Faculdade de Engenharia. Transmission Lines ELECTROMAGNETIC ENGINEERING MAP TELE 2007/2008

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1 Transmission ines EECTROMAGNETIC ENGINEERING MAP TEE 78

2 ast week eneral transmission line equations ( R jω)( G jωc) γ propaation constant and characteristic impedance finite transmission lines reflection coefficient stationary wave voltae maxima and minima, SWR Γ R jω G jωc transmission lines as circuit elements the Smith chart impedance and reflection coefficient determination admittance location of voltae and current maxima and minima EE 78 ines

3 Today the Smith chart review impedance matchin λ4 transformer reactive elements sinle-stub double-stub transients EE 78 ines 3

4 Smith chart Γ im x constant r constant Γ re EE 78 ines 4

5 Smith chart from: point in chart ( intersection of curves correspondin to r and x ) x Γ and θ Γ Γ im θ Γ Γ from: Γ r and x Γ re r EE 78 ines 5

6 Input impedance. draw the point correspondin to the normalied load impedance point P. draw the circle centered at the oriin with radius OP 3. draw the straiht line from O to P 4. draw the straiht line from O that corresponds to a rotation of l toward the enerator 5. intersection of this line with previous circle point P 6. obtain, where in is read from P in in Γ im P P Γ re EE 78 ines 6

7 Admittance j tan ( ) ( β' ) ' λ j tan( π ) ' j tan( β' ) 4 j tan( π ) ( λ 4) y ( λ ) 4 λ 36º Γ im λ 4 8º Γ re. draw y. rotate 8º EE 78 ines 7

8 Maxima and minima location voltae maxima where Γ( ' ) nπ voltae minima where Γ( ' ) ( n )π Γ im voltae maxima voltae minima Note:. maxima and minima where impedance is real. maxima (minima) points are separated by nl Γ re EE 78 ines 8

9 Problem EE 78 ines 9

10 EE 78 ines Impedance matchin lossless line: ( ) ( ) Γ Γ Γ Γ Γ Γ l j j j l j l j j j l j e e e e I e e e e β β β β β β β β and ( ) ( ) ( ) { } I P av * Re ( ) ( ) av l P θ θ β Γ Γ Γ Γ Γ cos ± Γ iven often real,

11 Impedance matchin P av ( ) Γ Γ Γ Γ cos( β l θ θ ) Γ ± Γ iven often Γ P av 8 ( Γ ) power delivered by the source is maximum when Γ line should be matched to the load EE 78 ines

12 Impedance matchin P av 8 ( Γ ) in ± matchin section iven matched line mismatched section (no reflections) (reflections) maximum power delivered by the source P av 8 maximum power delivered to the load EE 78 ines

13 l4 transformer ( ' ) j j tan tan ( β' ) ( β' ) λ j tan ( π ) ( π ) ' 4 j tan λ 4 transmission line with lenth l4 can be used to match impedances EE 78 ines 3

14 l4 adaptor mismatched line λ 4 matched line ( ) Note: must be real if both lines are lossless EE 78 ines 4

15 l4 adaptor complex λ 4 d matched line ( ) R R R EE 78 ines 5

16 l4 adaptor SWR R λ 4 d Γ SWR Γ Γ R Γ Γ R R Γ SWR SWR SWR matched section mismatched sections EE 78 ines 6

17 l4 adaptor Smith chart λ 4 R d R Γ im d r R r Γ re R EE 78 ines 7

18 l4 adaptor different frequency desin: f f p λ p v f p different frequency: f f d f p λ d v f d d λ p 4 R d R d λ d 4 R mismatched line Γ im Γ im d d r r Γ re Γ re EE 78 ines 8

19 Impedance matchin with reactive elements Note: elements in parallel use admittances matched line Y jb B a B Y Y Y jb a jx a d jb Y a j j tan tan ( βd) ( βd ) desin adaptor find X and d a X a > inductor X a ω X a < capacitor C ω X a such that Im{ Y } such that Re{ } Y EE 78 ines 9

20 Matchin with reactive elements Smith chart normalied admittances: d y Y jx a jb a Y jb y a jb a matchin condition: y Y Y jb a y y jb a y jb y y y jb a y jb b a b d such that Re( y ) EE 78 ines

21 Matchin with reactive elements Smith chart b a b d d such that Re( y ) y a jb a d Γ im d y y jb a y jb y P curve r intersected at two points Γ re two possible solutions P P y jb and P r b a jb X a ba EE 78 ines

22 Transmission lines as reactive elements short-circuited line l open-circuited line l ( l) j j tan tan ( βl) ( βl) ( ) j tan βl ( l) j tan( βl) j tan( βl) j tan ( βl ) input impedances are purely imainary STUBs short-circuited or open-circuited transmission lines can be used as reactive elements in impedance matchin EE 78 ines

23 ines as reactive elements input admittance short-circuited line l open-circuited line l y i y y i y Γ im l Γ im y i jb i y Γ re y Γ re y i jb i l EE 78 ines 3

24 Impedance matchin with a sinle stub stub in parallel y y y y y s y s imainary Re ( y ) ( y ) Re d matched line y Re( y ) Im ( ) Im( y ) y s y jb b s b y s jb s l desin:. find d such that Re( y ). find l such that Im( ) Im( y ) y s or open-circuited EE 78 ines 4

25 Impedance matchin with sinle stub Smith chart y y jb d d Γ im jb y y r y s jb l y Γ re jb l desin. find d such that y jb. find l such that y s jb Notes curve r intersected at two points two solutions (only one represented in fiure) for open-circuited stub, l is measured from y EE 78 ines 5

26 Impedance matchin with sinle stub Smith chart solution solution d Γ im jb l oc Γ im jb d y y l oc y Γ re Γ re y jb l sc l sc jb EE 78 ines 6

27 Impedance matchin with sinle stub summary d jb y y jb d y y s jb l jb l desin. draw y. find d such that y jb 3. find l such that y s jb problem: it is not always possible to place stub in the desired location EE 78 ines 7

28 Impedance matchin with double stub y 3 y y d when the location of the stub is predefined the transmission line should be matched usin a double stub y sb jb sb y sa jb sa l B l A for a iven d or opeb-circuited or open-circuited desin of adaptor obtain l A and l B EE 78 ines 8

29 Impedance matchin with double stub stubs in parallel y y y sa y y 3 y sb y s imainary Re( y ) Re( y ) Re ( y ) ( y ) 3 Re y 3 y sb jb sb y d y y sa jb sa rotatin d y y matched line y 3 l B l A y jb b sb b desin stub B from y or open-circuited or open-circuited { y } { y } b b b Im{ y } y Re jb b Im sa sa desin stub A from y EE 78 ines 9

30 Impedance matchin with double stub desin usin Smith chart How to find y and y? y y y sa ( y ) Re( ) Re y rotatin d y jb b sb b y y y in circle r Re{ y } y in circle r, rotated by d toward load y in circle r y 3 y d y Intersection of these two circles y y sb jb sb y sa jb sa l B l A EE 78 ines 3

31 Impedance matchin with double stub Smith chart find y Intersection of r Re{ y } and r, rotated by d toward load Important: y miht not exist r, rotated by d toward load d y y y 3 y d y y sb jb sb y sa jb sa y l B l A r Re{ y } EE 78 ines 3

32 Impedance matchin with double stub Smith chart find y rotate y by d toward enerator y 3 y sb jb sb y d y y sa jb sa y y y d y l B l A EE 78 ines 3

33 Impedance matchin with double stub Smith chart from y and y lenth of stubs y 3 y d y y jb { y } y Re jb y sb jb sb y sa jb sa b b sb b { } sa b Im y l B l A find l A and l B EE 78 ines 33

34 Impedance matchin with double stub desin y 3 y d y y sb jb sb y sa jb sa y y d l B l A y y desin. find y. find y 3. find lenth of stubs EE 78 ines 34

35 Impedance matchin with double stub summary y 3 y d y desin. draw y y sb jb sb y sa jb sa. draw circle r rotated by d toward load 3. find y as intersection of this circle with r Re{ } 4. find y rotatin y by d toward enerator 5. find l B from b sb b 6. find l A from b b Im{ } sa y y l B l A y y d Notes: y, y, y SA e y SB miht have two solutions for each stub it miht be necessary to consider solutions in short-curcuit and in open-circuit y y EE 78 ines 35

36 Impedance matchin with double stub load vs distance d find y intersection of r Re{ y } with r rotated by d toward load this intersection miht not be possible for iven loads and distances d r,rotated d y Is it possible to match the line under these conditions usin a double stub? r Re { } y first stub not located next to load EE 78 ines 36

37 Impedance matcthin with double stub load vs distance d locatin the stub at distance d from the load y 3 y d y y 4 d y sb jb sb y sa jb sa l B l A r,rotated d y y y y 4 r Re{ y 4 } d EE 78 ines 37

38 Impedance matchin final conclusions impedance matchin is only exact at the desin frequency only the main line is matched a iven method will not always work: l4 lines sometimes it is not possible to find lines with desired sinle stub the desired location for the stub miht not be accessible doule stub it miht not exist a solution for y and it miht not be possible to locate the stub at a distance d from the load EE 78 ines 38

40 Problem assume short-circuited stubs EE 78 ines 4

42 Transients in lossless lines with resistive loads lossless lines R v C C ± R t R I R l at a iven location of the line, the voltae (current) at a iven instant is equal to the sum of all the voltae (current) waves that have reached that location incident reflected at the load need to know the amplitude and location of all waves as function of time reflected at source EE 78 ines 4

43 Amplitude of voltae waves t reflection coefficient at the load Γ R R R R reflection coefficient R R Γ at the enerator R R ± R l R I R amplitudes: st wave R R R nd wave Γ 3 rd wave Γ Γ Γ incident reflected at load 4 th wave Γ 5 th wave ( ) 3 Γ Γ Γ ( Γ Γ ) reflected at enerator M EE 78 ines 43

44 ocation of voltae waves t propaation velocity propaation time (end to end) v l T v ± R R I R l times of arrival: st wave load, at t T incident nd wave source, at t T 3 rd wave load, at t 3T reflected at load 4 th wave source, at t 4T 5 th wave load, at t 5T reflected at source M EE 78 ines 44

45 oltae reflection diaram voltae reflection diaram raphical representation of the voltae waves t 3 5T ± R t l R I R 4T T 3 3T l T v R R R Γ Γ Γ Γ ( ) Γ Γ Γ T ( Γ Γ ) 3 Γ M l EE 78 ines 45

46 oltae in the line at a iven instant t R I t 3 5T ± R R 4T 3 l T 3T t t ( ) T l l EE 78 ines 46

47 oltae at a point in the line t R I ± R R t 3 l 5T 4T 3 t 5 t 4 ( ) 3 T 3T t 3 t l T t t t t t t t EE 78 ines 47

48 oltae at the load t R I ± R R t 3 l 5T 3 4T 3 3 3T T T l T 3T 5T t EE 78 ines 48

49 Final voltae in the line t R I ( t ) 3 3 ± R R R R R Γ Γ Γ Γ l t 3 3 5T ( ) Γ Γ Γ 4T ( Γ Γ ) 3 Γ M T 3T [ ] ( t ) ( Γ ) Γ Γ ( Γ Γ ) ( Γ Γ ) 3 l T EE 78 ines 49

50 Final voltae in the line t [ ] ( t ) ( Γ ) Γ Γ ( Γ Γ ) ( Γ Γ ) 3 ± R R I R if Γ Γ < l ( t ) Γ Γ Γ R R R ( t ) Γ Γ R R R R R R R R R R R 4T T t 3 3 5T 3T T final value does not depend on R l EE 78 ines 5