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# RF Module Design - [Chapter 3] Linearity

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### RF Module Design - [Chapter 3] Linearity

1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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= ∆ = − = − ∆ 34/49
35. 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. Department of Electronic Engineering, NTUT ( ) ( )2-toneIMR dB 2 3 inIIP P= ∆ = − ( )2 3 outOIP P= − • Intermodulation Ratio (IMR) ∆ 35/49
36. 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= − − = − Department of Electronic Engineering, NTUT36/49
37. 37. Unequal Input Tone Power Department of Electronic Engineering, NTUT ( ) 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 37/49
38. 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 Department of Electronic Engineering, NTUT x 38/49
39. 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 Department of Electronic Engineering, NTUT Signal Generator Combiner DUT Spectrum Analyzer 39/49
40. 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= + − − = − Department of Electronic Engineering, NTUT P1dB: very useful result! OIP3: 40/49
41. 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 Department of Electronic Engineering, NTUT 1G 2G 3G 41/49
42. 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 Department of Electronic Engineering, NTUT 1G 42/49
43. 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 Department of Electronic Engineering, NTUT 1G 2G 43/49
44. 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 Department of Electronic Engineering, NTUT44/49
45. 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 − = = = = + + + ∏ ∑ ⋯ Department of Electronic Engineering, NTUT ( ) ( )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 = + + + + ⋅ ⋯ ⋯ ⋯ ⋯ 45/49
46. 46. Example (I) • Calculate the cascaded OIP3 of the following stages. Department of Electronic Engineering, NTUT 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. 47. Example (II) Department of Electronic Engineering, NTUT 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 47/49
48. 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 Department of Electronic Engineering, NTUT48/49
49. 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. Department of Electronic Engineering, NTUT49/49