7. Transmission line analysis

Transcription

1 7. Transmission line analysis Dr. Rakhesh Singh Kshetrimayum 1

2 7.1 Introduction Transmission line analysis Introduction High reactance effect Radiation effect Size effect Telegrapher s equations Wave equation Ideal Terminated Distributed element concept Lossless line Lossy line Ideal Terminated Line junction Line λ/4 transformer impedance Fig. 7.1 Transmission line Smith chart Impedance Admittance

3 7.1 Introduction High reactance effect Consider a 1-V ac source is connected to a 5 Ω load by a small copper wire of 1 mm radius Assume that the dc resistance of the wire is R=1m Ω and inductance of L=.1 µh At 1 GHz, inductive reactance is jx L =jωl 683 Ω and hence all the ac signal will die out in the wire itself The load will not receive any signal Hence we need special devices which will take these signals from the source to the load 3

4 7.1 Introduction Radiation effect An accelerating or decelerating charge radiates electromagnetic energy Besides the energy radiated from a current carrying conductor depends on the frequency of current flowing You might have observed this when you study Herz dipole (an infinitesimally current carrying element) S avg = 1 I 3 dl sinθ β rˆ Watt / 4πr ωε ( ) m 4

5 7.1 Introduction Hence the radiation power loss is directly proportional to the square of the frequency of the ac current flowing So there will be high loss of power We definitely cannot use open wires for transferring energy or signal 5

6 7.1 Introduction Introduction What is a transmission line? A structure, which can guide electrical energy from one point to another Generally, a transmission line is a two parallel conductor system one end of which is connected to a source and the other end is connected to a load Examples: coaxial cables waveguides microstrip lines 6

7 7.1 Introduction Fig. 7. (a) Transmission lines examples (b) General transmission line structure 7

8 7.1 Introduction Two conductor systems could support transverse electromagnetic (TEM) waves Both electric and magnetic fields are perpendicular (transverse) to the direction of the propagation It is guided wave between these two conductors Hence the radiation losses are minimized What is microwave frequency? 3 MHz to 3 GHz (λ=1m to 1 cm) Nowadays it is meant for frequency up to 3 GHz (λ=1cm) 8

9 7.1 Introduction Size effect Size of commonly used lump elements like capacitor, inductor and resistor are of the order of cm Now this size is comparable to the microwave wavelength Hence the phase {βl=(πf/c)l} of the electrical signal might vary along the length of the device For instance, consider a parallel plate capacitor We assume capacitor conductor plate is an equipotential surface 9

10 7.1 Introduction But this is not true at microwave frequencies Besides radiation also increases the problem So we cannot use such capacitors at high frequencies So we will see later that a section of a transmission line could be used as an inductor or resistor or series/shunt RLC resonators If we increase the frequency of operation of a circuit, Usually we require temporal analysis at low frequency we can t neglect space in the circuit analysis due to size effect 1

11 7.1 Introduction 7.1. Causal effect What is causal effect? EM wave requires a finite time to travel along an electrical circuit Since no EM wave can travel with infinite velocity (What is the maximum speed?) A finite time delay between the 'cause' and the effect Also known as the causal effect in physics When is this effect important? 11

12 7.1 Introduction If the time period of the EM wave or signal (T=1/f) >> the transit time (t r ), we may ignore this effect 1 l v T >> tr >> >> l λ>> l f v f Causal effect becomes important when the length of the line (l) becomes comparable to the wavelength (λ) As the frequency increases, the wavelength reduces, and the Causal effect becomes more evident 1

13 7.1 Introduction Distributed vs lumped elements To overcome the effect of transit time or causality or size effect (more appropriate to use), one can chop off the transmission line into small sections such that for each section, this causality effect is minuscule At high frequencies, the circuit elements cannot be defined for the whole transmission line instead it has to be defined for a unit length of the line The circuit elements are not located at a point of the line but are distributed all along the length 13

14 7.1 Introduction Analysis of a transmission lines must be carried out using the concept of distributed elements not as lumped elements as we used to do from our previous circuit analysis at the low frequencies But, we can still employ lump element analysis of transmission lines by chopping off small sections of the line so that the Causal effect is negligible in the chopped off sections 14

15 7. Telegrapher s equations 7..1 Lumped element circuit model Per unit length parameters: L=Series inductance per unit length C=Shunt capacitance per unit length R=Series resistance per unit length G= Shunt conductance per unit length 15

16 7. Telegrapher s equations z Fig. 7.3 (a) Sub-section of length z of a general transmission line and its (b) lumped element equivalent circuit V(z, t),i(z, t) R z L z G z C z V(z + z,t),i(z + z,t) 16

17 7. Telegrapher s equations L represents the self-inductance of the two conductors (magnetic energy storage) C is due to the close proximity of two conductors (electric energy storage) R is due to the finite conductivity of the two conductors (power loss due to finite conductivity of metallic conductors) G is due to dielectric loss in the material between the conductors (power dissipation in lossy dielectric) 17

18 7. Telegrapher s equations 7.. Telegrapher s equations Let the voltage at the input be V and current at the input be I Due to voltage drop in the series arm, the output voltage will be different from the input voltage, say V+ V Due to current through the capacitance and the conductance, the output current will be different from the input current, say I+ I 18

19 7. Telegrapher s equations Applying Kirchoff s voltage law (KVL) and Kirchoff s current law (KCL) δi(z, t) v(z, t) R zi(z, t) L z v(z + z, t) = δt δ v(z + z, t) i(z, t) G zv(z + z, t) C z i(z + z, t) = δt 19

20 7. Telegrapher s equations Dividing the above two equations by Δz and taking the limit Δz (What is its implications?) δv(z, t) δi(z, t) = Ri(z, t) L δ z δ t δi(z, t) δv(z, t) = Gv(z, t) C δz δt Telegrapher s Equations

21 7. Telegrapher s equations 7..3 Wave propagation For time-harmonic signals, telegrapher s equation reduces to dv(z) dz = (R + jωl)i(z) di ( z) dz = -( G + jωc) V ( z) 1

22 7. Telegrapher s equations It is similar to Maxwell s curl equations, hence, we can get wave equations d V(z) γ dz d I(z) γ dz V(z) I(z) = = γ = α + jβ = ( R + jωl)( G + jωc)

23 Transmission line analysis Traveling wave solutions for the above two equations are V(z) = V e + V e + γz γz I( z) = I e + I e + γ z Point to be noted: current or voltage is wave which is a function of both space and time unlike the low frequency counter-parts (where is the time dependence?) γ z 3

24 7. Telegrapher s equations Physical interpretations Wave phase has two components: time phase (ωt) and space phase (βz) Since βz is the phase of the wave as function of z, β represents phase change per unit length of the transmission line for a traveling wave phase constant (unit is radians per meter) Re{V + e αz e jωt jβz } = Re{ V + e jφ e αz e jωt jβz } = V + e αz cos(ωt βz + φ) 4

25 7. Telegrapher s equations For a positive α, the amplitude exponentially decreases as a function of z V + e α z α represents attenuation of the wave on the transmission line attenuation constant of the line (unit is Nepers per meter, 1 Neper= 8.68dB) γ I(z) = V e V e R + jωl + γz γz R + jω L R + jωl = = γ G + jωc V I V + = = + I 5

26 7. Telegrapher s equations The characteristic impedance of a transmission line is defined as the ratio of positively traveling voltage wave to current wave at any point on the line Now for a wave the distance over which the phase changes by ϖ is called the wavelength 'λ phase change per unit length β=ϖ/λ v p πf = λ f = = β ω β 6

27 7.3 Lossless line Ideal lossless line L z V(z, t),i(z, t) C z V(z + z,t),i(z + z,t) Fig. 7.4 Lumped element equivalent circuit of a sub-section of length z of a lossless transmission line (R=G=) 7

28 7.3 Lossless line γ = α + jβ = jω LC β = ω LC α = = L C + jβz jβz = V(z) = V e + V e + V jβz V jβz I(z) = e e π π λ = = β ω LC v p ω ω = = = β ω LC 1 LC 8

29 7.3 Lossless line 7.3. Terminated lossless lines,β Fig. 7.4 (b) A lossless transmission line terminated with load impedance L 9

30 7.3 Lossless line Reflection coefficient At the load, z=, L = + ( ) V + V = o o ( ) V I V + o V o V L = Γ = + L V + V V + ( ) ( jβz jβz ) I z = e Γe ( ) ( jβz jβz ) z = V e + Γ e + jβl V e Γ l = = Γ e Γ( ) = + + jβl V e jβl z = l ( ) ( ) L L + 3

31 7.3 Lossless line Power flow and return loss Time average power flow along the line at the point z, V Savg = Re V z I z = Re 1 Γ e + Γ e Γ * ( ) ( ) { * * } β z β z S avg + 1 V = Γ { 1 } 31

32 7.3 Lossless line When the load is mismatched, not all of the available power from the generator is delivered to the load, this loss is called Return loss (RL) and is defined in db as Standing wave ratio (SWR) RL = log Γ ( ) = + + Γ j z V z V 1 e β jβz e 1 = V max = V + 1+ Γ = V min = V + 1 Γ jβz e 1 VSWR = V V max min = Γ Γ 3

33 7.3 Lossless line What is BW? For acceptable value of VSWR = within the operating frequency region of a device also known as bandwidth (BW) Γ VSWR = = = ; RL = log VSWR = 9.54 Return loss (RL) should be higher than 9.54, which is approximately 1 db RL 1 db has become an acceptable definition for BW of many devices 33

34 7.3 Lossless line ISWR = I I V max min I ( ) max max PSWR = = PSWR ISWR = VSWR V min I min PL Pi Pr VSWR 1 4VSWR For VSWR=, ( ) = = 1 Γ = 1 = P P VSWR + 1 VSWR + 1 i i ( ) only 89% of the incident power reaches the load 34

35 7.3 Lossless line + l = z V ( l) = V 1+ Γe l + λ Point to be noted: jβl λ jβ l+ + jβl λ V l = V + Γ e = V + Γ e = V l shortest distance between two successive maxima (or minima) is not λ but λ/, it is very important to realize this since in your experiment on Frequency and Wavelength measurements, this is a major mistake most of you make ( ) 35

36 7.3 Lossless line l + λ 4 λ jβ l jβl 1 1 λ V l = V + Γ e = V Γe 4 the distance between adjacent maximum and minimum is λ/ Transmission line impedance equation A certain value of load impedance at the end of a particular transmission line is transformed into another value of impedance at the input of the line impedance transformer 36

37 7.3 Lossless line Transmission line impedance equation jβl L jβl + jβl jβl e + e ( ) V e e V l + Γ L + in = = = + jβl jβl I ( l ) V e Γe jβl L jβl e e L + jβl jβl jβl jβl jβl jβl ( L + ) e + ( L ) e L ( e + e ) + ( e e ) jβl jβl jβl jβl jβl jβl ( L + ) e ( L ) e ( e + e ) + L ( e e ) = = ( ) ( ) ( cos β ) + ( sin β ) L cos l j sin l L j tan( ) β + β + βl = = l j L tan( ) L l + j βl 37

38 7.3 Lossless line Fig. 7.5 Transmission line impedance in,β 38

39 7.3 Lossless line Quarter-wave transformer λ λ λ λ π λ π λ π l = + n β l = β + nβ = + n = + nπ 4 4 λ 4 λ π tan ( β ) = tan + nπ = L + j tan( βl) j tan( βl) in = = = + j tan( βl) j tan( βl) l L in L L L How to do impedance matching for a complex load using quarter-wave transformer? = 39

40 7.3 Lossless line Special cases of lossless terminated lines Terminated in a short circuit + j tan( βl) = = j tan( βl) sc L in + jl tan( βl) Terminated in open circuit + j tan( βl) = = j cot( βl) oc L in + jl tan( βl) Terminated with matched load Γ = + = 4

41 7.3 Lossless line Another important observation is that if we measure open and short circuit input impedances of a lossless transmission line and multiply those two values and take the square root what we have is the characteristic impedance of the line (one of the methods for finding the characteristic impedance of a given line in laboratory) 41

42 7.3 Lossless line Reflection and transmission at the transmission line junction Γ τ Fig. 7.7 Junction of two transmission line with different characteristic impedance 4

43 7.3 Lossless line For z<, characteristic impedance ; z>, characteristic impedance 1 and the junction of the two transmission lines is at z= At the junction, looking from z< towards the right, it sees an infinite transmission line of characteristic impedance 1 and hence it is equivalent to L = 1 for the transmission line z< Assuming ζ is the transmission coefficient and IL is insertion loss in db 43

44 7.3 Lossless line Γ = < ( ) ( jβz jβz ) V z = V e + Γ z z e + jβz > V(z) = V τe z = IL τ = = log τ 1+ Γ = =

45 7.4 Lossy lines One type of metal loss is I R loss In transmission lines, the resistance of the conductors is never equal to zero except for superconductors Whenever current flows through one of these conductors, some energy is dissipated in the form of heat 45

46 7.4 Lossy lines Another type of loss is due to skin effect Current in the center of the wire becomes smaller and most of the electron flows on the wire surface When the frequency applied is in the GHz range, the electron movement in the center is so small that the center of the wire could be removed without any noticeable effect on the current 46

47 7.4 Lossy lines Note that the effective cross-sectional area decreases as the frequency increases Since resistance is inversely proportional to the cross-sectional area (R=ρl/A), the resistance will increase as the frequency is increased Also, since power loss increases as resistance increases, power losses increase with an increase in frequency because of the skin effect 47

48 7.4 Lossy lines Dielectric losses result from the heating effect on the dielectric material between the conductors Power from the source is used in heating the dielectric The heat produced is dissipated into the surrounding medium When there is no potential difference between two conductors, the atoms in the dielectric material between them are normal and the orbits of the electrons are circular 48

49 7.4 Lossy lines When there is a potential difference between two conductors, the orbits of the electrons change The excessive negative charge on one conductor repels electrons on the dielectric toward the positive conductor and thus distorts the orbits of the electrons A change in the path of electrons requires more energy, introducing a power loss 49

50 7.4 Lossy lines Induction losses occur when the electromagnetic field about a conductor cuts through any nearby metallic object and a current is induced in that object As a result, power is dissipated in the object and is lost 5

51 7.4 Lossy lines Radiation losses occur because some magnetic lines of force about a conductor do not return to the conductor when the cycle alternates These lines of force are projected into space as radiation, and these results in power losses That is, power is supplied by the source, but is not available to the load 51

52 7.4 Lossy lines Ideal lossy line characteristics γ = α + jβ = ( R + jωl)( G + jωc) = jω LC Low loss case R 1 + ωl G RG j ωc ω LC R << ωl,g << ωc RG << ω LC 1 C L 1 R α R G G + = + L C R G = (jωl)(jω C) ( + 1)( + 1) jωl jωc 1 R G γ jω LC 1 j + ωl ωc β ω LC R + jωl L = G + jωc C 5

53 7.4 Lossy lines 7.4. Terminated lossy lines, γ V I + ( ) ( +γl γl l = V e + Γe ) + +γl γl V ( ) ( e Γe ) l = Fig. 7.8 (a) A lossy transmission line terminated with load impedance L Γ V γl ( ) γl αl jβl αl jβl l = = Γ( ) e = Γ( ) e e = Γe e V + e e + γl 53

54 7.4 Lossy lines γl L γl + γl γl e + e V e e V( l) + Γ L + in = = = + γl γl I( l) V e Γe γl L γl e e L + γl γl γl γl γl γl ( L + ) e + ( L ) e L ( e + e ) + ( e e ) γl γl γl γl γl γl ( L + ) e ( L ) e ( e + e ) + L ( e e ) = = ( γ ) + ( γ ) ( γ ) + ( γ ) L cosh l sinh l L tanh + γl = = cosh l L sinh l + L tanh γl 54

55 7.4 Lossy lines + + γl γl γl γl { l l } ( ) ( ) V e Γe = Re V e + Γe 1 Pin = Re V( )I ( ) + * * * * * 1 V = Re Γ + Γ Γ { } γ l+ γ l γ l γ l γ l+ γ l γ l γ l e e e e + αl * jβl jβl αl What happens to P loss when α increases? 1 + V α l α l 1 = Re{ e Γ e + Γe Γ e } { } = e Γ e V + 1 V P = ( 1 Γ ) 1 + V αl L ( 1) ( αl Ploss = Pin PL = e + Γ e + 1) [ ] * 55

56 7.4 Lossy lines Introduction to electromagnetic resonators: in in λ 4 λ, γ = α + jβ L =, γ = α + jβ L = in in Fig. 7.8 (b) Series RLC resonant circuit (c) Tank or shunt RLC resonant circuit (d) O.C. terminated transmission line of length λ/4 and (e) S.C. terminated transmission line of lengthλ/ 56

57 7.4 Lossy lines Microwave/electromagnetic resonators are used in many applications: filters, oscillators, frequency meters, tuned amplifiers, etc. Its operations are very similar to the series and parallel RLC resonant circuits 57

58 7.4 Lossy lines We will review the series and parallel RLC ciruits and discuss the implementation of the microwave resonators using distributive elements such as microstrip line, rectangular and circular waveguides, etc. Series RLC resonant circuits Consider the series RLC resonator The input impedance in is given by = R + jwl + in 1 jwc 58

59 7.4 Lossy lines The average complex power delivered to the resonator is The average power dissipated by the resistor is 1 Ploss = I R 59

60 7.4 Lossy lines The time-averaged energy stored in the inductor is (recall the energy stored in the inductor) W m Similarly, the time-averaged energy stored in the capacitor is = 1 4 L I W e 1 1 C I 1 I C Vc = = = 4 4 w C 4 w C 6

61 7.4 Lossy lines The input impedance can then be expressed as follows: in At resonance, ( ) Ploss + jw W in m - W e P = = R ( ) \ P = P + jw W - W in loss m e R the average stored magnetic and electric energies are equal, therefore, we have W m P = = 1 I loss = W e in R 61

62 7.4 Lossy lines Hence, the resonance frequency is defined as The quality factor is defined as the product of the angular frequency and the ratio of the average energy stored to energy loss per second 6

63 7.4 Lossy lines Q is a measure of loss of a resonant circuit, lower loss implies higher Q and high Q implies narrower bandwidth As R increases, power loss increases and quality factor decreases Let us see what the approximate in near resonance The input impedance can be rewritten in the following form: 63

64 7.4 Lossy lines Near by the resonance The above form is useful for finding equivalent circuit near the resonance, for example, we can find out the resistance at resonance and so as L 64

65 7.4 Lossy lines Half power fractional bandwidth When the real power delivered to the circuit is half that of the resonance, occurs when in = R 65

66 7.4 Lossy lines Shunt RLC Resonant Circuits Now let us turn our attention to the parallel RLC resonator The input impedance is equal to 66

67 7.4 Lossy lines The average complex power delivered to the resonator is Pin = 1 V I * = 1 V V The average power dissipated by the resistor is * * in P loss = 1 V R 67

68 7.4 Lossy lines The time-averaged energy stored in the inductor is (recall the energy stored in the inductor) 1 1 V 1 V L W m = L I = L = 4 w L w Similarly, the time-averaged energy stored in the capacitor is L W c = 1 C V 4 68

69 7.4 Lossy lines The input impedance can then be expressed as follows: At resonance, in Pin = = I ( ) P + jw W - W 1 I loss m c the average stored magnetic and electric energies are equal, therefore, we have (same results as in series RLC ) ( ) \ P = P + jw W - W in loss m c w = 1 LC 69

70 7.4 Lossy lines The quality factor, however, is different Contrary to series RLC, the Q of the parallel RLC increases as R increases 7

71 7.4 Lossy lines Similar to series RLC, we can derive an approximate expression for parallel RLC near resonance 71

72 7.4 Lossy lines As in the series case, the half-power bandwidth is given by = in R 7

73 7.4 Lossy lines We discuss the use of transmission lines to realize the RLC resonator For a resonator, we are interested in Q and therefore, we need to consider lossy transmission lines Short-circuited λ/ line Note that tanh(a+b) =(tanh A + tanh B)/(1+ tanh A tanh B) 73

74 7.4 Lossy lines tan( x) = jx jx [ e e ] / ( j) jx jx [ e + e ] / tanh( x) = x x [ e e ] / x x [ e + e ] / Consider the transmission line equation For a short-circuited line 74

75 7.4 Lossy lines Our goal here is to compare the above equation with input impedance of Series or shunt RLC resonant circuit near resonance so that we can find out the corresponding R, L and C For a length l=λ/ of the transmission line, assuming a TEM line so that 75

76 7.4 Lossy lines β = ω µε = ω / v p l = λ / = πv p / ω o βl ωl ωol ω l = = + = π + v v v p ωπ ω p p o For low-loss transmission lines, αl is small, hence tanβl = tan( π + ωπ ) tan( ) ω = ωπ ω ωπ ω o o o 76

77 7.4 Lossy lines Note that the loss is usually very small and therefore, the input impedance can be rewritten as: 77

78 7.4 Lossy lines This equation can be compared favorably with the input impedance of a series RLC resonant circuit near the resonance It behaves like a series RLC resonator with 78

79 7.4 Lossy lines As α increases, Q decreases which is according to our expectation Open-Circuited λ/4 Line For a lossy line of length l with propagation constant γ and characteristic impedance, we can find the input impedance for a load of L as follows: 79

80 7.4 Lossy lines For o.c., For l = λ / 4 = πv p / ( ω o ) βl ωl ω ol ω l π ωπ = = + = + v v v ω p p p o 8

81 7.4 Lossy lines Knowing that tan d = d when d is small The input impedance can be written as, 81

82 7.4 Lossy lines This equation can be compared favorably with the input impedance of a series RLC resonant circuit near the resonance It behaves like a series RLC resonator with 8

83 7.4 Lossy lines As α increases, Q decreases which is according to our expectation We can extend this analysis for a s.c. λ/4 lines, o.c. λ/ lines and so on 83

84 7.5 Smith chart Impedance Smith chart Smith chart is basically a graphical representation of transmission line impedance transformation formula: + j tan( βl) L in = + j L tan( βl ) 84

85 7.5 Smith chart If we represent this in x-y coordinates with x as real part and y as imaginary part of L and then it becomes a semi-infinite plane, not practical We know that the modulus of reflection coefficient ( Γ ) is always less than or equal to 1 And there is one to one correspondence between Γ and in in 85

86 7.5 Smith chart Γ ( l) = we will draw ( l) in ( l) + in in + Γ l = = Γ normalized constant resistance and constant reactance contours in 1 ( ) 1 ( l) in the reflection coefficient plane which is a circle of Γ 1 A movement of d distance along the transmission line e j is equivalent to β d change in the reflection plane 86

87 7.5 Smith chart Distance in movement in terms of wavelength is given in the circumference of the circle It could be either towards load (WTL) or source (WTG) At first glance, Smith chart looks intimidating with so many contours of constant resistance and reactance 87

88 7.5 Smith chart Fig Smith chart 88

89 7.5 Smith chart Smith chart as a polar plot of Γ (o.c. open circuit and s.c. short circuit) It can be simply interpreted as a polar plot of Γ θ Γ = Γ e, 18 θ 18 jθ 89

90 7.5 Smith chart The real utility of Smith chart lies in the fact that we can read the corresponding normalized impedance value of Γ from the constant reactance and resistance contours 1+ Γ 1+ Γ + jγ = = R + jx = = in r i in in in 1 Γ 1 Γr jγi 9

91 7.5 Smith chart constant resistance circles Rin 1 Γr + ( Γ i ) = Rin + 1 Rin + 1 constant reactance circles ( Γ 1) r Γ = i X in X in 91

92 7.5 Smith chart Constant resistance circles (WTG Wavelength towards generator and WTL Wavelength towards load) WTG R in =.5-1 R in =1 +1 R in = WTL 9

93 7.5 Smith chart Constant reactance circles of an impedance smith chart X in =1 X in =.5 X in = X in =.5 X in = X in = 1 93

94 7.5 Smith chart In many applications, transmission line and impedances are connected in parallel (shunt), then, the admittance analysis is more convenient than the impedance analysis 1 1 L YL Y Y YL 1 YL YL 1 Γ = = = = = 1 1 L + + Y + YL 1+ YL YL + 1 Y Y L 94

95 7.5 Smith chart Rules for conversion of impedance (say L at N) to admittance (say Y L at N ) Γ 95

96 7.5 Smith chart The admittance smith chart is therefore obtained by rotating the impedance Smith chart by π and replacing r by g and x by b Since it is just a matter of rotation, there is no need to have separate Smith charts for impedance and admittance Although r and x can be interchanged with g and b respectively and a point (r,x) and (g,b) will have the same spatial location on the Smith chart for r=g and x=b, 96

97 7.5 Smith chart But, the physical interpretation corresponding to the two will not be identical Upper half of the impedance Smith chart with +jx represent inductive loads whereas +jb represents capacitive load on the admittance Smith chart Point B on impedance Smith chart represents s.c. whereas point B on admittance Smith chart represent o.c. 97

98 7.5 Smith chart Interchange on location of o.c./s.c. and location of VSWR on an impedance Smith chart Inductive/Capacitive B D C A Capacitive/Inductive 98

99 7.5 Smith chart Point A on impedance Smith chart which represents o.c. whereas point A on admittance Smith chart which represents s.c. Note that the distance between o.c. and s.c. is λ/4 99

100 7.6 Summary Introduction Transit time effect Distributed element concept Wave equation d V(z) γ V(z) = dz d I(z) γ I(z) = dz γ = α + jβ = ( R + jωl)( G + jωc) V I + = = + I 1 V Telegrapher s equations δv(z, t) δi(z, t) = Ri(z, t) L δz δt δi(z, t) δv(z, t) = Gv(z, t) C δz δt = L C Ideal γ = α + jβ = jω LC V(z) = V e + V e + jβz jβz + V jβz V jβz I(z) = e e Terminated + jβz jβz ( z) = V ( e + Γe ) + V jβz jβz ( ) z = ( e Γe ) V I V VSWR = V max min 1+ Γ = 1 Γ Transmission line analysis Lossless line λ/4 transformer = L in Line impedance + j tan( βl) L in = + j L tan( βl ) τ = Lossy line Ideal β ω LC R + jωl L = G + jωc C Impedance 1 C L 1 R α R + G = + G L C Line junction 1+ Γ = = Smith chart Admittance Terminated Fig. 7.1 Transmission line in a nutshell V I + +γl γl ( l) = V ( e + Γe ) + +γl γl V ( ) ( e Γe ) l = in = L+ tanh γl + tanh γl L ( ) ( )