Notes 18 Green s Functions

Transcription

1 ECE 638 Fall 017 David R. Jackso Notes 18 Gree s Fuctios Notes are from D. R. Wilto, Dept. of ECE 1

2 Gree s Fuctios The Gree's fuctio method is a powerful ad systematic method for determiig a solutio to a problem with a kow forcig fuctio o the RHS. The Gree s fuctio is the solutio to a poit or impulse forcig fuctio. It is similar to the idea of a impulse respose i circuit theory. George Gree ( ) Gree's Mill i Seito (Nottigham), Eglad, the mill owed by Gree's father. The mill was reovated i 1986 ad is ow a sciece cetre.

3 Gree s Fuctios Cosider the followig secod-order liear differetial equatio: 1 d d P( x) u( x) Q( x) u( x) f( x) w( x + = ) dx dx or u = f ( f is a forcig fuctio.) where 1 d d P x + w x dx dx Q( x) 3

4 Gree s Fuctios Problem to be solved: u = f u( b) u a = 0, = 0 ( Dirichlet BCs) or u ( b) u a = 0, = 0 ( Neuma BCs) 4

5 Gree s Fuctios (cot.) We ca thik of the forcig fuctio f (x) as beig broke ito may small rectagular pieces. Usig superpositio, we add up the solutio from each small piece. Each small piece ca be represeted as a delta fuctio i the limit as the width approaches ero. f( x) f ( x) ( ) ( ) f( x i ) x x x i f( x ) xδ x x i i Note: The fuctio becomes a δ fuctio i the limit as x 0. 1, x x /, x / x = x 0, otherwise ( ) a x x i b x 5

6 Gree s Fuctios (cot.) The Gree s fuctio G(x, x ) is defied as the solutio with a delta-fuctio at x = x for the RHS. G xx, = δ x x The solutio to the origial differetial equatio (from superpositio) is the Note: b (, ) u x = f x G x x dx a G( x,x ) N b u x f x xg xx, f x G xx, dx i= 1 i i a a x a x (, ) solutio from sigle pulse ( ) G xx x x x i i cetered at 6

7 Gree s Fuctios (cot.) There are two geeral methods for costructig Gree s fuctios. Method 1: Fid the solutio to the homogeous equatio to the left ad right of the delta fuctio, ad the eforce boudary coditios at the locatio of the delta fuctio. Au1 x, x x G( xx, ) = Bu x, x x Au1 ( x) a x Bu ( x) b x The fuctios u 1 ad u are solutios of the homogeous equatio. The Gree s fuctio is assumed to be cotiuous. The derivative of the Gree s fuctio is allowed to be discotiuous. 7

8 Gree s Fuctios (cot.) Method : Use the method of eigefuctio expasio. Eigevalue problem: ψ = λψ or ψ ( a) ψ ( b) = 0, = 0 ( Dirichlet BCs) ( a) ψ ( b) ψ = 0, = 0 ( Neuma BCs) We the have: G xx = aψ x, Note: The eigefuctios are orthogoal. 8

9 Gree s Fuctios (cot.) lim Method 1 Itegrate both sides over the delta fuctio: x ε 1 d dg G dx = lim P ( x) + Q ( x) G dx w( x) dx dx x + ε + ε 0 ε 0 x ε x ε Au1 ( x) a x Bu ( x) b x (, ) = δ ( ) G xx x x x P x dg( x, x ) P x dg( x, x ) + ε = = lim δ ( x x ) dx = 1 w( x ) dx dx ε 0 x ε Au1 x, x x G( xx, ) = Bu x, x x ( ) dg( x, x ) dg( x, x ) w x = dx dx P x + ( ) ( ) w x Bu ( x ) Au 1( x ) = P x 9

10 Gree s Fuctios (cot.) Method 1 (cot.) Au1 ( x) Bu ( x) Au1 x, x x G( xx, ) = Bu x, x x a x b x Also, we have (from cotiuity of the Gree s fuctio): ( ) = ( ) Au x Bu x 1 10

11 Gree s Fuctios (cot.) Method 1 (cot.) Au1 ( x) Bu ( x) We the have: ( ) ( ) w x u 1( x ) u ( x ) A P x u1( x ) u( x ) = B 0 a x b x where A, B w x u x w x u1 x = = P x Wu [ 1, u] P x Wu [ 1, u] W[ u, u ] = W x = u ( x ) u ( x ) u ( x ) u ( x ) (Wroskia)

12 Gree s Fuctios (cot.) Method 1 (cot.) Au1 ( x) Bu ( x) a x b x We the have: W ( x ) w x u x u x P x G( xx, ) = w( x ) u x u x P( x ) 1 < W ( x ) 1, >, x x x x 1

13 Gree s Fuctios (cot.) Method The Gree s fuctio is expaded as a series of eigefuctios. G xx = a x, ψ =4 a =3 x = =1 b x where ψ = λψ The eigefuctios correspodig to distict eigevalues are orthogoal (from Sturm-Liouville theory). 13

14 Gree s Fuctios (cot.) Method (cot.) G xx, = δ x x = ( ) aψ x δ x x = ( ) a ψ x δ x x = ( ) a λψ x δ x x Multiply both sides by ψ m *(x) w(x) ad the itegrate from a to b. =4 a =3 x = =1, = ( ), a λ ψ x ψ x δ x x ψ x m m *, = ( ) ( ) a λ ψ x ψ x w x ψ x m m *, = ( ) ( ) a λ ψ x ψ x w x ψ x m m m m m b Orthogoality x Delta-fuctio property 14

15 Gree s Fuctios (cot.) Method (cot.) =3 =4 =1 = Hece a m = * ( ) ψ m ( x ) ( x) ( x) w x λ ψ ψ m m m a x b x Therefore, we have * w x ψ (, ) x G xx = ψ λ ψ x ψ x x Note : b ψ x ψ x ψ x w x dx a 15

16 Applicatio: Trasmissio Lie A short-circuited trasmissio lie with a distributed curret source: x The distributed curret source is a surface curret. d = 0 I I = h + V ( ) - s d s = [ A/m] I J sx Gree s fuctio: d s = δ ( ) = 1A I I s 16

17 Applicatio: Trasmissio Lie (cot.) A illustratio of the Gree s fuctio: + - = G(, ) V = 0 I = h I s = 1A = = The total voltage due to the distributed curret source is the h d (, ) V = I s G d 0 17

18 Applicatio: Trasmissio Lie (cot.) Telegrapher s equatios for a distributed curret source: dv = jωli d di d = Is jωcv d L = iductace/meter C = capacitace/meter (Please see the Appedix.) Take the derivative of the first ad substitute from the secod: dv d = jωl I ( d ) s jωcv Note: j is used istead of i here. 18

19 Applicatio: Trasmissio Lie (cot.) Hece or dv d + kv = ( j ω L ) I s d 1 dv d + kv = Is jωl d Therefore: (, ) 1 dg (, ) δ ( ) + kg = jωl d where k = ω LC 19

20 Applicatio: Trasmissio Lie (cot.) 1 dv d + kv = Is jωl d Compare with: 1 d d P( ) u( ) Q( ) u( ) f( ) w( + = ) d d Therefore: = ω = = w j L, P 1, Q k 0

21 Applicatio: Trasmissio Lie (cot.) Method 1 = 0 I = h + V( ) = G(, ) - 1A = The geeral solutio of the homogeeous equatio is: = = V Au A k 1 1 si = = ( ) si V Bu B k h Homogeeous equatio: dv + 0 kv = d 1

22 Applicatio: Trasmissio Lie (cot.) + V( ) = G(, ) - = 0 I = h 1A = The Gree s fuctio is: W ( ) w u u P G(, ) = w( ) u u P( ) 1 < W ( ) 1 >,, = u1 si k = ( ) u si k h

23 Applicatio: Trasmissio Lie (cot.) + V( ) = G(, ) - The fial form of the Gree s fuctio is: ( k( h) ) si ( k ) W( ) ( k ) si ( k( h) ) W( ) si ( jωl), < G(, ) = si ( jωl), > = 0 I = h 1A = where ( ) W = Wu [ 1, u ] = k si k cos k h cos k si k h = ksi kh 3

24 Applicatio: Trasmissio Lie (cot.) Method = 0 I = h + V( ) = G(, ) - 1A = The eigevalue problem is: 1 d ψ k ψ λψ jωl + = d This may be writte as d ψ = d λψ where λ jωl λ + k 4

25 Applicatio: Trasmissio Lie (cot.) We the have: d ψ d ( h) λψ ψ ψ =, 0 = = 0 The solutio is: ψ ( ) = si ( λ ) λ = π h Note : b π h ψ ψ ψ ψ ω ω h ( ) ( ) = ( ) * ( ) w( ) d = si ( j L) d = ( j L) a h 0 5

26 Applicatio: Trasmissio Lie (cot.) We the have: * w ψ (, ) G = ψ λ ψ ψ where λ ψ ( λ k ) jωl jωl h π = w h ( ) = 1 1 π = = k si h ψ ψ ω ( ) ( ) = ( j L) jωl 6

27 Applicatio: Trasmissio Lie (cot.) The fial solutio is the: π si h π = ω = 1 π k h G(, ) ( j L) si h h 7

28 Applicatio: Trasmissio Lie (cot.) Summary = 0 I = h + V( ) = G(, ) - 1A = k = ω LC G(, ) = ( ω ) ( ω ) ( k( h) ) si ( k ) ksi ( kh ) ( k ) si ( k( h) ) k si ( kh) si j L, < si j L, > π si h π G(, ) = ( jωl) si h = 1 π h k h 8

29 Applicatio: Trasmissio Lie (cot.) Other possible Gree s fuctios for the trasmissio lie: We solve for the Gree s fuctio that gives us the curret I () due to the 1A parallel curret source. We solve for the Gree s fuctio givig the voltage due a 1V series voltage source istead of a 1A parallel curret source. We solve for the Gree s fuctio givig the curret to due a 1V series voltage source istead of a 1A parallel curret source. 9

30 Appedix This appedix presets a derivatio of the telegrapher s equatios with distributed sources. i - d vs + L + + v - d i C s - - i + v + Allow for distributed sources 30

31 Appedix (cot.) + d v v = vs L i + i t ( d ) s 0: v = v d s L i t i - d vs + L + + v - d i C s - - i + v + 31

32 Appedix (cot.) + d = s ( ) i i i C v t + 0: i = i d s C v t i - d vs + L + + v - d i C s - - i + v + 3

33 Appedix (cot.) I the phasor domai: v i d i = vs L t d v = is C t t jω dv d = V s j ω LI d di d = I d s jωcv 33