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Transmission Lines We have obtained the following solutions for the steady-state voltage and current phasors in a transmission line: Loss-less line V (z) V e jz V e j z 1 I (z) V e jz V e j z Z0 Lossy line V (z) V ez V ez 1 I (z) V ez V ez Z0 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. © Amanogawa, 2000 – Digital Maestro Series 65 Transmission Lines 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. ZR New Space Coordinate z d 0 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. © Amanogawa, 2000 – Digital Maestro Series 66 Transmission Lines 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 jd jd V (d) V e Lossy line d d V e 1 I (d) V e jd V e jd Z0 V (d) V e V e 1 I (d) V ed V e d Z0 At the load (d = 0) we have, for both cases, V (0) V V 1 I (0) V V Z0 © Amanogawa, 2000 – Digital Maestro Series 67 Transmission Lines For a given load impedance ZR , the load boundary condition is V (0) ZR I (0) Therefore, we have ZR V V V V Z0 from which we obtain the voltage load reflection coefficient V ZR Z0 R Z Z V R 0 © Amanogawa, 2000 – Digital Maestro Series 68 Transmission Lines We can introduce this result into the transmission line equations as Loss-less line Lossy line V e jd 2 j d I (d) e 1 R Z0 V (d) V e jd 1 R e2 j d V e d 2 d I (d) e 1 R Z0 V (d) V ed 1 R e2 d At each line location we define a Generalized Reflection Coefficient (d) R e2 j d (d) R e2 d and the line equations become V (d) V e jd 1 (d) V (d) V ed 1 (d) V e jd I (d) 1 (d) Z0 V ed I (d) 1 (d) Z0 © Amanogawa, 2000 – Digital Maestro Series 69 Transmission Lines We define the line impedance as 1 (d ) V (d ) Z0 Z( d ) 1 (d ) I (d ) A simple circuit diagram can illustrate the significance of line impedance and generalized reflection coefficient: Req = (d) d © Amanogawa, 2000 – Digital Maestro Series ZR Zeq=Z(d) 0 70 Transmission Lines 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 (d) Req ( d ) ZReq Z0 ZReq Z0 If the total length of the line is L, the input impedance is obtained from the formula for the line impedance as 1 ( L) Vin V ( L) Z0 Zin 1 ( L) Iin I ( L) The input impedance is the equivalent impedance representing the entire line terminated by the load. © Amanogawa, 2000 – Digital Maestro Series 71 Transmission Lines 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 so that ( j L R)( j C G) R G 1 j L j C 1 j L j C R G RG j LC 1 j L j C 2 LC The last term under the square root can be neglected, because it is the product of two very small quantities. © Amanogawa, 2000 – Digital Maestro Series 72 Transmission Lines What remains of the square root can be expanded into a truncated Taylor series 1 R G j LC 1 2 j L j C 1 C L R G j LC 2 L C so that 1 C L R G 2 L C © Amanogawa, 2000 – Digital Maestro Series LC 73 Transmission Lines 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 R j L L Z0 G j C C and the phase velocity associated to the wave propagation is 1 vp LC 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. © Amanogawa, 2000 – Digital Maestro Series 74 Transmission Lines Again, we obtain the loss-less transmission line if we assume R0 G0 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 L Z0 C while the propagation constant has no attenuation term ( j L)( j C ) j LC j The loss-less line does not dissipate power, because = 0. © Amanogawa, 2000 – Digital Maestro Series 75 Transmission Lines For all cases, the line impedance was defined as 1 (d) V (d) Z0 Z(d) 1 (d) I (d) By including the appropriate generalized reflection coefficient, we can derive alternative expressions of the line impedance: A) Loss-less line 1 Re2 jd ZR jZ0 tan( d) Z0 Z(d) Z0 2 jd jZR tan( d) Z0 1 Re B) Lossy line (including low-loss) 1 R e2 d ZR Z0 tanh( d) Z0 Z(d) Z0 2 d ZR tanh( d) Z0 1 Re © Amanogawa, 2000 – Digital Maestro Series 76 Transmission Lines 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 P(d , t ) Re V (d) I * (d) 2 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 ! © Amanogawa, 2000 – Digital Maestro Series 77 Transmission Lines The generator, the input impedance, the input voltage and the input current determine the power injected at the transmission line input. Iin Zin Vin VG ZG Zin ZG VG Vin Generator © Amanogawa, 2000 – Digital Maestro Series Zin Line 1 Iin VG ZG Zin 1 * Pin Re Vin Iin 2 78 Transmission Lines The time-average power reaching the load of the transmission line is given by 1 P(d=0 , t ) Re V (0) I * (0) 2 * 1 1 Re V 1 R V 1 R 2 Z0* 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 P line Pin P(d 0 , t ) Remember that the internal impedance of the generator dissipates part of the total power generated. © Amanogawa, 2000 – Digital Maestro Series 79 Transmission Lines 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. Zin Load Generator ZG VG ZG Z*in © Amanogawa, 2000 – Digital Maestro Series Transmission line ZR for maximum power transfer 80 Transmission Lines 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. The characteristic impedance of the loss-less line is real and we can express the power flow as 1 P(d , t ) Re V (d) I* (d) 2 1 Re V e j d 1 Re j 2 d 2 1 * j d j 2 d * (V ) e 1 Re Z0 1 1 2 2 2 V V R 2 Z0 2 Z0 Incident wave © Amanogawa, 2000 – Digital Maestro Series Reflected wave 81 Transmission Lines In the case of low-loss lines, the characteristic impedance is again real, and the time-average power flow along the line is given by 1 P(d , t ) Re V (d) I * (d) 2 1 Re V e d e j d 1 Re2 d 2 1 * d j d 2 d * (V ) e e 1 Re Z0 1 1 2 2 2 d 2 2 d V e V e R 2 Z0 2 Z0 Incident wave © Amanogawa, 2000 – Digital Maestro Series Reflected wave 82 Transmission Lines 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 V e L V e d V input inside the line load 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 V R e L V R e d V R input inside the line load © Amanogawa, 2000 – Digital Maestro Series 83 Transmission Lines For a general lossy line the characteristic impedance is complex, and the time-average power is 1 P(d , t ) Re V (d) I * (d) 2 1 Re V e d e j d 1 (d) 2 * * * d j d Y0 (V ) e e 1 (d) G0 2 2 d G0 2 2 d 2 R V e V e 2 2 B0 V © Amanogawa, 2000 – Digital Maestro Series 2 e2 d Im((d)) 84 Transmission Lines We have introduced for convenience the characteristic admittance of the line Y0 1 G0 jB0 Z0 since a complex characteristic impedance would appear at denominator in the expression for the power. Note that for a low-loss transmission line the characteristic impedance is approximately real and B0 0 The previous result for the low-loss line can be readily recovered from the time-average power for the general lossy line. © Amanogawa, 2000 – Digital Maestro Series 85 Transmission Lines To completely specify the transmission line problem, we still have to determine the value of V+ from the input boundary condition. The load boundary condition imposes the shape of the interference pattern of voltage and current along the line. The input boundary condition, linked to the generator, imposes the scaling for the interference patterns. We have Zin Vin V ( L) VG ZG Zin or with 1 (L) Zin Z0 1 (L) ZR jZ0 tan( L) Zin Z0 jZR tan( L) Z0 loss - less line ZR Z0 tanh( L) Zin Z0 ZR tanh( L) Z0 lossy line © Amanogawa, 2000 – Digital Maestro Series 86 Transmission Lines For a loss-less transmission line: V (L) V e j L 1 (L) V e j L (1 Re j 2 L ) ! 1 Zin V VG ZG Zin e j L (1 Re j 2 L ) For a lossy transmission line: V (L) V e L 1 (L) V e L 1 Re2 d ! 1 Zin V VG ZG Zin e L (1 Re2 L ) © Amanogawa, 2000 – Digital Maestro Series 87 Transmission Lines 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: Z01 Z02 ZR 1 Z01 © Amanogawa, 2000 – Digital Maestro Series Zin Zin Z01 1 Zin Z01 88 Transmission Lines Special Cases ZR 0 (SHORT CIRCUIT) ZR = 0 Z0 The load boundary condition due to the short circuit is V (0) = 0 V (d 0) V e j 0 (1 R e j 2 0 ) V (1 R ) 0 © Amanogawa, 2000 – Digital Maestro Series R 1 89 Transmission Lines Since R V V V V We can write the line voltage phasor as V (d) V e j d V e j d V e j d V e j d V ( e j d e j d ) 2 jV sin( d) © Amanogawa, 2000 – Digital Maestro Series 90 Transmission Lines For the line current phasor we have 1 (V e j d V e j d ) I (d) Z0 1 (V e j d V e j d ) Z0 V j d (e e j d ) Z0 2V cos( d) Z0 The line impedance is given by 2 jV sin( d) V (d) jZ0 tan( d) Z(d) I (d) 2V cos( d) / Z0 © Amanogawa, 2000 – Digital Maestro Series 91 Transmission Lines The time-dependent values of voltage and current are obtained as V (d, t ) Re[V (d) e j t ] Re[2 j | V | e j sin( d) e j t ] 2| V |sin( d) Re[ j e j( t ) ] 2| V |sin( d) Re[ j cos(t ) sin(t )] 2| V |sin( d) sin(t ) I (d, t ) Re[ I (d) e j t ] Re[2 | V | e j cos( d) e j t ] / Z0 2| V |cos( d) Re[ e j ( t ) ] / Z0 2| V |cos( d) Re[ (cos(t ) j sin(t )] / Z0 | V | cos( d) cos(t ) 2 Z0 © Amanogawa, 2000 – Digital Maestro Series 92 Transmission Lines The time-dependent power is given by P(d, t ) V (d, t ) I (d, t ) | V |2 sin( d) cos( d) sin(t ) cos(t ) 4 Z0 | V |2 sin(2 d) sin (2t 2) Z0 and the corresponding time-average power is 1 T P(d, t ) P(d, t ) dt T 0 | V |2 1 T sin(2 d) sin (2t 2) 0 Z0 T 0 © Amanogawa, 2000 – Digital Maestro Series 93 Transmission Lines ZR (OPEN CIRCUIT) ZR Z0 The load boundary condition due to the open circuit is I (0) = 0 V j 0 I (d 0) e (1 R e j 2 0 ) Z0 V (1 R ) 0 Z0 © Amanogawa, 2000 – Digital Maestro Series R 1 94 Transmission Lines Since R V V V V We can write the line current phasor as 1 (V e j d V e j d ) I (d) Z0 1 (V e j d V e j d ) Z0 2 jV V j d j d (e ) sin( d) e Z0 Z0 © Amanogawa, 2000 – Digital Maestro Series 95 Transmission Lines For the line voltage phasor we have V (d) (V e j d V e j d ) (V e j d V e j d ) V ( e j d e j d ) 2V cos( d) The line impedance is given by 2V cos( d) V (d) Z0 j Z(d) tan( d) I (d) 2 jV sin( d) / Z0 © Amanogawa, 2000 – Digital Maestro Series 96 Transmission Lines The time-dependent values of voltage and current are obtained as V (d, t ) Re[V (d) e j t ] Re[2 | V | e j cos( d) e j t ] 2| V |cos( d) Re[ e j( t ) ] 2| V |cos( d) Re[ (cos(t ) j sin(t )] 2| V |cos( d) cos( t ) I (d, t ) Re[ I (d) e j t ] Re[2 j | V | e j sin( d) e j t ] / Z0 2| V |sin( d) Re[ j e j ( t ) ] / Z0 2| V |sin( d) Re[ j cos( t ) sin(t )] / Z0 | V | sin( d) sin(t ) 2 Z0 © Amanogawa, 2000 – Digital Maestro Series 97 Transmission Lines The time-dependent power is given by P(d, t ) V (d, t ) I (d, t ) | V |2 cos( d) sin( d) cos(t ) sin(t ) 4 Z0 | V |2 sin(2 d) sin (2t 2) Z0 and the corresponding time-average power is 1 T P(d, t ) P(d, t ) dt T 0 | V |2 1 T sin(2 d) sin (2t 2) 0 Z0 T 0 © Amanogawa, 2000 – Digital Maestro Series 98 Transmission Lines ZR Z0 (MATCHED LOAD) Z0 ZR = Z0 The reflection coefficient for a matched load is ZR Z0 Z0 Z0 R 0 ZR Z0 Z0 Z0 no reflection! The line voltage and line current phasors are V (d) V e j d (1 R e2 j d ) V e j d V j d V (1 R e2 j d ) I( d ) e e j d Z0 Z0 © Amanogawa, 2000 – Digital Maestro Series 99 Transmission Lines The line impedance is independent of position and equal to the characteristic impedance of the line V (d) V e j d Z0 Z(d) I (d) V j d e Z0 The time-dependent voltage and current are V (d, t ) Re[| V | e j e j d e j t ] | V | Re[e j( t d ) ] | V | cos(t d ) I (d, t ) Re[| V | e j e j d e j t ] / Z0 | V | | | V j ( t d ) ] cos(t d ) Re[e Z0 Z0 © Amanogawa, 2000 – Digital Maestro Series 100 Transmission Lines The time-dependent power is | | V cos(t d ) P(d, t ) | V | cos(t d ) Z0 | V |2 cos2 (t d ) Z0 and the time average power absorbed by the load is 1 t | V |2 cos2 (t d ) dt P(d) T 0 Z0 | V |2 2 Z0 © Amanogawa, 2000 – Digital Maestro Series 101 Transmission Lines ZR jX (PURE REACTANCE) ZR = j X Z0 The reflection coefficient for a purely reactive load is ZR Z0 jX Z0 R ZR Z0 jX Z0 ( jX Z0 )( jX Z0 ) ( jX Z0 )( jX Z0 ) © Amanogawa, 2000 – Digital Maestro Series X 2 Z02 Z02 X 2 2j XZ0 Z02 X 2 102 Transmission Lines In polar form R R exp( j) where R 2 2 X Z0 4 X 2 Z02 2 2 Z02 X 2 Z02 X 2 2 2 2 2 Z0 X 1 2 Z02 X 2 1 2 XZ0 tan X 2 Z2 0 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 and returns powers to the line without dissipation. © Amanogawa, 2000 – Digital Maestro Series 103 Transmission Lines 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 2 f 2 L Zin j Z0 tan L j Z0 tan L j Z0 tan vp At a specified frequency f, any reactance value can be obtained by changing the length of the line from 0 to /2. Since the tangent function is periodic, the behavior of the impedance will repeat identically for each line increment of length /2. Note that a similar periodic behavior is obtained when the length of the transmission line is fixed and the frequency of operation is changed. © Amanogawa, 2000 – Digital Maestro Series 104 Transmission Lines Shorted transmission line – Fixed frequency L L 0 0 L L 4 4 L 4 2 L 2 3 L 2 4 3 L 4 3 L 4 Z in 0 s h o rt c irc u it Im Z in 0 in d u c ta n c e Z in o p e n c irc u it c a p a c ita n c e Im Z in 0 Z in 0 s h o rt c irc u it Im Z in 0 in d u c ta n c e Z in o p e n c irc u it Im Z in 0 c a p a c ita n c e © Amanogawa, 2000 – Digital Maestro Series 105 Z(L)/Zo = j tan( L) Transmission Lines Impedance of a short circuited transmission line (fixed frequency, variable length) 40 30 20 inductive Normalized Input Impedance 10 inductive inductive 0 -10 capacitive cap. -20 -30 -40 0 100 200 300 400 500 L [deg] Line Length L © Amanogawa, 2000 – Digital Maestro Series 106 Z(L)/Zo = j tan( L) Transmission Lines Impedance of a short circuited transmission line (fixed length, variable frequency) 40 30 20 inductive Normalized Input Impedance 10 inductive inductive 0 -10 capacitive cap. -20 -30 -40 0 100 vp / (4L) 200 vp / (2L) 300 3vp / (4L) 400 vp / L 500 5vp / (4L) [deg] f Frequency of operation © Amanogawa, 2000 – Digital Maestro Series 107 Transmission Lines For a transmission line of length L terminated by an open circuit, the input impedance is again purely imaginary Z0 j Zin j tan L Z0 Z0 j 2 2 f tan L tan L v p We can use the open circuited line to realize any reactance, but starting from a capacitive value when the line length is very short. The reactance varies linearly with frequency in the case of lumped elements since X L (inductance) or 1 (capacitance) X C This is not the case when the reactance is realized with a section of transmission line. © Amanogawa, 2000 – Digital Maestro Series 108 Transmission Lines Open transmission line – Fixed frequency L L 0 0 L L 4 4 L 4 2 L 2 3 L 2 4 3 L 4 3 L 4 Z in o p e n c irc u it Im Z in 0 c a p a c ita n c e Z in 0 s h o rt c irc u it Im Z in 0 in d u c ta n c e Z in o p e n c irc u it Im Z in 0 c a p a c ita n c e Z in 0 s h o rt c irc u it Im Z in 0 in d u c ta n c e © Amanogawa, 2000 – Digital Maestro Series 109 Normalized Input Impedance Z(L)/ Zo = - j cotan( L) Transmission Lines Impedance of an open circuited transmission line (fixed frequency, variable length) 40 30 20 inductive inductive inductive 10 0 -10 capacitive capacitive capacitive -20 -30 -40 0 100 200 300 400 500 [deg] L Line Length L © Amanogawa, 2000 – Digital Maestro Series 110 Normalized Input Impedance Z(L)/ Zo = - j cotan( L) Transmission Lines Impedance of an open circuited transmission line (fixed length, variable frequency) 40 30 20 inductive inductive inductive 10 0 -10 capacitive capacitive capacitive -20 -30 -40 0 100 vp / (4L) 200 vp / (2L) 300 3vp / (4L) 400 vp / L 500 5vp / (4L) [deg] f Frequency of operation © Amanogawa, 2000 – Digital Maestro Series 111 Transmission Lines 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: I L1 short circuit Z0 L2 V Zin1 Zin1 j Z0 tan L 1 © Amanogawa, 2000 – Digital Maestro Series Z0 short circuit Zin2 Zin2 j Z0 tan L 2 112 Transmission Lines 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) 1 1 0 j Z0 tan r L1 j Z0 tan r L 2 or r r tan L1 tan L 2 vp vp with 2 r r r vp Since the tangent is a periodic function, there is a multiplicity of possible resonant angular frequencies r that satisfy the condition above. The solutions can be found by using a numerical procedure. © Amanogawa, 2000 – Digital Maestro Series 113