# 1.1 Mathematical description transm. lines: dimensions and el. properties are identical Uniform all planes transverse to the direction of propagation.

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1 1.1 Mathematical description transm. lines: dimensions and el. properties are identical Uniform all planes transverse to the direction of propagation. at of the transm. line need not be negligible compared Length the wavelength. Extension of a.c. circuit theory to DIS- to TRIBUTED circuits. resistance inductance capacitance Dene conductance and per G ; L ; C for uniform transm. line. ; R length: unit Explain dierence between R and G Microwave Circuit Design I cb. Pejcinovic 1

2 related to dimensions and conductivity of (metallic) conductors R G related to loss tangent of the insulating material between L related to magnetic ux linking the conductors C related to charge on conductors Fig. 3-1 (analogy to twin lead or coax) represents a \small" section of the transm. line small is z to the wavelength relative Microwave Circuit Design I cb. Pejcinovic 2

3 Kircho's laws for V and I for a small segment (sketch) z)i + (L V (R + (V + V ) (1) I = (I + I) + (V + V )(G z) + (C + V ) division by z gives: V (V + V ) z I (I + I) z I + = = (V + V )G + + V Microwave Circuit Design I cb. Pejcinovic (2) (3) (4)

4 Taking the limit z! = R I and = V G @z Analogous to the eqs. we had for E and H. Combining produces: V V C 2 + (R C + G L L + R G V 2 and similar for I. For perfect conductor R = perfect insulator G = Microwave Circuit Design I cb. Pejcinovic 4 (5)

5 = V C 2 and 2 = I 2 are wave eqs and their solutions are \EM wave"-like These along both directions. Both current and voltage waves. traveling In general solutions will satisfy f (t p L C z). at (z ) f (t) = V and at another z = z 1 where Look f = = p p L C z 1 ). By the time t = L C z 1 the value inside (t f = V is same as for z = ) only dierence is time delay of p () the C z. 1 L Microwave Circuit Design I cb. Pejcinovic 5

6 The velocity of propagation of such wave is = 1 v = z d t Note that f (t + 1 z p L 1 z C = 1 p C (8) L p C travels in negative direction. z) L I and V travel with the same velocity. EM waves we found that E/H was constant: same here but for of we have instead CHARACTERISTIC IMPEDANCE Z = V I = Microwave Circuit Design I cb. Pejcinovic 6 v uuu L u ut C (9)

7 1.2 Transient behavior Charging of an innite transmission line. (g. 3-2) of sequential current increases capacitance charging current Idea etc. Instantaneous changes impossible due to L and C. increase V and I generate each other and travel with the same velocity v. innite ) only + directed wave. Since V/I is constant only Line 1) is seen at all times. ) innite line presents Z load Z(= to the battery and it keeps drawing current. Note that the steady state at the input is reach immediately. Microwave Circuit Design I cb. Pejcinovic 7

8 Finite length: open circuit. Fig t = ;V = 1V appears at input of transm. line i.e. line At like and innite line ) input impedance = Z at t =. behaves current I in = V G =(R G + Z ) = 2=2 = :1A and input Input voltage V in = 1V. These stay constant until later (when reected wave arrives). propagation velocity v = m=s time it takes the wave For get to the end is t = l=v = 4=2 1 8 = 2ns. to 3-3 a) shows the situation at half the time it takes to travel Fig. the end and 3-3 b) is for time just before reaching the end. to Microwave Circuit Design I cb. Pejcinovic 8

9 Something must happen at the end of the line: constant ratio V + =I + = Z must be preserved for the traveling wave Z = 1 since the end is open-circuited. of traveling waves at O-C makes it possible to satisfy Reection requirements. both voltage and current at the load we have V L = V + + V and For L = I + I. Since I L = ) I + = I and V = I Z = I I + Z = V +. Finally V L = 2V + = 2V. reection coecient = V =V + = I =I + and is =1 for Dene case. O-C Microwave Circuit Design I cb. Pejcinovic 9

10 Current becomes zero as I wave returns and V goes to 2V. the generator end V =I = Z = 1 is the same as the At of the generator so no need for additional require- impedance ments to be satised ) no more reections. Resistive termination 3-4. part a) is as before. At the end of the line Z 6= R L Fig. Ohm's law is not satised if we also want to keep the Z and constant. Solution: reections must occur at load end. L = V + + V ; I L = I + I = (V + V )=Z. Ohm's law V L = I L R L ) V Microwave Circuit Design I cb. Pejcinovic 1

11 RL = V + + V + V 1 L or L = Z R L + Z = L 1 Z V In steady-state transm. line is a short circuit. Analogy with for TEM waves (= ( )=( + ). RL + Z (1) For resistance is real but it is complex for other terminations. = 1 for O-C and = 1 for S-C (short circuit). RL = Z no reections occur since Ohms law is automatically For satised. Microwave Circuit Design I cb. Pejcinovic 11

12 Multiple reections Both R L and R G 6= Z! see Fig t= generator sees Z (as before) ) I + = 9=3 = :3A At V + = 3V. R L has no inuence. and 2ns there is a reection on load end: L = (25 1)=125 After :6 resulting in V = V + = 18V. = voltage V total = V + + V = 12V. Similarly for reected Total total currents we have I = I + = :18A and Itot = IL = and :3 + :18 = :48A two combined waves reach source end and there is another These (R G 6= Z ): G = (R G Z )=(R G +Z ) = V refl =V inc = reection I refl =I inc = 1=3. Microwave Circuit Design I cb. Pejcinovic 12

13 To nd the reected part V ref = G V = 6V ; etc. the incident (incoming) wave is V = 18V which gives Now = 6V and new total voltage Vtot = 12 6 = 6V Vref reection at the load end this time the incident wave Another 6V so that V ref = 6 (:6) = 3:6V for the new V tot = is 6 + 3:6 = 9:6V. of reected waves get smaller with each reection Amplitudes (why?). d.c. value V = 9 R L =(R G + R L ) = 1V. Fig. 3-6 shows Final time evolution of the input voltage. the Microwave Circuit Design I cb. Pejcinovic 13

14 diagram: voltage and current as functions of time and Bewley's Fig Construct constant z diagram or nd value of position. V or I for xed t. xed z (=2m) intersections represent arrivals of wave fronts; For summed up; forward current waves added but reected voltages ones subtracted. t: sum of voltages at any z above the line gives V variation Fixed z (same for I) with occurs: period T=8ns (2 x round trip) or f = 1=T = Ringing For v = ) f = v and = 16m or l = =4. 12:5MHz. result: d.c. source produces high frequency oscillations. Interesting If both -s were unity ) continuous oscillations i.e. resonant circuit. Microwave Circuit Design I cb. Pejcinovic 14

15 1.3 Sinusoidal excitation Equations become (using phasor notation): d = (R + j!l ) ^I = Z ^I (11) ^V dz di = (G + j!c ) ^V = Y ^V (12) dz As for EM waves these are combined: 2 d = Z ^V ^V ) ^V = ^V + e z + ^V e z = V + + V (13) 2 Y dz As before is (complex) propagation constant. = p Y = Z s (R + j!l )(G + j!c ) = + j (14) Microwave Circuit Design I cb. Pejcinovic 15

16 For current we have: = ^I + e z ^I e +z = ^I + ^I (15) ^I impedance (voltage to current ratio fro traveling Characteristic waves) = V + Z I + V = I = v uuu Z u ut Y = v uuu u ut R + j!l G + j!c (16) and currents are phasor quantities and depend on conditions Voltages on load and generator end. Power ow : P = Re( ^V ^I ) = IV cos pf Example 3-1. Microwave Circuit Design I cb. Pejcinovic 16

17 Low-loss lines <<!L and G!C. Almost always At high frequency: R range. MW in true Z = v uuu L u ut R G Z ; + 2Z C! p L C v = Conclusions: 1 p L 2 Np/length (17) = p (18) C L c C Eqs. for Z ;v;; the same as for lossless lines. Microwave Circuit Design I cb. Pejcinovic 17

18 Since Z is practically real the average power ow in a traveling wave at any point z is product of rms voltage and current P + = V + I + and P = V I. frequency signals ) nite R ;G introduce attenuation For single waves (e z for power it's square). propagating on to EM waves in lossy dielectric. Reason: there is a wave Similar with sinusoidal waves but now it is guided. associated Microwave Circuit Design I cb. Pejcinovic 18

19 Coaxial transmission lines Fig. 3-8 shows E and H eld distribution. = V E ln b a r Average power ow: P = Z = I H 2r S ~ E ~ H d~s (19) (2) cylindrical coordinates: d~s = r d dr (d~s is element normal In the plane of paper i.e. cross-sectional area; sketch) to P = Z Z 2 b a I V 2 2r (why these limits?) ln b a 1 1 C A r ddr = V I (21) Microwave Circuit Design I cb. Pejcinovic 19

20 power ow associated with the voltage and current Interpretation: is just another way of viewing EM propagation in the region waves between conductors! From early on: C = 2 r l= ln(b=a). Divide by l to get C. Similarly L = r l ln(b=a)=(2). v = Z = 1 p C = L v uuu L u ut C = v uuu 1 = c p (familiar?) (22) p r r r r u t r 4 2 r ln b a = 6 v uuu u r t r ln b a Microwave Circuit Design I cb. Pejcinovic 2 (23)

21 1.4 Terminated transm. lines Discussion of various loads on transm. lines. view: tr. line terminated in its Z has no reections Alternative based on innite line picture. Conclusion: line terminated in Z is equivalent to innitely long line and has no reections. its impedance = Z. Input Fig. 3{9 ^Vin; ^Iin at point z=: ^Vin = Z ^ ^VG V G ^I = Z + ZG reections present ) ^V in = ^V + ; ^I in = ^I +. No ZG + Z (24) Microwave Circuit Design I cb. Pejcinovic 21

22 ^V ; ^I still vary along the line: = ^V + ez ^I = ^I + ez (25) ^V At the load end this gives (z=l): ^VL = Z ZG + Z ^VGe l 6 l ^IL = ^VG ZG + Z e l 6 l (26) the power absorbed by load? P L = V L I L (rms values) What's Z is real. since PL = Z V G ZG + Z 2 e 2l (27) magnitude needed since Z G may be complex Microwave Circuit Design I cb. Pejcinovic 22

23 General case (with reections) If Z L 6= Z ) reections. Starting point is = ^V + ez + ^V e+z = ^V + + ^V ^I = ^I + ez ^I e+z (28) ^V coe. was dened only at the load or generator end Reection no need for such a restriction ) dene it anywhere on line: (z) = V or I re. V or I = incident At load end we have: L = e l ^V + e l = ^V ^V ^I ^V = + ^V ^I + (29) + e 2l (3) ^V Microwave Circuit Design I cb. Pejcinovic 23

24 convenience: transform coordinated so that zero is at load For and + direction is towards generator (Fig. 3{1). New end coordinate: d=l-z. Voltage wave somewhere on the tr. line: + V el = ^V e d {z } 64 gen.!load L e d {z } + load!gen (31) ^I = ^I + e l " e l L e d# (32) Microwave Circuit Design I cb. Pejcinovic 24

25 at any point is: = Le 2d = Le 2d 6 2d = jlj e 2d 6 (L2d)(33) Input wave attenuated by e d on the way to load; Interpretation: shifted by l. After reection attenuated by another e d phase ) total=e 2d. Only a fraction reected as determined by L. input terminals the mag. of re. wave is j L je 2l. Phase On = 2l but L adds its own ) total = L 2l. shift = ; jj is independent of d but phase angle is still function For d. of Microwave Circuit Design I cb. Pejcinovic 25

26 = ) ^V L = ^V + el [1 + L ] ^IL = ^I + el [1 L ] (34) d ) Z L = ^VL L Z L Z = = Z + ZL where Y L = 1=Z L 1 + L or = Z L 1 ^IL ZL Z ZL 1 ZL + 1 also L = 1 Y L L = 1 + L = (35) Z L Y L (36) to equations in transient case. Dierence is in steady Analogous a.c. analysis ) valid for any complex load. Also merger state a.c. circuit theory (impedance) and wave theory (re. coecient). of Microwave Circuit Design I cb. Pejcinovic 26

27 Extend the concept to any point on the tr. line: Z = 1 (d) and Y = Y 1 (d) ^V 1 + (d) Z = ^I 1 + (d) Represents total Z or Y to the right of point d. (37) means standing waves. When Z L 6= Z ) L 6= and Reection waves exist. standing the short circuit (S-C) load: same as wave reection of Take conductor. see g. 2{24 2{25. Successive minima are perfect =2 apart I shifted by =4 relative to V. Microwave Circuit Design I cb. Pejcinovic 27

28 General case: ^V is sum of phasor quantities ^V + + ^V. case: two counter-rotating vectors of xed magnitude Lossless 2{28). When in phase! maximum (g. V max = V + + jjv + = (1 + jj)v + (38) =4 away there is a minimum (phasors 18 out of phase) V min = V + jjv + = (1 jj)v + (39) Standing wave ratio = max SWR = V 1 + jj min V 1 jj jj = SWR 1 and + 1 SWR Microwave Circuit Design I cb. Pejcinovic 28 (4)

29 For lossless line SWR=const. but not otherwise (not useful). example 3.2 Special Cases Short circuit: Z L = ) L = 1; SWR = 1. At d= A) V=; other nulls occur at multiple of =2. (z=l) I is maximum at the load (d=) its rms value is 2I +. V and I are displace by =4 at d = =4 we have Z! 1 Since short circuit is transformed into open circuit!!! i.e. Open circuit: Y L = ; L = 1; SWR = 1. See g. 3{11 B) standing waves. At d= I=; Vmax = 2V + and at d = =4 for we have Z = i.e. O-C becomes S-C!!! Microwave Circuit Design I cb. Pejcinovic 29

30 C) Reactive loads: ZL = jx ) L = j X 1 X = +1 {z } For?? j X + 1 = arctan X (41) L = 16 9 ; for X = 1 {z } )?? either case jj = 1;SWR! 1. ) L = In reactance cannot absorb power ) all power in incident Interpret: wave must be reected (analogous to S-C and O-C in this Consequence: jlj < 1 only if ZL has a resistive component. respect). 3{12 for SW pattern (ZL = jz ); rst voltage null is 3=8 Fig. from d= away Microwave Circuit Design I cb. Pejcinovic 3

31 D) Resistive loads: L R L Z = real (42) ) Z + RL For RL = Z ) no reections. R L > Z ) L > V max and I min are at the load. (why?) For = RL=Z. SWR R L < Z ) L < and V min ;I min at the load. SWR = For L Z=R SWR = R L =Z or Z =R L whichever is > 1. j L j < 1 i.e. some power is absorbed. Microwave Circuit Design I cb. Pejcinovic 31

32 1.5 Power ow Generally P = Re( ^V ^I ) = V I cos pf of from generator to load ^V = ^V + ez + ^V ez = Direction ow + + ) and ^I = ^I + + ^I = I (1 ) (1 ^V power becomes The + )(1 ) ^V + ^I + # "(1 P = Re = Re (1 + )(1 ) 2 ^V + V + Z 6 64 (1 + jj+ 2 ) jv + j 2 Re = Z (43) Low-loss high freq. ) Z is real ) is imaginary. Microwave Circuit Design I cb. Pejcinovic 32

33 = (1 jj 2 ) jv + j 2 Z P = (1 jj 2 )P + = P + jj 2 P + = P + P Z real net power ow at any point on the line is the difference For between the power in the forward wave P + and in the reected wave P. Note: is function of position. How about input terminals: = P + in (1 j inj 2 ) P L = P + L (1 j Lj 2 ) (44) Pin Remember that + and - signs indicate traveling waves. + in = ( P magn: } { z + ) 2 =Z ; P + L = V +2 L =Z = P + ine 2l (45) V Microwave Circuit Design I cb. Pejcinovic 33

34 incident power at the load is input power attenuated along i.e. line of length l. For lossless lines ) P + in = P + L = P + the power ow is position dependent and we need to determine NET + at any point. P real Z ^V + ; ^I + are in phase ) P + = V +2 For Z V + 2 e 2z = + and its rms value V + are obtained from Kircho's laws at ^V input: = ^Vin + ^IinZG = ( ^V + + in ^V +) + ZG ^VG Microwave Circuit Design I cb. Pejcinovic 34 Z + in ^V + ^V Z (46)

35 + = ^V ^VG Z (ZG + Z )(1 GLe 2l ) where L = (Z G Z )=(Z G + Z ) ^V G ;Z G ;Z L (circuit) and Z ;;l (line) we can nd V +. Given Alternative form: + = ^V (1 G ) ^VG G in ) = 2(1 ^VG (1 G ) 2(1 G L e 2l ) V + known P can be determined. With Microwave Circuit Design I cb. Pejcinovic 35 (47) (48)

36 dierent approaches: a) multiple reection b) steady state Two circuit. What's the relationship? a.c. 1) Either L = or G = so no multiple reections case Initial forward traveling wave from voltage divider Z G +Z occur. ^V + = ^VGZ =(ZG + Z ). Same as in eq. (47). ) 2) Multiple reections. ^V + represents phasor sum of all the case forward traveling waves at input (z=). wave: ^V + (1) = ^V G Z =(Z G + Z ) Initial one: ^V + (2) = G L e 2l ^VG Z =(Z G + Z ) 2nd one: ^V + (3) = ( G L e 2l ) 2 ^V G Z =(Z G + Z ) 3rd etc. Microwave Circuit Design I cb. Pejcinovic 36

37 result is the sum of all of the above; limit of the series is Final j L G e 2l j < 1) is (1 L G e 2l ) 1. This leads to the (when same expression as in a.c. theory (eq. (47)). to power calculation: P + = V +2 e 2z =Z. Now plug in Back for V + expr. + = V 2 G P 4Z (eq. 48)) 1 G so for z= we have + in = V 2 G P 4Z 1 Gin 1 G G in {z } 1 2 e 2z (49) 2 2l e L = Microwave Circuit Design I cb. Pejcinovic 37 (5)

38 concept: available power from source P A = V 2 G=(4R G ) Useful R G = Re(Z G ) where + in = P A P 1 j G j 2 j1 Ginj 2 (51) Finally what is the NET input power? = + (1 j 2 G (1 j )(1 j 2 j in j 2 ) A P = ) inj in P Pin Power deliver to the load: j1 G in j (52) (1 j G j 2 )(1 j L j 2 ) PL = PA 2l j 2 e 2l (53) GLe j1 Microwave Circuit Design I cb. Pejcinovic 38

39 or for lossless line: (1 jgj 2 )(1 jlj 2 ) Pin = P L = P A j2lj2 (54) e L G j1 In this case j in j = j L j then in = L e j2l that the input is matched ZG = Z then G = and PL Assume of frequency and phase angle l ( =!=v). independent for matched input ^V + = ^V G =2 and P + in = P A = Therefore V 2 G=(4Z ) so that Pin = P A (1 j in j 2 ) and P L = P A e 2l (1 j L j 2 ) (55) Microwave Circuit Design I cb. Pejcinovic 39

40 line generator end matched; if load is matched as well Lossless all of PA is delivered to load PL = PA. ) Z G 6= Z it does not follow that matching the load will For P L = P A (see eq. 54). Maximizing power transfer more produce complicated. Example 3-3. Return loss: LR = 1 log(p + =P ) db. Since P = jj 2 P + = log 1 LR 2 db jj (56) lossless line L R is the same anywhere (i.e. jj is independent For position). of Microwave Circuit Design I cb. Pejcinovic 4

41 line means position dependent L R but it is easily determined Lossy from jj = j L je 2d. For input we have in = j L je 2l 6 L 2l so that L R;in = L R;load + 2(8:686l) loss: used only when Z G = Z. Measures reduction in Reection power due to mismatch of load impedance (Z L 6= Z ) load Ref. loss = 1 log aka mismatch loss. A e 2l =P } { z L = Z ) PL(Z L ) PL(Z {z } =P A e 2l (1j L j 2 ) db = 1 log Microwave Circuit Design I cb. Pejcinovic 41 1 j L j 2 (57) 1

42 1.6 Impedance transformation direction; impedance anywhere on line replaces everything Reference to the right. Starting point: 1 + Z Z Z + L e 2d 1 = = 1 1 L e 2d (58) some manipulation (note: bar in e.g. Z = Z=Z is for After values): normalized Z L + Z tanh d Z = Z tanh d ; Z = L Z + Z L + tanh d Z + Z L tanh d 1 Microwave Circuit Design I cb. Pejcinovic 42 (59)

43 or for admitance Y L + Y tanh d Y = Y tanh d ; Y = L Y + Y YL + tanh d + Y L tanh d 1 Input impedance: set d=l; what about d=? If is neglected Z L + jz tan d Z = Z tan d ; Z = L jz + Z and equivalent for Y-s Example 3-4. L + j tan d Z + j Z L tan d 1 Microwave Circuit Design I cb. Pejcinovic 43 (6) (61)

44 changes in Z; standing waves are the cause. No SW ) Dramatic impedance transformation (Z L = Z ) Z = Z anywhere). no happens at LF? l ) tan l so that Z = Z L for What and low loss line. LF Fixed length ) freq. dependence of Z since l =!l=v = 2l= Excellent for impedance transf. but bad for broadband design! =2 line: l = n=2 then Z in = Z L (substitute l = n into eq. 59 ). Impedance repeats itself every =2!! Microwave Circuit Design I cb. Pejcinovic 44

45 Z G = Z L both real (e.g. 2 Ohms). Tr. line has Z = 1 Take and l = =2. SWR = Z =R = 5 but for this lenght of tr. Ohms Z in = Z L = 2 Ohms. All available power is delivered down line lossless line to Z L. Despite SW max. power is delivered to the the load. =4 line: l = (2n 1)=4. Here tan l! 1 and Z in = Z 2 =Z L or normalized Z in = 1= Z L. Large Z L! small Z in and inverse. Inductive Z L! capacitive Z in (and inverse) Z L behaves as a series resonant circuit Z in is like parallel res. If circuit. Microwave Circuit Design I cb. Pejcinovic 45

46 ZL real it behaves like a transformer with turns ratio nt = For SWR) for Z L < Z. But for Z L > Z ;n t < 1. 5(= for matching a resistive load to generator (to deliver all Useful power). For Z G real Z in must be real and Z in = R G or available Z = p R G R L. Since line is assumed lossless all power is delivered to the load. Problem: works only at one (narrow range of) frequency. Microwave Circuit Design I cb. Pejcinovic 46

47 Short circuit: obviously Zin = jxin = jz tan l. See g For l < =2 (or l < =4) Z in is inductive. X in =! = Z tan(l)=! Z l=v = L l Leq = } {z for l<:8 (62) that L eq is freq. dependent. Think of this circuit as one turn note a coil (L). of l = n=2 and odd multiples S-C behaves like parallel Near circuit. resonant l = n S-C behaves like a series resonant circuit. see g. Near Microwave Circuit Design I cb. Pejcinovic 47

48 number of resonances. Note on lenght: go for the shortest Innite... one circuit: use admitances Y L = and Y in = jb in = jy tan l Open Z in = jz cot l. Fig. 3-16; or For l < =4 = Y tan(l)=![f ] Y l=v Ceq {z } l<:8 = l=(z v) = C l (63) again it is freq. dependent; it looks like a parallel plate capacitor. l = = 2l= (i.e. l = =2) we have Y in = (O-C) but For l = (2n 1)=4 Y in = 1 (S-C). for Microwave Circuit Design I cb. Pejcinovic 48

49 1) near l = (2n1)=2 O-C behaves like series resonant circuit 2) near l = n O-C looks like parallel resonant circuit. Microwave Circuit Design I cb. Pejcinovic 49