This document discusses nonlinear effects in RF transceiver module design. It begins by outlining the causes of nonlinear distortion, including internal and external interference effects. It then analyzes specific nonlinear effects like 1-dB compression point, second-order intercept point, and third-order intercept point. The document examines these effects for both single-tone and two-tone input signals. Nonlinear characteristics are evaluated using concepts like intercept points and two-tone intermodulation distortions. Linear and nonlinear amplifier classes are also introduced.
This presentation covers noise performance of Continuous wave modulation systems; It explains modelling of white noise , noise figure of DSB-SC, SSB, AM, FM system
Electronic circuits include a diode, a BJT (Bipolar Junction Transistor), and a FET (Field Effect Transistor). In this presentation, we briefly summarize the most important characteristics of the diode, transistor, amplifier, filter, and mixer.
Basic blocks to understand RFFE Architecture. how Analog front end and Digital front is different. Basic components like Filter, Mixer, Power Amplifier, circulator, Duplexer, LNA and demodulator working is explained. It can held to design your own front end as RF link budget has been explained in well manner. what to do to avoid saturation of PA?
In the case of class A amplifier, we have observed that the transistor conducts for
the full cycle of the input signal i.e. the conduction angle is 180◦. Although
the transistor conducts for the full cycle of the input signal, the power conversion
efficiency is poor in class A amplifier. In addition to that, a great deal of
distortion is introduced by the nonlinearity in the dynamic transfer characteristic
of the transistor. The power conversion efficiency can be improved by biasing
the transistor at cut off point on VCE axis and a great deal of the distortion
due to nonlinearity in dynamic transfer characteristic may be eliminated by
the push-pull configuration of the transistor as discussed in next section
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
We have compiled the most important slides from each speaker's presentation. This year’s compilation, available for free, captures the key insights and contributions shared during the DfMAy 2024 conference.
About
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Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
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Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)
RF Module Design - [Chapter 3] Linearity
1. RF Transceiver Module Design
Chapter 3 Nonlinear Effects
李健榮 助理教授
Department of Electronic Engineering
National Taipei University of Technology
2. Outline
• Nonlinear Effects on an RF Signal
• Analysis of 1-dB-Compression Point (P1dB)
• Analysis of Second-Order Intercept Point (IP2)
• Analysis of Third-Order Intercept Point (IP3)
• Nonlinear Effect of a Cascaded System
• Nonlinear Effect on a Digitally-Modulated Signal
Department of Electronic Engineering, NTUT2/49
3. Nonlinear Effects
• The distortion of an RF transceiver are resulted from internal
interferences and external interferences.
1) The internal interferences are generated from the nonlinear
effect of its own devices.
2) The external interference are from outside the transceiver
and intercepted by the antenna or EM coupling.
3) Internal distortion is primarily generated from power
amplifier.
Department of Electronic Engineering, NTUT3/49
4. Power Amplifier Categories
• Linear Amplifier: Class A, B, AB, and C
Classified in terms of current conduction angle
CEv
,maxCEVkneeV QV
,maxCI
Ci
QI
A
AB
BC
Biased Transistor
Input Matching Output Matching
Department of Electronic Engineering, NTUT4/49
5. Linear Amplifier
Normalized DSi
A
C B AB
0 π 2π
tω
Class Duty Cycle Theoretical Efficiency Linearity
A 100% 50% Excellent
B 50% 78.5% Moderate
AB 50~100% 50~78.5% In-Between Class-A and -B
C 0~50% 100% Poor
Department of Electronic Engineering, NTUT5/49
6. Nonlinear Amplifier
• Constant-envelop, nonlinear or switching-mode amplifier
• Class D, E, F, S :
Transistor is driven in switching mode, theoretical efficiency 100%.
Department of Electronic Engineering, NTUT
DDV
dcL
pC
0L 0C jX
LRS
t
DSiDSv
6/49
7. Amplifier AM/AM and AM/PM Distortion
• Modulated Input signal:
• Distorted Output signal:
( ) ( ) ( )( )cosin cv t A t t tω φ= +
( ) ( ) ( ) ( )( ), cos ,out cv t B f A t t f Aω φ θ= + +
outP 40
0
40−
80−
20
0
20−
40−
OutputPower(dBm)
PhaseShift
Input Power (dBm)
10− 5− 0 5 10 15 20 25
Class A
AB
C AB
A
C
AM/AM Distortion AM/PM Distortion
Department of Electronic Engineering, NTUT
( )inv t ( )outv t
7/49
8. Nonlinear Memoryless Device (I)
• An input-output relationship of a nonlinear memoryless
device can be represented as
( ) ( ) ( ) ( ) ( )2 3 4
0 1 2 3 4out in in in inv t v t v t v t v tα α α α α= + + + + +⋯
( )inv t ( )outv t
inV
outV
linear
nonlinear
small signal
large signal
linear output
distorted output
f
f
Perfect sinusoid
Harmonics
Department of Electronic Engineering, NTUT8/49
9. Nonlinear Memoryless Device (II)
Coefficients αi are depending on
1) DC bias, RF characteristics of the active device used in the circuit.
2) Magnitude vin of the signal.
3) When Pin < P1dB (linear region), all can be treated as constant.
• Assume the input and output impedance of the circuit are ,
and ,respectively. Considering a CW input signal with the
voltage ,the input available power is
( )inv t ( )outv t
( ) sin 2in in cv t V f tπ= ( ) ( )2
2in c in in cP f V Z f=
Department of Electronic Engineering, NTUT
( )inZ f
( )outZ f
( ) ( ) ( ) ( ) ( )2 3 4
0 1 2 3 4out in in in inv t v t v t v t v tα α α α α= + + + + +⋯
9/49
10. Small-signal Power Gain (Linear Gain)
• For linear operation
where Pin is the available input power and G1 is the available small-signal
power gain, which equals to
( ) ( )1 1 sin 2out in in cv t v t V tα α π= =
( )
( )
2 2 2 2
2 21
1 1
1 1 1
2 2 2
in cout in in in
out in
out out in out out c
Z fV V V Z
P P
Z Z Z Z Z f
α
α α= = = =
( )
( )120log 10log in c
out in
out c
Z f
P P
Z f
α= + + ( ) ( ) ( )1 dBmout c in cP f P f G= +
( )
( )1 120log 10log in c
out c
Z f
G
Z f
α= +
( ) sin 2in in cv t V f tπ=
Department of Electronic Engineering, NTUT
( ) ( ) ( ) ( ) ( )2 3 4
0 1 2 3 4out in in in inv t v t v t v t v tα α α α α= + + + + +⋯
( )inv t ( )outv t
Assume , we have .( ) ( )in c out cZ f Z f= 1 120logG α=
10/49
11. Linear Amplification
( ) ( )dBmin cP f
1G
1
1
( ) ( )dBmout cP f
( ) ( )dBmin cP f
1G
( ) ( )dBmout cP f
inP
cf
f
f
1out inP P G= +
Department of Electronic Engineering, NTUT
( )inv t ( )outv t
11/49
12. Third-order Effect
• For a single-tone input signal,
• α3 < 0 gives gain compression phenomenon
• α3 > 0 gives gain enhancement phenomenon
( ) 1cosinv t A tω=
( ) ( )3 3
1 1 3 1cos cosoutv t A t A tα ω α ω= +
3 3
1 3 1 3 1
3 1
cos cos3
4 4
A A t A tα α ω α ω
= + +
Out-of-band Distortion (3rd Harmonic)
3rd-order effect
In-band Distortion
3rd-order effect
Desired Signal
linear effect
( )inv t ( )outv t
( ) ( ) ( )3
1 3out in inv t v t v tα α= +
Department of Electronic Engineering, NTUT12/49
13. 1 dB-Compression Point
• When the input signal becomes stronger, the output signal will
not grow proportionally but with a slower rate. It is a
saturation phenomena.
1 dB
1dBOP
G
1dBIP
( )out cP f
( ) ( )dBmin cP f
1
1
• When the actual output power is 1 dB less than
the linear extrapolated power, it reaches the 1-
dB gain compression point. At this point, the
input power is called the input 1-dB-
compressed power (IP1dB), the output power is
called the output 1-dB-compressed power
(OP1dB) ,and the gain is called the 1-dB-
compressed gain (G1dB).
Department of Electronic Engineering, NTUT
( ) 3 3
1 3 1 3 1
3 1
cos cos3
4 4
outv t A A t A tα α ω α ω
= + +
α3 < 0
13/49
14. Analysis of 1dB-Compression Point (I)
• At P1dB , the output power is compressed 1 dB, i.e.,
• The input voltage magnitude at P1dB as
3
11 1dB 3 1dB
20
1 1dB
3
4 0.891 10
A A
A
α α
α
−+
= =
( )
3
1 1dB 3 1dB
desired+distorted
desired 1 1dB
3
410log 20log 1 dB
A AP
P A
α α
α
+
= = −
1
1dB
3
0.145A
α
α
=
( )
2
1dB 1 1
1dB
3 3
1
10log 30 10log 0.0725 30 18.6 10log dBm
2 in in in
A
IP
R R R
α α
α α
= + = + = +
( )
2
3
3 31 1dB 3 1dB
1 1
1dB
3 3
3
1 0.05754
10log 30 10log 30 17.6 10log dBm
2 out in out
A A
OP
R R R
α α
α α
α α
+ = + = + = +
Department of Electronic Engineering, NTUT
( ) ( )21
1 1 1
3
17.6 10log 1 dBmdB
out
IP G
R
α
α
α
= + ⋅ = + −
14/49
15. Analysis of 1dB-Compression Point (II)
1G
( )dBminP
cf
cf
1out inP P G= +
( )1dB 1 1out in inP P G P G= + = + −
1out inP P G= +
Department of Electronic Engineering, NTUT15/49
16. Measurement of P1dB
• By network analyzer in the power sweep mode:
Obtain small signal gain and .
• By spectrum analyzer :
Test various input signal power level to measurement the output power spectral
content to obtain output v.s. input power curve.
1 120logG α= 1dBG
Department of Electronic Engineering, NTUT
Network Analyzer
Amplifier
Signal Generator
Amplifier
Spectrum Analyzer
16/49
17. Distortion Characterization (I)
• Amplifier input-output relation:
• If only one signal is present, the undesired components will
be harmonics of the fundamental, but, if there are more
signals at input, signals will be produced with frequencies
that are mathematical combinations of the frequencies of the
input signals, called intermodulation products (IMPs) or
intermods. It is instructive to study the results when there are
two input signals (although we will eventually consider large
numbers of signals).
( ) ( ) ( ) ( ) ( )2 3 4
0 1 2 3 4out in in in inv t v t v t v t v tα α α α α= + + + + +⋯
Department of Electronic Engineering, NTUT17/49
18. Distortion Characterization (II)
• Characterized by 1-dB gain compression, IPs , 2-tone
intermodulation distortions (IMDs)
1cosinv A tω=
,1 1cosout ov G A tω=
,2 2 1cos2outv A tα ω=
,3 3 1cos3outv A tα ω=
Department of Electronic Engineering, NTUT
Single-tone excitation
Nonlinear Harmonics
1f
f
1f
f
12 f 13 f 14 f
18/49
19. Distortion Characterization (III)
Designed Amplifier
1f 2f
f
1f 2f
f
1 22 f f− 2 12 f f−
1f 2f
f
1 22 f f− 2 12 f f−
1f 2f
f
1 22 f f− 2 12 f f−
IMD from AM/AM distortion
IMD from AM/PM distortion
Department of Electronic Engineering, NTUT
Two-tone excitation
Nonlinear
IM
Products
• Characterized by 1-dB gain compression, IPs , 2-tone IMDs
19/49
20. Intercept Points
• The nonlinear properties can be described by the concept of
intercept points (IPs). The input intercept point (IIPn) is a
fictitious input power where the desired output signal
component equals in amplitude the undesired component.
( )out nP f
( )out cP f
( ) ( )dBmin cP f
IIPn1dBIP
OIPn
1dBOP
1 dB
1
1 1
n
OutputPower(dBm)
Department of Electronic Engineering, NTUT20/49
21. Second-Order Nonlinear Effect (I)
• Single-tone excitation:
• For the inclusion of only the linear term and the second term,
the output voltage is
( ) sin 2in cv t A tπ=
( )
( )
2
2
in c
in c
A
P f
Z f
=
( ) ( ) ( ) ( ) ( )
22
1 2 1 2sin 2 sin 2out in in c cv t v t v t A f t A f tα α α π α π= + = +
( ) ( )
2
22
1 2sin 2 sin 2
2
c c
A
A f t A f t
α
α π α π= + −
2 2
2 1 2
1 1
sin cos2
2 2
c cA A t A tα α ω α ω= + −
Out-of-band Distortion
2nd-order effect
DC Offset
2nd-order effect
Desired Signal
linear effect
Department of Electronic Engineering, NTUT
( )in cZ f
( )inv t ( )outv t
cf
f
0
21/49
22. Second-Order Nonlinear Effect (II)
• Two-tone Excitation: ( ) 1 2sin sininv t A t B tω ω= +
( ) ( ) ( )
2
1 1 2 2 1 2sin sin sin sinoutv t A t B t A t B tα ω ω α ω ω= + + +
( ) [ ]2 2
2 1 1 1 2
1
sin sin
2
A B A t B tα α ω α ω
= + + +
( ) ( )2 1 2 2 1 2cos cosAB t AB tα ω ω α ω ω+ − + +
2 2
2 1 2 2
1 1
cos2 cos2
2 2
A t B tα ω α ω
+ − −
2 1f f−0 1f 2f 12 f 22 f1 2f f+
a b
c
e
d
fg
g : DC term
a, b : linear term
c : IM (down beating)
d : IM (up beating)
e, f : 2nd harmonic
Department of Electronic Engineering, NTUT
a bg
c d
e f
22/49
23. Linear and 2nd-order Effects
• Linear effect:
A superscript (1) of denotes that the power content contributed from the first-
order term (linear term).
• 2nd-order effect:
( )
( ) ( )
( )
( )
1
120log 10log in c
out c in c
out c
Z f
P f P f
Z f
α= + +
( )
( ) ( ) ( )1
1 dBmout c in cP f P f G= +
( )1
outP
Department of Electronic Engineering, NTUT
Linear Gain
( )
( )
( ) ( )
( )
( )
( )
( )
2
2
2
2 222
2 2 2 2
2 2
1
1 1 12
2
2 2 2 2 2 2 2
in c in c
out c in
out c in c out c out c
A
Z f Z fA
P f P
Z f Z f Z f Z f
α
α α
= = =
( )
( )
( )
2
220log 3 2 dBm 10log
2
in c
in
out c
Z f
P
Z f
α= − + +
( )
( ) ( ) ( )2
22 2 dBmout c in cP f G P f= +
( )
( )
( )
2
2 2dB 20log 3 10log
2
in c
out c
Z f
G
Z f
α= − +
Slope of 2
23/49
24. Second-Order Intercept Point
6 dB
6dB
IM2
2nd harmonic
Fundamental
Fundamental input power (dBm)
Outputpower(dBm)
6dB
6 dB
• The 2nd-order products increase twice
as fast as the desired fundamental, the
straight lines cross. At the crossing
point, either for the intermod or the
harmonic, the fundamental and the
2nd-order product have equal output
powers.
• Since the slopes of the straight lines
are known, these crossing points,
called intercept points (IPs), define
the 2nd-order products at low levels.
OIP2H
OIP2IM
IIP2IM IIP2H
6 dB
• Typically, the larger of the input or
output intercept points is specified; so
amplifiers use OIPs and mixers use
IIPs. Some may even add the power
of the two fundamentals, increasing
the value of the IP by 3 dB.
6dB
Department of Electronic Engineering, NTUT24/49
25. Example
• For an amplifier with 21 dB linear gain and the OIP2H is at 17
dBm, find the output 2nd harmonic power when the
fundamental output signal power is −8 dBm.
( )12 2 dBmH HOIP IIP G= +
OIP2H = 17 dBm
2nd harmonic
Fundamental
Fundamental input power (dBm)
Outputpower(dBm)
IP2H
−8 dBm
25dB
25dB
−33 dBm
−29 dBm −4 dBm
(IIP2H )
( )17 2 21 dBmHIIP= +
( )2 4 dBmHIIP = −
( ) ( ) ( ) ( )2 2 dBmout c out c H out cP f P f OIP P f= − −
[ ] ( )8 17 8 33 dBm= − − + = −
Department of Electronic Engineering, NTUT25/49
26. Unequal Input Tone Power
Department of Electronic Engineering, NTUT
( ) ( ) [ ] ( ) ( )2 2 2 2
2 1 1 1 2 2 1 2 2 1 2 2 1 2 2
1 1 1
sin sin cos cos cos2 cos2
2 2 2
outv t A B A t B t AB t AB t A t B tα α ω α ω α ω ω α ω ω α ω α ω
= + + + + − + + + − −
( ) 1 2sin sininv t A t B tω ω= +
• If the amplitude of only one input signal changes, the harmonic of the changing signal
will change by twice as many dB as does the input, but the other harmonic will be
unaffected. The IM amplitudes change by the sum of the changes in the two input
signals; so, if only one fundamental changes, the IMs will change by the same amount.
2IIP1dBIP
2OIP
1dBOP
1f 2f
,i AP
,i BP
2 1f f−0 1f 2f 12 f 22 f1 2f f+
, , 1o A i AP P G= + , , 1o B i BP P G= +
δ
δ
2δ
δδ
26/49
27. Half-IF Interference (I)
• Input signal with two sinusoidal signals at f2 and f2/2
( ) 2 2
1
sin sin
2
inv t A t B tω ω= +
( )
2
1 2 2 2 2 2
1 1
sin sin sin
2 2
outv t A t B t B tα ω ω α ω ω
= + + +
( )2 2 2
2 1 2 2 2 1 2 2 2 2 2
1 1 1 1 3
sin cos sin cos cos
2 2 2 2 2
A B A t AB t B t A t AB tα α ω α ω α ω α ω α ω
= + + + + − +
Out-of-band Distortion
2nd-order effect
In-band Distortion
2nd-order effect
Desired Signal
linear effect
DC Offset
2nd-order effect
Department of Electronic Engineering, NTUT
2 1f f−
0 2
1
2
f
f = 2f
22 f1 2f f+
12 f
27/49
28. Half-IF Interference (II)
2IIP1dBIP
2OIP
1dBOP
2
1
2
f 2f
,i AP
,i BP
2 1f f−0 1f 2f 12 f 22 f1 2f f+
, , 1o A i AP P G= +
, , 1o B i BP P G= +
2
1
2
f 2f 22 f
,o AP
,o BP
Department of Electronic Engineering, NTUT
2
1
2
f
f ≠
2
1
2
f
f =
28/49
29. Half-IF Rejection
•
where S is the sensitivity or minimum detectable power, CR is the capture ratio,
which is the ratio of the desired signal and the second-order distortion when the
receiver fails to demodulate the signal.
( )
1
Half-IF Rejection 2
2
IIP S CR= − −
Department of Electronic Engineering, NTUT
2IIP1dBIP
2OIP
1dBOP
1G
CR
S
( )2out cP f
( )out cP f
( ) ( )dBmin cP f
Half-IF rejection
(IMR)
2IIP S−
2IIP S CR− −
29/49
30. Measurement of IP2 (I)
• Mixer: use single-tone cw test
( )2 dBmIFOIP P= ∆ +
( )12 2 dBmRFIIP OIP G P= − = ∆ +LOf RFf
RFP
LOP
IFP
IFf 2 IFf
( )dB∆
Department of Electronic Engineering, NTUT
Spectrum Analyzer
30/49
31. Measurement of IP2 (II)
• Amplifier : use two-tone cw test
( ) ( ), ,
1
2 3 dBm
2
A B o A o BOIP P P= ∆ + ∆ + + +
( )12 2 dBmIIP OIP G= −
,i AP ,i BP
1f 2f
2 1f f−0 1f 2f 12 f 22 f1 2f f+
,o AP
,o BP
A∆
B∆
Department of Electronic Engineering, NTUT
Signal Generator
Combiner
DUT
Spectrum Analyzer
31/49
32. Third-Order Nonlinear Effect (I)
• Consider only the first-order and the third-order effect of a
nonlinear device, i.e., .
• Single-tone excitation:
The input signal contains only a sinusoidal signal , where its available
power can be obtained as .
• In-band and out-of-band distortions
The output voltage becomes
3
1 3out in inv v vα α= +
1cosiv A tω=
( )2
2in inP A Z=
Department of Electronic Engineering, NTUT
3 3
1 1 3 1cos cosoutv A t A tα ω α ω= +
3 3
1 3 1 3 1
3 1
cos cos3
4 4
A A t A tα α ω α ω
= + +
( ) ( )
( ) ( )1 3 3
1 1 1 3 1cos cos3V V t V tω ω= + +
Out-of-band Distortion
3rd-order effect
In-band Distortion
3rd-order effect
Desired Signal
linear effect
3rd harmonic
32/49
33. Third-Order Nonlinear Effect (II)
• Gain Compression or Enhancement:
At f1, the amplified linear-term signal has been mixed with the third-order term
If α3 < 0 , the linear gain is compressed, otherwise, it is enhanced
( ) 3
1 1 3 1
3
cos
4
outv f A A tα α ω
= +
3 0α >
( ) ( )dBmin cP f
3 0α <
1
1
Department of Electronic Engineering, NTUT33/49
34. Third-Order Nonlinear Effect (III)
• Two-tone excitation:
Department of Electronic Engineering, NTUT
( ) 1 2 1 2sin sin ,inv t A t B tω ω ω ω= + <
i : DC term
a, b : linear term(desired signal)
+inband distortion
c , d : IM3, adjacent band distortion
e, f : 3rd harmonics
g, h : out of band distortion
( ) ( ) ( )3
1 3out in inv t v t v tα α= +
2 2 3 3
3 3 1 3 1 1 3 2
3 3 9 9
cos cos
2 2 4 4
A B AB A A t B B tα α α α ω α α ω
= + + + + +
( ) ( )2 2 3 3
3 1 2 3 2 1 3 1 3 2
3 3 1 1
cos 2 cos 2 cos3 cos3
4 4 4 4
A B t AB t A t B tα ω ω α ω ω α ω α ω+ − + − + +
( ) ( )2 2
3 1 2 3 1 2
3 3
cos 2 cos 2
4 4
A B t AB tα ω ω α ω ω+ + + +
a bi
c d fe
g h
c g
fe
d
a b
h
1 22 f f−
0 1f 2f 13 f 23 f
1 22 f f+2 12 f f− 1 22f f+
( ) ( )2-toneIMR 2 3 2 3in outIIP P OIP P= ∆ = − = −
∆
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35. Third-order Intercept Point
10 dB
10dB
IM3
3rd harmonic
Fundamental
Fundamental input power (dBm)
Outputpower(dBm)
4.77dB
4.77 dB
OIP3H
OIP3IM
IIP3IM IIP3H
4.77 dB
9.54dB
• The slopes for the 3rd-order products
are steeper than 2nd-order products
since they represent cubic
nonlinearities rather than squares. IMs
and harmonics change 3 dB for each
dB change in the inputs and
fundamental outputs.
• Since the slopes of the straight lines
are known, these crossing points,
called intercept points (IPs), define
the 3rd-order products at low levels.
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( ) ( )2-toneIMR dB 2 3 inIIP P= ∆ = −
( )2 3 outOIP P= −
• Intermodulation Ratio (IMR)
∆
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36. Example
• For an amplifier with 9 dB linear gain and the OIP3IM is at 21
dBm, find the output IM3 power when the fundamental input
signal power for each signal is −4 dBm.
( )13 3 dBmIM IMOIP IIP G= + OIP3IM = 21 dBm
IM3
Fundamental
Fundamental input power in each signal (dBm)
Outputpower(dBm)
IP3IM
5 dBm
16dB
32dB
−27 dBm
−4 dBm 12 dBm
(IIP3IM )
( )21 3 9 dBmIMIIP= +
( )3 12 dBmIMIIP =
( ) ( ) ( )3 2 3 dBmIM out c IM out cP P f OIP P f= − −
( ) ( )5 2 21 5 27 dBm= − − = −
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37. Unequal Input Tone Power
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( ) 1 2 1 2sin sin ,inv t A t B tω ω ω ω= + <
( ) ( ) ( )3 2 2 3 3
1 3 3 3 1 3 1 1 3 2
3 3 9 9
cos cos
2 2 4 4
out in inv t v t v t A B AB A A t B B tα α α α α α ω α α ω
= + = + + + + +
( ) ( )2 2 3 3
3 1 2 3 2 1 3 1 3 2
3 3 1 1
cos 2 cos 2 cos3 cos3
4 4 4 4
A B t AB t A t B tα ω ω α ω ω α ω α ω+ − + − + +
( ) ( )2 2
3 1 2 3 1 2
3 3
cos 2 cos 2
4 4
A B t AB tα ω ω α ω ω+ + + +
3IIP1dBIP
3OIP
1dBOP
1f 2f
,i AP
,i BP
δ
0 1f 2f 13 f 23 f
, , 1o A i AP P G= + , , 1o B i BP P G= +
δ
2δδδ 2δ
3δ
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38. Third-order Intermodulation Rejection
• From triangular A-B-C, we have
• From D-E-F, which has slope of 3, we have
• From the relations, we can obtain
third-order intermodulation rejection
1G S IMR CR x+ + = +
13 3 3 3
3
x
IIP S IMR OIP IIP G
+ − − = = +
( )
1
2 3 2
2
IMR IIP S CR= − −
3IIP
3OIP
1dBOP
( ) ( )1 dBminP f
S1G
IMR
CR
A
D
B E C F
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x
38/49
39. Measurement of the IP3
• Amplifier : use two-tone cw test
( ), ,
1
3
2
i A i BOIP P P= ∆ + +
1f 2f
,i AP ,i BP
B∆
1f 2f1 22 f f− 2 12 f f−
A∆
,o AP
,o BP
0
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Signal Generator
Combiner
DUT
Spectrum Analyzer
39/49
40. Relationship Between Products
• IMs may be predictable from harmonics:
IM2s are 6 dB higher than the 2nd-order harmonics
IM3s are 9.54 dB greater than the 3rd-order harmonics
IP3H exceeds the IP3IM by 4.77 dB
• In addition, we may be able to relate the −1-dB compression
level to the IP3:
( )
3
1 1dB 3 1dB
desired+distorted
desired 1 1dB
3
410log 20log 1 dB
A A
P
P A
α α
α
+
= = − 23
1dB
1
3
0.10875
4
A
α
α
=
3
3, 1 3, 3 3,
3
4
OIP IM IIP IM IIP IMA A Aα α= = 2 1
3,
3
4
3
IIP IMA
α
α
=
2
1dB 1dB
2
3,
0.10875 9.64 dB
3IIP IM IM
A IP
A IIP
= = = −
( )1 3 1 9.64 dB 3 10.64 dBdB IM IMOP IIP G OIP= + − − = −
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P1dB:
very useful result!
OIP3:
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41. Cascaded System (I)
• We take a three-stage system as an example of cascaded IP3
and then extend to an N-stage system.
inP 1C 2C 3C
1I 2I′ 3I′
3I′′2I′′
3I′′′
1st stage 2nd stage 3rd stage
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1G 2G 3G
41/49
42. Cascaded System (II)
1 1inC P G=
( )
3
1
1 2
13
inP G
I
IIP
=
2
1 1
1
3
in
C IIP
I P
=
inP
1C
1I
1st stage 2nd stage 3rd stage
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1G
42/49
43. Cascaded System (III)
2 1 2 1 2inC C G P G G= =
( )
3
1 2
2 1 2 2
13
inP G G
I I G
IIP
′ = =
( ) ( )
3 33
1 21 2
2 2 2
2 23 3
inP G GC G
I
IIP IIP
′′ = =
3 3 3
1 2 1 2
2 2 2
2 13 3
in inP G G P G G
I I I
IIP IIP
′ ′′= + = + 2
2
2 2 1
2 1
1
1
3 3
in
C
I G
P
IIP IIP
=
+
inP 1C 2C
1I
2I′
2I′′
1st stage 2nd stage 3rd stage
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1G 2G
43/49
44. Cascaded System (IV)
3 1 2 3inC P G G G=
( )
3
1 2
3 2 3 32
13
inP G G
I I G G
IIP
′ ′= =
( )
2
2
3 1 2 1
3 3 3 1 2 3
3 2 1
1
3 3 3
in
G G G
I I I I P G G G
IIP IIP IIP
′ ′′= + + = + +
( )
3 3
1 2
3 2 3 32
23
inP G G
I I G G
IIP
′′ ′′= =
( ) ( )
3 3 3 3
2 3 1 2 3
3 2 2
3 33 3
inC G P G G G
I
IIP IIP
′′′= =
3 1 2 3
2 3
1 2 33 2 1 2 1
3 2 1
1
1
33 3 3
tot in
intot
in
tot
C C G G G P
P G G GI IG G G
P IIPIIP IIP IIP
= = =
+ +
1 2 1
3 2 1
1 1
3 3 3 3tot
G G G
IIP IIP IIP IIP
= + +
inP 1C 2C 3C
1I 2I′ 3I′
3I′′2I′′
3I′′′
1st stage 2nd stage 3rd stage
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45. Cascaded System (V)
• IIP3 of a N-Stage System
• The above equation shows that the IIP3 of an inter-stage is
reduced by a factor of the previous stage subtotal gain. It
means, the back-end stage will enter saturation first.
• OIP3 of a N-Stage System
1
1 1 1 2
1 1 2 3
1 1
3 3 3 3 3
n
kN
k
ntot n
G
G G G
IIP IIP IIP IIP IIP
−
=
=
= = + + +
∏
∑ ⋯
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( ) ( )1 2 3 2 3 4 3
1 1 1 1 1 1
3 3 3 3 3 3tot T tot T N N N NOIP G IIP G IIP G G G IIP G G G IIP G IIP
= = + + + +
⋅ ⋅ ⋅
⋯
⋯ ⋯
( ) ( ) ( )2 3 1 3 4 2 4 5 3
1 1 1 1
3 3 3 3N N N NG G G OIP G G G OIP G G G OIP OIP
= + + + +
⋅
⋯
⋯ ⋯ ⋯
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46. Example (I)
• Calculate the cascaded OIP3 of the following stages.
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21 dBm+ ∞ 25 dBm+
10 dB 3 dB− 10 dB
3OIP
Gain
21 dBm+ ∞ 25 dBm+
15 dB 3 dB− 10 dB
3OIP
Gain
stage 1 stage 2 stage3
Gain (dB) 10 -3 10
OIP3 (dBm) 21 100 25
IIP3 (dBm) 11 103 15
Gain (linear) 10 0.5011872 10
OIP3(linear, mW) 125.89254 1E+10 316.22777
IIP3(linear, mW) 12.589254 1.995E+10 31.622777
1/IIP3cas (linear) 0.2379221
IIP3cas (linear) 4.2030556
IIP3cas (dBm) 6.2356514
OIP3cas(dBm) 23.235651
stage 1 stage 2 stage3
Gain (dB) 15 -3 10
OIP3 (dBm) 21 100 25
IIP3 (dBm) 6 103 15
Gain (linear) 31.622777 0.5011872 10
OIP3(linear, mW) 125.89254 1E+10 316.22777
IIP3(linear, mW) 3.9810717 1.995E+10 31.622777
1/IIP3cas (linear) 0.7523759
IIP3cas (linear) 1.3291229
IIP3cas (dBm) 1.2356514
OIP3cas(dBm) 23.235651
46/49
47. Example (II)
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21 dBm+ ∞ 25 dBm+
10 dB 3 dB− 10 dB
3OIP
Gain
21 dBm+ ∞ 25 dBm+
10 dB 3 dB− 15 dB
3OIP
Gain
stage 1 stage 2 stage3
Gain (dB) 10 -3 10
OIP3 (dBm) 21 100 25
IIP3 (dBm) 11 103 15
Gain (linear) 10 0.5011872 10
OIP3(linear, mW) 125.89254 1E+10 316.22777
IIP3(linear, mW) 12.589254 1.995E+10 31.622777
1/IIP3cas (linear) 0.2379221
IIP3cas (linear) 4.2030556
IIP3cas (dBm) 6.2356514
OIP3cas(dBm) 23.235651
stage 1 stage 2 stage3
Gain (dB) 10 -3 15
OIP3 (dBm) 21 100 25
IIP3 (dBm) 11 103 10
Gain (linear) 10 0.5011872 31.622777
OIP3(linear, mW) 125.89254 1E+10 316.22777
IIP3(linear, mW) 12.589254 1.995E+10 10
1/IIP3cas (linear) 0.5806201
IIP3cas (linear) 1.7222967
IIP3cas (dBm) 2.3610797
OIP3cas(dBm) 24.36108
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48. Spectrum Regrowth
• How do we estimate ACPR of a modulated RF signal from 2-
tone measurement
( )
3
2-tone 6 10log dBc
4
m
ACPR IMR
A B
= − +
+
where
3 2 mod
2 3 2 2
24 8
m
m m m
A
− − = +
2
mod
2
4
m
m
B
−
=
m denotes number of tones
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49. Summary
• In this chapter, 2nd-order and 3rd-order nonlinear effects were
introduced. These nonlinearities will result in harmonics and
intermodulation distortions in frequency domain.
• The distortion can be easily defined using frequency-domain
parameters related to signal power. It is easier to qualify the
distortion by frequency components than time-domain
waveforms. The nonlinearities can be described by P1dB and
intercept points.
• The cascaded formula was also derived to show that the IIP3
of an inter-stage is reduced by a factor of the previous stage
subtotal gain. It means, the back-end stage will enter saturation
first.
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