf1 + nβf2 + βnf1+mf2)kLsec (2.20)

and the phase shift for the wave arriving at the load

ϕPIMf,k = (mβf1 + nβf2 − βmf1+nf2)kLsec + (K + 1)Lsecβmf1+nf2 . (2.21)

Vg

Vnl

kLsec

RL

(K+1-k)Lseck = 1:K

PIMr=n1-m2+n,-m PIMf=n1-m2+3

Figure 2.11: Step four of four wave mixing.

Equations (2.20) and (2.21) describe the propagation aspects of the PIMproducts. Losses in the system can be accounted for in the model by introduc-ing attenuation constant α. The carrier tones and PIM products thus attenuatewith e−αLαf , where L is the physical length travelled by the wave. It is im-portant to note that the attenuation of the carrier tones must be accounted forbefore PIM current is calculated, due to nonlinearity of the phenomenon.

The discussion so far has been focused on accounting the phase behavior ofdistributed nonlinearity, with the choice of the current-voltage relation still leftopen. Here, we make use of the model previously developed for a single dis-crete nonlinearity to calculate the values of PIM currents. Such an approach is


valid provided that other nonlinear resistors do not significantly load the volt-age, as only a single nonlinear resistor is considered in calculation of scatteredPIM waves. This is typically a valid assumption, as PIM is a weakly nonlinearphenomenon [32]. The final equations for forward and reverse PIM voltagesare eq. (2.22) and eq. (2.23), where the current is calculated with the integralrelation (2.16), represented by eq. (2.24) where the the I(V ) relation is givenby eq. (2.3) for positive a3,k and eq. (2.4) for negative a3,k.

VPIM,r =K∑k=1

IPIM,kRgejϕPIMr,ke−kLsecαPIM (2.22)

VPIM,f =K∑k=1

IPIM,kRLejϕPIMf,ke−(N+1−n)LsecαPIM (2.23)

IPIM,k = IPIM(V1e−kLsecαf1 , V2e

−kLsecαf2 , a1,k, a3,k) (2.24)

2.2.1 Verification and examplesThemodel was first verified through comparison with a harmonic balance sim-ulation. A nonlinear lossless dispersionless line was implemented in AWRwith a series of coaxial cables with β = ω

c, where c is the speed of light, and

nonlinear resistors a1 = 2000 and a3 = 106, presented in Fig. 2.12. TheLsec ischosen such that a phase shift of 1 degree at f1 is produced for a single coaxialelement. A subcircuit with 6 resistor-transmission line sections was createdto more easily obtain larger phase shifts. The results obtained from simula-tion and the model developed in the thesis are presented in Fig. 2.13a andFig. 2.13b. Both forward and reverse PIM powers, calculated by the model,match well the values obtained through harmonic balance simulation for boththird and fifth order products.

Figure 2.12: Implementation of nonlinear transmission line in AWR.


0 0.2 0.4 0.6 0.8 1

Line length/m

-140

-130

-120

-110

-100

-90

-80

-70

PIM

3 le

ve

l/d

Bm Reverse PIM

Forward PIM

Reverse PIM AWR

Forward PIM AWR

(a)

0 0.2 0.4 0.6 0.8 1

Line length/m

-210

-200

-190

-180

-170

-160

-150

-140

PIM

5 le

ve

l/d

Bm

Reverse PIM

Forward PIM

Reverse PIM AWR

Forward PIM AWR

(b)

Figure 2.13: a): Lower PIM3 power vs transmission line length calculated bymodel (lines) and AWR simulation (markers) b): Lower PIM5 power vs trans-mission line length calculated bymodel (lines) and AWR simulation (markers)

Forward and reverse PIM

By looking at the behavior of PIM products on Fig. 2.13, we see that thereis a strong difference between the behavior of forward and reverse PIM prod-ucts. While the forward PIM grows when the length of the line is increased,the reverse PIM has nulls when the line length is half of wavelength of thecorresponding PIM product

L = λPIM/2 =2π

mβf1 + nβf2. (2.25)

This is a direct result of the four-wave mixing process. By looking at equationsfor phase of a single section,

ϕPIMf,k = (mβf1 + nβf2 − βmf1+nf2)kLsec + (K + 1)Lsecβmf1+nf2 , (2.26)

ϕPIMr,k = (mβf1 + nβf2 + βmf1+nf2)kLsec, (2.27)

we see that they can be further simplified if we account for the following prop-erty of a dispersionless line

βmf1+nf2 = mβf1 + nβf2 . (2.28)

Equations (2.26) and (2.27) then simplify to

ϕPIMf,k = (K + 1)Lsecβmf1+nf2 , (2.29)

ϕPIMr,k = 2(mβf1 + nβf2)Lseck, (2.30)


for the forward and reverse PIM respectively. The always growing nature offorward PIM can be explained by eq. (2.29). Since all individual componentshave identical phase shift, they constructively interfere regardless of the lengthof the line. On the other hand, the phase shift of a single cell is different forevery cell in the case of reverse PIM, indicated by its dependence on k inequation (2.30). When the length of line is λPIM/2, each section has a pairsection where the phase is opposite, resulting in destructive interference and acomplete cancellation of the reverse PIM product.

The four-wave mixing process has interesting implications on the proper-ties of the spectrum. In the case of the lumped model, the spectrum is sym-metric (upper and lower PIM of the same order have the same power) as themodel’s properties are independent of the frequency. However, due to the factthat phase shifts for a given length are a function of frequency through β(f),this is not the case for distributed nonlinearity or when multiple sources arepresent. Only reverse PIM is affected due to its periodic cancellation whichhas different periods for lower and upper PIM, described by eq. (2.25). Anexample depicting this behavior is presented in Fig. 2.14.

1.75 1.8 1.85 1.9 1.95

Frequency/GHz

-200

-150

-100

-50

0

50

Pow

er

spectr

um

/dB

m

(a)

1.75 1.8 1.85 1.9 1.95

Frequency/GHz

-200

-150

-100

-50

0

50

Pow

er

spectr

um

/dB

m

(b)

Figure 2.14: Power spectres of a nonlinear transmission line with line lengthLline = 317.2 mm. The dark blue lines are PIM products and light blue linesare carrier tones. a): Power spectrum at the generator (reverse PIM). b): Powerspectrum at the load (forward PIM). The parameters are a1 = 2000, a3 = 106,β(f) = 2πf/c0, Lsecβ(fPIM3l) = 1, P = 44 dBm.

Effect of losses on distributed PIM

When lossess are accounted for in the calculation, the behavior of forward andreverse PIM changes, particularly when line lengths become appreciably long,


as can be seen on Fig. 2.15. The forward PIM reaches a peak and then slopesdown, as both the PIM wave and the carrier waves attenuate. Furthermore,the reverse PIM does not exhibit nulls at longer line lengths of kλPIM/2 as inthe lossless case. This is due to the difference in amplitudes between the twocomponents λPIM/2 apart, caused by different lengths (and thus attenuations)the waves had to travel.

0 1 2 3 4

Line length/m

-130

-120

-110

-100

-90

-80

PIM

3 p

ow

er/

dB

m

Reverse PIM3

Forward PIM3

(a)

0 1 2 3 4

Line length/m

-200

-190

-180

-170

-160

PIM

5 p

ow

er/

dB

m

Reverse PIM5

Forward PIM5

(b)

Figure 2.15: a): Lower PIM3 power vs transmission line length calculated bymodel b): Lower PIM5 power vs transmission line length calculated by model(lines). α = 1, a1 = 2000, a3 = 106, β(f) = 2πf/c0, Lsecβ(fPIM3l) = 1,P = 44 dBm.

Effect of dispersion on forward PIM

The behavior of PIM on a dispersive transmission line has not really been stud-ied to the authors’ knowledge. On a real transmission line, waves experiencedispersion if the mode propagating is not TEM, for example in the case ofmicrostrip lines (due to the presence dielectric-air interface) and hollow rect-angular waveguides (where the propagating mode is typically TE10). In thelatter case, the dispersion is especially pronounced if the cutoff frequency isclose to the propagating frequency. Thus, the example studied here is a hol-low lossless rectangular waveguide. The propagation constant describing thedispersion for TE10 modes is

β(f) =2πf

c0

√1−

(fcf

)2. (2.31)

This results in forward PIM exhibiting the same four-wave mixing process asthe reverse PIM, although with a much longer period. This can be calculated


by calculating the length at which the phases first reach 360

L2π,f =2π

mβf1 + nβf2 − βfPIM

(2.32)

for the forward and

L2π,r =2π

mβf1 + nβf2 + βfPIM

(2.33)

for the reverse PIM. Using the equations, we obtain 20.69 m and 0.27 m forforward and reverse lower PIM3 in Fig. 2.17a, respectively. The periods forlower PIM5, presented in Fig. 2.17b, are calculated to be 5.43 m for forwardand 0.35 m for reverse PIM. All calculated periods are in excellent agreementwith the nulls observed in the model results.

Since the forward PIM experiences four-wavemixing, its spectrum also be-comes asymmetric. This is clearly seen on example of a spectrum presentedin Fig. 2.16 and when comparing the plots for lower PIM (Fig. 2.17) to up-per PIM(Fig. 2.18). The forward upper PIM has a larger maximum and largerperiod L2π,f than lower PIM, which is due to the structure becoming less dis-persive at higher frequencies, causing the denominator in eq. (2.32) to tendto zero. Furthermore, the reverse lower PIM has a larger maximum and pe-riod than upper PIM, as the denominator in eq. (2.33) becomes larger withfrequency. This difference is increased with dispersion, as βfPIM

goes to zeroat fc.

1.75 1.8 1.85 1.9 1.95

Frequency/GHz

-200

-150

-100

-50

0

50

Po

we

r sp

ectr

um

/dB

m

(a)

1.75 1.8 1.85 1.9 1.95

Frequency/GHz

-200

-150

-100

-50

0

50

Po

we

r sp

ectr

um

/dB

m

(b)

Figure 2.16: a): Power spectrum at the generator (reverse PIM). b): Powerspectrum at the load (forward PIM). The β(f) is taken to be eq. (2.31) withfc = 1.7 GHz. The parameters are Lline = 22.95 m, a1 = 2000, a3 = 106,Lsecβ(fPIM3l) = 1, P = 44 dBm.


0 5 10 15 20 25

Line length/m

-120

-100

-80

-60

-40Low

er

PIM

3 p

ow

er/

dB

m Reverse PIM3l

Forward PIM3l

0 0.5 1-140

-120

-100

-80

(a)

0 5 10 15 20 25

Line length/m

-200

-180

-160

-140

-120

Low

er

PIM

5 p

ow

er/

dB

m Reverse PIM5l

Forward PIM5l

0 0.5 1-200

-180

-160

-140

(b)

Figure 2.17: a): Lower PIM3 power vs transmission line length calculatedby model b): Lower PIM5 power vs transmission line length calculated bymodel (lines). The inset depicts first 1 m of the figure. The β(f) is taken to beeq. (2.31) with fc = 1.7 GHz. The line parameters are a1 = 2000, a3 = 106,Lsecβ(fPIM3l) = 1, P = 44 dBm.

0 5 10 15 20 25

Line length/m

-120

-100

-80

-60

-40

Upper

PIM

3 p

ow

er/

dB

m Reverse PIM3u

Forward PIM3u

0 0.5 1-140

-120

-100

-80

(a)

0 5 10 15 20 25

Line length/m

-200

-180

-160

-140

-120

Upper

PIM

5 p

ow

er/

dB

m Reverse PIM5u

Forward PIM5u

0 0.5 1-200

-180

-160

-140

(b)

Figure 2.18: a): Upper PIM3 power vs transmission line length calculated bymodel b): Upper PIM5 power vs transmission line length calculated by model(lines). The inset depicts first 1 m of the figure. The β(f) is taken to beeq. (2.31) with fc = 1.7 GHz. The line parameters are a1 = 2000, a3 = 106,Lsecβ(fPIM3l) = 1, P = 44 dBm.

Chapter 3

Measurements and results

The models developed in previous chapter were so far only verified throughharmonic balance simulations. However, such verification only shows the ap-propriateness of assumptions and numerical procedures used. To fully eval-uate the model, measurements are needed. The fundamental idea for verifi-cation is to measure multiple orders of PIM and obtain coefficients a1 and a3through fit of PIM3 vs power measurements. Then, the model can be used topredict higher order products which can be compared to the measurements. Ifthe agreement is good, the model is appropriate and the coefficients a1 and a3give sufficient description of the nonlinearity.

A transition from stripline to microstrip, seen in Fig. 3.2, was chosen asthe experimental setup as it consists from several typical sources, such as con-nectors and the transition, which are rarely studied in the literature. The transi-tion consists of an right angle shape connector (Rosenberger 32K201-400L5),a 70 mm long stripline, a transition from stripline to microstrip, 49 mm longmicrostrip line and a straight connector (Rosenberger 32K145-400L5). Tosimplify the problem, only connectors and transitions are modelled as a PIMsource. The contributions from transmission lines are neglected, as literatureshows the reverse PIM level of transmission lines is typically much lower [22,34] than connectors [1, 6]. Thus, the equivalent model of the transition, pre-sented in Fig. 3.3, is modelled by three nonlinear resistors with three differentcoefficients a1 and a3, to be characterized, and two phase shifts accountingfor the electrical length of transmission lines. However, this still gives us 3x2parameters for a single set of PIM3 measurements. Therefore, the connectorsmust first be characterized independently with a back to back transition. Then,the known connector a1 and a3 coefficients can be used in the model and theunknown coefficients of the transition, a1t and a3t, can be extracted.

32

CHAPTER 3. MEASUREMENTS AND RESULTS 33

Themeasurements were performedwithAnritsu PIMMasterMW82119A.The device measures reverse lower PIM and allows for measurements of up to7th order in two power ranges of 20-37 dBm and 37-46 dBm. The frequen-cies of the two carrier tones are f1 = 1822 MHz and f1 = 1859 MHz andare chosen such that they can be used for PIM products fPIM3 = 1785 MHz,fPIM5 = 1748 MHz, fPIM7 = 1711 MHz. Repeatability of results is ensuredwith a careful procedure of connecting and disconnecting the device under test.The connectors are cleaned with compressed air before each mounting. Thisis very important as the screwing may cause small metal particles to be cre-ated, increasing the overall PIM generation. Additionally, the connectors aremounted with a 0.9 Nm torque screw, ensuring a good electrical connectionand stability and repeatability of measurements [16].

Figure 3.1: Calibration of PIM Master MW82119A. A low PIM load is con-nected and the calibration procedure is currently running.

34 CHAPTER 3. MEASUREMENTS AND RESULTS

The measurement setup is calibrated when first started or switching amongmeasuring different PIM products. Recalibration is needed when the deviceexperiences significant temperature drift. Calibration procedure starts by con-necting a low PIM 50 Ω load, as presented in Fig. 3.1. Then, the calibrationroutine in the device is run. After the calibration is complete, the low PIM loadis unscrewed and screwed back and the noise floor is measured. Under idealconditions, the PIM level of the linear load is non detectable and the measuredPIM level is -140 dBm. However, sometimes this is difficult to achieve so thecalibration is accepted when the measured noise floor is below -120 dBm at46 dBm power of carrier signals.

Figure 3.2: Picture of stripline to microstrip transition to be characterized.

Vg

Vnl

RL

IVnl

a1m, a3m a1t, a3t a1s, a3s

Vnl

Figure 3.3: Equivalent PIM model for the transition.

3.1 Connector characterizationTo first determine the PIM contribution of individual connectors, two differentback to back transitions were constructed. Since any back to back transition


uses the same connector on both sides, the equivalent model only has a sin-gle set of a parameters to be determined. Thus, the equivalent model for bothconsists of two identical nonlinear resistors and a phase shift representing thetransmission line, as depicted on Fig. 3.4. The phase shift and losses are mea-sured by a VNA and the a coefficients are extracted through a PIM3 versuscarrier power fit. The equation used for fitting is

IPIM = |I(P1, P2, a1, a3)+GPIMI(P1+g1, P2+g2, a1, a3)ej(mϕ1+nϕ2+ϕPIM )|,

(3.1)where the phase shifts ϕ and losses G and g are calculated from VNA mea-surements by

ϕk = 6 S2,1(fk), gk = 20 log10 |S2,1(fk)|, Gk = |S2,1(fk)|. (3.2)

Vg

Vnl

RL

I

a1, a3

Vnl

a1, a3

Figure 3.4: Equivalent PIM model for back to back transitions, used for char-acterizing the two connectors.

3.1.1 Right angle connectorsThree back to back transitions were constructed to characterize right angleconnectors. The transmission line is a 70 mm long stripline, built from RogersRO4003C. The S-parameters of the individual transitions can be seen on Fig. 3.6.All transitions are matched below -20 dB and exhibit similar insertion loss ofabout 0.2 dB. However, the first built transition (S1) exhibits a 6.1 degree dif-ference in phase shift over the structure when compared to transitions S2 andS3. The difference is likely due to manufacturing variances.

The PIM vs power curve was measured for all three samples, as presentedon Fig 3.5a. The measurement results can be seen on Fig. 3.7. While the mea-sured PIM3 level in all samples is remarkably similar, the PIM5 level exhibits


(a) (b)

Figure 3.5: Measurements of PIM level for back to back transition. a):Stripline back to back, for characterizing right angle connector. b): Microstripback to back, for characterizing straight connector.

a greater variance. This could be due to manufacturing differences, measure-ment error or an indication of noise floor presence.

The PIM3 power vs input carrier tone power was fitted to the model repre-sented by eq. (3.1) and coefficients a1 and a3 were obtained. The individual fitsand predicted higher order PIM products can be seen on Figs. 3.8, 3.9 and 3.10for first, second and third sample respectively. The fits match very well to themeasurements presented. However, the higher order modes are not accuratelypredicted by the model, with a difference of about 19 dB, indicating the non-linearity is not well described by the model. The corresponding coefficientscan be seen on Tab. 3.1. By using them in the model of a single nonlinearresistor, we can predict the contribution to PIM power of a single right-angleconnector. This was done in Fig. 3.11 for all three sets of parameters.

a1,s a3,sSample 1 (stripline 1, S1) 1.060 · 103 1.555 · 108

Sample 2 (stripline 2, S2) 1.008 · 103 1.382 · 108

Sample 3 (stripline 3, S3) 1.170 · 103 1.489 · 108

Table 3.1: Extracted a parameters for a right-angle connector.


1.7 1.75 1.8 1.85 1.9

Frequency/GHz

-40

-20

0

|S11|/dB

1.7 1.75 1.8 1.85 1.9

Frequency/GHz

-0.4

-0.2

0

|S21|/dB

1.7 1.75 1.8 1.85 1.9

Frequency/GHz

-40

-20

0

S21/d

eg.

S1

S2

S3

Figure 3.6: S-parameter measurements of stripline back to back transition withright angle connectors. S1, S2 and S3 represent first, second and third striplineback-to-back transition sample, respectively.


25 30 35 40 45 50

Carrier input power/dBm

-130

-120

-110

-100

-90

-80

-70P

IM p

ow

er/

dB

m PIM3 Stripline 1

PIM3 Stripline 2

PIM3 Stripline 3

PIM5 Stripline 1

PIM5 Stripline 2

PIM5 Stripline 3

Figure 3.7: PIM versus carrier power measurements of stripline back to backtransition with right angle connectors.

25 30 35 40 45


-140

-130

-120

-110

-100

-90

-80

-70

-60

PIM

pow

er/

dB

m

PIM3 stripline 1 measurements

PIM3 model fit




Figure 3.8: Model fit of PIM versus carrier power measurements for striplineback to back transition (sample 1) with right angle connectors.


25 30 35 40 45


-140

-130

-120

-110

-100

-90

-80

-70

-60

PIM

pow

er/

dB

m


PIM3 model fit





25 30 35 40 45


-140

-130

-120

-110

-100

-90

-80

-70

-60

PIM

pow

er/

dB

m


PIM3 model fit






25 30 35 40 45


-140

-130

-120

-110

-100

-90

-80

-70

PIM

pow

er/

dB

m

PIM3 parameters 1

PIM5 parameters 1

PIM7 parameters 1

PIM3 parameters 2

PIM5 parameters 2

PIM7 parameters 2

PIM3 parameters 3

PIM5 parameters 3

PIM7 parameters 3

Figure 3.11: Extracted PIM vs power produced by a single right-angle con-nector for parameters in Table 3.1.


3.1.2 Straight connectorsTo characterize straight connectors, three back to back transitionswith a 49mmlong microstrip line were built. As before, the substrate chosen was RO4003C.The S-parameters of the samples can be seen on Fig. 3.12. All samples arematched below -15 dB and have insertion loss of about 0.2 dB. Furthermore,the third sample experiences a phase deviation of about 3.6 degrees, likely dueto manufacturing variances.

1.7 1.75 1.8 1.85 1.9

Frequency/GHz

-15

-10

-5

0

|S1

1|/dB M1

M2

M3

1.7 1.75 1.8 1.85 1.9

Frequency/GHz

-0.4

-0.2

0

|S2

1|/dB

1.7 1.75 1.8 1.85 1.9

Frequency/GHz

120

140

S2

1/d

eg.

Figure 3.12: S-parameter measurements of microstrip back to back transitionswith straight connectors. M1, M2 and M3 represent first, second and thirdmicrostrip back-to-back transition sample, respectively.

The PIM vs power curve was measured for all three samples, as presentedon Fig 3.5b. The measurement results can be seen on Fig. 3.13. The sampleshave a similar PIM vs power slope for all powers, with first sample having anapproximately 5 dB lower PIM3 power measured. The offset can be explainedby a combination of manufacturing differences and measurement error.


As before, the coefficients a1 and a3 were obtained through a fit of PIM3power vs input carrier tone power. The plots of the fits and predicted higherorder PIM products can be seen on Figs. 3.14, 3.15 and 3.16 for first, secondand third sample respectively. In all measurements, the predicted PIM5 curvehas a similar slope, but a level shift of about 5 dB is observed for samples 2and 3. Unfortunately, the measurements of PIM7 were not possible due to thepresence of the noise floor. The extracted coefficients are seen on Table 3.2.They were used to calculate individual contribution of a single connector inFig. 3.17.

25 30 35 40 45 50


-140

-130

-120

-110

-100

-90

-80

-70

PIM

po

we

r/d

Bm

PIM3 Microstrip 1

PIM3 Microstrip 2

PIM3 Microstrip 3

PIM5 Microstrip 1

PIM5 Microstrip 2

PIM5 Microstrip 3

Figure 3.13: PIM versus carrier power measurements of stripline back to backtransition with right angle connectors.

a1,m a3,mSample 1 (micro 1, M1) 3.471 · 102 9.655 · 105

Sample 2 (micro 2, M2) 2.743 · 102 5.712 · 105

Sample 3 (micro 3, M3) 1.668 · 102 8.4724 · 104

Table 3.2: Extracted a parameters for a straight connector.


25 30 35 40 45


-140

-130

-120

-110

-100

-90

-80

-70

-60

PIM

po

we

r/d

Bm

PIM3 micro 1 measurements

PIM3 model fit




Figure 3.14: Model fit of PIM versus carrier power measurements for mi-crostrip back to back transition (sample 1) with straight connectors.

25 30 35 40 45


-140

-130

-120

-110

-100

-90

-80

-70

-60

PIM

po

we

r/d

Bm


PIM3 model fit






25 30 35 40 45


-140

-130

-120

-110

-100

-90

-80

-70

-60P

IM p

ow

er/

dB

mPIM3 micro 3 measurements

PIM3 model fit





25 30 35 40 45


-140

-130

-120

-110

-100

-90

-80

-70

-60

PIM

po

we

r/d

Bm

PIM3 parameters 1

PIM5 parameters 1

PIM7 parameters 1

PIM3 parameters 2

PIM5 parameters 2

PIM7 parameters 2

PIM3 parameters 3

PIM5 parameters 3

PIM7 parameters 3

Figure 3.17: Extracted PIM vs power produced by a single right-angle con-nector for parameters in Table 3.2.


3.2 Microstrip to stripline transitionNow that the coefficients of the two connectors are known (Tables 3.1 and 3.2),the transition frommicrostrip to stripline can be characterized. Two transitionswere built and their S-parametersmeasurements can be seen on Fig. 3.18. Bothtransitions are matched below -15 dB and have insertion loss of approximately0.4 dB. The phase shifts over the structures are very similar with only a slightvariation.

1.7 1.75 1.8 1.85 1.9

Frequency/GHz

-15

-10

-5

0

|S11|/dB T1

T2

1.7 1.75 1.8 1.85 1.9

Frequency/GHz

-0.4

-0.2

0

|S21|/dB

1.7 1.75 1.8 1.85 1.9

Frequency/GHz

150

200

S21/d

eg.

Figure 3.18: S-parameter measurements of stripline to microstrip transition.T1 and T2 represent first and second stripline to microstrip transition sample,respectively.


Since there are three sources, the fitting equation becomes

IPIM = |I(P1, P2, a1,s, a3,s)+

GPIM,sI(P1 + g1,s, P2 + g2,s, a1,t, a3,t)ejϕtran+

+GPIM,tI(P1 + g1,t, P2 + g2,t, a1,m, a3,m)ejϕt|, (3.3)

where ak,s and ak,m are already known from Tab. 3.1 and Tab. 3.2. As before,GPIM,s (linear) and g1,s (in dB) are estimated losses (including the connectorlosses) of the stripline section and GPIM,t and g1,t of the overall transition.The phase

ϕt = m6 S2,1,T (f1) + n6 S2,1,T (f2) + 6 S2,1,T (fPIM) (3.4)

is readily obtained from the transition S-parameters. However, the phase

ϕtran = mϕ1,s + nϕ2,s + ϕPIM,s, (3.5)

which represents propagation from a single right angle connector to the transi-tion junction needs to be obtained. The phase shift has two contributions, theconnector and the transmission line. The phase shift of stripline is obtainedthrough a simulation in CST Microwave Studio [40]. The phase shift of theconnector is obtained by subtracting the simulated stripline phase shift fromthe phase shift of the corresponding back-to-back measurement. The remain-ing difference is attributed to phase shifts over the connectors and is halved toobtain phase shift over a single connector.

The PIM versus power measurements are presented on Fig. 3.19. Bothtransitions exhibit a higher PIM level, allowing for measurements of 7th orderproducts. A noticable decrease in slope of PIM3 at powers larger than 43 dBmis seen and can be attributed to saturation of the receiver of the measurementsetup. Thus, these measurements are not considered when fitting. Further-more, a kink in PIM5 measurement of transition 1 can be seen at 37 dBm.This kink persisted even when recalibrating and remeasuring the sample.

The transition was characterized by fitting the model from eq. (3.3) to thePIM3 data. The resulting plots can be seen on Fig. 3.20 for sample 1 andFig. 3.21 for sample 2. Both plots show excellent agreement of model pre-diction and measurements, with the fifth and even seventh order PIM productmatching in both slope and level. Similar results are obtained regardless whichcoefficients are taken from Tab. 3.1 and 3.2. The extracted coefficients are tab-ulated in Tab. 3.3.


25 30 35 40 45 50


-130

-120

-110

-100

-90

-80

-70

-60

PIM

pow

er/

dB

m

PIM3 transition 1

PIM3 transition 2

PIM5 transition 1

PIM5 transition 2

PIM7 transition 1

PIM7 transition 2

Figure 3.19: PIM versus carrier power measurements of stripline to microstriptransition with right angle connectors.

25 30 35 40 45 50


-130

-120

-110

-100

-90

-80

-70

-60

PIM

pow

er/

dB

m

PIM3 measurements

PIM3 model fit

PIM5 measurements


PIM7 measurements


Figure 3.20: Model fit of PIM versus carrier power measurements for striplineto microstrip transition (sample 1).


25 30 35 40 45 50


-130

-120

-110

-100

-90

-80

-70

-60

PIM

pow

er/

dB

mPIM3 measurements

PIM3 model fit

PIM5 measurements


PIM7 measurements


Figure 3.21: Model fit of PIM versus carrier power measurements for striplineto microstrip transition (sample 2).

25 30 35 40 45


-140

-130

-120

-110

-100

-90

-80

-70

-60

PIM

po

we

r/d

Bm

PIM3 transition 1

PIM5 transition 1

PIM7 transition 1

PIM3 transition 2

PIM5 transition 2

PIM7 transition 2

Figure 3.22: Extracted PIM vs power produced by the transition for parametersin Table 3.3.


a1 a3Sample 1 (transition 1, T1) 2.632 · 102 5.547 · 106

Sample 2 (transition 2, T2) 3.198 · 102 9.55 · 106

Table 3.3: Extracted a parameters for the transition. The connector coefficientsused in the model are a1,m = 3.471 ·102, a3,m = 9.655 ·105, a1,s = 1.060 ·103

a3,s = 1.555 · 108.

3.3 DiscussionThe measurement procedure was challenging, in spite of the device being easyto set up and calibrate. The measurements are very sensitive and deviations ofabout ±1 dB were observed by just screwing and unscrewing the device. Fur-thermore, to reduce residual PIM in the system, no cables were used. As thelow PIM load is very heavy, the device was thus lain down as seen on Fig. 3.5and the connectors were aligned with a variable height table. Nevertheless,occasionally the mechanical connection of connectors was not properly made,resulting in a discrepancy in measured PIM level. For this reason, the mea-surements taken were checked multiple times to ensure a consistent PIM level.Moreover, due to the low power of the signals measured, the measurements be-come very noisy when approaching 25 dBm of input power. Thus, the lowerend of the measurements has a bigger measurement uncertainty.

The model developed seems to be able to produce a good fit for the en-tire PIM3 vs input power measurement, regardless of the device measured.Furthermore, in the case of straight and right-angled connector as well as theoverall transition it estimates the higher order PIM products to a good degree ofaccuracy, as evident from Figs. 3.15 and 3.20. This implies that the model de-scribes the nonlinearity properties well. However, its ability to predict higherorder products occasionally fails, as seen in the case of right-angle connectoron Figs. 3.8, 3.9 and 3.10. It could be that the model series chosen here is in-appropriate for this connector or that higher orders, such as a5 should be takeninto account in chosen I(V ) curve. However, this introduces new challengesin fitting as more parameters could result in too many degrees of freedom.Additionally, it is possible that a distributed model might perform better asthe connectors have a notable phase shift (about 50 at 1.8 GHz). Thus, it ispossible that a distributed model could be more appropriate in this case.

The characterization procedure has been shown to work and excellent re-sults have been obtained in Figs. 3.20 and 3.21. However, it is important to notethat this is likely due to the fact that the PIM level of the stripline to microstrip


transition is much higher. In fact, the unknown PIM source to be characterizedmust always be stronger than other sources. The reason for this is that, whilethe characterization procedure attempts to remove presence of other sources,the small differences between the model and reality will always remain due toimperfect modelling and measurement errors. Thus, if the source to be char-acterized is not dominant, it is possible that the fitting done fits these errorsrather than the intermodulation produced by the component itself.

Chapter 4

Future work and conclusion

4.1 ConclusionIn this master thesis, the topic of passive intermodulation modelling was ex-plored.

First, a literature survey was conducted. Based on current understandingof the field, a generic nonlinear resistor model was chosen as it is more flexibleand general than physical based models. The model accounts for the loadingeffects between nonlinear component and the linear resistor, used for termina-tion in the measurements. Additionally, three different methods of calculat-ing intermodulation products were developed and tested. The first, expandingI(Vg) into a Taylor series was found to be unstable in certain cases and thusunreliable. The second, where the I(Vg) is transformed via the FFT was foundto be too slow. The third, where orthogonality is used to extract only a singlespectral component was found to be both numerically stable and computation-ally fast.

Second, the discrete model was expanded to include multiple sources andsignal propagation. This model allows for modelling of distributed structures,such as waveguides and filters. Furthermore, the differences between for-ward and reverse PIM were qualitatively explained. Additionally, it was foundthrough the model that in heavily dispersive waveguides, the forward PIM ex-periences similar cancellation as the reverse PIM. Equation for line lengths atwhich these cancellations occur were derived.

Finally, the developed model was verified through PIM vs power measure-ments. Three different components were characterized by fitting the modelparameters to PIM3 versus power curve, extracting the components’ approx-imate current-voltage relations. In all cases, it was found to be a compact

51

52 CHAPTER 4. FUTURE WORK AND CONCLUSION

description of PIM3 vs power curve for a wide range of powers 25-46 dB. Fur-thermore, it was able to predict with a good degree of accuracy the power ofhigher order products for two out of three components.

4.2 Future workThere is still more work to be done in understanding passive intermodulation.A survey of the literature shows that in many cases, the origin of nonlinearityis not completely known or explored. For example, further comparisons areneeded to establish when electrothermal PIM is present in the system, partic-ularly on transmission lines [9, 13]. Furthermore, an overview of componentsand the physical sources of PIM generation could be very useful to a new re-searcher or an engineer choosing among components.

Additionally, most measurements in the literature measure PIM at a singlefrequency. More measurements are needed to better understand the frequencybehavior of PIM generation and to discern the sources. Particularly, there is alack of measurements in the Ka band, which will be used in 5G. For this highfrequencies, new measurement setups would need to be developed.

Moreover, commonlymissing from all papers is the phase behavior of PIM.The reason for this is that such measurements are not easy to do with the dy-namic range required for detecting PIM signals. Nevertheless, knowing thephase is crucial for understanding the type (resistive, inductive or capacitive)of the nonlinearity. Thus, one of the possible research lines is to use the setuppresented in [29, 28] to measure phase of PIM products.

Finally, there is little exploration of how metasurfaces compare to tradi-tional structures in terms of PIM performance. While there is some evidencethat the metasurfaces generate similar PIM levels [33], more studies are re-quired to elucidate PIM performance of novel structures.

Bibliography

[1] Justin JHenrie, Andrew JChristianson, andWilliam JChappell. “Linear–nonlinear interaction and passive intermodulation distortion”. In: IEEETransactions onMicrowave Theory and Techniques 58.5 (2010), pp. 1230–1237.

[2] C Vicente and H.L Hartnagel. “Passive-intermodulation analysis be-tween rough rectangular waveguide flanges”. eng. In: IEEE Transac-tions on Microwave Theory and Techniques 53.8 (2005), pp. 2515–2525. issn: 0018-9480.

[3] C Vicente et al. “Experimental Analysis of Passive Intermodulation atWaveguide Flange Bolted Connections”. eng. In: IEEE Transactions onMicrowave Theory and Techniques 55.5 (2007), pp. 1018–1028. issn:0018-9480.

[4] Xiaolong Zhao et al. “Analytic Passive IntermodulationModel for FlangeConnection Based on Metallic Contact Nonlinearity Approximation”.eng. In: IEEE Transactions on Microwave Theory and Techniques 65.7(2017), pp. 2279–2287. issn: 0018-9480.

[5] Kai Zhang, Tuanjie Li, and Jie Jiang. “Passive intermodulation of con-tact nonlinearity on microwave connectors”. In: IEEE Transactions onElectromagnetic Compatibility 60.2 (2018), pp. 513–519.

[6] Qiuyan Jin et al. “Modeling of Passive Intermodulation With Electri-cal Contacts in Coaxial Connectors”. eng. In: IEEE Transactions onMicrowave Theory and Techniques 66.9 (2018), pp. 4007–4016. issn:0018-9480.

[7] Farrokh Arazm and Frank A Benson. “Nonlinearities in Metal ContactsatMicrowave Frequencies”. eng. In: IEEE Transactions on Electromag-netic Compatibility EMC-22.3 (1980), pp. 142–149. issn: 0018-9375.

53

54 BIBLIOGRAPHY

[8] Jonathan RWilkerson et al. “Electro-thermal theory of intermodulationdistortion in lossy microwave components”. In: IEEE Transactions onMicrowave Theory and Techniques 56.12 (2008), pp. 2717–2725.

[9] Jonathan R Wilkerson et al. “Distributed passive intermodulation dis-tortion on transmission lines”. In: IEEE Transactions on MicrowaveTheory and Techniques 59.5 (2011), pp. 1190–1205.

[10] Jonathan R Wilkerson et al. “Passive intermodulation distortion in an-tennas”. In: IEEETransactions on Antennas and Propagation 63.2 (2015),pp. 474–482.

[11] Jonathan R Wilkerson, Kevin G Gard, and Michael B Steer. “Electro-thermal passive intermodulation distortion in microwave attenuators”.In: 2006 European Microwave Conference. IEEE. 2006, pp. 157–160.

[12] E Rocas et al. “Passive Intermodulation Due to Self-Heating in PrintedTransmission Lines”. eng. In: IEEE Transactions onMicrowave Theoryand Techniques 59.2 (2011), pp. 311–322. issn: 0018-9480.

[13] Dmitry Kozlov et al. “Practical Mitigation of Passive Intermodulationin Microstrip Circuits”. eng. In: IEEE Transactions on ElectromagneticCompatibility PP.99 (2018), pp. 1–10. issn: 0018-9375.

[14] A.J Christianson, J.J Henrie, and W.J Chappell. “Higher Order Inter-modulation Product Measurement of Passive Components”. eng. In:IEEE Transactions on Microwave Theory and Techniques 56.7 (2008),pp. 1729–1736. issn: 0018-9480.

[15] Y Yamamoto and N Kuga. “Short-Circuit Transmission Line Methodfor PIM Evaluation of Metallic Materials”. eng. In: IEEE Transactionson Electromagnetic Compatibility 49.3 (2007), pp. 682–688. issn: 0018-9375.

[16] Justin Henrie, Andrew Christianson, and William J Chappell. “Predic-tion of passive intermodulation from coaxial connectors in microwavenetworks”. In: IEEE transactions on microwave theory and techniques56.1 (2008), pp. 209–216.

[17] J Henrie, A Christianson, and W.J Chappell. “Engineered Passive Non-linearities for Broadband Passive Intermodulation Distortion Mitiga-tion”. eng. In: IEEEMicrowave andWireless Components Letters 19.10(2009), pp. 614–616. issn: 1531-1309.

BIBLIOGRAPHY 55

[18] T. Dahm and D. J. Scalapino. “Theory of intermodulation in a super-conducting microstrip resonator”. eng. In: Journal of Applied Physics81.4 (1997), pp. 2002–2009. issn: 0021-8979.

[19] C Collado, J Mateu, and J.M O’Callaghan. “Analysis and simulation ofthe effects of distributed nonlinearities in microwave superconductingdevices”. eng. In: IEEE Transactions on Applied Superconductivity 15.1(2005), pp. 26–39. issn: 1051-8223.

[20] Kazunori Yamanaka et al. “Intermodulation distortion characteristicsof a 4GHz-band HTS microstrip-disk filter”. eng. In: Physica C: Su-perconductivity and its applications 445-448.1-2 (2006), pp. 998–1002.issn: 0921-4534.

[21] M. I. Salkola. “Intermodulation response of superconducting filters”.eng. In: Journal of Applied Physics 98.2 (2005). issn: 0021-8979.

[22] Alexey P Shitvov, Dmitry S Kozlov, and Alexander G Schuchinsky.“Nonlinear characterization for microstrip circuits with low passive in-termodulation”. In: IEEE Transactions onMicrowave Theory and Tech-niques 66.2 (2018), pp. 865–874.

[23] D.EZelenchuk et al. “Passive Intermodulation in Finite Lengths of PrintedMicrostrip Lines”. eng. In: IEEE Transactions on Microwave Theoryand Techniques 56.11 (2008), pp. 2426–2434. issn: 0018-9480.

[24] G Macchiarella, G.B Stracca, and L Miglioii. “Experimental study ofpassive intermodulation in coaxial cavities for cellular base stations du-plexers”. eng. In: 34th European Microwave Conference, 2004. Vol. 2.IEEE, 2004, pp. 981–984. isbn: 1580539920.

[25] The PIM Source. Using Distance-to-PIM for Faster PIM Fault Identifi-cation. 2014.url:https://anritsu.typepad.com/thepimsource/2014/10/using-distance-to-pim-for-faster-pim-fault-identification-.html (visited on 10/10/2019).

[26] S Hienonen et al. “Near-field scanner for the detection of passive inter-modulation sources in base station antennas”. eng. In: IEEE Transac-tions on Electromagnetic Compatibility 46.4 (2004), pp. 661–667. issn:0018-9375.

[27] S Hienonen, P Vainikainen, and A.V Raisanen. “Sensitivity measure-ments of a passive intermodulation near-field scanner”. eng. In: IEEEAntennas and Propagation Magazine 45.4 (2003), pp. 124–129. issn:1045-9243.

56 BIBLIOGRAPHY

[28] YeMing et al. “Relative phase measurement of passive intermodulationproducts”. In: 2015 12th IEEE International Conference on ElectronicMeasurement & Instruments (ICEMI). Vol. 2. IEEE. 2015, pp. 1072–1076.

[29] AaronWalker, Michael Steer, and Kevin Gard. “Simple, broadband rel-ative phasemeasurement of intermodulation products”. In: 65th ARFTGConference Digest, 2005. Spring 2005. IEEE. 2005, 5–pp.

[30] Justin Henrie, Andrew Christianson, and William J Chappell. “Linear-nonlinear interaction’s effect on the power dependence of nonlinear dis-tortion products”. In: Applied Physics Letters 94.11 (2009), p. 114101.

[31] DEZelenchuk et al. “Discrimination of passive intermodulation sourcesonmicrostrip lines”. In:Proc. Sixth Int.Workshop onMultipactor, Coronaand Passive intermodulation in Space RF Hardware, Valencia, Spain.2008, pp. 1–4.

[32] JordiMateu et al. “Third-order intermodulation distortion and harmonicgeneration inmismatchedweakly nonlinear transmission lines”. In: IEEETransactions onMicrowave Theory and Techniques 57.1 (2009), pp. 10–18.

[33] Aleksey P Shitvov et al. “Effects of geometrical discontinuities on dis-tributed passive intermodulation in printed lines”. In: IEEE Transac-tions on Microwave Theory and Techniques 58.2 (2010), pp. 356–362.

[34] AP Shitvov et al. “Passive intermodulation in printed lines: Effects oftrace dimensions and substrate”. In: IET microwaves, antennas & prop-agation 3.2 (2009), pp. 260–268.

[35] Alexey P Shitvov and Alexander G Schuchinsky. “Distributed passiveintermodulation generation in coplanar waveguide transmission lines”.In: 2012 IEEE/MTT-S InternationalMicrowave SymposiumDigest. IEEE.2012, pp. 1–3.

[36] IreneOrtiz De Saracho Pantoja. “Study on Passive Intermodulation (PIM)inMicrowave Filters”.MA thesis. Sweden: KTHRoyal Institute of Tech-nology, 2018.

[37] Qiuyan Jin et al. “Modeling of Passive Intermodulation in Connec-tors With Coating Material and Iron Content in Base Brass”. eng. In:IEEE Transactions on Microwave Theory and Techniques 67.4 (2019),pp. 1346–1356. issn: 0018-9480.

BIBLIOGRAPHY 57

[38] Wolfram Research. Wolfram Mathematica. 2020. url: https : / /www.wolfram.com/mathematica/ (visited on 04/09/2020).

[39] Qiuyan Jin et al. “Behavior modeling of passive intermodulation dis-tortion with multiple nonlinear sources”. In: Microwave and OpticalTechnology Letters 60.9 (2018), pp. 2182–2185. issn: 0895-2477.

[40] Dassault systemes. CST Studio Suite. 2020. url: https://www.3ds.com/products-services/simulia/products/cst-studio-suite/ (visited on 04/09/2020).

www.kth.seTRITA-EECS-EX-2020:290

Date post:	22-Nov-2021
Category:	Documents
Upload:	others
View:	15 times
Download:	0 times

Modelling of passive intermodulation in RF systems

Documents