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D-tutorial.pdf
1.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 65 We have obtained the following solutions for the steady-state voltage and current phasors in a transmission line: Loss-less line Lossy line ( ) z z z z 0 (z) 1 (z) j j j j V V e V e I V e V e Z β β β β + − − + − − = + = − ( ) z z z z 0 (z) 1 (z) V V e V e I V e V e Z γ γ γ γ + − − + − − = + = − Since V (z) and I (z) are the solutions of second order differential (wave) equations, we must determine two unknowns, V + and V− , which represent the amplitudes of steady-state voltage waves, travelling in the positive and in the negative direction, respectively. Therefore, we need two boundary conditions to determine these unknowns, by considering the effect of the load and of the generator connected to the transmission line.
2.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 66 Before we consider the boundary conditions, it is very convenient to shift the reference of the space coordinate so that the zero reference is at the location of the load instead of the generator. Since the analysis of the transmission line normally starts from the load itself, this will simplify considerably the problem later. We will also change the positive direction of the space coordinate, so that it increases when moving from load to generator along the transmission line. z d 0 ZR New Space Coordinate
3.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 67 We adopt a new coordinate d = − z, with zero reference at the load location. The new equations for voltage and current along the lossy transmission line are Loss-less line Lossy line ( ) d d d d 0 (d) 1 (d) j j j j V V e V e I V e V e Z β β β β + − − + − − = + = − ( ) d d d d 0 (d) 1 (d) V V e V e I V e V e Z γ γ γ γ + − − + − − = + = − At the load (d = 0) we have, for both cases, ( ) 0 (0) 1 (0) V V V I V V Z + − + − = + = −
4.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 68 For a given load impedance ZR , the load boundary condition is (0) (0) R V Z I = Therefore, we have ( ) 0 R Z V V V V Z + − + − + = − from which we obtain the voltage load reflection coefficient 0 0 R R R Z Z V Z Z V Γ − + − = = +
5.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 69 We can introduce this result into the transmission line equations as Loss-less line Lossy line ( ) ( ) d 2 d d 2 d 0 (d) 1 (d) 1 j j R j j R V V e e V e I e Z β β β β Γ Γ + − + − = + = − ( ) ( ) d 2 d d 2 d 0 (d) 1 (d) 1 R R V V e e V e I e Z γ γ γ γ Γ Γ + − + − = + = − At each line location we define a Generalized Reflection Coefficient 2 d (d) j R e− β Γ = Γ 2 d (d) R e− γ Γ = Γ and the line equations become ( ) ( ) d d 0 (d) 1 (d) (d) 1 (d) j j V V e V e I Z β β Γ Γ + + = + = − ( ) ( ) d d 0 (d) 1 (d) (d) 1 (d) V V e V e I Z γ γ Γ Γ + + = + = −
6.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 70 We define the line impedance as A simple circuit diagram can illustrate the significance of line impedance and generalized reflection coefficient: Zeq=Z(d) ΓReq = Γ(d) d 0 ZR 0 (d) 1 (d) (d) (d) 1 (d) V Z Z I + Γ = = − Γ
7.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 71 If you imagine to cut the line at location d, the input impedance of the portion of line terminated by the load is the same as the line impedance at that location “before the cut”. The behavior of the line on the left of location d is the same if an equivalent impedance with value Z(d) replaces the cut out portion. The reflection coefficient of the new load is equal to Γ(d) If the total length of the line is L, the input impedance is obtained from the formula for the line impedance as The input impedance is the equivalent impedance representing the entire line terminated by the load. 0 (L) 1 (L) (L) 1 (L) in in in V V Z Z I I + Γ = = = − Γ 0 0 ( ) Req Req Req Z Z d Z Z − Γ = Γ = +
8.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 72 An important practical case is the low-loss transmission line, where the reactive elements still dominate but R and G cannot be neglected as in a loss-less line. We have the following conditions: L R C G ω >> ω >> so that 2 ( )( ) 1 1 1 j L R j C G R G j L j C j L j C R G RG j LC j L j C LC γ ω ω ω ω ω ω ω ω ω ω = + + = + + ≈ + + − The last term under the square root can be neglected, because it is the product of two very small quantities.
9.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 73 What remains of the square root can be expanded into a truncated Taylor series 1 1 2 1 2 R G j LC j L j C C L R G j LC L C γ ω ω ω ω ≈ + + = + + so that 1 2 C L R G LC L C α β ω = + =
10.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 74 The characteristic impedance of the low-loss line is a real quantity for all practical purposes and it is approximately the same as in a corresponding loss-less line 0 R j L L Z G j C C + ω = ≈ + ω and the phase velocity associated to the wave propagation is 1 p v LC ω = ≈ β BUT NOTE: In the case of the low-loss line, the equations for voltage and current retain the same form obtained for general lossy lines.
11.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 75 Again, we obtain the loss-less transmission line if we assume 0 0 R G = = This is often acceptable in relatively short transmission lines, where the overall attenuation is small. As shown earlier, the characteristic impedance in a loss-less line is exactly real 0 L Z C = while the propagation constant has no attenuation term ( )( ) j L j C j LC j γ = ω ω = ω = β The loss-less line does not dissipate power, because α = 0.
12.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 76 For all cases, the line impedance was defined as 0 (d) 1 (d) (d) (d) 1 (d) V Z Z I + Γ = = − Γ By including the appropriate generalized reflection coefficient, we can derive alternative expressions of the line impedance: A) Loss-less line 2 d 0 0 0 2 d 0 tan( d) 1 (d) tan( d) 1 j R R j R R Z jZ e Z Z Z jZ Z e − β − β + β + Γ = = β + − Γ B) Lossy line (including low-loss) 2 d 0 0 0 2 d 0 tanh( d) 1 (d) tanh( d) 1 R R R R Z Z e Z Z Z Z Z e − γ − γ + γ + Γ = = γ + − Γ
13.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 77 Let’s now consider power flow in a transmission line, limiting the discussion to the time-average power, which accounts for the active power dissipated by the resistive elements in the circuit. The time-average power at any transmission line location is { } * 1 (d, ) Re (d) (d) 2 P t V I 〈 〉 = This quantity indicates the time-average power that flows through the line cross-section at location d. In other words, this is the power that, given a certain input, is able to reach location d and then flows into the remaining portion of the line beyond this point. It is a common mistake to think that the quantity above is the power dissipated at location d !
14.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 78 The generator, the input impedance, the input voltage and the input current determine the power injected at the transmission line input. { } * 1 1 Re 2 in in G G in in G G in in in in Z V V Z Z I V Z Z P V I = + = + 〈 〉 = Vin Iin Generator Line VG ZG Zin
15.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 79 The time-average power reaching the load of the transmission line is given by the general expression { } ( ) ( ) ( ) * * * 0 1 (d=0, ) Re (0) (0) 2 1 1 Re 1 1 2 R R P t V I V V Z Γ Γ + + 〈 〉 = = + − This represents the power dissipated by the load. The time-average power absorbed by the line is simply the difference between the input power and the power absorbed by the load (d 0, ) 〈 〉 = 〈 〉 − 〈 = 〉 line in P P P t In a loss-less transmission line no power is absorbed by the line, so the input time-average power is the same as the time-average power absorbed by the load. Remember that the internal impedance of the generator dissipates part of the total power generated.
16.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 80 It is instructive to develop further the general expression for the time-average power at the load, using Z0=R0+jX0 for the characteristic impedance, so that 0 0 0 0 0 * * 2 2 2 0 0 0 0 0 0 1 Z R jX R jX Z Z Z R X Z + + = = = + Alternatively, one may simplify the analysis by introducing the line characteristic admittance 0 0 0 0 1 Y G jB Z = = + It may be more convenient to deal with the complex admittance at the numerator of the power expression, rather than the complex characteristic impedance at the denominator.
17.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 81 * * 0 2 0 0 2 0 2 0 0 2 0 1 Re 1 1 (d=0, ) Re 1 1 2 Re 1 Re Im 1 Re Im 2 Re 2 R R R R R R R P t V V Z V R jX j j Z V R jX Z + + + + − Γ 〈 〉 = +Γ −Γ = = + + Γ + Γ − Γ Γ = + + 2 2 2 2 0 0 2 0 2 2 0 0 0 2 0 Im 2Im Re 1 2Im 2 2 Im 2 R R R R R R j V R jX j Z V R R X Z + + + Γ + Γ = + − Γ + Γ = = − Γ − Γ
18.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 82 Equivalently, using the complex characteristic admittance: * 0 * 2 0 0 2 0 0 1 Re 1 (d=0, ) Re 1 1 2 Re 1 Re Im 1 Re Im 2 Re 2 R R R R R R R P t V Y V V G jB j j V G jB + + + + − Γ 〈 〉 = +Γ −Γ = = − + Γ + Γ − Γ Γ = − + 2 2 2 2 0 0 2 2 0 0 0 Im 2Im Re 1 2Im 2 2 Im 2 R R R R R R j V G jB j V G G B + + + Γ + Γ = − − Γ + Γ = = − Γ + Γ
19.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 83 The time-average power, injected into the input of the transmission line, is maximized when the input impedance of the transmission line and the internal generator impedance are complex conjugate of each other. Z Z G in = * Zin ZR ZG Transmission line Load Generator for maximum power transfer VG
20.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 84 The characteristic impedance of the loss-less line is real and we can express the power flow, anywhere on the line, as ( ) ( ) * d 2 d * * d 2 d 0 2 2 2 0 0 1 (d, ) Re{ (d) (d) } 2 1 Re 1 2 1 ( ) 1 1 1 2 2 j j R j j R R P t V I V e e V e e Z V V Z Z β β β β Γ Γ Γ Incident Reflected w wave ave + − + − − + + 〈 〉 = = + − = − This result is valid for any location, including the input and the load, since the transmission line does not absorb any power.
21.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 85 In the case of low-loss lines, the characteristic impedance is again real, but the time-average power flow is position dependent because the line absorbs power. { } ( ) { ( ) * d d 2 d * * d d 2 d 0 2 2 2 2 d 2 d 0 0 1 (d, ) Re (d) (d) 2 1 Re 1 2 1 ( ) 1 1 1 2 2 j R j R R P t V I V e e e V e e e Z V e V e Z Z α β γ α β γ α α Γ Γ Γ Reflec Inciden ted wave t wave + − + − − + + − 〈 〉 = = + − = −
22.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 86 Note that in a lossy line the reference for the amplitude of the incident voltage wave is at the load and that the amplitude grows exponentially moving towards the input. The amplitude of the incident wave behaves in the following way input inside the line L d load V e V e V + α + α + ⇔ ⇔ The reflected voltage wave has maximum amplitude at the load, and it decays exponentially moving back towards the generator. The amplitude of the reflected wave behaves in the following way input inside the line lo L d ad R R R V e V e V + −α + −α + Γ Γ ⇔ ⇔ Γ
23.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 87 For a general lossy line the power flow is again position dependent. Since the characteristic impedance is complex, the result has an additional term involving the imaginary part of the characteristic admittance, B0, as { } ( ) { ( ) } * d d * * * d d 0 2 2 2 2 d 2 d 0 0 2 2 d 0 1 (d, ) Re (d) (d) 2 1 Re 1 (d) 2 ( ) 1 (d) 2 2 Im( (d)) j j R P t V I V e e Y V e e G G V e V e B V e α β α β α α α Γ Γ Γ Γ + + − + + − + 〈 〉 = = + − = − +
24.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 88 For the general lossy line, keep in mind that * 0 0 0 0 0 0 0 0 * 2 2 2 0 0 0 0 0 0 0 0 0 0 2 2 2 2 0 0 0 0 1 Z R jX R jX Y G jB Z Z Z R X Z R X G B R X R X − − = = = = = + + − = = + + Recall that for a low-loss transmission line the characteristic impedance is approximately real, so that 0 0 B ≈ and 0 0 0 1 Z G R ≈ ≈ . The previous result for the low-loss line can be readily recovered from the time-average power for the general lossy line.
25.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 89 To completely specify the transmission line problem, we still have to determine the value of V+ from the input boundary condition. ¾The load boundary condition imposes the shape of the interference pattern of voltage and current along the line. ¾The input boundary condition, linked to the generator, imposes the scaling for the interference patterns. We have 0 1 (L) (L) 1 (L) in in G in G in Z V V V Z Z Z Z Γ Γ with + = = = + − 0 0 0 0 0 0 tan( L) tan( L) tanh( L) tanh( L) R in R R in R Z jZ Z Z jZ Z Z Z Z Z Z Z β β γ γ loss - less line lossy o lin r e + = + + = +
26.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 90 For a loss-less transmission line: [ ] L L 2 L (L) 1 (L) (1 ) j j j R V V e V e e β β β Γ Γ + + − = + = + L 2 L 1 (1 ) in G j j G in R Z V V Z Z e e β β Γ + − ⇒ = + + For a lossy transmission line: [ ] ( ) L L 2 d (L) 1 (L) 1 R V V e V e e γ γ γ Γ Γ + + − = + = + L 2 L 1 (1 ) in G G in R Z V V Z Z e e γ γ Γ + − ⇒ = + +
27.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 91 In order to have good control on the behavior of a high frequency circuit, it is very important to realize transmission lines as uniform as possible along their length, so that the impedance behavior of the line does not vary and can be easily characterized. A change in transmission line properties, wanted or unwanted, entails a change in the characteristic impedance, which causes a reflection. Example: Γ1 ZR Z01 Z02 Z01 Zin Γ1 in 01 in 01 Z Z Z Z = − +
28.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 92 Special Cases 0 R Z → (SHORT CIRCUIT) The load boundary condition due to the short circuit is V (0) = 0 0 2 0 (d 0) (1 ) (1 ) 0 1 j j R R R V V e e V + β − β + ⇒ = = + Γ = + Γ = ⇒ Γ = − ZR = 0 Z0
29.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 93 Since R V V V V − + − + Γ = ⇒ = − We can write the line voltage phasor as d d d d d d (d) ( ) 2 sin( d) j j j j j j V V e V e V e V e V e e jV + β − − β + β + − β + β − β + = + = − = − = β
30.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 94 For the line current phasor we have d d 0 d d 0 d d 0 0 1 (d) ( ) 1 ( ) ( ) 2 cos( d) j j j j j j I V e V e Z V e V e Z V e e Z V Z + β − − β + β + − β + β − β + = − = + = + = β The line impedance is given by 0 0 (d) 2 sin( d) (d) tan( d) (d) 2 cos( d)/ V jV Z jZ I V Z + + β = = = β β
31.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 95 The time-dependent values of voltage and current are obtained as ( ) (d, ) Re[ (d) ] Re[2 | | sin( d) ] 2| |sin( d) Re[ ] 2| |sin( d) Re[ cos( ) sin( )] 2| |sin( d)sin( ) j t j j t j t V t V e j V e e V j e V j t t V t ω + θ ω + ω +θ + + = = β = β ⋅ = β ⋅ ω + θ − ω + θ = − β ω + θ 0 ( ) 0 0 0 (d, ) Re[ (d) ] Re[2 | | cos( d) ]/ 2| |cos( d) Re[ ]/ 2| |cos( d) Re[(cos( ) sin( )]/ | | 2 cos( d)cos( ) j t j j t j t I t I e V e e Z V e Z V t j t Z V t Z ω + θ ω + ω +θ + + = = β = β ⋅ = β ⋅ ω + θ + ω + θ = β ω + θ
32.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 96 The time-dependent power is given by 2 0 2 0 (d, ) (d, ) (d, ) | | 4 sin( d)cos( d)sin( )cos( ) | | sin(2 d)sin (2 2 ) P t V t I t V t t Z V t Z + + = ⋅ = − β β ω + θ ω + θ = − β ω + θ and the corresponding time-average power is 0 2 0 0 1 (d, ) (d, ) | | 1 sin(2 d) sin (2 2 ) 0 T T P t P t dt T V t Z T + = = − β ω + θ = ∫ ∫
33.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 97 R Z → ∞ (OPEN CIRCUIT) The load boundary condition due to the open circuit is I (0) = 0 0 2 0 0 0 (d 0) (1 ) (1 ) 0 1 j j R R R V I e e Z V Z + β − β + ⇒ = = − Γ = − Γ = ⇒ Γ = ZR → ∞ Z0
34.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 98 Since R V V V V − + − + Γ = ⇒ = We can write the line current phasor as d d 0 d d 0 d d 0 0 1 (d) ( ) 1 ( ) 2 ( ) sin( d) j j j j j j I V e V e Z V e V e Z V jV e e Z Z + β − − β + β + − β + + β − β = − = − = − = β
35.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 99 For the line voltage phasor we have d d d d d d (d) ( ) ( ) ( ) 2 cos( d) j j j j j j V V e V e V e V e V e e V + β − − β + β + − β + β − β + = + = + = + = β The line impedance is given by 0 0 (d) 2 cos( d) (d) (d) tan( d) 2 sin( d)/ V V Z Z j I jV Z + + β = = = − β β
36.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 100 The time-dependent values of voltage and current are obtained as ( ) (d, ) Re[ (d) ] Re[2 | | cos( d) ] 2| |cos( d) Re[ ] 2| |cos( d) Re[(cos( ) sin( )] 2| |cos( d)cos( ) j t j j t j t V t V e V e e V e V t j t V t ω + θ ω + ω +θ + + = = β = β ⋅ = β ⋅ ω + θ + ω + θ = β ω + θ 0 ( ) 0 0 0 (d, ) Re[ (d) ] Re[2 | | sin( d) ]/ 2| |sin( d) Re[ ]/ 2| |sin( d) Re[ cos( ) sin( )]/ | | 2 sin( d)sin( ) j t j j t j t I t I e j V e e Z V j e Z V j t t Z V t Z ω + θ ω + ω +θ + + = = β = β ⋅ = β ⋅ ω + θ − ω + θ = − β ω + θ
37.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 101 The time-dependent power is given by 2 0 2 0 (d, ) (d, ) (d, ) | | 4 cos( d)sin( d)cos( )sin( ) | | sin(2 d)sin (2 2 ) P t V t I t V t t Z V t Z + + = ⋅ = = − β β ω + θ ω + θ = − β ω + θ and the corresponding time-average power is 0 2 0 0 1 (d, ) (d, ) | | 1 sin(2 d) sin (2 2 ) 0 T T P t P t dt T V t Z T + = = − β ω + θ = ∫ ∫
38.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 102 0 R Z Z = (MATCHED LOAD) The reflection coefficient for a matched load is 0 0 0 0 0 0 0 R R R Z Z Z Z Z Z Z Z − − Γ = = = + + no reflection! The line voltage and line current phasors are d 2 d d d 2 d d 0 0 (d) (1 ) ( ) (1 ) j j j R j j j R V V e e V e V V I d e e e Z Z + β − β + β + + β − β β = + Γ = = − Γ = ZR = Z0 Z0
39.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 103 The line impedance is independent of position and equal to the characteristic impedance of the line d 0 d 0 (d) (d) (d) j j V V e Z Z I V e Z + β + β = = = The time-dependent voltage and current are d ( d ) d 0 ( d ) 0 0 (d, ) Re[| | ] | | Re[ ] | | cos( d ) (d, ) Re[| | ]/ | | | | Re[ ] cos( d ) j j j t j t j j j t j t V t V e e e V e V t I t V e e e Z V V e t Z Z + θ β ω + ω +β +θ + + θ β ω + + ω +β +θ = = ⋅ = ω + β + θ = = ⋅ = ω + β + θ
40.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 104 The time-dependent power is 0 2 2 0 | | (d, ) | | cos( d ) cos( d ) | | cos ( d ) V P t V t t Z V t Z + + + = ω + β + θ ω + β + θ = ω + β + θ and the time average power absorbed by the load is 2 2 0 0 2 0 1 | | (d) cos ( d ) | | 2 t V P t dt T Z V Z + + = ω + β + θ = ∫
41.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 105 R Z jX = (PURE REACTANCE) The reflection coefficient for a purely reactive load is 0 0 0 0 2 2 0 0 0 0 2 2 2 2 0 0 0 0 ( )( ) 2 ( )( ) R R R Z Z jX Z Z Z jX Z jX Z jX Z X Z XZ j jX Z jX Z Z X Z X − − Γ = = = + + − − − = = + + − + + ZR = j X Z0
42.
Transmission Lines © Amanogawa,
2006 – Digital Maestro Series 106 In polar form exp( ) R R j Γ = Γ θ where ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 0 0 0 2 2 2 2 2 2 2 2 2 0 0 0 1 0 2 2 0 4 1 2 tan R X Z Z X X Z Z X Z X Z X XZ X Z − − + Γ = + = = + + + θ = − The reflection coefficient has unitary magnitude, as in the case of short and open circuit load, with zero time average power absorbed by the load. Both voltage and current are finite at the load, and the time-dependent power oscillates between positive and negative values. This means that the load periodically stores power and then returns it to the line without dissipation.
43.
Transmission Lines © Amarcord,
2006 – Digital Maestro Series 107 Reactive impedances can be realized with transmission lines terminated by a short or by an open circuit. The input impedance of a loss-less transmission line of length L terminated by a short circuit is purely imaginary ( ) 0 0 0 2 2 tan L tan L tan L in p f Z j Z j Z j Z v π π = β = = λ For a specified frequency f, any reactance value (positive or negative!) can be obtained by changing the length of the line from 0 to λ/2. An inductance is realized for L λ/4 (positive tangent) while a capacitance is realized for λ/4 L λ/2 (negative tangent). When L = 0 and L = λ/2 the tangent is zero, and the input impedance corresponds to a short circuit. However, when L = λ/4 the tangent is infinite and the input impedance corresponds to an open circuit.
44.
Transmission Lines © Amarcord,
2006 – Digital Maestro Series 108 Since the tangent function is periodic, the same impedance behavior of the impedance will repeat identically for each additional line increment of length λ/2. A similar periodic behavior is also obtained when the length of the line is fixed and the frequency of operation is changed. At zero frequency (infinite wavelength), the short circuited line behaves as a short circuit for any line length. When the frequency is increased, the wavelength shortens and one obtains an inductance for L λ/4 and a capacitance for λ/4 L λ/2, with an open circuit at L = λ/4 and a short circuit again at L = λ/2. Note that the frequency behavior of lumped elements is very different. Consider an ideal inductor with inductance L assumed to be constant with frequency, for simplicity. At zero frequency the inductor also behaves as a short circuit, but the reactance varies monotonically and linearly with frequency as X L = ω (always aninductance)
45.
Transmission Lines © Amarcord,
2006 – Digital Maestro Series 109 Short circuited transmission line – Fixed frequency L L L L L L L L = = = → ∞ = = = → ∞ 0 0 0 4 0 4 4 2 0 2 0 2 3 4 0 3 4 3 4 0 Z Z Z Z Z Z Z Z in in in in in in in in short circuit inductance open circuit capacitance short circuit inductance open circuit capacitance λ λ λ λ λ λ λ λ λ λ Im Im Im Im k p k p k p k p … L
46.
Transmission Lines © Amarcord,
2006 – Digital Maestro Series 110 β Impedance of a short circuited transmission line (fixed frequency, variable length) Line Length L L θ [deg] inductive capacitive inductive inductive cap. -40 -30 -20 -10 0 10 20 30 40 0 200 300 400 500 100 2π/β= λ 5π/(2β)= 5λ/4 3π/(2β)= 3λ/4 π/β =λ/2 π/(2β)= λ/4 Normalized Input Impedance Z(L)/Zo = j tan( L)
47.
Transmission Lines © Amarcord,
2006 – Digital Maestro Series 111 β θ [deg] inductive capacitive inductive inductive cap. -40 -30 -20 -10 0 10 20 30 40 0 200 300 400 500 100 vp / (4L) Impedance of a short circuited transmission line (fixed length, variable frequency) vp / (2L) 3vp / (4L) vp / L 5vp / (4L) f Frequency of operation Normalized Input Impedance Z(L)/Zo = j tan( L)
48.
Transmission Lines © Amarcord,
2006 – Digital Maestro Series 112 For a transmission line of length L terminated by an open circuit, the input impedance is again purely imaginary ( ) 0 0 0 2 tan L 2 tan L tan L in p Z Z Z Z j j j f v = − = − = − π β π λ We can also use the open circuited line to realize any reactance, but starting from a capacitive value when the line length is very short. Note once again that the frequency behavior of a corresponding lumped element is different. Consider an ideal capacitor with capacitance C assumed to be constant with frequency. At zero frequency the capacitor behaves as an open circuit, but the reactance varies monotonically and linearly with frequency as 1 X C = ω (always a capacitance)
49.
Transmission Lines © Amarcord,
2006 – Digital Maestro Series 113 Open circuit transmission line – Fixed frequency L L L L L L L L = → ∞ = = = → ∞ = = 0 0 4 0 4 0 4 2 0 2 2 3 4 0 3 4 0 3 4 0 Z Z Z Z Z Z Z Z in in in in in in in in open circuit capacitance short circuit inductance open circuit capacitance short circuit inductance λ λ λ λ λ λ λ λ λ λ Im Im Im Im k p k p k p k p … L
50.
Transmission Lines © Amarcord,
2006 – Digital Maestro Series 114 capacitive capacitive inductive inductive inductive θ [deg] β capacitive L Line Length L 0 π/(2β) = λ/4 5π/(2β) = 5λ/4 2π/β = 2λ 3π/(2β) = 3λ/4 π/β = λ/2 -40 -30 -20 -10 0 10 20 30 40 0 200 400 500 300 100 Impedance of an open circuited transmission line (fixed frequency, variable length) Normalized Input Impedance Z(L)/ Zo = - j cotan( L)
51.
Transmission Lines © Amarcord,
2006 – Digital Maestro Series 115 capacitive capacitive inductive inductive inductive θ [deg] β 0 capacitive vp / (4L) vp / (2L) 3vp / (4L) vp / L 5vp / (4L) f Frequency of operation -40 -30 -20 -10 0 10 20 30 40 0 200 400 500 300 100 Impedance of an open circuited transmission line (fixed length, variable frequency) Normalized Input Impedance Z(L)/ Zo = - j cotan( L)
52.
Transmission Lines © Amarcord,
2006 – Digital Maestro Series 116 It is possible to realize resonant circuits by using transmission lines as reactive elements. For instance, consider the circuit below realized with lines having the same characteristic impedance: ( ) ( ) 1 0 1 2 0 2 tan L tan L in in Z j Z Z j Z = β = β I V L2 L1 short circuit short circuit Zin1 Zin2 Z0 Z0
53.
Transmission Lines © Amarcord,
2006 – Digital Maestro Series 117 The circuit is resonant if L1 and L2 are chosen such that an inductance and a capacitance are realized. A resonance condition is established when the total input impedance of the parallel circuit is infinite or, equivalently, when the input admittance of the parallel circuit is zero ( ) ( ) 0 2 0 1 1 1 0 tan L tan L r r j Z j Z + = β β or 1 2 2 tan L tan L r r r r p p r p v v v ω ω π ω = − β = = λ with Since the tangent is a periodic function, there is a multiplicity of possible resonant angular frequencies ωr that satisfy the condition above. The values can be found by using a numerical procedure to solve the trascendental equation above.
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