ECE 451 Advanced Microwave Measurements. Requirements of Physical Channels

Transcription

1 ECE 451 Advanced Microwave Measurements Requirements of Physical Channels Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois ECE 451 Jose Schutt Aine 1

2 Issues Frequency and time limitations Minimum phase characteristics Reality Stability Causality Passivity ECE 451 Jose Schutt Aine 2

3 Complex Plane An arbitrary network s transfer function can be described in terms of its s-domain representation s is a complex number s = + j The impedance (or admittance) or transfer function of networks can be described in the s domain as T() s m as m a s... asa n s b s bsb m1 m1 1 0 n1 n ECE 451 Jose Schutt Aine 3

4 T() s Transfer Functions m as m a s... asa n s b s bsb m1 m1 1 0 n1 n The coefficients a and b are real and the order m of the numerator is smaller than or equal to the order n of the denominator A stable system is one that does not generate signal on its own. For a stable network, the roots of the denominator should have negative real parts 4 ECE 451 Jose Schutt Aine 4

5 Transfer Functions The transfer function can also be written in the form T() s a m sz sz... sz 1 2 sp sp... sp 1 2 m m Z 1, Z 2, Z m are the zeros of the transfer function P 1, P 2, P m are the poles of the transfer function For a stable network, the poles should lie on the left half of the complex plane 5 ECE 451 Jose Schutt Aine 5

6 Fourier Transform Pairs a re (t): real part of even time-domain function a ie (t): imaginary part of even time-domain function a ro (t): real part of odd time-domain function a io (t): imaginary part of odd time-domain function at () a () t ja () t a () t ja () t re ie ro io In the frequency domain accounting for all the components, we can write: A RE (w): real part of even function in the frequency domain A IE (w): imaginary part of even function in the frequency domain A RO (w): real part of odd function in the frequency domain A IO (w): imaginary part of odd function in the frequency domain A A ja A ja RE IE RO IO ECE 451 Jose Schutt Aine 6

7 Fourier Transform Pairs We also have the Fourier-transform-pair relationships: Time Domain : at ( ) a ( t) ja ( t) a ( t) ja ( t) Freq Domain : re ie ro io A A ja A ja RE IE RO IO B S A ja A ja RE IE RO IO In the time domain, this corresponds to: bt () st ()* are() t aro() t j aie() t aio() t ECE 451 Jose Schutt Aine 7

8 a () t a () t 0 which implies that: A A 0 ie Fourier Transform Pairs We now impose the restriction that in the time domain, the function must be real. As a result, io The Fourier-transform pair relationship then becomes: Time Domain : at ( ) a ( t) a ( t) Freq Domain : re ro IE A A ja The frequency-domain relations reduce to: RE B S ARE ja IO IO RO ECE 451 Jose Schutt Aine 8

9 In summary, the general relationship is: Time Domain : at ( ) a ( t) ja ( t) a ( t) ja ( t) Freq Domain : Time Domain : at ( ) a ( t) ja () t a ( t) ja ( t) Freq Domain : Fourier Transform Pairs But for a real system: re ie ro io re ie ja A A A ja RE IE ro RO A A ja A ja RE IE RO IO io IO ECE 451 Jose Schutt Aine 9

10 Causality Violations NON-CAUSAL Z( f) Ro f jl Near (blue) and Far (red) end responses of lossy TL CAUSAL Z( f) Ro f jro f jl ECE 451 Jose Schutt Aine 10

11 Consider a function h(t) ht ( ) 0, t0 ht () h() t h() t e Causality Principle Every function can be considered as the sum of an even function and an odd function o 1 he () t h() t h( t) 2 1 ho () t h() t h( t) 2 Even function Odd function h () t o he (), t t 0 he (), t t 0 h ( t) sgn( t) h ( t) o e ECE 451 Jose Schutt Aine 11

12 ht () h() t sgn() th() t e 1 H( f) He( f ) * He( f ) j f H ( f) H ( f) jhˆ ( f) e Hilbert Transform In frequency domain this becomes e e Imaginary part of transfer function is related to the real part through the Hilbert transform Hˆ ( f ) is the Hilbert transform of H ( f ) e 1 1 x( ) xˆ( t) x()* t d t t e ECE 451 Jose Schutt Aine 12

13 Discrete Hilbert Transform Imaginary part of transfer function can be recovered from the real part through the Hilbert transform If frequency domain data is discrete, use discrete Hilbert Transform (DHT)* H( f ) n fˆ k 2 2 n n odd even fn, k even k n fn, k odd k n *S. C. Kak, "The Discrete Hilbert Transform", Proceedings of the IEEE, pp , April ECE 451 Jose Schutt Aine 13

14 HT for Via: 1 MHz 20 GHz Actual Transform Actual Transform 0.3 Real(S11) - Via 0.5 Imag(S11) - Via Actual Actual Actual Transform A Actual Transform A Actual is red, HT is blue 1 Real(S12) - Via 1 Imag(S12) - Via Real(S12) Imag(S12) Observation: Poor agreement (because frequency range is limited) A A ECE 451 Jose Schutt Aine 14

15 Example: 300 KHz 6 GHz Actual Transform Real Part of S11 Actual Transform Imaginary Part of S Real(S11) 0 Imag(S11) Actual Transform A Real(S12) Actual Transform A Imag(S12) Actual is red, HT is blue Real(S12) 0 Imag(S12) A Observation: Good agreement A ECE 451 Jose Schutt Aine 15

16 Microstrip Line S11 ECE 451 Jose Schutt Aine 16

17 Microstrip Line S21 ECE 451 Jose Schutt Aine 17

18 Discontinuity S11 ECE 451 Jose Schutt Aine 18

19 Discontinuity S21 ECE 451 Jose Schutt Aine 19

20 Backplane S11 ECE 451 Jose Schutt Aine 20

21 Backplane S21 ECE 451 Jose Schutt Aine 21

22 Minimum Phase System A linear, time invariant system is said to be minimumphase if the system and its inverse are causal and stable. Any stable and causal rational transfer function can be decomposed into a minimum phase system and an all pass system written as s H () s M() s P() s e where M(s)P(s)=N(s) M(s) is a minimum phase system, and P(s) is an all-pass system. The minimum phase system is not only causal and stable, but also has causal and stable inverse. That is, all the poles and zeros of the minimum phase system are in the left half of the complex plane. ECE 451 Jose Schutt Aine 22

23 Minimum Phase System The all-pass system has constant magnitude over the whole frequency range. The poles and zeros of the all pass system are symmetric about the imaginary axis in the complex plane. For any stable and causal rational function, the poles are all in the left half of the complex plane due to the stability constraints ECE 451 Jose Schutt Aine 23

24 Minimum Phase System However, the zeros can be anywhere in the complex plane except that they should be symmetric about the real axis so that real coefficients of the rational function are guaranteed. For a rational function with zeros in the right half plane, the zeros can be reflected to the left half plane. ECE 451 Jose Schutt Aine 24

25 Minimum Phase System Im[s] Im[s] Im[s] x o o x o o pole-zero cancellation o x o o Re[s] x = X Re[s] o x o x o Re[s] N () s M () s P () s ij ij Therefore, the original system is decomposed into two systems: a minimum phase system with poles of the original system and the reflected zeros, and an all pass system with poles at where the reflected zeros are and zeros in the original right half plane. ij ECE 451 Jose Schutt Aine 25

26 HT of Minimum Phase System s ij H ( s) M ( s) P ( s) e ij ij ij j ij P ( j) e 1 ij s j H ( j) M ( j) ij ij The phase of a minimum phase system can be completely determined by its magnitude via the Hilbert transform 2 U( ) U( ) arg[ Mij ( )] d HTU ( )( ) 0 U( ) ln M ( ) ln H ( ) ij ij ECE 451 Jose Schutt Aine 26

27 Enforcing Causality in TL The complex phase shift of a lossy transmission line l Rj L Gj Cl X e e is non causal We assume that e e e e j( ) ( ) j LCl Rj L Gj Cl is minimum phase non causal ECE 451 Jose Schutt Aine 27

28 Enforcing Causality in TL j l HT e e HT e ( ) ( ) ' ln ln ( ) e j ' ( ) ( ) e is minimum phase and causal ' X e e e ' j ( ) ( ) j LCl is the causal phase shift of the TL In essence, we keep the magnitude of the propagation function of the TL but we calculate/correct for the phase via the Hilbert transform. ECE 451 Jose Schutt Aine 28