3636:. If the incident wave has been totally reflected and the line is lossless, there will be complete cancellation at the nodes with zero signal present there despite the ongoing transmission of waves in both directions. The points where the waves are in phase are anti-nodes and represent a peak in amplitude. Nodes and anti-nodes alternate along the line and the combined wave amplitude varies continuously between them. The combined (incident plus reflected) wave appears to be standing still on the line and is called a
977:
5919:
1700:
3141:
5630:. This instrument makes use of the reflection phenomenon to make many different measurements at microwave frequencies. One use is that VSWR and node position can be used to calculate the impedance of a test component terminating the slotted line. This is a useful method because measuring impedances by directly measuring voltages and currents is difficult at these frequencies.
612:, at which point the current in the line is zero (by the definition of an open circuit). Since charge continues to arrive at the end of the line through the incident current, but no current is leaving the line, then conservation of electric charge requires that there must be an equal and opposite current into the end of the line. Essentially, this is
1975:
5675:
1490:
2955:
3617:
3627:
For a transmission line carrying sinusoidal waves, the phase of the reflected wave is continually changing with distance, with respect to the incident wave, as it proceeds back down the line. Because of this continuous change there are certain points on the line that the reflection will be in phase
898:
the step transient will be absorbed in the generator internal impedance and there will be no further reflections. This counter-intuitive doubling of voltage may become clearer if the circuit voltages are considered when the line is so short that it can be ignored for the purposes of analysis. The
835:
to each other so that after the step has been reflected, twice the incident voltage appears across the output terminals of the line. As the reflection proceeds back up the line the reflected voltage continues to add to the incident voltage and the reflected current continues to subtract from the
1091:
The reflection from a short-circuited line can be described in similar terms to that from an open-circuited line. Just as in the open circuit case where the current must be zero at the end of the line, in the short circuit case the voltage must be zero since there can be no volts across a short
1238:
For a line terminated in its own characteristic impedance there is no reflection. By definition, terminating in the characteristic impedance has the same effect as an infinitely long line. Any other impedance will result in a reflection. The magnitude of the reflection will be smaller than the
557:
The incident wave travelling down the line is not affected in any way by the open circuit at the end of the line. It cannot have any effect until the step actually reaches that point. The signal cannot have any foreknowledge of what is at the end of the line and is only affected by the local
4158:
3602:
Reflections in more complex scenarios, such as found on a network of cables, can result in very complicated and long lasting waveforms on the cable. Even a simple overvoltage pulse entering a cable system as uncomplicated as the power wiring found in a typical private home can result in an
6347:
6148:
5065:
1829:
5914:{\displaystyle Z_{\mathrm {in} }={\frac {V_{\mathrm {T} }}{I_{\mathrm {T} }}}=Z_{0}{\frac {(1+{\mathit {\Gamma }}\,)\cosh(\gamma \,x)+(1-{\mathit {\Gamma }}\,)\,\sinh(\gamma \,x)}{(1-{\mathit {\Gamma }}\,)\,\cosh(\gamma \,x)+(1+{\mathit {\Gamma }}\,)\,\sinh(\gamma \,x)}}}
3151:
A discontinuity, or mismatch, somewhere along the length of the line results in part of the incident wave being reflected and part being transmitted onward in the second section of line as shown in figure 5. The reflection coefficient in this case is given by
2766:
5614:
1695:{\displaystyle V_{\mathrm {r} }=V_{\mathrm {o} }-V_{\mathrm {i} }=2\,V_{\mathrm {i} }{\frac {Z_{\mathrm {L} }}{Z_{\mathrm {0} }+Z_{\mathrm {L} }}}-V_{\mathrm {i} }=V_{\mathrm {i} }{\frac {Z_{\mathrm {L} }-Z_{\mathrm {0} }}{Z_{\mathrm {L} }+Z_{\mathrm {0} }}}}
1391:
3136:{\displaystyle \theta ={\begin{cases}\pi -2\,\arctan {\frac {X_{\mathrm {L} }}{R_{\mathrm {0} }}}&{\mbox{if }}{X_{\mathrm {L} }}>0\\-\pi -2\,\arctan {\frac {X_{\mathrm {L} }}{R_{\mathrm {0} }}}&{\mbox{if }}{X_{\mathrm {L} }}<0\\\end{cases}}}
5445:
the first point is an anti-node and thereafter they will alternate. For terminations that are not purely resistive the spacing and alternation remain the same but the whole pattern is shifted along the line by a constant amount related to the phase of
5356:
3394:
4876:
1011:
appears at the generator output terminals as in figure 3. The same situation pertains if a very short transmission line is inserted between the generator and the open circuit. If, however, a longer line with a characteristic impedance of
779:. None of this energy is dissipated in the line or its termination and it must go somewhere. The only available direction is back up the line. Since the reflected current is equal in magnitude to the incident current, it must also be so that
2873:
4016:
870:
the reflected step arrives at the generator end and the condition of double voltage and zero current will pertain there also as well as all along the length of the line. If the generator is matched to the line with an impedance of
2118:
547:
1789:
3508:
3237:
4551:
3588:
4027:
6576:
5633:
VSWR is the conventional means of expressing the match of a radio transmitter to its antenna. It is an important parameter because power reflected back into a high power transmitter can damage its output circuitry.
5217:
3911:
3607:
as they are known persist for far longer than the original pulse and their waveforms bears little obvious resemblance to the original disturbance, containing high frequency components in the tens of MHz range.
6203:
1482:
704:
6012:
1206:
As the reflection travels back up the line, the two voltages subtract and cancel, while the currents will add (the reflection is double negative - a negative current traveling in the reverse direction), the
2377:
6428:
4967:
3772:
1970:{\displaystyle {\mathit {\Gamma }}={\frac {V_{\mathrm {r} }}{V_{\mathrm {i} }}}={\frac {I_{\mathrm {r} }}{I_{\mathrm {i} }}}={\frac {Z_{\mathrm {L} }-Z_{\mathrm {0} }}{Z_{\mathrm {L} }+Z_{\mathrm {0} }}}}
6004:
4556:
Because of the absolute value bars, the general case analytical solution is tiresomely complicated, but in the case of lossless lines (or lines that are short enough that the losses can be neglected)
2419:
4782:
4479:
4324:
3632:
of the two waves will add. There will be other points where the two waves are in anti-phase and will consequently subtract. At these latter points the amplitude is at a minimum and they are known as
2046:
399:
1201:
1146:
2291:
826:
777:
2674:
1059:
at the output. But after an interval, a reflected transient will return from the end of the line with the "information" that the line is actually unterminated, and the voltage will become
5527:
5619:
for a lossless line; the expression for the current standing wave ratio (ISWR) is identical in this case. For a lossy line the expression is only valid adjacent to the termination; VSWR
5669:
and will be a function of the length of the line. The value of this impedance can be found by dividing the expression for total voltage by the expression for total current given above:
2242:
2201:
1239:
magnitude of the incident wave if the terminating impedance is wholly or partly resistive since some of the energy of the incident wave will be absorbed in the resistance. The voltage (
5443:
5401:
2920:
1303:
2159:
946:
can be represented as in figure 2. That is, the generator can be represented as an ideal voltage generator of twice the voltage it is to deliver and an internal impedance of
5468:
5091:
2662:
2528:
2504:
2002:
1727:
6503:
5248:
5515:
5147:
3844:
3426:
3289:
2629:
2480:
1821:
1423:
1295:
1266:
738:, to drive the reflected current down the line. This reflected voltage must exist by reason of conservation of energy. The source is supplying energy to the line at a rate of
736:
643:
431:
316:
3812:
2333:
868:
610:
3297:
6361:
which is a short length of line terminated in a short circuit (or it can be an open circuit). This produces a purely imaginary impedance at its input, that is, a reactance
265:), applied to one end of a lossless line, and consider what happens when the line is terminated in various ways. The step will be propagated down the line according to the
4793:
6466:
6195:
4598:
4959:
4919:
6171:
5948:
4574:
3709:
3664:
2947:
287:
190:
4899:
4618:
5667:
5118:
2600:
2573:
2451:
1037:
971:
924:
896:
576:
458:
1081:
1009:
239:
3623:
on a transmission line with an open-circuit load (top), and a short-circuit load (bottom). Black dots represent electrons, and the arrows show the electric field.
89:
in power lines. Although this article is limited to describing reflections on conducting lines, this is essentially the same phenomenon as optical reflections in
2777:
5240:
4939:
3934:
3684:
3260:
1057:
944:
336:
263:
210:
39:
3942:
5642:
The input impedance looking into a transmission line which is not terminated with its characteristic impedance at the far end will be something other than
2051:
465:
1735:
3434:
139:
to be set up on the line. Conversely, standing waves are an indication that reflections are present. There is a relationship between the measures of
3158:
4490:
3936:
is measured in the reverse direction back up the line and the voltage is increasing closer to the source. Likewise the reflected voltage is given by
2123:
The physical interpretation of this is that the reflection cannot be greater than the incident wave when only passive elements are involved (but see
1092:
circuit. Again, all of the energy must be reflected back up the line and the reflected voltage must be equal and opposite to the incident voltage by
4153:{\displaystyle V_{\mathsf {T}}=V_{\mathsf {i}}+V_{\mathsf {r}}=V_{\mathsf {iL}}\,\left(e^{\gamma \,x}+{\mathit {\Gamma }}\,e^{-\gamma \,x}\right)~.}
3515:
6944:
6511:
5155:
6949:
3399:
Another kind of discontinuity is caused when both sections of line have an identical characteristic impedance but there is a lumped element,
6342:{\displaystyle Z_{\mathrm {in} }=Z_{0}{\frac {Z_{\mathrm {L} }+j\,Z_{0}\tan(\beta \,\ell )}{Z_{0}+j\,Z_{\mathrm {L} }\tan(\beta \,\ell )}}}
3852:
6143:{\displaystyle Z_{\mathrm {in} }=Z_{0}{\frac {Z_{\mathrm {L} }+Z_{0}\tanh(\gamma \,\ell )}{Z_{0}+Z_{\mathrm {L} }\tanh(\gamma \,\ell )}}}
46:; an instrument used to locate the position of faults on lines from the time taken for a reflected wave to return from the discontinuity.
1428:
651:
5060:{\displaystyle -2\,\operatorname {\mathcal {I_{m}}} \{\,{\mathit {\Gamma }}\,\}=\tan \left({\frac {\,4\,\pi \,}{\lambda }}\,x\right)~.}
6437:
129:
2339:
6918:
6903:
6881:
6866:
6851:
6836:
6367:
2925:
showing that all the incident wave is reflected, and none of it is absorbed in the termination, as is to be expected from a pure
5953:
2382:
4630:
4335:
4172:
3717:
2010:
1039:
and noticeable end-to-end delay is inserted, the generator – being initially matched to the impedance of the line – will have
344:
1157:
1102:
70:
in its characteristic impedance. This can happen, for instance, if two lengths of dissimilar transmission lines are joined.
2250:
785:
1208:
2761:{\displaystyle {\mathit {\Gamma }}={\frac {j\,X_{\mathrm {L} }-R_{\mathrm {0} }}{j\,X_{\mathrm {L} }+R_{\mathrm {0} }}}}
6954:
5518:
1297:), may be calculated by replacing the output of the line with an equivalent generator (figure 4) and is given by
741:
59:
6357:
There are two structures that are of particular importance which use reflected waves to modify impedance. One is the
5609:{\displaystyle {\mathsf {VSWR}}={\frac {1+\left|{\mathit {\Gamma }}\,\right|}{1-\left|{\mathit {\Gamma }}\,\right|}}}
1093:
613:
6934:
6582:
156:
6618:
1386:{\displaystyle V_{\mathrm {o} }=2\,V_{\mathrm {i} }{\frac {Z_{\mathrm {L} }}{Z_{\mathrm {0} }+Z_{\mathrm {L} }}}}
266:
43:
2212:
2171:
6433:
By suitable choice of length, the stub can be used in place of a capacitor, an inductor or a resonant circuit.
5406:
5364:
2884:
63:
3616:
1219:
For the general case of a line terminated in some arbitrary impedance it is usual to describe the signal as a
5626:
VSWR and the positions of the nodes are parameters that can be directly measured with an instrument called a
2138:
2538:
85:
lines, and indeed, in telecommunications are generally made from copper, but other metals are used, notably
38:
5351:{\displaystyle x=0,~~{\tfrac {1}{4}}\lambda ,~~{\tfrac {1}{2}}\lambda ,~~{\tfrac {3}{4}}\lambda ,~\dots ~.}
6939:
6598:
140:
67:
6720:
5449:
5072:
2634:
2509:
2485:
1983:
1708:
6471:
4621:
78:
5481:
3389:{\displaystyle T={\frac {V_{\mathrm {t} }}{V_{\mathrm {i} }}}={\frac {2\,Z_{02}}{\,Z_{02}+Z_{01}\,\,}}}
5123:
3817:
3594:
Similar expressions can be developed for a series element, or any electrical network for that matter.
3402:
3265:
2605:
2456:
1797:
1399:
1271:
1242:
976:
712:
619:
407:
292:
155:
There are several approaches to understanding reflections, but the relationship of reflections to the
4871:{\displaystyle -2\,\operatorname {\mathcal {I_{m}}} \{\,{\mathit {\Gamma }}\,\}=\tan(2\,\beta \,x)~.}
3644:
2926:
2665:
2296:
839:
581:
55:
2970:
4164:
2124:
144:
132:. The special cases of open circuit and short circuit lines are of particular relevance to stubs.
17:
6608:
6586:
6358:
242:
125:
105:
6443:
3780:
6176:
4579:
6914:
6899:
6877:
6862:
6847:
6832:
4944:
4904:
74:
51:
33:
6156:
5927:
4559:
3649:
2932:
272:
162:
4884:
4603:
1232:
1224:
5645:
5096:
3603:
oscillatory disturbance as the pulse is reflected to and from multiple circuit ends. These
2868:{\displaystyle |jX_{\mathrm {L} }-R_{\mathrm {0} }|=|jX_{\mathrm {L} }+R_{\mathrm {0} }|\,}
2578:
2551:
2429:
2127:
for an example where this condition does not hold). For the special cases described above,
1015:
949:
902:
874:
561:
436:
4484:
The voltage nodes (current nodes are not at the same locations) and anti-nodes occur when
1062:
990:
215:
3428:, at the discontinuity. For the example shown (figure 6) of a shunt lumped element,
4011:{\displaystyle V_{\mathsf {r}}={\mathit {\Gamma }}\,V_{\mathsf {iL}}\,e^{-\gamma \,x}~.}
3689:
5225:
4924:
3919:
3669:
3633:
3245:
1042:
929:
321:
248:
195:
2004:
is a complex function but the above expression shows that the magnitude is limited to
6928:
6603:
3637:
3620:
2531:
2113:{\displaystyle \operatorname {Re} (Z_{\mathrm {L} }),\operatorname {Re} (Z_{0})>0}
542:{\displaystyle i_{\mathrm {i} }={\frac {v_{\mathrm {i} }}{Z_{0}}}=I\,u(\kappa \,t-x)}
136:
1784:{\displaystyle {\mathit {\Gamma }}={\frac {\,V_{\mathrm {r} }\,}{V_{\mathrm {i} }}}}
6613:
5627:
5093:
is purely real when the termination is short circuit or open circuit, or when both
3503:{\displaystyle {\mathit {\Gamma }}={\frac {-Z_{0}}{\,Z_{0}+2\,Z_{\mathrm {L} }\,}}}
3777:
However, it is often more convenient to work in terms of distance from the load (
6721:
IEEE Standard 587 Applicability to
Adjustable Frequency Control (Surge Voltages)
6628:
6623:
3232:{\displaystyle {\mathit {\Gamma }}={\frac {\,Z_{02}-Z_{01}\,\,}{Z_{02}+Z_{01}}}}
117:
113:
109:
90:
4546:{\displaystyle {\frac {\partial \left|V_{\mathsf {T}}\right|}{\partial x}}=0~.}
3583:{\displaystyle T={\frac {2\,Z_{\mathrm {L} }}{\,Z_{0}+2\,Z_{\mathrm {L} }\,}}}
121:
5620:
4787:
and the partial differential of the magnitude of this yields the condition,
3629:
1228:
98:
94:
86:
5149:
are purely resistive. In those cases the nodes and anti-nodes are given by
6571:{\displaystyle Z_{\mathrm {in} }={\frac {{Z_{0}}^{2}}{Z_{\mathrm {L} }}}}
5212:{\displaystyle \tan \left(\,{\frac {4\,\pi \,}{\lambda }}\,x\right)=0~,}
616:
in operation. This equal and opposite current is the reflected current,
6889:
Microwave
Filters, Impedance-Matching Networks, and Coupling Structures
58:
back in the opposite direction when the travelling signal encounters a
5623:
approaches unity with distance from the termination or discontinuity.
124:. However, the reflection phenomenon can be useful in such devices as
3906:{\displaystyle V_{\mathrm {i} }=V_{\mathsf {iL}}\,e^{\gamma \,x}\,\!}
82:
6153:
As before, when considering just short pieces of transmission line,
104:
Reflections cause several undesirable effects, including modifying
3615:
1477:{\displaystyle V_{\mathrm {i} }+V_{\mathrm {r} }=V_{\mathrm {o} }}
975:
699:{\displaystyle i_{\mathrm {r} }={\frac {v_{\mathrm {r} }}{Z_{0}}}}
159:
is particularly enlightening. A simple example is a step voltage,
37:
1220:
6874:
Microwave circuit theory and foundations of microwave metrology
4624:. The voltage equation then reduces to trigonometric functions
2372:{\displaystyle \operatorname {Re} ({\mathit {\Gamma }})<1\,}
558:
characteristics of the line. However, if the line is of length
6423:{\displaystyle X_{\mathrm {in} }=Z_{0}\tan(\beta \,\ell )\,\!}
3643:
The incident wave can be characterised in terms of the line's
987:
However, if the generator is left open circuit, a voltage of
5968:
5879:
5836:
5794:
5752:
5592:
5566:
5455:
5078:
5001:
4985:
4981:
4827:
4811:
4807:
4731:
4682:
4430:
4387:
4273:
4226:
4116:
3963:
3440:
3164:
2895:
2680:
2515:
2491:
2397:
2354:
2218:
2177:
2144:
2021:
1989:
1835:
1741:
1714:
433:, can be found by dividing by the characteristic impedance,
6505:
this will produce the inverse of its terminating impedance
3129:
5999:{\displaystyle (1+{\mathit {\Gamma }}\,)\cosh(\gamma \,x)}
2414:{\displaystyle \operatorname {Im} ({\mathit {\Gamma }})=0}
4777:{\displaystyle V_{\mathsf {T}}=V_{\mathsf {iL}}\,\left~,}
4474:{\displaystyle I_{\mathsf {T}}=I_{\mathsf {iL}}\,\left~.}
4319:{\displaystyle V_{\mathsf {T}}=V_{\mathsf {iL}}\,\left~.}
3767:{\displaystyle V_{\mathrm {i} }=V\,e^{-\gamma \,x'}\,\!}
2041:{\displaystyle \left|{\mathit {\Gamma }}\,\right|\leq 1}
5521:(VSWR) and is related to the reflection coefficient by
394:{\displaystyle v_{\mathrm {i} }=V\,u(\kappa \,t-x)\,\!}
6197:
and the expression reduces to trigonometric functions
5319:
5295:
5271:
3262:, can be defined to describe the portion of the wave,
3100:
3020:
1196:{\displaystyle i_{\mathrm {r} }=-i_{\mathrm {i} }\,\!}
1141:{\displaystyle v_{\mathrm {r} }=-v_{\mathrm {i} }\,\!}
6514:
6474:
6446:
6370:
6206:
6179:
6159:
6015:
5956:
5930:
5678:
5648:
5530:
5484:
5452:
5409:
5367:
5251:
5228:
5158:
5126:
5099:
5075:
4970:
4947:
4927:
4907:
4887:
4796:
4633:
4606:
4582:
4562:
4493:
4338:
4175:
4030:
3945:
3922:
3855:
3820:
3783:
3720:
3692:
3672:
3652:
3518:
3437:
3405:
3300:
3268:
3248:
3161:
2958:
2935:
2887:
2780:
2677:
2637:
2608:
2581:
2554:
2534:
of the reflected wave relative to the incident wave.
2512:
2488:
2459:
2432:
2385:
2342:
2299:
2286:{\displaystyle Z_{\mathrm {L} }=R_{\mathrm {L} }\,\!}
2253:
2215:
2174:
2141:
2054:
2013:
1986:
1832:
1800:
1738:
1711:
1493:
1431:
1402:
1306:
1274:
1245:
1160:
1105:
1065:
1045:
1018:
993:
952:
932:
905:
877:
842:
821:{\displaystyle v_{\mathrm {r} }=v_{\mathrm {i} }\,\!}
788:
744:
715:
654:
622:
584:
564:
468:
439:
410:
347:
324:
295:
275:
251:
218:
198:
165:
2530:
is complex, this is to be interpreted as a shift in
899:
equivalent circuit of a generator matched to a load
6911:
Protection of
Electronic Circuits from Overvoltages
6827:Bowick, Christopher; Ajluni, Cheryl; Blyler, John,
4163:It is often convenient to express this in terms of
3814:) and the incident voltage that has arrived there (
3291:, that it is transmitted in the forward direction:
6570:
6497:
6460:
6422:
6341:
6189:
6165:
6142:
5998:
5942:
5913:
5661:
5608:
5509:
5462:
5437:
5395:
5350:
5234:
5211:
5141:
5112:
5085:
5059:
4953:
4933:
4913:
4893:
4870:
4776:
4612:
4592:
4568:
4545:
4473:
4318:
4152:
4010:
3928:
3905:
3838:
3806:
3766:
3703:
3678:
3658:
3582:
3502:
3420:
3388:
3283:
3254:
3231:
3135:
2941:
2914:
2867:
2760:
2656:
2623:
2594:
2567:
2522:
2498:
2474:
2445:
2413:
2371:
2327:
2285:
2236:
2195:
2153:
2112:
2040:
1996:
1969:
1815:
1783:
1721:
1694:
1476:
1417:
1385:
1289:
1260:
1195:
1140:
1075:
1051:
1031:
1003:
965:
938:
918:
890:
862:
820:
771:
730:
698:
637:
604:
570:
541:
452:
425:
393:
330:
310:
281:
257:
233:
204:
184:
66:of the line, or if the far end of the line is not
6419:
5950:, the length of the line and dividing through by
3902:
3763:
3242:In a similar manner, a transmission coefficient,
2324:
2282:
2233:
2192:
2150:
1227:. The impedance is consequently represented as a
1192:
1137:
817:
578:the step will arrive at the open circuit at time
390:
772:{\displaystyle v_{\mathrm {i} }i_{\mathrm {i} }}
6440:. As its name suggests, this is a line exactly
2506:must be purely real. In the general case when
1223:traveling down the line and analyse it in the
836:incident current. After a further interval of
81:lines. Such lines are loosely referred to as
8:
6581:Both of these structures are widely used in
5007:
4995:
4833:
4821:
4329:Similarly, the total current on the line is
6896:Introduction to high power pulse technology
6887:Matthaei, G.; Young, L.; Jones, E. M. T.,
4021:The total voltage on the line is given by
2237:{\displaystyle {\mathit {\Gamma }}=-1\,\!}
2196:{\displaystyle {\mathit {\Gamma }}=+1\,\!}
1425:must be the exact amount required to make
6559:
6558:
6548:
6541:
6536:
6533:
6520:
6519:
6513:
6487:
6473:
6450:
6445:
6418:
6411:
6393:
6376:
6375:
6369:
6329:
6310:
6309:
6304:
6292:
6279:
6261:
6256:
6243:
6242:
6235:
6229:
6212:
6211:
6205:
6183:
6178:
6158:
6130:
6111:
6110:
6097:
6084:
6066:
6052:
6051:
6044:
6038:
6021:
6020:
6014:
5989:
5973:
5967:
5966:
5955:
5929:
5901:
5888:
5884:
5878:
5877:
5858:
5845:
5841:
5835:
5834:
5816:
5803:
5799:
5793:
5792:
5773:
5757:
5751:
5750:
5738:
5732:
5716:
5715:
5704:
5703:
5697:
5684:
5683:
5677:
5653:
5647:
5597:
5591:
5590:
5571:
5565:
5564:
5550:
5532:
5531:
5529:
5502:
5495:
5494:
5485:
5483:
5454:
5453:
5451:
5438:{\displaystyle R_{\mathrm {L} }>R_{0}}
5429:
5415:
5414:
5408:
5396:{\displaystyle R_{\mathrm {L} }<R_{0}}
5387:
5373:
5372:
5366:
5318:
5294:
5270:
5250:
5227:
5188:
5181:
5177:
5171:
5170:
5157:
5132:
5131:
5125:
5104:
5098:
5077:
5076:
5074:
5042:
5035:
5031:
5027:
5024:
5006:
5000:
4999:
4998:
4984:
4979:
4978:
4977:
4969:
4946:
4926:
4906:
4886:
4855:
4851:
4832:
4826:
4825:
4824:
4810:
4805:
4804:
4803:
4795:
4762:
4755:
4742:
4736:
4730:
4729:
4717:
4704:
4691:
4687:
4681:
4680:
4670:
4664:
4654:
4653:
4639:
4638:
4632:
4605:
4586:
4581:
4561:
4509:
4508:
4494:
4492:
4459:
4452:
4439:
4435:
4429:
4428:
4409:
4396:
4392:
4386:
4385:
4375:
4369:
4359:
4358:
4344:
4343:
4337:
4304:
4297:
4284:
4278:
4272:
4271:
4250:
4237:
4231:
4225:
4224:
4212:
4206:
4196:
4195:
4181:
4180:
4174:
4133:
4126:
4121:
4115:
4114:
4105:
4101:
4091:
4081:
4080:
4066:
4065:
4051:
4050:
4036:
4035:
4029:
3996:
3989:
3984:
3974:
3973:
3968:
3962:
3961:
3951:
3950:
3944:
3921:
3901:
3895:
3891:
3886:
3876:
3875:
3861:
3860:
3854:
3826:
3825:
3819:
3782:
3762:
3751:
3744:
3739:
3726:
3725:
3719:
3691:
3671:
3651:
3576:
3569:
3568:
3563:
3551:
3546:
3537:
3536:
3531:
3525:
3517:
3496:
3489:
3488:
3483:
3471:
3466:
3458:
3448:
3439:
3438:
3436:
3411:
3410:
3404:
3382:
3381:
3375:
3362:
3357:
3349:
3344:
3338:
3326:
3325:
3314:
3313:
3307:
3299:
3274:
3273:
3267:
3247:
3220:
3207:
3200:
3199:
3193:
3180:
3175:
3172:
3163:
3162:
3160:
3112:
3111:
3106:
3099:
3088:
3087:
3076:
3075:
3069:
3062:
3032:
3031:
3026:
3019:
3008:
3007:
2996:
2995:
2989:
2982:
2965:
2957:
2934:
2915:{\displaystyle |{\mathit {\Gamma }}|=1\,}
2911:
2900:
2894:
2893:
2888:
2886:
2864:
2859:
2852:
2851:
2837:
2836:
2824:
2816:
2809:
2808:
2794:
2793:
2781:
2779:
2748:
2747:
2733:
2732:
2727:
2715:
2714:
2700:
2699:
2694:
2688:
2679:
2678:
2676:
2647:
2646:
2641:
2636:
2614:
2613:
2607:
2586:
2580:
2559:
2553:
2514:
2513:
2511:
2490:
2489:
2487:
2465:
2464:
2458:
2437:
2431:
2396:
2395:
2384:
2368:
2353:
2352:
2341:
2323:
2317:
2304:
2298:
2281:
2274:
2273:
2259:
2258:
2252:
2232:
2217:
2216:
2214:
2191:
2176:
2175:
2173:
2149:
2143:
2142:
2140:
2095:
2069:
2068:
2053:
2026:
2020:
2019:
2012:
1988:
1987:
1985:
1957:
1956:
1942:
1941:
1928:
1927:
1913:
1912:
1905:
1893:
1892:
1881:
1880:
1874:
1862:
1861:
1850:
1849:
1843:
1834:
1833:
1831:
1806:
1805:
1799:
1772:
1771:
1765:
1758:
1757:
1752:
1749:
1740:
1739:
1737:
1713:
1712:
1710:
1682:
1681:
1667:
1666:
1653:
1652:
1638:
1637:
1630:
1623:
1622:
1608:
1607:
1590:
1589:
1575:
1574:
1562:
1561:
1555:
1548:
1547:
1542:
1529:
1528:
1514:
1513:
1499:
1498:
1492:
1467:
1466:
1452:
1451:
1437:
1436:
1430:
1408:
1407:
1401:
1373:
1372:
1358:
1357:
1345:
1344:
1338:
1331:
1330:
1325:
1312:
1311:
1305:
1280:
1279:
1273:
1251:
1250:
1244:
1191:
1184:
1183:
1166:
1165:
1159:
1136:
1129:
1128:
1111:
1110:
1104:
1069:
1064:
1044:
1023:
1017:
997:
992:
957:
951:
931:
910:
904:
882:
876:
852:
841:
816:
809:
808:
794:
793:
787:
762:
761:
750:
749:
743:
721:
720:
714:
688:
677:
676:
670:
660:
659:
653:
628:
627:
621:
594:
583:
563:
526:
516:
502:
491:
490:
484:
474:
473:
467:
444:
438:
416:
415:
409:
389:
376:
366:
353:
352:
346:
323:
301:
300:
294:
274:
250:
217:
197:
169:
164:
2929:. There is, however, a change of phase,
2129:
709:there must also be a reflected voltage,
50:A signal travelling along an electrical
6639:
2154:{\displaystyle {\mathit {\Gamma }}\,\!}
1794:and substituting in the expression for
5542:
5539:
5536:
5533:
5517:at anti-nodes and nodes is called the
5496:
4658:
4655:
4640:
4510:
4363:
4360:
4345:
4200:
4197:
4182:
4085:
4082:
4067:
4052:
4037:
3978:
3975:
3952:
3880:
3877:
3830:
3827:
6658:
6656:
6654:
6652:
7:
3916:The negative sign is absent because
1268:) across the terminating impedance (
1211:situation to the open circuit case.
926:to which it is delivering a voltage
27:Electrical waves in return direction
6913:, Courier Dover Publications, 2002
5463:{\displaystyle {\mathit {\Gamma }}}
5086:{\displaystyle {\mathit {\Gamma }}}
2657:{\displaystyle j\,X_{\mathrm {L} }}
2523:{\displaystyle {\mathit {\Gamma }}}
2499:{\displaystyle {\mathit {\Gamma }}}
1997:{\displaystyle {\mathit {\Gamma }}}
1722:{\displaystyle {\mathit {\Gamma }}}
6560:
6524:
6521:
6498:{\displaystyle \beta \ell =\pi /2}
6438:quarter wave impedance transformer
6380:
6377:
6311:
6244:
6216:
6213:
6112:
6053:
6025:
6022:
5717:
5705:
5688:
5685:
5416:
5374:
5133:
4522:
4497:
3862:
3727:
3570:
3538:
3490:
3412:
3327:
3315:
3275:
3113:
3077:
3033:
2997:
2838:
2795:
2734:
2701:
2648:
2615:
2466:
2275:
2260:
2070:
1943:
1914:
1894:
1882:
1863:
1851:
1807:
1773:
1759:
1668:
1639:
1624:
1609:
1591:
1563:
1549:
1530:
1515:
1500:
1468:
1453:
1438:
1409:
1374:
1346:
1332:
1313:
1281:
1252:
1185:
1167:
1130:
1112:
810:
795:
763:
751:
722:
678:
661:
629:
492:
475:
417:
354:
302:
25:
18:Discontinuity (transmission lines)
6846:, McGraw-Hill Professional, 2001
6798:Connor, pp. 32–35, Matthaei
5510:{\displaystyle |V_{\mathsf {T}}|}
2548:Another special case occurs when
5142:{\displaystyle Z_{\mathrm {L} }}
3839:{\displaystyle V_{\mathsf {iL}}}
3421:{\displaystyle Z_{\mathrm {L} }}
3284:{\displaystyle V_{\mathrm {t} }}
2624:{\displaystyle Z_{\mathrm {L} }}
2475:{\displaystyle Z_{\mathrm {L} }}
1816:{\displaystyle V_{\mathrm {r} }}
1418:{\displaystyle V_{\mathrm {r} }}
1290:{\displaystyle Z_{\mathrm {L} }}
1261:{\displaystyle V_{\mathrm {o} }}
731:{\displaystyle v_{\mathrm {r} }}
638:{\displaystyle i_{\mathrm {r} }}
426:{\displaystyle i_{\mathrm {i} }}
311:{\displaystyle v_{\mathrm {i} }}
5403:the first point is a node, for
3686:, and distance from the source
3628:with the incident wave and the
2328:{\displaystyle Z_{0}=R_{0}\,\!}
863:{\displaystyle t=\ell /\kappa }
605:{\displaystyle t=\ell /\kappa }
6945:Telecommunications engineering
6415:
6405:
6333:
6323:
6283:
6273:
6134:
6124:
6088:
6078:
5993:
5983:
5974:
5957:
5905:
5895:
5885:
5868:
5862:
5852:
5842:
5825:
5820:
5810:
5800:
5783:
5777:
5767:
5758:
5741:
5503:
5486:
4859:
4845:
4759:
4749:
4708:
4698:
4688:
4671:
4456:
4446:
4436:
4419:
4413:
4403:
4393:
4376:
4301:
4291:
4254:
4244:
2901:
2889:
2860:
2825:
2817:
2782:
2402:
2392:
2359:
2349:
2101:
2088:
2076:
2061:
536:
520:
386:
370:
228:
222:
212:is the height of the step and
179:
173:
1:
2949:, in the reflection given by
2125:negative resistance amplifier
1705:The reflection coefficient,
6950:Distributed element circuits
6662:Pai & Zhang, pages 89–96
6861:, Edward Arnold Ltd., 1972
6719:Term originally defined in
6583:distributed element filters
6436:The other structure is the
5519:voltage standing wave ratio
5474:Voltage standing wave ratio
54:will be partly, or wholly,
6971:
6844:Practical antenna handbook
6461:{\displaystyle \lambda /4}
3807:{\displaystyle x=\ell -x'}
2536:
2482:are purely resistive then
289:and the incident voltage,
31:
6898:, World Scientific, 1995
6619:Time-domain reflectometry
6190:{\displaystyle j\,\beta }
4941:to be solved in terms of
4593:{\displaystyle j\,\beta }
44:time-domain reflectometer
4954:{\displaystyle \lambda }
4914:{\displaystyle \lambda }
4901:in terms of wavelength,
3147:Discontinuity along line
831:These two voltages will
338:on the line is given by
64:characteristic impedance
6894:Pai, S. T.; Zhang, Qi,
6166:{\displaystyle \gamma }
5943:{\displaystyle x=\ell }
4569:{\displaystyle \gamma }
3659:{\displaystyle \gamma }
2942:{\displaystyle \theta }
2539:Reflection phase change
1094:Kirchhoff's voltage law
614:Kirchhoff's current law
282:{\displaystyle \kappa }
185:{\displaystyle V\,u(t)}
79:electrically conducting
6599:Attenuation distortion
6572:
6499:
6462:
6424:
6343:
6191:
6167:
6144:
6000:
5944:
5915:
5663:
5610:
5511:
5464:
5439:
5397:
5352:
5236:
5213:
5143:
5114:
5087:
5061:
4955:
4935:
4915:
4895:
4894:{\displaystyle \beta }
4872:
4778:
4614:
4613:{\displaystyle \beta }
4594:
4570:
4547:
4475:
4320:
4154:
4012:
3930:
3907:
3840:
3808:
3768:
3705:
3680:
3660:
3624:
3584:
3504:
3422:
3390:
3285:
3256:
3233:
3137:
2943:
2916:
2869:
2762:
2658:
2625:
2596:
2569:
2524:
2500:
2476:
2447:
2415:
2373:
2329:
2287:
2238:
2197:
2155:
2114:
2042:
1998:
1971:
1817:
1785:
1723:
1696:
1478:
1419:
1387:
1291:
1262:
1197:
1142:
1077:
1053:
1033:
1005:
984:
983:Open circuit generator
967:
940:
920:
892:
864:
822:
773:
732:
700:
639:
606:
572:
543:
454:
427:
404:The incident current,
395:
332:
312:
283:
267:telegrapher's equation
259:
235:
206:
186:
141:reflection coefficient
130:impedance transformers
73:This article is about
47:
6909:Standler, Ronald B.,
6731:Standler, pages 74–76
6573:
6500:
6463:
6425:
6344:
6192:
6168:
6145:
6001:
5945:
5916:
5664:
5662:{\displaystyle Z_{0}}
5611:
5512:
5465:
5440:
5398:
5353:
5237:
5214:
5144:
5115:
5113:{\displaystyle Z_{0}}
5088:
5062:
4956:
4936:
4916:
4896:
4873:
4779:
4622:phase change constant
4615:
4595:
4571:
4548:
4476:
4321:
4155:
4013:
3931:
3908:
3841:
3809:
3769:
3706:
3681:
3661:
3619:
3585:
3505:
3423:
3391:
3286:
3257:
3234:
3138:
2944:
2917:
2870:
2763:
2659:
2631:is purely imaginary (
2626:
2597:
2595:{\displaystyle R_{0}}
2570:
2568:{\displaystyle Z_{0}}
2525:
2501:
2477:
2448:
2446:{\displaystyle Z_{0}}
2416:
2374:
2330:
2288:
2239:
2198:
2156:
2115:
2043:
1999:
1972:
1818:
1786:
1724:
1697:
1479:
1420:
1388:
1292:
1263:
1198:
1143:
1078:
1054:
1034:
1032:{\displaystyle Z_{0}}
1006:
979:
968:
966:{\displaystyle Z_{0}}
941:
921:
919:{\displaystyle Z_{0}}
893:
891:{\displaystyle Z_{0}}
865:
823:
774:
733:
701:
640:
607:
573:
571:{\displaystyle \ell }
544:
455:
453:{\displaystyle Z_{0}}
428:
396:
333:
313:
284:
260:
236:
207:
187:
41:
6512:
6472:
6444:
6368:
6204:
6177:
6157:
6013:
5954:
5928:
5676:
5646:
5528:
5482:
5450:
5407:
5365:
5249:
5226:
5156:
5124:
5097:
5073:
4968:
4945:
4925:
4905:
4885:
4794:
4631:
4604:
4580:
4560:
4491:
4336:
4173:
4165:hyperbolic functions
4028:
3943:
3920:
3853:
3818:
3781:
3718:
3690:
3670:
3650:
3645:propagation constant
3516:
3435:
3403:
3298:
3266:
3246:
3159:
2956:
2933:
2885:
2778:
2675:
2664:), that is, it is a
2635:
2606:
2579:
2552:
2544:Reactive termination
2510:
2486:
2457:
2430:
2383:
2340:
2297:
2251:
2213:
2172:
2139:
2052:
2011:
1984:
1830:
1798:
1736:
1709:
1491:
1429:
1400:
1304:
1272:
1243:
1158:
1103:
1076:{\displaystyle 2\,V}
1063:
1043:
1016:
1004:{\displaystyle 2\,V}
991:
950:
930:
903:
875:
840:
786:
742:
713:
652:
620:
582:
562:
466:
437:
408:
345:
322:
293:
273:
249:
234:{\displaystyle u(t)}
216:
196:
163:
6789:Connor, pages 13–14
6758:Connor, pages 31–32
6740:Connor, pages 28–31
6697:Connor, pages 30–31
6173:can be replaced by
4576:can be replaced by
1215:Arbitrary impedance
145:standing wave ratio
106:frequency responses
6955:Transmission lines
6767:Engen, pages 73–76
6609:Fresnel reflection
6587:impedance matching
6568:
6495:
6458:
6420:
6339:
6187:
6163:
6140:
5996:
5940:
5911:
5659:
5606:
5507:
5460:
5435:
5393:
5348:
5328:
5304:
5280:
5232:
5209:
5139:
5110:
5083:
5057:
4951:
4931:
4911:
4891:
4868:
4774:
4610:
4590:
4566:
4543:
4471:
4316:
4150:
4008:
3926:
3903:
3836:
3804:
3764:
3704:{\displaystyle x'}
3701:
3676:
3656:
3625:
3580:
3500:
3418:
3386:
3281:
3252:
3229:
3133:
3128:
3104:
3024:
2939:
2912:
2865:
2758:
2654:
2621:
2592:
2565:
2520:
2496:
2472:
2443:
2411:
2369:
2325:
2283:
2234:
2193:
2151:
2110:
2038:
1994:
1967:
1813:
1781:
1719:
1692:
1474:
1415:
1383:
1287:
1258:
1193:
1138:
1087:Short circuit line
1073:
1049:
1029:
1001:
985:
963:
936:
916:
888:
860:
818:
769:
728:
696:
635:
602:
568:
539:
450:
423:
391:
328:
308:
279:
255:
243:unit step function
231:
202:
182:
135:Reflections cause
75:signal reflections
48:
6935:Electronic design
6891:McGraw-Hill 1964.
6872:Engen, Glenn F.,
6859:Wave Transmission
6842:Carr, Joseph J.,
6829:RF Circuit Design
6646:Carr, pages 70–71
6566:
6468:in length. Since
6337:
6138:
5909:
5723:
5604:
5344:
5338:
5327:
5317:
5314:
5303:
5293:
5290:
5279:
5269:
5266:
5235:{\displaystyle x}
5222:which solves for
5205:
5186:
5053:
5040:
4934:{\displaystyle x}
4864:
4770:
4539:
4529:
4467:
4312:
4146:
4004:
3929:{\displaystyle x}
3679:{\displaystyle V}
3666:, source voltage
3578:
3498:
3384:
3333:
3255:{\displaystyle T}
3227:
3103:
3095:
3023:
3015:
2756:
2424:
2423:
1965:
1900:
1869:
1779:
1690:
1598:
1381:
1052:{\displaystyle V}
939:{\displaystyle V}
694:
553:Open circuit line
508:
331:{\displaystyle x}
269:at some velocity
258:{\displaystyle t}
205:{\displaystyle V}
157:conservation laws
52:transmission line
34:signal reflection
16:(Redirected from
6962:
6816:
6809:
6803:
6796:
6790:
6787:
6781:
6774:
6768:
6765:
6759:
6756:
6750:
6747:
6741:
6738:
6732:
6729:
6723:
6717:
6711:
6704:
6698:
6695:
6689:
6682:
6676:
6669:
6663:
6660:
6647:
6644:
6577:
6575:
6574:
6569:
6567:
6565:
6564:
6563:
6553:
6552:
6547:
6546:
6545:
6534:
6529:
6528:
6527:
6504:
6502:
6501:
6496:
6491:
6467:
6465:
6464:
6459:
6454:
6429:
6427:
6426:
6421:
6398:
6397:
6385:
6384:
6383:
6348:
6346:
6345:
6340:
6338:
6336:
6316:
6315:
6314:
6297:
6296:
6286:
6266:
6265:
6249:
6248:
6247:
6236:
6234:
6233:
6221:
6220:
6219:
6196:
6194:
6193:
6188:
6172:
6170:
6169:
6164:
6149:
6147:
6146:
6141:
6139:
6137:
6117:
6116:
6115:
6102:
6101:
6091:
6071:
6070:
6058:
6057:
6056:
6045:
6043:
6042:
6030:
6029:
6028:
6006:reduces this to
6005:
6003:
6002:
5997:
5972:
5971:
5949:
5947:
5946:
5941:
5920:
5918:
5917:
5912:
5910:
5908:
5883:
5882:
5840:
5839:
5823:
5798:
5797:
5756:
5755:
5739:
5737:
5736:
5724:
5722:
5721:
5720:
5710:
5709:
5708:
5698:
5693:
5692:
5691:
5668:
5666:
5665:
5660:
5658:
5657:
5615:
5613:
5612:
5607:
5605:
5603:
5602:
5598:
5596:
5595:
5577:
5576:
5572:
5570:
5569:
5551:
5546:
5545:
5516:
5514:
5513:
5508:
5506:
5501:
5500:
5499:
5489:
5469:
5467:
5466:
5461:
5459:
5458:
5444:
5442:
5441:
5436:
5434:
5433:
5421:
5420:
5419:
5402:
5400:
5399:
5394:
5392:
5391:
5379:
5378:
5377:
5357:
5355:
5354:
5349:
5342:
5336:
5329:
5320:
5315:
5312:
5305:
5296:
5291:
5288:
5281:
5272:
5267:
5264:
5241:
5239:
5238:
5233:
5218:
5216:
5215:
5210:
5203:
5196:
5192:
5187:
5182:
5172:
5148:
5146:
5145:
5140:
5138:
5137:
5136:
5119:
5117:
5116:
5111:
5109:
5108:
5092:
5090:
5089:
5084:
5082:
5081:
5066:
5064:
5063:
5058:
5051:
5050:
5046:
5041:
5036:
5025:
5005:
5004:
4991:
4990:
4989:
4988:
4960:
4958:
4957:
4952:
4940:
4938:
4937:
4932:
4920:
4918:
4917:
4912:
4900:
4898:
4897:
4892:
4877:
4875:
4874:
4869:
4862:
4831:
4830:
4817:
4816:
4815:
4814:
4783:
4781:
4780:
4775:
4768:
4767:
4763:
4741:
4737:
4735:
4734:
4686:
4685:
4663:
4662:
4661:
4645:
4644:
4643:
4619:
4617:
4616:
4611:
4599:
4597:
4596:
4591:
4575:
4573:
4572:
4567:
4552:
4550:
4549:
4544:
4537:
4530:
4528:
4520:
4519:
4515:
4514:
4513:
4495:
4480:
4478:
4477:
4472:
4465:
4464:
4460:
4434:
4433:
4391:
4390:
4368:
4367:
4366:
4350:
4349:
4348:
4325:
4323:
4322:
4317:
4310:
4309:
4305:
4283:
4279:
4277:
4276:
4236:
4232:
4230:
4229:
4205:
4204:
4203:
4187:
4186:
4185:
4159:
4157:
4156:
4151:
4144:
4143:
4139:
4138:
4137:
4120:
4119:
4110:
4109:
4090:
4089:
4088:
4072:
4071:
4070:
4057:
4056:
4055:
4042:
4041:
4040:
4017:
4015:
4014:
4009:
4002:
4001:
4000:
3983:
3982:
3981:
3967:
3966:
3957:
3956:
3955:
3935:
3933:
3932:
3927:
3912:
3910:
3909:
3904:
3900:
3899:
3885:
3884:
3883:
3867:
3866:
3865:
3845:
3843:
3842:
3837:
3835:
3834:
3833:
3813:
3811:
3810:
3805:
3803:
3773:
3771:
3770:
3765:
3761:
3760:
3759:
3732:
3731:
3730:
3710:
3708:
3707:
3702:
3700:
3685:
3683:
3682:
3677:
3665:
3663:
3662:
3657:
3589:
3587:
3586:
3581:
3579:
3577:
3575:
3574:
3573:
3556:
3555:
3544:
3543:
3542:
3541:
3526:
3509:
3507:
3506:
3501:
3499:
3497:
3495:
3494:
3493:
3476:
3475:
3464:
3463:
3462:
3449:
3444:
3443:
3427:
3425:
3424:
3419:
3417:
3416:
3415:
3395:
3393:
3392:
3387:
3385:
3383:
3380:
3379:
3367:
3366:
3355:
3354:
3353:
3339:
3334:
3332:
3331:
3330:
3320:
3319:
3318:
3308:
3290:
3288:
3287:
3282:
3280:
3279:
3278:
3261:
3259:
3258:
3253:
3238:
3236:
3235:
3230:
3228:
3226:
3225:
3224:
3212:
3211:
3201:
3198:
3197:
3185:
3184:
3173:
3168:
3167:
3142:
3140:
3139:
3134:
3132:
3131:
3119:
3118:
3117:
3116:
3105:
3101:
3096:
3094:
3093:
3092:
3082:
3081:
3080:
3070:
3039:
3038:
3037:
3036:
3025:
3021:
3016:
3014:
3013:
3012:
3002:
3001:
3000:
2990:
2948:
2946:
2945:
2940:
2921:
2919:
2918:
2913:
2904:
2899:
2898:
2892:
2874:
2872:
2871:
2866:
2863:
2858:
2857:
2856:
2843:
2842:
2841:
2828:
2820:
2815:
2814:
2813:
2800:
2799:
2798:
2785:
2767:
2765:
2764:
2759:
2757:
2755:
2754:
2753:
2752:
2739:
2738:
2737:
2722:
2721:
2720:
2719:
2706:
2705:
2704:
2689:
2684:
2683:
2668:. In this case,
2663:
2661:
2660:
2655:
2653:
2652:
2651:
2630:
2628:
2627:
2622:
2620:
2619:
2618:
2601:
2599:
2598:
2593:
2591:
2590:
2575:is purely real (
2574:
2572:
2571:
2566:
2564:
2563:
2529:
2527:
2526:
2521:
2519:
2518:
2505:
2503:
2502:
2497:
2495:
2494:
2481:
2479:
2478:
2473:
2471:
2470:
2469:
2452:
2450:
2449:
2444:
2442:
2441:
2420:
2418:
2417:
2412:
2401:
2400:
2378:
2376:
2375:
2370:
2358:
2357:
2334:
2332:
2331:
2326:
2322:
2321:
2309:
2308:
2292:
2290:
2289:
2284:
2280:
2279:
2278:
2265:
2264:
2263:
2243:
2241:
2240:
2235:
2222:
2221:
2202:
2200:
2199:
2194:
2181:
2180:
2160:
2158:
2157:
2152:
2148:
2147:
2130:
2119:
2117:
2116:
2111:
2100:
2099:
2075:
2074:
2073:
2047:
2045:
2044:
2039:
2031:
2027:
2025:
2024:
2003:
2001:
2000:
1995:
1993:
1992:
1976:
1974:
1973:
1968:
1966:
1964:
1963:
1962:
1961:
1948:
1947:
1946:
1935:
1934:
1933:
1932:
1919:
1918:
1917:
1906:
1901:
1899:
1898:
1897:
1887:
1886:
1885:
1875:
1870:
1868:
1867:
1866:
1856:
1855:
1854:
1844:
1839:
1838:
1822:
1820:
1819:
1814:
1812:
1811:
1810:
1790:
1788:
1787:
1782:
1780:
1778:
1777:
1776:
1766:
1764:
1763:
1762:
1750:
1745:
1744:
1729:, is defined as
1728:
1726:
1725:
1720:
1718:
1717:
1701:
1699:
1698:
1693:
1691:
1689:
1688:
1687:
1686:
1673:
1672:
1671:
1660:
1659:
1658:
1657:
1644:
1643:
1642:
1631:
1629:
1628:
1627:
1614:
1613:
1612:
1599:
1597:
1596:
1595:
1594:
1581:
1580:
1579:
1568:
1567:
1566:
1556:
1554:
1553:
1552:
1535:
1534:
1533:
1520:
1519:
1518:
1505:
1504:
1503:
1483:
1481:
1480:
1475:
1473:
1472:
1471:
1458:
1457:
1456:
1443:
1442:
1441:
1424:
1422:
1421:
1416:
1414:
1413:
1412:
1396:The reflection,
1392:
1390:
1389:
1384:
1382:
1380:
1379:
1378:
1377:
1364:
1363:
1362:
1351:
1350:
1349:
1339:
1337:
1336:
1335:
1318:
1317:
1316:
1296:
1294:
1293:
1288:
1286:
1285:
1284:
1267:
1265:
1264:
1259:
1257:
1256:
1255:
1233:complex function
1225:frequency domain
1202:
1200:
1199:
1194:
1190:
1189:
1188:
1172:
1171:
1170:
1147:
1145:
1144:
1139:
1135:
1134:
1133:
1117:
1116:
1115:
1082:
1080:
1079:
1074:
1058:
1056:
1055:
1050:
1038:
1036:
1035:
1030:
1028:
1027:
1010:
1008:
1007:
1002:
972:
970:
969:
964:
962:
961:
945:
943:
942:
937:
925:
923:
922:
917:
915:
914:
897:
895:
894:
889:
887:
886:
869:
867:
866:
861:
856:
827:
825:
824:
819:
815:
814:
813:
800:
799:
798:
778:
776:
775:
770:
768:
767:
766:
756:
755:
754:
737:
735:
734:
729:
727:
726:
725:
705:
703:
702:
697:
695:
693:
692:
683:
682:
681:
671:
666:
665:
664:
644:
642:
641:
636:
634:
633:
632:
611:
609:
608:
603:
598:
577:
575:
574:
569:
548:
546:
545:
540:
509:
507:
506:
497:
496:
495:
485:
480:
479:
478:
459:
457:
456:
451:
449:
448:
432:
430:
429:
424:
422:
421:
420:
400:
398:
397:
392:
359:
358:
357:
337:
335:
334:
329:
318:, at some point
317:
315:
314:
309:
307:
306:
305:
288:
286:
285:
280:
264:
262:
261:
256:
240:
238:
237:
232:
211:
209:
208:
203:
191:
189:
188:
183:
21:
6970:
6969:
6965:
6964:
6963:
6961:
6960:
6959:
6925:
6924:
6831:, Newnes, 2011
6824:
6819:
6815:, pages 434–435
6810:
6806:
6802:, pages 595–605
6797:
6793:
6788:
6784:
6775:
6771:
6766:
6762:
6757:
6753:
6749:Connor, page 29
6748:
6744:
6739:
6735:
6730:
6726:
6718:
6714:
6705:
6701:
6696:
6692:
6683:
6679:
6670:
6666:
6661:
6650:
6645:
6641:
6637:
6595:
6554:
6537:
6535:
6515:
6510:
6509:
6470:
6469:
6442:
6441:
6389:
6371:
6366:
6365:
6355:
6305:
6288:
6287:
6257:
6238:
6237:
6225:
6207:
6202:
6201:
6175:
6174:
6155:
6154:
6106:
6093:
6092:
6062:
6047:
6046:
6034:
6016:
6011:
6010:
5952:
5951:
5926:
5925:
5824:
5740:
5728:
5711:
5699:
5679:
5674:
5673:
5649:
5644:
5643:
5640:
5638:Input impedance
5589:
5585:
5578:
5563:
5559:
5552:
5526:
5525:
5490:
5480:
5479:
5476:
5448:
5447:
5425:
5410:
5405:
5404:
5383:
5368:
5363:
5362:
5247:
5246:
5224:
5223:
5173:
5169:
5165:
5154:
5153:
5127:
5122:
5121:
5100:
5095:
5094:
5071:
5070:
5026:
5023:
5019:
4980:
4966:
4965:
4943:
4942:
4923:
4922:
4903:
4902:
4883:
4882:
4806:
4792:
4791:
4722:
4718:
4669:
4665:
4649:
4634:
4629:
4628:
4602:
4601:
4578:
4577:
4558:
4557:
4521:
4504:
4500:
4496:
4489:
4488:
4374:
4370:
4354:
4339:
4334:
4333:
4264:
4260:
4217:
4213:
4211:
4207:
4191:
4176:
4171:
4170:
4122:
4097:
4096:
4092:
4076:
4061:
4046:
4031:
4026:
4025:
3985:
3969:
3946:
3941:
3940:
3918:
3917:
3887:
3871:
3856:
3851:
3850:
3821:
3816:
3815:
3796:
3779:
3778:
3752:
3740:
3721:
3716:
3715:
3693:
3688:
3687:
3668:
3667:
3648:
3647:
3614:
3600:
3564:
3547:
3545:
3532:
3527:
3514:
3513:
3484:
3467:
3465:
3454:
3450:
3433:
3432:
3406:
3401:
3400:
3371:
3358:
3356:
3345:
3340:
3321:
3309:
3296:
3295:
3269:
3264:
3263:
3244:
3243:
3216:
3203:
3202:
3189:
3176:
3174:
3157:
3156:
3149:
3127:
3126:
3107:
3097:
3083:
3071:
3047:
3046:
3027:
3017:
3003:
2991:
2966:
2954:
2953:
2931:
2930:
2883:
2882:
2847:
2832:
2804:
2789:
2776:
2775:
2743:
2728:
2723:
2710:
2695:
2690:
2673:
2672:
2642:
2633:
2632:
2609:
2604:
2603:
2582:
2577:
2576:
2555:
2550:
2549:
2546:
2541:
2508:
2507:
2484:
2483:
2460:
2455:
2454:
2433:
2428:
2427:
2381:
2380:
2379:
2338:
2337:
2313:
2300:
2295:
2294:
2293:
2269:
2254:
2249:
2248:
2211:
2210:
2170:
2169:
2137:
2136:
2091:
2064:
2050:
2049:
2018:
2014:
2009:
2008:
1982:
1981:
1952:
1937:
1936:
1923:
1908:
1907:
1888:
1876:
1857:
1845:
1828:
1827:
1801:
1796:
1795:
1767:
1753:
1751:
1734:
1733:
1707:
1706:
1677:
1662:
1661:
1648:
1633:
1632:
1618:
1603:
1585:
1570:
1569:
1557:
1543:
1524:
1509:
1494:
1489:
1488:
1462:
1447:
1432:
1427:
1426:
1403:
1398:
1397:
1368:
1353:
1352:
1340:
1326:
1307:
1302:
1301:
1275:
1270:
1269:
1246:
1241:
1240:
1217:
1179:
1161:
1156:
1155:
1124:
1106:
1101:
1100:
1089:
1061:
1060:
1041:
1040:
1019:
1014:
1013:
989:
988:
953:
948:
947:
928:
927:
906:
901:
900:
878:
873:
872:
838:
837:
804:
789:
784:
783:
757:
745:
740:
739:
716:
711:
710:
684:
672:
655:
650:
649:
623:
618:
617:
580:
579:
560:
559:
555:
498:
486:
469:
464:
463:
440:
435:
434:
411:
406:
405:
348:
343:
342:
320:
319:
296:
291:
290:
271:
270:
247:
246:
214:
213:
194:
193:
161:
160:
153:
97:reflections in
36:
28:
23:
22:
15:
12:
11:
5:
6968:
6966:
6958:
6957:
6952:
6947:
6942:
6937:
6927:
6926:
6923:
6922:
6907:
6892:
6885:
6870:
6857:Connor, F.R.,
6855:
6840:
6823:
6820:
6818:
6817:
6804:
6791:
6782:
6769:
6760:
6751:
6742:
6733:
6724:
6712:
6699:
6690:
6677:
6664:
6648:
6638:
6636:
6633:
6632:
6631:
6626:
6621:
6616:
6611:
6606:
6601:
6594:
6591:
6579:
6578:
6562:
6557:
6551:
6544:
6540:
6532:
6526:
6523:
6518:
6494:
6490:
6486:
6483:
6480:
6477:
6457:
6453:
6449:
6431:
6430:
6417:
6414:
6410:
6407:
6404:
6401:
6396:
6392:
6388:
6382:
6379:
6374:
6354:
6351:
6350:
6349:
6335:
6332:
6328:
6325:
6322:
6319:
6313:
6308:
6303:
6300:
6295:
6291:
6285:
6282:
6278:
6275:
6272:
6269:
6264:
6260:
6255:
6252:
6246:
6241:
6232:
6228:
6224:
6218:
6215:
6210:
6186:
6182:
6162:
6151:
6150:
6136:
6133:
6129:
6126:
6123:
6120:
6114:
6109:
6105:
6100:
6096:
6090:
6087:
6083:
6080:
6077:
6074:
6069:
6065:
6061:
6055:
6050:
6041:
6037:
6033:
6027:
6024:
6019:
5995:
5992:
5988:
5985:
5982:
5979:
5976:
5970:
5965:
5962:
5959:
5939:
5936:
5933:
5922:
5921:
5907:
5904:
5900:
5897:
5894:
5891:
5887:
5881:
5876:
5873:
5870:
5867:
5864:
5861:
5857:
5854:
5851:
5848:
5844:
5838:
5833:
5830:
5827:
5822:
5819:
5815:
5812:
5809:
5806:
5802:
5796:
5791:
5788:
5785:
5782:
5779:
5776:
5772:
5769:
5766:
5763:
5760:
5754:
5749:
5746:
5743:
5735:
5731:
5727:
5719:
5714:
5707:
5702:
5696:
5690:
5687:
5682:
5656:
5652:
5639:
5636:
5621:asymptotically
5617:
5616:
5601:
5594:
5588:
5584:
5581:
5575:
5568:
5562:
5558:
5555:
5549:
5544:
5541:
5538:
5535:
5505:
5498:
5493:
5488:
5475:
5472:
5457:
5432:
5428:
5424:
5418:
5413:
5390:
5386:
5382:
5376:
5371:
5359:
5358:
5347:
5341:
5335:
5332:
5326:
5323:
5311:
5308:
5302:
5299:
5287:
5284:
5278:
5275:
5263:
5260:
5257:
5254:
5231:
5220:
5219:
5208:
5202:
5199:
5195:
5191:
5185:
5180:
5176:
5168:
5164:
5161:
5135:
5130:
5107:
5103:
5080:
5068:
5067:
5056:
5049:
5045:
5039:
5034:
5030:
5022:
5018:
5015:
5012:
5009:
5003:
4997:
4994:
4987:
4983:
4976:
4973:
4950:
4930:
4910:
4890:
4879:
4878:
4867:
4861:
4858:
4854:
4850:
4847:
4844:
4841:
4838:
4835:
4829:
4823:
4820:
4813:
4809:
4802:
4799:
4785:
4784:
4773:
4766:
4761:
4758:
4754:
4751:
4748:
4745:
4740:
4733:
4728:
4725:
4721:
4716:
4713:
4710:
4707:
4703:
4700:
4697:
4694:
4690:
4684:
4679:
4676:
4673:
4668:
4660:
4657:
4652:
4648:
4642:
4637:
4609:
4589:
4585:
4565:
4554:
4553:
4542:
4536:
4533:
4527:
4524:
4518:
4512:
4507:
4503:
4499:
4482:
4481:
4470:
4463:
4458:
4455:
4451:
4448:
4445:
4442:
4438:
4432:
4427:
4424:
4421:
4418:
4415:
4412:
4408:
4405:
4402:
4399:
4395:
4389:
4384:
4381:
4378:
4373:
4365:
4362:
4357:
4353:
4347:
4342:
4327:
4326:
4315:
4308:
4303:
4300:
4296:
4293:
4290:
4287:
4282:
4275:
4270:
4267:
4263:
4259:
4256:
4253:
4249:
4246:
4243:
4240:
4235:
4228:
4223:
4220:
4216:
4210:
4202:
4199:
4194:
4190:
4184:
4179:
4161:
4160:
4149:
4142:
4136:
4132:
4129:
4125:
4118:
4113:
4108:
4104:
4100:
4095:
4087:
4084:
4079:
4075:
4069:
4064:
4060:
4054:
4049:
4045:
4039:
4034:
4019:
4018:
4007:
3999:
3995:
3992:
3988:
3980:
3977:
3972:
3965:
3960:
3954:
3949:
3925:
3914:
3913:
3898:
3894:
3890:
3882:
3879:
3874:
3870:
3864:
3859:
3832:
3829:
3824:
3802:
3799:
3795:
3792:
3789:
3786:
3775:
3774:
3758:
3755:
3750:
3747:
3743:
3738:
3735:
3729:
3724:
3699:
3696:
3675:
3655:
3621:Standing waves
3613:
3612:Standing waves
3610:
3599:
3596:
3592:
3591:
3572:
3567:
3562:
3559:
3554:
3550:
3540:
3535:
3530:
3524:
3521:
3511:
3492:
3487:
3482:
3479:
3474:
3470:
3461:
3457:
3453:
3447:
3442:
3414:
3409:
3397:
3396:
3378:
3374:
3370:
3365:
3361:
3352:
3348:
3343:
3337:
3329:
3324:
3317:
3312:
3306:
3303:
3277:
3272:
3251:
3240:
3239:
3223:
3219:
3215:
3210:
3206:
3196:
3192:
3188:
3183:
3179:
3171:
3166:
3148:
3145:
3144:
3143:
3130:
3125:
3122:
3115:
3110:
3098:
3091:
3086:
3079:
3074:
3068:
3065:
3061:
3058:
3055:
3052:
3049:
3048:
3045:
3042:
3035:
3030:
3018:
3011:
3006:
2999:
2994:
2988:
2985:
2981:
2978:
2975:
2972:
2971:
2969:
2964:
2961:
2938:
2923:
2922:
2910:
2907:
2903:
2897:
2891:
2876:
2875:
2862:
2855:
2850:
2846:
2840:
2835:
2831:
2827:
2823:
2819:
2812:
2807:
2803:
2797:
2792:
2788:
2784:
2769:
2768:
2751:
2746:
2742:
2736:
2731:
2726:
2718:
2713:
2709:
2703:
2698:
2693:
2687:
2682:
2650:
2645:
2640:
2617:
2612:
2589:
2585:
2562:
2558:
2545:
2542:
2517:
2493:
2468:
2463:
2440:
2436:
2422:
2421:
2410:
2407:
2404:
2399:
2394:
2391:
2388:
2367:
2364:
2361:
2356:
2351:
2348:
2345:
2335:
2320:
2316:
2312:
2307:
2303:
2277:
2272:
2268:
2262:
2257:
2245:
2244:
2231:
2228:
2225:
2220:
2208:
2204:
2203:
2190:
2187:
2184:
2179:
2167:
2163:
2162:
2146:
2134:
2121:
2120:
2109:
2106:
2103:
2098:
2094:
2090:
2087:
2084:
2081:
2078:
2072:
2067:
2063:
2060:
2057:
2037:
2034:
2030:
2023:
2017:
1991:
1978:
1977:
1960:
1955:
1951:
1945:
1940:
1931:
1926:
1922:
1916:
1911:
1904:
1896:
1891:
1884:
1879:
1873:
1865:
1860:
1853:
1848:
1842:
1837:
1809:
1804:
1792:
1791:
1775:
1770:
1761:
1756:
1748:
1743:
1716:
1703:
1702:
1685:
1680:
1676:
1670:
1665:
1656:
1651:
1647:
1641:
1636:
1626:
1621:
1617:
1611:
1606:
1602:
1593:
1588:
1584:
1578:
1573:
1565:
1560:
1551:
1546:
1541:
1538:
1532:
1527:
1523:
1517:
1512:
1508:
1502:
1497:
1470:
1465:
1461:
1455:
1450:
1446:
1440:
1435:
1411:
1406:
1394:
1393:
1376:
1371:
1367:
1361:
1356:
1348:
1343:
1334:
1329:
1324:
1321:
1315:
1310:
1283:
1278:
1254:
1249:
1216:
1213:
1204:
1203:
1187:
1182:
1178:
1175:
1169:
1164:
1149:
1148:
1132:
1127:
1123:
1120:
1114:
1109:
1088:
1085:
1072:
1068:
1048:
1026:
1022:
1000:
996:
960:
956:
935:
913:
909:
885:
881:
859:
855:
851:
848:
845:
829:
828:
812:
807:
803:
797:
792:
765:
760:
753:
748:
724:
719:
707:
706:
691:
687:
680:
675:
669:
663:
658:
631:
626:
601:
597:
593:
590:
587:
567:
554:
551:
550:
549:
538:
535:
532:
529:
525:
522:
519:
515:
512:
505:
501:
494:
489:
483:
477:
472:
447:
443:
419:
414:
402:
401:
388:
385:
382:
379:
375:
372:
369:
365:
362:
356:
351:
327:
304:
299:
278:
254:
230:
227:
224:
221:
201:
181:
178:
175:
172:
168:
152:
151:Specific cases
149:
137:standing waves
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6967:
6956:
6953:
6951:
6948:
6946:
6943:
6941:
6940:Signal cables
6938:
6936:
6933:
6932:
6930:
6920:
6919:0-486-42552-5
6916:
6912:
6908:
6905:
6904:981-02-1714-5
6901:
6897:
6893:
6890:
6886:
6883:
6882:0-86341-287-4
6879:
6875:
6871:
6868:
6867:0-7131-3278-7
6864:
6860:
6856:
6853:
6852:0-07-137435-3
6849:
6845:
6841:
6838:
6837:0-08-055342-7
6834:
6830:
6826:
6825:
6821:
6814:
6808:
6805:
6801:
6795:
6792:
6786:
6783:
6779:
6773:
6770:
6764:
6761:
6755:
6752:
6746:
6743:
6737:
6734:
6728:
6725:
6722:
6716:
6713:
6710:, pages 34–35
6709:
6703:
6700:
6694:
6691:
6687:
6681:
6678:
6674:
6668:
6665:
6659:
6657:
6655:
6653:
6649:
6643:
6640:
6634:
6630:
6627:
6625:
6622:
6620:
6617:
6615:
6612:
6610:
6607:
6605:
6604:Antenna tuner
6602:
6600:
6597:
6596:
6592:
6590:
6588:
6584:
6555:
6549:
6542:
6538:
6530:
6516:
6508:
6507:
6506:
6492:
6488:
6484:
6481:
6478:
6475:
6455:
6451:
6447:
6439:
6434:
6412:
6408:
6402:
6399:
6394:
6390:
6386:
6372:
6364:
6363:
6362:
6360:
6352:
6330:
6326:
6320:
6317:
6306:
6301:
6298:
6293:
6289:
6280:
6276:
6270:
6267:
6262:
6258:
6253:
6250:
6239:
6230:
6226:
6222:
6208:
6200:
6199:
6198:
6184:
6180:
6160:
6131:
6127:
6121:
6118:
6107:
6103:
6098:
6094:
6085:
6081:
6075:
6072:
6067:
6063:
6059:
6048:
6039:
6035:
6031:
6017:
6009:
6008:
6007:
5990:
5986:
5980:
5977:
5963:
5960:
5937:
5934:
5931:
5924:Substituting
5902:
5898:
5892:
5889:
5874:
5871:
5865:
5859:
5855:
5849:
5846:
5831:
5828:
5817:
5813:
5807:
5804:
5789:
5786:
5780:
5774:
5770:
5764:
5761:
5747:
5744:
5733:
5729:
5725:
5712:
5700:
5694:
5680:
5672:
5671:
5670:
5654:
5650:
5637:
5635:
5631:
5629:
5624:
5622:
5599:
5586:
5582:
5579:
5573:
5560:
5556:
5553:
5547:
5524:
5523:
5522:
5520:
5491:
5478:The ratio of
5473:
5471:
5430:
5426:
5422:
5411:
5388:
5384:
5380:
5369:
5345:
5339:
5333:
5330:
5324:
5321:
5309:
5306:
5300:
5297:
5285:
5282:
5276:
5273:
5261:
5258:
5255:
5252:
5245:
5244:
5243:
5229:
5206:
5200:
5197:
5193:
5189:
5183:
5178:
5174:
5166:
5162:
5159:
5152:
5151:
5150:
5128:
5105:
5101:
5054:
5047:
5043:
5037:
5032:
5028:
5020:
5016:
5013:
5010:
4992:
4974:
4971:
4964:
4963:
4962:
4948:
4928:
4908:
4888:
4865:
4856:
4852:
4848:
4842:
4839:
4836:
4818:
4800:
4797:
4790:
4789:
4788:
4771:
4764:
4756:
4752:
4746:
4743:
4738:
4726:
4723:
4719:
4714:
4711:
4705:
4701:
4695:
4692:
4677:
4674:
4666:
4650:
4646:
4635:
4627:
4626:
4625:
4623:
4607:
4587:
4583:
4563:
4540:
4534:
4531:
4525:
4516:
4505:
4501:
4487:
4486:
4485:
4468:
4461:
4453:
4449:
4443:
4440:
4425:
4422:
4416:
4410:
4406:
4400:
4397:
4382:
4379:
4371:
4355:
4351:
4340:
4332:
4331:
4330:
4313:
4306:
4298:
4294:
4288:
4285:
4280:
4268:
4265:
4261:
4257:
4251:
4247:
4241:
4238:
4233:
4221:
4218:
4214:
4208:
4192:
4188:
4177:
4169:
4168:
4167:
4166:
4147:
4140:
4134:
4130:
4127:
4123:
4111:
4106:
4102:
4098:
4093:
4077:
4073:
4062:
4058:
4047:
4043:
4032:
4024:
4023:
4022:
4005:
3997:
3993:
3990:
3986:
3970:
3958:
3947:
3939:
3938:
3937:
3923:
3896:
3892:
3888:
3872:
3868:
3857:
3849:
3848:
3847:
3822:
3800:
3797:
3793:
3790:
3787:
3784:
3756:
3753:
3748:
3745:
3741:
3736:
3733:
3722:
3714:
3713:
3712:
3697:
3694:
3673:
3653:
3646:
3641:
3639:
3638:standing wave
3635:
3631:
3622:
3618:
3611:
3609:
3606:
3597:
3595:
3565:
3560:
3557:
3552:
3548:
3533:
3528:
3522:
3519:
3512:
3485:
3480:
3477:
3472:
3468:
3459:
3455:
3451:
3445:
3431:
3430:
3429:
3407:
3376:
3372:
3368:
3363:
3359:
3350:
3346:
3341:
3335:
3322:
3310:
3304:
3301:
3294:
3293:
3292:
3270:
3249:
3221:
3217:
3213:
3208:
3204:
3194:
3190:
3186:
3181:
3177:
3169:
3155:
3154:
3153:
3146:
3123:
3120:
3108:
3089:
3084:
3072:
3066:
3063:
3059:
3056:
3053:
3050:
3043:
3040:
3028:
3009:
3004:
2992:
2986:
2983:
2979:
2976:
2973:
2967:
2962:
2959:
2952:
2951:
2950:
2936:
2928:
2908:
2905:
2881:
2880:
2879:
2853:
2848:
2844:
2833:
2829:
2821:
2810:
2805:
2801:
2790:
2786:
2774:
2773:
2772:
2749:
2744:
2740:
2729:
2724:
2716:
2711:
2707:
2696:
2691:
2685:
2671:
2670:
2669:
2667:
2643:
2638:
2610:
2587:
2583:
2560:
2556:
2543:
2540:
2535:
2533:
2461:
2438:
2434:
2408:
2405:
2389:
2386:
2365:
2362:
2346:
2343:
2336:
2318:
2314:
2310:
2305:
2301:
2270:
2266:
2255:
2247:
2246:
2229:
2226:
2223:
2209:
2207:Short circuit
2206:
2205:
2188:
2185:
2182:
2168:
2165:
2164:
2135:
2132:
2131:
2128:
2126:
2107:
2104:
2096:
2092:
2085:
2082:
2079:
2065:
2058:
2055:
2035:
2032:
2028:
2015:
2007:
2006:
2005:
1958:
1953:
1949:
1938:
1929:
1924:
1920:
1909:
1902:
1889:
1877:
1871:
1858:
1846:
1840:
1826:
1825:
1824:
1802:
1768:
1754:
1746:
1732:
1731:
1730:
1683:
1678:
1674:
1663:
1654:
1649:
1645:
1634:
1619:
1615:
1604:
1600:
1586:
1582:
1576:
1571:
1558:
1544:
1539:
1536:
1525:
1521:
1510:
1506:
1495:
1487:
1486:
1485:
1463:
1459:
1448:
1444:
1433:
1404:
1369:
1365:
1359:
1354:
1341:
1327:
1322:
1319:
1308:
1300:
1299:
1298:
1276:
1247:
1236:
1234:
1230:
1226:
1222:
1214:
1212:
1210:
1180:
1176:
1173:
1162:
1154:
1153:
1152:
1125:
1121:
1118:
1107:
1099:
1098:
1097:
1095:
1086:
1084:
1070:
1066:
1046:
1024:
1020:
998:
994:
982:
978:
974:
958:
954:
933:
911:
907:
883:
879:
857:
853:
849:
846:
843:
834:
805:
801:
790:
782:
781:
780:
758:
746:
717:
689:
685:
673:
667:
656:
648:
647:
646:
624:
615:
599:
595:
591:
588:
585:
565:
552:
533:
530:
527:
523:
517:
513:
510:
503:
499:
487:
481:
470:
462:
461:
460:
445:
441:
412:
383:
380:
377:
373:
367:
363:
360:
349:
341:
340:
339:
325:
297:
276:
268:
252:
244:
225:
219:
199:
176:
170:
166:
158:
150:
148:
146:
142:
138:
133:
131:
127:
123:
119:
115:
111:
107:
102:
100:
96:
92:
88:
84:
80:
76:
71:
69:
65:
61:
60:discontinuity
57:
53:
45:
40:
35:
30:
19:
6910:
6895:
6888:
6876:, IET, 1992
6873:
6858:
6843:
6828:
6812:
6807:
6799:
6794:
6785:
6777:
6772:
6763:
6754:
6745:
6736:
6727:
6715:
6707:
6702:
6693:
6688:, pages 8–10
6685:
6680:
6672:
6667:
6642:
6614:Lecher lines
6580:
6435:
6432:
6356:
6353:Applications
6152:
5923:
5641:
5632:
5628:slotted line
5625:
5618:
5477:
5360:
5221:
5069:
4880:
4786:
4555:
4483:
4328:
4162:
4020:
3915:
3776:
3642:
3626:
3604:
3601:
3593:
3398:
3241:
3150:
2924:
2877:
2770:
2547:
2425:
2166:Open circuit
2122:
1979:
1793:
1704:
1395:
1237:
1218:
1205:
1150:
1090:
1083:as before.
986:
980:
832:
830:
708:
645:, and since
556:
403:
154:
134:
118:overvoltages
114:transmitters
103:
72:
49:
29:
6629:Smith Chart
6624:Space cloth
4881:Expressing
2133:Termination
1980:In general
122:power lines
91:fibre-optic
6929:Categories
6822:References
6780:, page 182
6675:, pages 34
6589:networks.
3605:ring waves
2537:See also:
2426:When both
1231:dependant
245:with time
108:, causing
99:waveguides
93:lines and
68:terminated
32:See also:
6811:Matthaei
6706:Matthaei
6684:Matthaei
6671:Matthaei
6635:Citations
6485:π
6479:ℓ
6476:β
6448:λ
6413:ℓ
6409:β
6403:
6331:ℓ
6327:β
6321:
6281:ℓ
6277:β
6271:
6185:β
6161:γ
6132:ℓ
6128:γ
6122:
6086:ℓ
6082:γ
6076:
5987:γ
5981:
5969:Γ
5938:ℓ
5899:γ
5893:
5880:Γ
5856:γ
5850:
5837:Γ
5832:−
5814:γ
5808:
5795:Γ
5790:−
5771:γ
5765:
5753:Γ
5593:Γ
5583:−
5567:Γ
5456:Γ
5340:…
5331:λ
5307:λ
5283:λ
5184:λ
5179:π
5163:
5079:Γ
5038:λ
5033:π
5017:
5002:Γ
4993:
4972:−
4949:λ
4921:, allows
4909:λ
4889:β
4853:β
4843:
4828:Γ
4819:
4798:−
4753:β
4747:
4732:Γ
4727:−
4702:β
4696:
4683:Γ
4608:β
4588:β
4564:γ
4523:∂
4498:∂
4450:γ
4444:
4431:Γ
4407:γ
4401:
4388:Γ
4383:−
4295:γ
4289:
4274:Γ
4269:−
4248:γ
4242:
4227:Γ
4131:γ
4128:−
4117:Γ
4103:γ
3994:γ
3991:−
3964:Γ
3893:γ
3794:−
3791:ℓ
3749:γ
3746:−
3654:γ
3630:amplitude
3452:−
3441:Γ
3187:−
3165:Γ
3067:
3057:−
3054:π
3051:−
2987:
2977:−
2974:π
2960:θ
2937:θ
2927:reactance
2896:Γ
2802:−
2708:−
2681:Γ
2666:reactance
2516:Γ
2492:Γ
2398:Γ
2390:
2355:Γ
2347:
2227:−
2219:Γ
2178:Γ
2161:relation
2145:Γ
2086:
2059:
2033:≤
2022:Γ
1990:Γ
1921:−
1836:Γ
1742:Γ
1715:Γ
1646:−
1601:−
1522:−
1229:frequency
1177:−
1122:−
858:κ
850:ℓ
600:κ
592:ℓ
566:ℓ
531:−
524:κ
381:−
374:κ
277:κ
112:power in
95:microwave
87:aluminium
56:reflected
6593:See also
3801:′
3757:′
3698:′
3598:Networks
3102:if
3022:if
110:overload
6776:Bowick
4620:is the
981:Fig. 3.
241:is the
192:(where
62:in the
6917:
6902:
6880:
6865:
6850:
6835:
6813:et al.
6800:et al.
6778:et al.
6708:et al.
6686:et al.
6673:et al.
5343:
5337:
5316:
5313:
5292:
5289:
5268:
5265:
5204:
5052:
4863:
4769:
4600:where
4538:
4466:
4311:
4145:
4003:
3590:
3510:
3064:arctan
2984:arctan
2771:Since
2602:) and
83:copper
3711:, by
3634:nodes
2878:then
2532:phase
2048:when
126:stubs
6915:ISBN
6900:ISBN
6878:ISBN
6863:ISBN
6848:ISBN
6833:ISBN
6585:and
6359:stub
6119:tanh
6073:tanh
5978:cosh
5890:sinh
5847:cosh
5805:sinh
5762:cosh
5423:>
5381:<
5361:For
5120:and
4441:sinh
4398:cosh
4286:sinh
4239:cosh
3121:<
3041:>
2453:and
2363:<
2105:>
1221:wave
1209:dual
1151:and
143:and
128:and
116:and
6400:tan
6318:tan
6268:tan
5242:at
5160:tan
5014:tan
4840:tan
4744:sin
4693:cos
3846:).
833:add
120:on
77:on
6931::
6651:^
5470:.
4961::
3640:.
3377:01
3364:02
3351:02
3222:01
3209:02
3195:01
3182:02
2387:Im
2344:Re
2083:Re
2056:Re
1823:,
1484:,
1235:.
1096::
973:.
147:.
101:.
42:A
6921:.
6906:.
6884:.
6869:.
6854:.
6839:.
6561:L
6556:Z
6550:2
6543:0
6539:Z
6531:=
6525:n
6522:i
6517:Z
6493:2
6489:/
6482:=
6456:4
6452:/
6416:)
6406:(
6395:0
6391:Z
6387:=
6381:n
6378:i
6373:X
6334:)
6324:(
6312:L
6307:Z
6302:j
6299:+
6294:0
6290:Z
6284:)
6274:(
6263:0
6259:Z
6254:j
6251:+
6245:L
6240:Z
6231:0
6227:Z
6223:=
6217:n
6214:i
6209:Z
6181:j
6135:)
6125:(
6113:L
6108:Z
6104:+
6099:0
6095:Z
6089:)
6079:(
6068:0
6064:Z
6060:+
6054:L
6049:Z
6040:0
6036:Z
6032:=
6026:n
6023:i
6018:Z
5994:)
5991:x
5984:(
5975:)
5964:+
5961:1
5958:(
5935:=
5932:x
5906:)
5903:x
5896:(
5886:)
5875:+
5872:1
5869:(
5866:+
5863:)
5860:x
5853:(
5843:)
5829:1
5826:(
5821:)
5818:x
5811:(
5801:)
5787:1
5784:(
5781:+
5778:)
5775:x
5768:(
5759:)
5748:+
5745:1
5742:(
5734:0
5730:Z
5726:=
5718:T
5713:I
5706:T
5701:V
5695:=
5689:n
5686:i
5681:Z
5655:0
5651:Z
5600:|
5587:|
5580:1
5574:|
5561:|
5557:+
5554:1
5548:=
5543:R
5540:W
5537:S
5534:V
5504:|
5497:T
5492:V
5487:|
5431:0
5427:R
5417:L
5412:R
5389:0
5385:R
5375:L
5370:R
5346:.
5334:,
5325:4
5322:3
5310:,
5301:2
5298:1
5286:,
5277:4
5274:1
5262:,
5259:0
5256:=
5253:x
5230:x
5207:,
5201:0
5198:=
5194:)
5190:x
5175:4
5167:(
5134:L
5129:Z
5106:0
5102:Z
5055:.
5048:)
5044:x
5029:4
5021:(
5011:=
5008:}
4996:{
4986:m
4982:I
4975:2
4929:x
4866:.
4860:)
4857:x
4849:2
4846:(
4837:=
4834:}
4822:{
4812:m
4808:I
4801:2
4772:,
4765:]
4760:)
4757:x
4750:(
4739:)
4724:1
4720:(
4715:j
4712:+
4709:)
4706:x
4699:(
4689:)
4678:+
4675:1
4672:(
4667:[
4659:L
4656:i
4651:V
4647:=
4641:T
4636:V
4584:j
4541:.
4535:0
4532:=
4526:x
4517:|
4511:T
4506:V
4502:|
4469:.
4462:]
4457:)
4454:x
4447:(
4437:)
4426:+
4423:1
4420:(
4417:+
4414:)
4411:x
4404:(
4394:)
4380:1
4377:(
4372:[
4364:L
4361:i
4356:I
4352:=
4346:T
4341:I
4314:.
4307:]
4302:)
4299:x
4292:(
4281:)
4266:1
4262:(
4258:+
4255:)
4252:x
4245:(
4234:)
4222:+
4219:1
4215:(
4209:[
4201:L
4198:i
4193:V
4189:=
4183:T
4178:V
4148:.
4141:)
4135:x
4124:e
4112:+
4107:x
4099:e
4094:(
4086:L
4083:i
4078:V
4074:=
4068:r
4063:V
4059:+
4053:i
4048:V
4044:=
4038:T
4033:V
4006:.
3998:x
3987:e
3979:L
3976:i
3971:V
3959:=
3953:r
3948:V
3924:x
3897:x
3889:e
3881:L
3878:i
3873:V
3869:=
3863:i
3858:V
3831:L
3828:i
3823:V
3798:x
3788:=
3785:x
3754:x
3742:e
3737:V
3734:=
3728:i
3723:V
3695:x
3674:V
3571:L
3566:Z
3561:2
3558:+
3553:0
3549:Z
3539:L
3534:Z
3529:2
3523:=
3520:T
3491:L
3486:Z
3481:2
3478:+
3473:0
3469:Z
3460:0
3456:Z
3446:=
3413:L
3408:Z
3373:Z
3369:+
3360:Z
3347:Z
3342:2
3336:=
3328:i
3323:V
3316:t
3311:V
3305:=
3302:T
3276:t
3271:V
3250:T
3218:Z
3214:+
3205:Z
3191:Z
3178:Z
3170:=
3124:0
3114:L
3109:X
3090:0
3085:R
3078:L
3073:X
3060:2
3044:0
3034:L
3029:X
3010:0
3005:R
2998:L
2993:X
2980:2
2968:{
2963:=
2909:1
2906:=
2902:|
2890:|
2861:|
2854:0
2849:R
2845:+
2839:L
2834:X
2830:j
2826:|
2822:=
2818:|
2811:0
2806:R
2796:L
2791:X
2787:j
2783:|
2750:0
2745:R
2741:+
2735:L
2730:X
2725:j
2717:0
2712:R
2702:L
2697:X
2692:j
2686:=
2649:L
2644:X
2639:j
2616:L
2611:Z
2588:0
2584:R
2561:0
2557:Z
2467:L
2462:Z
2439:0
2435:Z
2409:0
2406:=
2403:)
2393:(
2366:1
2360:)
2350:(
2319:0
2315:R
2311:=
2306:0
2302:Z
2276:L
2271:R
2267:=
2261:L
2256:Z
2230:1
2224:=
2189:1
2186:+
2183:=
2108:0
2102:)
2097:0
2093:Z
2089:(
2080:,
2077:)
2071:L
2066:Z
2062:(
2036:1
2029:|
2016:|
1959:0
1954:Z
1950:+
1944:L
1939:Z
1930:0
1925:Z
1915:L
1910:Z
1903:=
1895:i
1890:I
1883:r
1878:I
1872:=
1864:i
1859:V
1852:r
1847:V
1841:=
1808:r
1803:V
1774:i
1769:V
1760:r
1755:V
1747:=
1684:0
1679:Z
1675:+
1669:L
1664:Z
1655:0
1650:Z
1640:L
1635:Z
1625:i
1620:V
1616:=
1610:i
1605:V
1592:L
1587:Z
1583:+
1577:0
1572:Z
1564:L
1559:Z
1550:i
1545:V
1540:2
1537:=
1531:i
1526:V
1516:o
1511:V
1507:=
1501:r
1496:V
1469:o
1464:V
1460:=
1454:r
1449:V
1445:+
1439:i
1434:V
1410:r
1405:V
1375:L
1370:Z
1366:+
1360:0
1355:Z
1347:L
1342:Z
1333:i
1328:V
1323:2
1320:=
1314:o
1309:V
1282:L
1277:Z
1253:o
1248:V
1186:i
1181:i
1174:=
1168:r
1163:i
1131:i
1126:v
1119:=
1113:r
1108:v
1071:V
1067:2
1047:V
1025:0
1021:Z
999:V
995:2
959:0
955:Z
934:V
912:0
908:Z
884:0
880:Z
854:/
847:=
844:t
811:i
806:v
802:=
796:r
791:v
764:i
759:i
752:i
747:v
723:r
718:v
690:0
686:Z
679:r
674:v
668:=
662:r
657:i
630:r
625:i
596:/
589:=
586:t
537:)
534:x
528:t
521:(
518:u
514:I
511:=
504:0
500:Z
493:i
488:v
482:=
476:i
471:i
446:0
442:Z
418:i
413:i
387:)
384:x
378:t
371:(
368:u
364:V
361:=
355:i
350:v
326:x
303:i
298:v
253:t
229:)
226:t
223:(
220:u
200:V
180:)
177:t
174:(
171:u
167:V
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
