EE40458 Nonlinearity & Noise. Yet another lecture from the road

Transcription

1 EE40458 Nonlinearity & Noise Yet another lecture from the road

2 Recap & Perspec=ve An overview where have we been recently: General linear system theory S- parameters, but also equivalent small- signal models like the hybrid- π model, etc. Give linear response of circuit/system; can use superposi=on & Fourier analysis to determine output for arbitrary input signal Gain, phase at each frequency; no new frequencies, no signal components that were not present in the input signal Strictly applies only for systems governed by linear differen=al equa=ons (any order, but constant coefficients) Approximately applies to most systems if signals are small (equivalent to approxima=ng func=on with first two terms (constant plus linear) in Taylor series)

3 Recap & Perspec=ve (cont.) Overview (con=nued): Nonlinear effects Many important systems are not well approximated with a linear descrip=on Examples: almost any component under high power condi=ons (e.g. hea=ng); amplifiers driven with large inputs (or designed for switching- mode opera=on for high efficiency); devices for harmonic genera=on, mixing, or detec=on (nonlinear response is desired) Our approach: Taylor series expansion of transfer characteris=c, resul=ng in polynomial representa=on of response We simplified to neglect memory, history- dependent effects (e.g. hea=ng); assumed output depends only on instantaneous input value. More advanced approaches exist to handle this Conclusions: saw harmonic genera=on (plus DC shi^), intermodula=on products (e.g. sum & difference frequencies) Figures of merit: P 1dB (gain compression), P IP3 (third order intermodula=on)

4 Nonlinear Figures of Merit - Review Gain compression P 1dB : Single input tone Map output power (at input frequency) vs. power of input signal Linear theory: output power propor=onal to input power; nonlinear effects tend to cause satura=on: # v o! ( t) = k 1 A 1 3 k & % 3 A 2 (cos( ωt) $ 4 k 1 ' Note: input is power in single tone; output power counted is only the power at this frequency (i.e., the harmonic power is not included)

5 Nonlinear Figures of Merit - Review # v o! ( t) = k 1 A 1 3 k & % 3 A 2 (cos ωt Defini=on: Input 1 db compression point, $ 4 k 1 ' P 1dB, is input power at which output is 1 db below the linear case. Gain compression P 1dB : ( ) Note: some data sheets will report the output 1 db compression point (e.g. the y- axis, rather than x- axis). Depends on intended applica=on

6 Nonlinear Figures of Merit - Review Intermodula=on P IP3 : Saw complicated rela=onship with two input tones Figure of merit: for comparison of components P IP3! v o Two input tones, equal amplitude (for figure of merit); small enough that gain compression can be neglected: v in ( t) = A cos ω 1 t Output power at close- in intermodula=on product frequencies vs. power of input signal. Near ω 1 and ω 2, have: "# $ % + k 3 3 ( t) = k 1 A cos ω 1 t ( ) + cos( ω 2 t) ( ) + cos( ω 2 t)!" # $ 4 "# A3 cos( 2ω 2 ω 1 )t + cos( 2ω 1 ω 2 )t $ %

7 Nonlinear Figures of Merit - Review Intermodula=on P IP3 (con=nued): Purpose of P IP3 figure of merit: quan=fy rela=onship between P in, P out (at signal frequency), and intermodula=on products v in! v o Map power in desired frequencies (first term) to power in intermodula=on products (second term) ( t) = A!" cos( ω 1 t) + cos( ω 2 t) # $ ( t) = k 1 A "# cos( ω 1 t) + cos( ω 2 t) $ % + k "# A3 cos( 2ω 2 ω 1 )t + cos( 2ω 1 ω 2 )t $ % P in 1 2 A2 P in = power at each input tone P d 1 2 k 2 1 A 2 P in P im 1 2 k A6 2 9 = k 3 4 AP 3 in P d = power at each desired output tone P im = power at each intermod output tone

8 Nonlinear Figures of Merit - Review Intermodula=on graphically: Graph P d, P im vs. P in, usually on log- log scale (all powers in dbm) Reminder: dbm = 10*log10(P/1 mw) P in 1 2 A2 P d 1 2 k 2 1 A 2 P in P im 1 2 k A6 2 9 = k 3 4 AP 3 in P IP3 : intercept between linear (P d ) and intermod (P im ) terms Intercept is fic==ous ; in prac=ce, based on low power data (avoid gain compression); real lab data includes everything P d P im ( dbm) = Gain db ( ) + P in ( dbm) ( dbm) = offset + 3P in ( dbm)

9 Intermodula=on Analysis Measurement: Measure P d, P im at several (low) levels of P in Can easily separate P d, P im (on spectrum analyzer) because at different frequencies Analysis: Use slope of 1 for P d vs. P in (in dbm) Use slope of 3 for P im vs. P in (in dbm) Find intercept point; input IP3 (P IP3, IIP3) or output IP3 (OIP3) can be projected Intermodula=on Ra=o : Convenient rela=on: Relate expected intermodula=on products from (known) IIP3 and P in Find IIP3 given measured P im, P d IMR = P! im = P in # P d " P IP3 $ & % 2

10 Noise Linear and nonlinear analysis relates output to input s=mulus Linear: small signals; non- linear: large signals Circuits, systems also produce outputs independent of input: noise Ul=mately, noise limits our ability to resolve/recover/process very small signals Noise is fundamental cannot be eliminated; but can be managed Sources of noise: Thermal noise: random mo=on of carriers (electrons, holes) in resis=ve material Shot noise: cause by random =ming of events Current is made of up of flow of electrons, but they have some jiler in when they arrive; this generates shot noise Flicker or 1/f noise: trapping/detrapping, o^en defect or surface related; has ~1/f noise power spectral density

11 Thermal Noise Let s look at thermal noise in a resistor: e n = 0 e n 2 = 4kTBR Reminder: k=1.38x10-23 J/K (Boltzmann s const.), T (in Kelvin) Expect average voltage = 0 (no net flow, equal opposite flows) But variance 0; voltage variance propor=onal to noise power B: bandwidth of measurement how much noise power you see depends on how wide of a frequency range you look at Poten=al problem: noise power goes up as bandwidth increases no limit? Not really this conclusion comes from simplis=c assump=on on carrier sta=s=cs. But if T~300 K, is good approx. for frequencies to ~1 THz or so

12 Modeling Noise For analysis, need equivalent circuit for analysis One op=on: model noisy resistor as noiseless resistor with associated noise source (Thevenin or Norton op=ons): e n = 0, i n = 0 Can we get power from these sources? Yes, but Consider: e n 2 = 4kTBR, i n 2 = 4kTB R e n 2 Power available: Independent of R Power transfer? If both resistors at same temperature, net flow = 0 (equal/opposite flows). If at different temperatures, power from hot to cold (alempts to equilibrate the system). You knew that. 4R = ktb

13 Modeling Noise (cont.) How about complex impedances? One can show that: Z f ( ) = R( f ) + jx ( f ) No noise from reactances (no loss or dissipa=on, no noise) Numerical example: Noise voltage across at 1 MΩ resistor in bandwidth of 100 MHz (e.g. typical oscilloscope input) 4kT = 1.6x10-20 J (T=290 K) e n 2 e n 2 = 4kT R( f )df B = 4kTBR = = V 2 rms voltage : e n 2 = V Can see why oscilloscopes have minimum 2 mv/div scales anything smaller is just noise

14 Excess Noise Many components exhibit addi=onal noise, beyond the thermal contribu=ons O^en modeled as if it were thermal noise, but with modified parameters (i.e., fudge factors) Two common approaches: Circuit analysis approach (typical): Introduce R n as fudge factor, R noiseless is actual R value for circuit For pure thermal noise (no excess noise), R noiseless =R n System analysis approach: 2 e n = 4kT n BR Use T as fudge factor: T n no longer thermometer temperature; if excess noise present, T n >T Concept of noise temperature is common, and what we ll (mostly) use e n 2 = 4kTBR n

15 Noise Temperature Example An example: antenna At resonant frequency, antenna impedance (Zant) is resis=ve Zant = Rohmic + Rrad Rohmic: from loss in the conductors, a real resistance Rrad: accounts for conversion from input power to radiated power (think of antenna as broadcas=ng) Temperatures? Rohmic: at physical temperature of the antenna electrons bouncing around in conductors due to random thermal mo=on Rrad: at an effec=ve temperature, TA TA is fudge factor to allow us to make output noise of antenna match the power actually received Fun fact: Cosmic background radia=on 1978 Nobel prize measured TA ~ 3K, when expected to be 0 (dark sky)

16 Noise in Two- Port Networks So far, everything has just been about how much noise a one- port circuit makes. But two- port networks are more useful have inputs, outputs Basic idea: two- port network does some func=on (amplify, mix, etc), but also adds some noise Schema=cally: T E : effec=ve noise temperature of two- port network Is a func=on of Z s, frequency; characterizes the network, not Z s

17 Noise in Two- Port Networks, cont. To include effects of both two- port and termina=on, use opera=ng temperature, T op T op = T E +T s Adding temperatures: same as adding powers. Assumes no correla=on between noise sources Characteriza=on? Two- port network has s- parameters, plus T E

18 Noise Factor Another common way to characterize the noise added by a two- port network is the noise factor and noise figure Two equivalent defini=ons: Input SNR Defini=on #1: Noise Factor Output SNR Ts =T o =290K Result: F = S in Nin S out Nout = S in Nin G A S in Nout = N out G A N in

19 Noise Factor & Noise Figure Alterna=ve view: Defini=on #2: F Actual available noise power (output) Available noise power if two port was noiseless Result: F = N out G A N in = N A + G A N in G A N in Can be framed in terms of temperatures: F =1+ N A =1+ N A / G A =1+ kt EB G A N in kt O B =1+ T E So providing F is equivalent to providing T Technically, F=noise factor Noise Figure is more common; F converted into db NF = 10*log10(F) N in T o

20 Cau=on: From defini=on #1: Noise Figure Input SNR Noise Factor Output SNR Ts =T o =290K Looks like noise figure should be how much (in db) the SNR degrades because of the noise of the two port This is not strictly true: note that this is true only if T s =T o As we saw, T s can be an effec=ve temperature with no obvious connec=on to thermometer temperatures (e.g. if signal came from an antenna, etc) So noise figure should be thought of as a test- based metric In the lab, can test the SNR with T s =T o, and find NF In real systems, T s is almost never T o, so the actual SNR change can be quite different

21 Noise Figure and LNA Design How is this related to our LNA design approach? Recall: F = F min + 4R 2 Γ n S Γ opt 2 2 Z o 1+ Γ opt 1 ΓS ( ) This shows explicitly how F depends on Z s (Z s <- > Γ s ) This is important if you re doing detailed circuit design (e.g. of an amplifier to meet a specific noise figure target) But for system design or analysis, the Z s is usually already defined and fixed Block diagram- level interconnec=ons; not re- designing the individual components Example: making a system by interconnec=ng available 50 Ω components In this case, T E is sufficient (if T E is for the Z s in ques=on)

22 Noise Figure and Loss A special- case two- port is the matched alenuator Passive device just a resistor network. Typically designed to have input & output impedance matched to Z o, with a specified alenua=on (e.g., 3 db, 6 db, etc). What is it s noise figure? Since network is just passive, at output appears as Z o termina=on at physical temperature; N out = ktb But regular two- port equa=ons also apply: N out = ktb G A + G A N added Loss=1/Gain, so : T E = T att ( L 1) If T al =T o, then F=L F =1+ T E T O =1+ T att T O ( L 1)

23 System Noise Analysis Common situa=on: want to evaluate noise performance of a system consis=ng of several building blocks cascaded together Overall noise? Can show: T E = T E1 + T E 2 G A1 F =1+ T E T O = F 1 + F 2 1 G A1 Careful: G s are available power gains, T E s must be for actual impedances presented

24 Consider two amplifiers in cascade: System Noise Example And if we reverse the order of the amplifiers? Gain is the same, but what about noise? Note: usually want the lowest T E amplifier in front. But not always the gains also play a role

25 Receiver Sensi=vity For radio receivers, sensi=vity is limited by noise floor Define: minimum detectable signal (MDS) for given SNR O^en choose 0 db as the threshold (though other choices are possible depending on the system) Example: Results: NF = 8 db à F=6.31 à T E = 1540 K T op = 290 K K = 1830 K N out = k T op B G A ; S out = S in G A ; SNR out = S out N out = S in kt op B Seng SNR out =1 à S in = 5.3x10-17 W = dbm Effec=ve noise floor of the receiver; for reference, thermal noise at 290 K = dbm/hz

26 Noise Figure Measurement Basic idea: measure output noise power for two different source temperatures From this, can separate contribu=on from source and from two- port Y factor measurement: output powers N H =k(t E +T H )BG A N C =k(t E +T C )BG A Y = N H N C = T +T E H ; T E = T YT H C T E +T C Y 1

27 Noise Figure Measurement For hot and cold input termina=ons, could use resistors at different temperatures For best measurements, want largest possible difference between hot and cold temperatures (cryogenic resistor, hot resistor) Inconvenient in prac=ce, o^en use noise diode instead Diode off (no bias): room temperature resistor (thermal noise) Diode on (biased in reverse breakdown): avalanche breakdown process is very noisy, acts like hot resistor Characterized by the excess noise ra=o " ENR =10log10 T H 290 % $ ' # 290 & Typical ENR ~15 db; T H ~ 9461 K! (Solar surface ~6000 K). Much bigger temperature difference than possible using thermometer temperatures A caveat: we know that amplifier noise figure depends on Γ s : so if diode has different impedance in on and off states, measurement can be off o^en use low ENR diode to avoid this. Just a regular high ENR diode, followed by an alenuator

28 Dynamic Range For systems, very important considera=on Dynamic range: range of input signal amplitudes for which the system has acceptable performance System- level considera=ons dictate what counts as acceptable some systems can tolerate more distor=on than others, etc. Limited for large input signals by nonlineari=es Limited for small input signals by noise So we ll be combining the last few topics together

29 Dynamic Range Defini=on Basic no=onal system picture: With this framework in mind, can define dynamic range: DR = max. usable input power min. usable input power = P max P min Usually expressed in db since these are powers, 10 log 10 (DR)

30 Spur Free Dynamic Range One common (but not universal!) choice for acceptable performance: spur free dynamic range (SFDR) SFDR = DR f (same thing, different terminology) For SFDR: P min =minimum detectable signal (MDS) But what is the MDS? Smallest P in that will provide a specified SNR at the output. O^en this reference SNR=1 For SFDR: P max = input signal level at which 3 rd - order in- band products are equal to P min (=MDS) Easier to understand (I think) as a picture

31 Spur Free Dynamic Range SFDR in pictures: (think of 2- tone intermod measurement) P min =minimum detectable signal (MDS) P max = input signal level at which 3 rd - order in- band products are equal to MDS Basic idea: intermodula=on is never larger than the noise no spurs In microwave- speak, spur is short for spurious signal. Not poking horses.

32 Spur Free Dynamic Range Analy=cally: using IMR = P! im = P $ in # & P d " P IP3 % from our previous analysis, can work out SFDR, etc. Basic approach: consider IMR when P in = P max 2 IMR = P! im = P $ max # & P d " P IP3 % P im = P min G; P d = P max G IMR = P ming P max G = P! min = P max # P max " 2 P IP3 $ & % 2

33 Spur Free Dynamic Range Since it follows (just re- arranging) that: Remember: IMR = P G min P max G = P! min = P max # P max " P min = P 3 max ; P 2 max = P 1/3 2/3 min P IP3 P IP3 P IP3 P min comes from noise analysis, so is known $ & % 2 SFDR = P max /P min Final result:! SFDR = P IP3 # " P min $ & % 2/3

34 Spur Free Dynamic Range Careful: previous page was all in MKS (or similar) units Usually specify these things in db:! SFDR = P $ IP3 # & " P min % Becomes: 2/3 ( ) P min ( db) SFDR(dB) = 2 3 "# P IP3 db $ % Not complicated just be careful One final note: the book gets the same results, but from another path; Pozar does the analysis from the point of view of the output power (vs. input like done here).

35 Congratula=ons You made it to the end. We ll have a review session at our regular class period on Thursday come with your ques=ons about the final exam

36 Topics Covered This Year RF models lumped element RLC models of common components (l< λ/10) Behavior, resonanaces (series, parallel), some uses of these parasi=c phenomena Electromagne=c analysis of transmission line structures Maxwell s eq., boundary condi=ons, assump=ons made for solu=on Transmission- line models Deriva=on from lumped- element sec=ons, parameters (α, β, γ), significance of each Meaning and role of Z o, v p, λ, etc. Source, load mismatch effects; reflec=on coefficients, impedance transforms, VSWR, transla=on along lines, boundary condi=ons Power transmission Stub impedance/admilance; origin, uses Fun with Smith charts Lumped- element matching network design Distributed circuits λ/4 transformers Series- line matching networks, single- stub matching networks, double- stub Limita=ons, design procedures, detailed understanding Bandwidth effects in matching networks Network analysis S, Y, Z parameters matrix representa=ons, defini=ons, finding matrix elements from circuits Circuit analysis using matrix representa=ons Flow graphs, Mason s rule Generalized s- parameters

37 Topics Covered This Year (2) Amplifier design cases as considera=ons Simultaneous conjugate matching condi=ons maximum gain Design for specified gain; gain circles, trade- offs (e.g. for bandwidth) Design for noise figure; noise figure circles, interac=on with gain circles Detailed understanding and ability to design circuits Power gain defini=ons & use Opera=ng power gain (G P ), available power gain (G A ), transducer power gain (G T ) Defini=ons, significance of each What is each good for? Stability Meaning/significance of stability Source, load stability circles Interpreta=on of the circles k- Δ, µ tests and what they mean Uncondi=onal stability: defini=on, concepts Nonlinear effects Deriva=ons, founda=ons Gain compression Intermodula=on Defini=ons: P IP3, P 1dB, IMR, blocking, desensi=za=on, cross modula=on Input/output spectra, frequencies present, etc. Noise Thermal noise in resistors/passives Available noise Effec=ve noise temperature, noise resistance Noise in 2- ports: T E, F, NF Noise measurement (Y factor) Cascaded noise figure Receiver sensi=vity (MDS) Dynamic range (SFDR)