ECE 546 Lecture 19 X Parameters

Transcription

1 ECE 546 Lecture 19 X Parameters Spring 2018 Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois jesa@illinois.edu ECE 546 Jose Schutt Aine 1

2 References [1] J J. Verspecht and D. E. Root, ʺPolyharmonic Distortion Modeling,ʺ IEEE Magazine, June 2006, pp [2] D.E. Root, J. Verspecht, D. Sharrit, J. Wood, and A. Cognata, Broad band poly harmonic distortion (PHD) behavioral models from fast automated simulations and large signal vectorial network measurements, IEEE Trans. Microwave Theory Tech., vol. 53, no. 11, pp , Nov [3] D.E. Root, J. Verspecht, J. Horn, J. Wood, and M. Marcu, X Parameters, Cambridge University Press, ECE 546 Jose Schutt Aine 2

3 Scattering Parameters V1 a1 b1 I b1 S11a1 S12a2 a a2 b 1 b1 2 b2 S21a1 S22a I 2 2 Z Z o o B SA For a two-port For a general N-port N B B i SA Sij A Ak 0 i ij j j1 most successful behavioral models V a b j k j k1,..., N ECE 546 Jose Schutt Aine 3

4 X Parameters: Motivation S Parameters are a very powerful tool for signal integrity analysis. Today, X parameters are primarily used to characterize power amplifiers and nonlinear devices. Not yet applied to signal integrity. ECE 546 Jose Schutt Aine 4

5 X Parameters Purpose Characterize nonlinear behavior of devices and systems Advantages - Mathematically robust framework - Can handle nonlinearities - Instrument exists (NVNA) - Blackbox format vendor IP protection - Matrix format easy incorporation in CAD tools - X Parameters are a superset of S parameters ECE 546 Jose Schutt Aine 5

6 Challenges in HS Links High speed Serial channels are pushing the current limits of simulation. Models/Simulator need to handle current challenges Need to accurately handle very high data rates Simulate large number of bits to achieve low BER Non linear blocks with time variant systems Model TX/RX equalization All types of jitter: (random, deterministic, etc.) Crosstalk, loss, dispersion, attenuation, etc Handle and manage vendor specific device settings Clock data recovery (CDR) circuits These cannot be accurately modeled with S parameters ECE 546 Jose Schutt Aine 6

7 Motivation Limitation: S Parameters only work for linear systems. Many networks and systems are nonlinear Applications High-speed links, power amplifiers, mixed-signal circuits Existing Methods Load pull techniques IBIS models Models are flawed and incomplete ECE 546 Jose Schutt Aine 7

8 PHD Modeling Polyharmonic distortion (PHD) modeling is a frequency-domain modeling technique PHD model defines X parameters which form a superset of S parameters To construct PHD model, DUT is stimulated by a set of harmonically related discrete tones In stimulus, fundamental tone is dominant and higher-order harmonics are smaller ECE 546 Jose Schutt Aine 8

9 PHD Framework Signal is represented by a fundamental with harmonics Signals are periodic or narrowband modulated versions of a fundamental with harmonics Harmonic index: 0 for dc contribution, 1 for fundamental and 2 for second harmonic Power level, fundamental frequency can be varied to generate complete data for DUT ECE 546 Jose Schutt Aine 9

10 PHD Framework Stimulus A-waves are incident and B-waves are scattered Reference System Z C Default value is 50 ohm For a given port with voltage V and current I A V 2 Z I C B V Z I 2 C ECE 546 Jose Schutt Aine 10

11 PHD Framework F pm describes a time-invariant systemdelay in time domain corresponds to phase shift in frequency domain B e F A e, A e,..., A e, A e,... pm jm j j2 j j2 pm For phase normalization, define P j ( A11) e B F A, A P, A P,..., A P, A P,... P pm pm m ECE 546 Jose Schutt Aine 11

12 PHD Framework ECE 546 Jose Schutt Aine 12

13 Cross-Frequency Phase for Commensurate Tones Defined as the phase of each pseudowave when the fundamental, A 1,1, has zero phase. B 2,3 can be related to A 2,2 in magnitude and phase. A 1,1 A 2,2 B 2,3 ECE 546 Jose Schutt Aine 13

14 Time-Invariance Property of Nonlinear Scattering Function F ( A e, A (e ), A (e ),...) j j 2 j 3 pk, 1,1 1,2 1,3 F A A A j k pk, ( 1,1, 1,2, 1,3,...)(e ) Shifting all of the inputs by the same time means that different harmonic components are shifted by different phases. time delay f 0 180º phase shift 2f 0 360º phase shift 3f 0 540º phase shift ECE 546 Jose Schutt Aine 14

15 Defining Phase Reference B Can use time invariance to separate magnitude and phase dependence of one incident pseudowave. pk, pk, 1,1 1,2 1,3 F ( A, A, A,...) F ( A, A P, A P,...) P 2 3 pk, 1,1 1,2 1,3 Shifting reference to zero phase of A 1,1. k using A1,1 j A1,1 P e A 1,1 arg( ) ECE 546 Jose Schutt Aine 15

16 Commensurate Tones X-Parameter Formalism * Define X ( A, A P, A P,...) ( FB) 2 3 pk, 1,1 1,2 1,3 F ( A, A, A,...) P pk, 1,1 1,2 1,3 k B X ( A, A P, A P,...) P ( FB) 2 3 pk, pk, 1,1 1,2 1,3 Still difficult to characterize this nonlinear term. If only one incident pseudowave, A 1,1, is large then the other smaller inputs can be linearized about the largesignal response of F p,k to only A 1,1. *D. E. Root, et al., X-Parameters, k ECE 546 Jose Schutt Aine 16

17 PHD Framework Define variables A pm B pm port harmonic port harmonic Introduce multivariate complex function F pm such that B F A, A,..., A, A,... pm pm ECE 546 Jose Schutt Aine 17

18 Harmonic Superposition In many situations, there is only one dominant largesignal input component present. The harmonic frequency components are relatively small harmonic components can be superposed Harmonic superposition principle is key to PHD model ECE 546 Jose Schutt Aine 18

19 Nonanalytical Mapping* A nonlinearity described by: f ( x) x x 3 Signal is sum of main signal and additional perturbation term which is assumed to be small x() t x () t x() t o * see: J. Verspecht and D. E. Root, ʺPolyharmonic Distortion Modeling,ʺ IEEE Magazine, June 2006, pp ECE 546 Jose Schutt Aine 19

20 Case 1 Consider the signal x(t), given by the sum of a real dc component and a small tone at frequency f x () o t A A is real xt () jt * jt e e 2 is a small complex number The linear response in x(t) can be computed by y() t f x () t x() t f x () t o o ECE 546 Jose Schutt Aine 20

21 y() t f ' x () t x() t o For case 1, we evaluate the conductance nonlinearity f (x o ) at the fixed value x o =A 2 f ' A 3 A After substitution, we get Case 1 jt * jt 2 e e yt () 3A 2 The complex coefficient of term proportional to e jt is 3A 2 2 Linear input-output relationship ECE 546 Jose Schutt Aine 21

22 x t A t () cos o Case 2 Now, x o (t) is a periodically time-varying signal: xt () jt * jt e e 2 Evaluating the conductance nonlinearity at x o (t) gives 2 '( cos( )) 3 cos( ) f A t A t 3A 3A cos(2 t) ECE 546 Jose Schutt Aine 22

23 Case 2 We can evaluate (y(t)) to get: jt 2 jt 3A 3A e e yt () jt * jt e e 2 Now, we have terms proportional to e jt and e j3t and their complex conjugates. Restrict attention to complex term proportional to e jt ECE 546 Jose Schutt Aine 23

24 Case 2 The complex coefficient of term proportional to e jt is 3A 3A We observe that the output phasor at frequency is not just proportional to the input phasor at frequency but has distinct contributions to both and * * Linearization is not analytic In Fourier domain, we have: ˆ( ) 3 3 ˆ X ( ) Y A A e 2 jphase( ) ECE 546 Jose Schutt Aine 24

25 in which B K A P G pm H pm qn qn 11 m pq, mn 11 pq, mn 11 m n G A P Re( A P ) qn m n H A P Im( A P ) K,0,...,0 pm A11 Fpm A11 A pq, mn 11 PHD Derivation A pq, mn 11 F Re pm F Im n A qnp A11 pm n A qnp A11 qn,0,...,0,0,...,0 Spectral mapping is nonanalytic ECE 546 Jose Schutt Aine 25

26 Since we get n Re( A P ) qn n Im( A P ) qn B K A P pm PHD Derivation pm qn qn 11 m A P qn A P qn pq, mn 11 pq, mn 11 n G A P n H A P 2 conj A P m 2 j qn conj A P m A P qn A P qn n n qn n n 2 2 j conj A P qn conj A P qn n n ECE 546 Jose Schutt Aine 26

27 PHD Model 11, 11 B X A P X A P A ( FB) m ( S ) mn pm pm pq mn qn qn qn mn X A P conj A ( T) pq, mn 11 qn PHD Model Equation X A ( S ) p1, m1 11 K pm A11 ( T ) p m A 11 X A 1, ( S ) qn, 1,1 : X A pq, mn 11 ( T ) qn, 1,1 : X A pq, mn 11 G A jh A pq, mn 11 pq, mn 11 2 G A jh A pq, mn 11 pq, mn 11 2 ECE 546 Jose Schutt Aine 27

28 1-Tone X-Parameter Formalism * Incident Waves frequency frequency frequency B A 1,1 Large- Signal Approximates X ( A, A P, A P,...) ( FB) 2 3 pk, pk, 1,1 1,2 1,3 ( Nonlinear Mapping X (FB) p,k ( A 1,1,0,0,...) Simple Nonlinear Mapping S T X A X A ) ( ) * p, kql ;, ql, pkql, ;, ql, Nonanalytic Harmonic Superposition *J. Verspecht, et al., Linearization, Scattered Waves frequency frequency frequency ECE 546 Jose Schutt Aine 28

29 1-Tone X-Parameter Formalism * qn, lk qn, lk ( FB) k ( S) k l ( T ) * k l pk, pk, pkql, ;, ql, pkql, ;, ql, q1, l1 q1, l1 ( ql, ) (1,1) ( ql, ) (1,1) B X P X A P X A P Simple nonlinear map Linear harmonic map function of incident wave X parameters of type FB, S, and T fully characterize the nonlinear function. Depend on frequency large signal magnitude, A 1,1 DC bias output port *D. E. Root, et al., X-Parameters, Linear harmonic map function of conjugate of incident wave X ( S ) p, kql ;, output harmonic input port P A A input harmonic 1,1 1,1 ECE 546 Jose Schutt Aine 29

30 Excitation Design Excitation 1 Excitation 2 Excitation 3 Excitation 4 Each excitation will generate response with fundamental and all harmonics ECE 546 Jose Schutt Aine 30

31 X-Parameter Data File TOP: FILE DESCRIPTION! Created Fri Jul 30 07:44: ! Version = 2.0! HB_MaxOrder = 25! XParamMaxOrder = 12! NumExtractedPorts = 3! IDC_1=0 NumPts=1! IDC_2=0 NumPts=1! VDC_3=12 NumPts=1! ZM_2_1=50 NumPts=1! ZP_2_1=0 NumPts=1! AN_1_1=100e-03( dBm) NumPts=1! fund_1=[100 Hz->1 GHz] NumPts=4 ECE 546 Jose Schutt Aine 31

32 X-Parameter Data File MIDDLE: FORMAT DESCRIPTION BEGIN XParamData % fund_1(real) FV_1(real) FV_2(real) FI_3(real) FB_1_1(complex) % FB_1_2(complex) FB_1_3(complex) FB_1_4(complex) % FB_1_7(complex) FB_1_8(complex) FB_1_9(complex) % FB_1_12(complex) FB_2_1(complex) FB_2_2(complex) % FB_2_5(complex) FB_2_6(complex) FB_2_7(complex) % FB_2_10(complex) FB_2_11(complex) FB_2_12(complex) % T_1_1_1_1(complex) S_1_2_1_1(complex) T_1_2_1_1(complex) % S_1_4_1_1(complex) T_1_4_1_1(complex) S_1_5_1_1(complex) % T_1_6_1_1(complex) S_1_7_1_1(complex) T_1_7_1_1(complex) % S_1_9_1_1(complex) T_1_9_1_1(complex) S_1_10_1_1(complex)) % T_1_11_1_1(complex) S_1_12_1_1(complex) T_1_12_1_1(complex) % T_2_1_1_1(complex) S_2_2_1_1(complex) T_2_2_1_1(complex) % S_2_4_1_1(complex) T_2_4_1_1(complex) S_2_5_1_1(complex % T_2_6_1_1(complex) S_2_7_1_1(complex) T_2_7_1_1(complex) % S_2_9_1_1(complex) T_2_9_1_1(complex) S_2_10_1_1(complex) ECE 546 Jose Schutt Aine 32

33 X-Parameter Data File BOTTOM: DATA LISTING e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-14 Remarks Data is measured or generated from a harmonic balance simulator Data file can be very large ECE 546 Jose Schutt Aine 33

34 X-Parameter Relationship k kl kl * 11, 11, 11 b D a P S a P a T a P a ik ik ik jl jl ik jl jl ( jl, ) (1,1) P : Phase of a 11 D : B type X parameter ik Sik, jl : S type X parameter Tik, jl : T type X parameter ECE 546 Jose Schutt Aine 34

35 X Parameters of CMOS -1 x S11,11 - Amplitude (db) GHz 1 GHz T11,11 - Amplitude (db) GHz 1 GHz A11 (dbm) A11 (dbm) S21,11 - Amplitude (db) GHz 1 GHz T21,11 - Amplitude (db) GHz 1 GHz A11 (dbm) A11 (dbm) ECE 546 Jose Schutt Aine 35

36 X Parameters of CMOS S12,11 - Amplitude (db) GHz 1 GHz T12,11 - Amplitude (db) GHz 1 GHz A11 (dbm) A11 (dbm) S22,11 - Amplitude (db) GHz 1 GHz T22,11 - Amplitude (db) GHz 1 GHz A11 (dbm) A11 (dbm) ECE 546 Jose Schutt Aine 36

37 Large-Signal Reflection Microwave amplifier with fundamental frequency at 9.9 GHz ECE 546 Jose Schutt Aine 37

38 Compression and AM-PM Microwave amplifier with fundamental frequency at 9.9 GHz ECE 546 Jose Schutt Aine 38

39 T 22,11 Microwave amplifier with fundamental frequency at 9.9 GHz ECE 546 Jose Schutt Aine 39

40 Special Terms T-Type X Parameter Spectral mapping is non-analytical Real and imaginary parts in FD are treated differently Even and odd parts in TD are treated differently T involves non-causal component of signal Phase Term P P is phase of large-signal excitation (a 11 ) Contributions to B waves will depend on P In measurements, system must be calibrated for phase ECE 546 Jose Schutt Aine 40

41 Notation Change Define ( S ) S a X ( A ) ik, jl 11 ik, jl 11 ik ( T ) T a X ( A ) ik, jl 11 ik, jl 11 ( FB) D a X a 11 ik 11 ECE 546 Jose Schutt Aine 41

42 Handling Phase Term k kl kl * 11, 11, 11 b D a P S a P a T a P a ik ik ik jl jl ik jl jl ( jl, ) (1,1) Multiply through by P k * 11, 11, 11 bp D a S a P a T a P a k l l ik ik ik jl jl ik jl jl ( jl, ) (1,1) P where is the phase of a j 11 e we can always express the relationship in terms of modified power wave variables * 11, 11, 11 b D a S a a T a a ik ik ik jl jl ik jl jl ( jl, ) (1,1) where k b b P and a a P ik ik ik ik k ECE 546 Jose Schutt Aine 42

43 Handling R&I Components Because of non-analytical nature of spectral mapping, real and imaginary component interactions must be accounted for separately. we have b X X a r rr ri r b X X a i ir ii i where, X S T X S T rr r r ri i i, X S T X S T ir i i ii r r ECE 546 Jose Schutt Aine 43

44 Handling Phase Term Phase term can be accounted for by applying following transformations br Xrr Xriar b X X a i ir ii i ' cosb sinb b cos sin r Xrr Xri a a ' sinb cos b b Xir X ii sina cos i a ' a r ' ai in which ' br cosb sinbb r ' b i sinb cos b bi ' ar cosa sina a r ' a i sina cos a ai ECE 546 Jose Schutt Aine 44

45 X Matrix Construction Separate real and imaginary components Account for real-imaginary interactions Account for harmonic-to-harmonic contributions Account for harmonic-to-dc contributions Matrix size is 2mn 2mn m: number of harmonics n: number of ports ECE 546 Jose Schutt Aine 45

46 Matrix Formulation* size:2mn a1 a 2 a ap an a p a a a a a a (1) pr (1) pi (2) pr (2) pi ( m) pr ( m) pi We wish to use: b=xa b b b b b vector size is 2m m: number of harmonics n: number of ports (real vectors) *DC term not included 1 2 p n b p size:2 mn b b b b b b (1) pr (1) pi (2) pr (2) pi ( m) pr ( m) pi ECE 546 Jose Schutt Aine 46

47 X= X X X X 21 X22 Xpq Xn1 X n X pq (real matrix) *DC term not included Matrix Formulation* nn matrix size is 2mn 2mn m: number of harmonics n: number of ports size: 2m 2m X X X X X X X X X X X X X X X X X X (11) (11) (12) (12) (1 m) (1 m) pqrr pqri pqrr pqri pqrr pqri (11) (21) (21) (11) (21) (21) (12) (22) (22) (12) (22) (22) pqir pqrr pqir pqii pqri pqii pqir pqrr pqir pqii pqri pqii X X X X ( m1) ( m1) ( mm) ( mm) pqir pqii pqir pqii ECE 546 Jose Schutt Aine 47

48 X Matrix for 2-Port System* (2 harmonics) X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X (11) (11) (12) (12) (11) (11) (12) (12) 11rr 11ri 11rr 11ri 12rr 12ri 12rr 12ri (11) (11) (12) (12) (11) (11) (12) (12) 11ir 11ii 11ir 11ii 12ir 12ii 12ir 12ii (21) (21) (22) (22) (21) (21) (21) (21) 11rr 11ri 11rr 11ri 12rr 12ri X12rr X12ri (21) (21) (22) (22) (21) (21) (22) (22) 11ir 11ii 11ir 11ii 12ir 12ii 12ir 12ii (11) (11) (12) (12) (11) (11) (12) (12) 21rr 21ri 21rr 21ri 22rr 22ri 22rr 22ri (11) (11) (12) (12) (11) (11) (12) (12) 21ir 21ii 21ir 21ii 22ir X22ii X22ir X22ii (21) (21) (22) (22) (21) (21) (22) (22) 21rr 21ri 21rr 21ri 22rr 22ri 22rr 22ri (21) (21) (22) (22) (21) (21) (22) (22) 21ir 21ii 21ir 21ii 22ir 22ii 22ir 22ii (real matrix) For instance, X (12) 21ri is the contribution to the real part of the 1 st harmonic of the wave scattered at port 2 due to the imaginary part of the 2 nd harmonic of the wave incident port in port 1. *DC term not included ECE 546 Jose Schutt Aine 48

49 Polyharmonic Impedance Linear Impedance Polyharmonic Impedance Nonlinear Impedance - Time invariant - Linear - Scalar - Time invariant - Linear - Matrix - Time variant - Nonlinear - Function V ZI FD & TD [ V( f)] [ Z( f)][ I( f)] Vt () ZIt ( ()) FD only Model assumes that nonlinear effects are mild and are captured via harmonic superposition. ECE 546 Jose Schutt Aine 49

50 Polyharmonic Impedance 4-harmonic system in frequency domain: (1) (11) (12) (13) (14) (1) V Z Z Z Z I (2) (21) (22) (23) (24) (2) V Z Z Z Z I (3) (31) (32) (33) (34) (3) V Z Z Z Z I (4) (41) (42) (43) (44) (4) V Z Z Z Z I in time domain: v t v t v t v t v t (1) (2) (3) (4) () () () () () i t i t i t i t i t (1) (2) (3) (4) () () () () () ECE 546 Jose Schutt Aine 50

51 Polyharmonic Impedance Z o : Reference impedance matrix Z V I : Polyharmonic impedance matrix : Voltage vector : Current vector Describes interactions between harmonic Z= 1+X1-X -1 Zo components of voltage and current. V=ZI ECE 546 Jose Schutt Aine 51

52 Network Formulation Scattered waves b=xa Termination equations a=dv +Γb g Wave Solution a= 1-ΓX -1 Dvg Voltage Solution v= 1+Xa ECE 546 Jose Schutt Aine 52

53 Steady-State Simulations cubic term Time-Domain Response Vin Vout 40 X Parameter Volts time(ns) ADS ECE 546 Jose Schutt Aine 53

54 CMOS Driver/Receiver Channel Generate X parameters for composite system Power level: 20 dbm, frequency: 1 GHz Construct X matrix Combine with terminations for simulation ECE 546 Jose Schutt Aine 54

55 CMOS Driver/Receiver - Harmonics 8 6 DC+Fundamental Vin Vout Harmonics Vin Vout Volts 0 Volts time(ns) time(ns) Harmonics Vin Vout Harmonics Vin Vout Volts 0 Volts time(ns) time(ns) ECE 546 Jose Schutt Aine 55

56 Validation 8 6 Time-Domain Response Vin Vout 4 X Parameter Volts time(ns) ADS Vout, V Vin, V time, nsec ECE 546 Jose Schutt Aine 56

57 Cascading S-Parameter Blocks* A1(1) B1(1) A1(1) B1 (1) (1) [S] (1) [T] B2(1) A1(2) A2(1) B1(2) B2(1) A2 (1) = = A1(2) B1 (2) [S] [T] (2) (2) B2(2) A1(1) A2(2) B1(1) B2(2) A1(1) (2) (1) A2 [T] = transfer scattering parameters. B1 (T) [S] [T] B2(2) (T) (1) A2(2) = (2) [T] [T] A2(2) B2(2) Can disregard circuit behavior at internal node. *G. Gonzalez, Microwave Transistor Amplifiers: Analysis and Design, 2nd ed. Prentice-Hall, ECE 546 Jose Schutt Aine 57

58 Cascading X-Parameter Blocks* A1,k(1) B1,k(1) (1) [X] These equations at the internal node must always be satisfied: B1,k(1) = A1,k(2), A1,k(2) = B2,k(1) for all values of k. B2,k(1) A2,k(1) A1,k(1) B1,k(1) = = A1,k(2) B1,k(2) (T) [X] (2) [X] B2,k(2) A2,k(2) B2,k(2) A2,k(2) *D. E. Root, et al., X-Parameters, ECE 546 Jose Schutt Aine 58

59 Cascading X Parameters GOAL: Simulate complete channel by combining X- parameter blocks from different sources into a single composite X matrix. Vendor A Foundry In-House Vendor B In-House Vendor C Foundry X X-parameters of individual devices can be accurately cascaded within a harmonic balance simulator environment. ECE 546 Jose Schutt Aine 59

60 Volterra Series A linear causal system with memory can be described by the convolution representation y() t h x t d where x(t) is the input, y(t) is the output, and h(t) the impulse response of the system. A nonlinear system without memory can be described with a Taylor series as: yt () a () n n xt n1 where x(t) is the input and y(t) is the output. The an are Taylor series coefficients. ECE 546 Jose Schutt Aine 60

61 A Volterra series combines the above two representations to describe a nonlinear system with memory n 1 y( t) du... du g u,..., u x( t u ) Volterra Series 1 n n 1 n r n1 n! r1 y( t) 1 1! du du g ( u, u2) x( t u1) x( t u2) 2! du du du g ( u, u, u ) x( t u ) x( t u ) g ( u, u ) x( t u ) x( tu ) x( t u ) 2!... du g ( u ) x( t u ) impulse response higher-order impulse responses where x(t) is the input and y(t) is the output and the g n (u 1,,u n ) are called the Volterra kernels ECE 546 Jose Schutt Aine 61

62 1 1 y() t du g u x t u du du g 1! 2! u, u x tu x tu where the input x(t) is given by x() t exp j t exp j t 1 yt () dug 1 1 u1 e e 1! j t u j t u du du g u, u e e e e 2! Volterra Series Application to X parameters: Take order = 2 This can be written as j 1 tu1 j2 tu1 j 1tu2 j2tu2 ECE 546 Jose Schutt Aine 62

63 Volterra Series Define T e, T e j1t j2t 1 2 U U j 1u1 e j 1u2 U e j u e 2 2 U j u e 2 1 This gives yt () T dug u U T dug u U du du g u u TU T U TU T U 2!, or yt () I I I I I I ECE 546 Jose Schutt Aine 63

64 Volterra Series in which I T du g u U I TG f I T du g u U I TG f G 1 (f) is the Fourier transform of g 1 (u) evaluated at f ECE 546 Jose Schutt Aine 64

65 Volterra Series ! 2,, I T du du g u u U U T G f f 1 1 I TT du du g u u U U TTG f f,, ! I T T du du g u u U U TTG f f,, ! ! 2,, I T du du g u u U U T G f f G 2 (f 1, f 2 ) is the double Fourier transform of g(u,v) evaluated at (f 1, f 2 ) ECE 546 Jose Schutt Aine 65

66 So, G 1 (f) is the Fourier transform of g 1 (u) evaluated at f and G 2 (f 1, f 2 ) is the double Fourier transform of g(u,v) evaluated at (f 1, f 2 ) We can also express y(t) as: in which y t TG f TG f and yt y t y t 1 2 Volterra Series 1 y t TG f f TTG f f TTG f f TG f f 2! 2 2,,,, ECE 546 Jose Schutt Aine 66

67 If we take into account the respective amplitudes of the tone, we have x() t A exp j t A exp j t We can make the transformation T AT and T AT j t u j t u yt () dug u Ae Ae 1 2! Ae Volterra Series j t u , j t u j t u j t u 1 2 du du g u u A e A e Ae y1 t ATG f1 ATG f2 ATermH BTermH A B ECE 546 Jose Schutt Aine 67

68 Volterra Series y 2 t A1T1 G2 f1, f1 2! CTermHC AT ATG f f ATATG f f,, DTermH 2 2 AT 2 2G2 f2, f2 ETermHE If we choose f 2 = kf 1, then T 1 f 1 and T 2 kf 1. In general, 2 =k 1 so that if H A =A-Term contains terms in f 1 H B =B-Term contains terms in kf 1 H T1 f1, T2 kf C =C-Term contans terms in 2f 1 1 H D =D-Term contains terms in 2kf 1 H E =E-Term contains terms in (k+1)f 1 D ECE 546 Jose Schutt Aine 68

69 Volterra Series - First determine the X parameters of the system - Next, provide excitation a(t) at () Aexp j t Aexp j t Next, calculate b in phasor domain using X parameters b=xa - For each port, the scattered wave will include contributions from all harmonics bp HAHB HC HD HE ECE 546 Jose Schutt Aine 69

70 Volterra Series Finally, a relationship can be obtained to extract Volterra kernel Fourier transforms A-Term: G f 1 1 H A G f A 1 B-Term: 1 2 H B A 2 C-Term: G f f D-Term: 2H C 2 1, 1 2 A1 G f f 2H D 2 2, 2 2 A2 E-Term: G f f H E 2 1, 2 A1A 2 ECE 546 Jose Schutt Aine 70

71 Volterra Series TABLE OF VOLTERRA KERNEL TRANSFORMS Term Wave Coefficient Constant index k=1 k = 2 k = 3 k = 3 A-Term T 1 G 1 (f 1 ) A 1 f 1 f 1 f 1 f 1 f 1 B-Term T 2 G 1 (f 2 ) A 2 k f 1 f 1 2 f 1 3 f 1 4 f 1 C-Term T 1 2 G 2 (f 1,, f 1 ) A 12 /2! 2 f 1 2 f 1 2 f 1 2 f 1 2 f 1 D-Term T 2 2 G 2 (f 2,, f 2 ) A 22 /2! 2k f 1 2 f 1 4 f 1 6 f 1 8 f 1 E-Term T 1 T 2 G 2 (f 1,, f 2 ) A 1 A 2 (k+1) f 1 2 f 1 3 f 1 4 f 1 5 f 1 ECE 546 Jose Schutt Aine 71

72 Nonlinear Vector Network Analyzer (NVNA) NVNA instruments will gradually replace all VNAs ECE 546 Jose Schutt Aine 72

73 Nonlinear Vector Network Analyzer (NVNA) * *L. Betts, X-Parameters and NVNA, May 9, ECE 546 Jose Schutt Aine 73