1777:
1912:
523:
1984:
1405:
2012:
481:
1950:
1878:
1968:
1896:
2003:
1959:
1887:
347:
2260:
transmission line. Notice that as Δℓ is made smaller and smaller, both ΔL and ΔC decrease, but in the same proportion, so that the ratio ΔL/ΔC remains constant. So if we take the limit of Eq. (22.28) as ΔL and ΔC go to zero, we find that the characteristic impedance z0 is a pure resistance whose magnitude is √(ΔL/ΔC). We can also write the ratio ΔL/ΔC as L0/C0, where L0 and C0 are the inductance and capacitance of a unit length of the line; then we have
466:
2259:
If we imagine the line as broken up into small lengths Δℓ, each length will look like one section of the L-C ladder with a series inductance ΔL and a shunt capacitance ΔC. We can then use our results for the ladder filter. If we take the limit as Δℓ goes to zero, we have a good description of the
1836:
It should be borne in mind that the characteristics of the filter predicted by the image method are only accurate if the section is terminated with its image impedance. This is usually not true of the sections at either end, which are usually terminated with a fixed resistance. The further the
261:
that had been used previously. Campbell called his filters electric wave filters, but this term later came to mean any filter that passes waves of some frequencies but not others. Many new forms of wave filter were subsequently invented; an early (and important) variation was the
1792:
Several L-shape half-sections may be cascaded to form a composite filter. Like impedance must always face like in these combinations. There are therefore two circuits that can be formed with two identical L-shaped half-sections. Where a port of image impedance
1087:
1711:
415:, which is also an L-section but with half the element values of the full L-section. The image impedance of the half-section is dissimilar on the input and output ports: on the side presenting the series element it is equal to the mid-series
1618:
802:
1394:
1315:
1180:
373:(that is, a section from halfway through a series element to halfway through the next series element) will have the same image impedance on both ports due to symmetry. This image impedance is designated
1235:
973:
887:
1484:
2301:
1532:
2032:
Textbooks and design drawings usually show the unbalanced implementations, but in telecoms it is often required to convert the design to the balanced implementation when used with
668:
1833:
the section so formed is a T section. Further additions of half-sections to either of these section forms a ladder network which may start and end with series or shunt elements.
984:
1837:
section is from the end of the filter, the more accurate the prediction will become, since the effects of the terminating impedances are masked by the intervening sections.
1629:
607:
580:
324:
Some of the impedance terms and section terms used in this article are pictured in the diagram below. Image theory defines quantities in terms of an infinite cascade of
553:
189:
1623:
That is, the transmission is lossless in the pass-band with only the phase of the signal changing. Above the cut-off frequency, the transmission parameters are:
1543:
292:, but their subsequent use has been much more widespread than that. The design techniques used by Campbell have largely been superseded. However, the
741:
1326:
1250:
361:. The factor of two is introduced for mathematical convenience, since it is usual to work in terms of half-sections where it disappears. The
182:
1106:
727:, of the transmission line that would be formed by these infinitesimally small sections. It is also the image impedance of the section at
708:
as the size of the section (in terms of values of its components, such as inductances, capacitances, etc.) approaches zero, while keeping
175:
1092:
Given that the filter does not contain any resistive elements, the image impedance in the pass band of the filter is purely
229:
frequency response to within any prescribed limit with the addition of a sufficient number of sections. However, they are
2477:
2199:
1195:
911:
218:
835:
1440:
2447:
296:
used by
Campbell with the constant k is still in use today with implementations of modern filter designs such as the
250:
2263:
2467:
2218:
257:, was already making improvements to the design at this time. Campbell's filters were far superior to the simpler
2252:
222:
230:
1499:
249:. He published his work in 1922, but had clearly invented the filters some time before, as his colleague at
1765:
1082:{\displaystyle {\frac {1}{{Z_{\mathrm {i\Pi } }}^{2}}}={Y_{\mathrm {i\Pi } }}^{2}=Y^{2}+{\frac {1}{k^{2}}}}
100:
633:
281:
and stop band. It was only necessary to add more filter sections until the desired response was obtained.
258:
2040:
1537:
For the low-pass L-shape section, below the cut-off frequency, the transmission parameters are given by
340:
1776:
1761:
1757:
1753:
234:
117:
70:
65:
33:
1706:{\displaystyle \gamma =\alpha +i\beta =\cosh ^{-1}{\frac {\omega }{\omega _{c}}}+i{\frac {\pi }{2}}}
2472:
1425:
717:
2437:
217:
method. They are the original and simplest filters produced by this methodology and consist of a
1431:
585:
558:
111:
353:
The sections of the hypothetical infinite filter are made of series elements having impedance 2
2074:
613:
The building block of constant k filters is the half-section "L" network, composed of a series
522:
273:
and other simple filters of the time was that they could be designed for any desired degree of
246:
123:
83:
1721:
The presented plots of image impedance, attenuation and phase change correspond to a low-pass
266:
by Zobel who coined the term constant k for the
Campbell filter in order to distinguish them.
2417:
2248:
538:
366:
1404:
900:
1911:
1412:
of a constant k prototype low-pass filter for a single half-section showing attenuation in
728:
614:
439:
735:= 0 in the case of low-pass filters. For example, the pictured low-pass half-section has
8:
1983:
297:
828:
however, are both approaching zero, and from the formulae (below) for image impedances,
447:
2240:
2145:
2106:
2069:
443:
128:
40:
2011:
2443:
1409:
1186:
289:
210:
22:
2204:
Planar microwave engineering: a practical guide to theory, measurement, and circuits
2149:
2137:
2102:
2064:
1722:
1613:{\displaystyle \gamma =\alpha +i\beta =0+i\sin ^{-1}{\frac {\omega }{\omega _{c}}}}
1097:
480:
451:
325:
263:
95:
45:
2387:
2236:
2059:
1844:
362:
329:
214:
60:
50:
2093:
Campbell, G. A. (November 1922), "Physical Theory of the
Electric Wave-Filter",
797:{\displaystyle k={\sqrt {\frac {i\omega L}{i\omega C}}}={\sqrt {\frac {L}{C}}}}
690:
438:
Parts of this article or section rely on the reader's knowledge of the complex
343:, "L" refers to the specific filter shape which resembles inverted letter "L".
293:
2176:
2141:
1949:
1877:
2461:
2244:
424:, but on the side presenting the shunt element it is equal to the mid-shunt
225:
components. Historically, they are the first filters that could approach the
105:
55:
1967:
1741:
1895:
233:
for a modern design, the principles behind them having been superseded by
2223:
Electromagnetics for high-speed analog and digital communication circuits
2002:
1958:
1886:
1389:{\displaystyle Z_{\mathrm {iT} }=iL{\sqrt {\omega ^{2}-\omega _{c}^{2}}}}
1093:
285:
226:
161:
1764:
into high-pass, band-pass or band-stop types by application of suitable
1310:{\displaystyle Z_{\mathrm {iT} }=L{\sqrt {\omega _{c}^{2}-\omega ^{2}}}}
2396:
Microwave
Filters, Impedance-Matching Networks, and Coupling Structures
621:
386:" topology of a mid-series section. Likewise, the image impedance of a
333:
270:
254:
156:
151:
146:
1175:{\displaystyle {Z_{\mathrm {iT} }}^{2}=-(\omega L)^{2}+{\frac {L}{C}}}
1241:
346:
313:
309:
305:
465:
284:
The filters were designed by
Campbell for the purpose of separating
2033:
535:
of a constant k prototype low-pass filter is plotted vs. frequency
301:
278:
274:
2118:
Bray, p.62 gives 1910 as the start of
Campbell's work on filters.
1736:= 1 Ω. This is produced by a filter half-section with inductance
2403:
Theory and Design of
Uniform and Composite Electric Wave Filters
816:
can be made arbitrarily small while retaining the same value of
697:. A physical interpretation of k can be given by observing that
237:
which are more accurate in their prediction of filter response.
1417:
2202:(2004). "2.5. Driving-point impedance of Iterated Structure".
1320:
Above the cut-off frequency the image impedance is imaginary,
328:, and in the case of the filters being discussed, an infinite
1749:
1413:
433:. There are thus two variant ways of using a half-section.
369:
of a section will generally not be the same. However, for a
1760:
to the desired values. The low-pass prototype can also be
332:
of L-sections. Here "L" should not be confused with the
678:
469:
Constant k low-pass filter half section. Here inductance
16:
Type of electronic filter designed using the image method
2442:, New Delhi: Prentice Hall of India, pp. 544–563,
1100:. For example, for the pictured low-pass half-section,
2355:
Matthaei et al., pp.96-97, 412-413, 438-440, 727-729.
2266:
1632:
1546:
1502:
1443:
1329:
1253:
1198:
1109:
987:
914:
838:
744:
636:
588:
561:
541:
2420:, California Institute of Technology – HTML edition.
2412:
Feynman, Richard; Leighton, Robert; Sands, Matthew,
269:
The great advantage
Campbell's filters had over the
1434:for a general constant k half-section are given by
1230:{\displaystyle \omega _{c}={\frac {1}{\sqrt {LC}}}}
968:{\displaystyle {Z_{\mathrm {iT} }}^{2}=Z^{2}+k^{2}}
2295:
1725:section. The prototype has a cut-off frequency of
1705:
1612:
1526:
1478:
1388:
1309:
1229:
1174:
1081:
967:
882:{\displaystyle \lim _{Z,Y\to 0}Z_{\mathrm {i} }=k}
881:
796:
662:
627:. The "k" in "constant k" is the value given by,
601:
574:
547:
1523:
1479:{\displaystyle \gamma =\sinh ^{-1}{\frac {Z}{k}}}
905:The image impedances of the section are given by
555:. The impedance is purely resistive (real) below
2459:
2235:
840:
2296:{\displaystyle {\sqrt {\frac {L_{0}}{C_{0}}}}}
316:and multiple band filters are also possible.
277:rejection or steepness of transition between
183:
2430:For a simpler treatment of the analysis see,
2388:Innovation and the Communications Revolution
1839:
1716:
2210:
2180:, filed 30 April 1920, issued 23 Sept 1924.
681:. It is readily apparent that in order for
2394:Matthaei, G.; Young, L.; Jones, E. M. T.,
2391:, Institute of Electrical Engineers, 2002.
2219:"Section 9.2. An Infinite Ladder Network."
1399:
731:, in the case of band-pass filters, or at
190:
176:
2324:
2322:
2320:
2206:. Cambridge University Press. p. 44.
1788:low-pass constant-k filter half-sections.
1522:
484:Constant k band-pass filter half section.
2414:The Feynman Lectures on Physics, Vol. II
2216:
2092:
1842:
1775:
1527:{\displaystyle \gamma _{n}=n\gamma \,\!}
1426:Image impedance § Transfer function
1403:
582:, and purely reactive (imaginary) above
521:
479:
464:
677:will have units of impedance, that is,
300:. Campbell gave constant k designs for
2460:
2439:Network Theory: Analysis and Synthesis
2317:
2192:
2128:White, G. (January 2000), "The Past",
2031:
2435:
2127:
1870:
1867:
1864:
1771:
1922:
1852:
663:{\displaystyle k^{2}={\frac {Z}{Y}}}
357:and shunt elements with admittance 2
245:Constant k filters were invented by
2198:
13:
2424:
2107:10.1002/j.1538-7305.1922.tb00386.x
2010:
2001:
1982:
1966:
1957:
1948:
1910:
1894:
1885:
1876:
1732:= 1 rad/s and a nominal impedance
1339:
1336:
1263:
1260:
1121:
1118:
1096:and in the stop band it is purely
1033:
1030:
1004:
1001:
926:
923:
894:
867:
345:
14:
2489:
2405:, Bell System Technical Journal,
901:Image impedance § Derivation
2367:
2358:
2349:
2340:
2331:
2308:
2254:The Feynman Lectures on Physics
1244:, the image impedance is real,
2229:
2183:
2164:
2155:
2121:
2112:
2086:
1150:
1140:
853:
319:
1:
2379:
1928:
1856:
460:
66:Optimum "L" (Legendre) filter
712:at its initial value. Thus,
7:
2418:Section 6. A ladder network
2416:, Chapter 22. AC Circuits,
2053:
2038:
1185:The transition occurs at a
602:{\displaystyle \omega _{c}}
575:{\displaystyle \omega _{c}}
403:" topology. Half of such a
10:
2494:
2478:Electronic filter topology
2364:Matthaei et al., pp.65-68.
2337:Matthaei et al., pp.61-62.
1996:X Section (mid-Π-Derived)
1993:X Section (mid-T-Derived)
1811:, the section is called a
1423:
898:
341:electronic filter topology
240:
2217:Niknejad, Ali M. (2007).
2172:Multiple-band wave filter
2028:
2024:
2009:
2000:
1995:
1992:
1981:
1976:
1965:
1956:
1947:
1942:
1939:
1936:
1931:
1909:
1904:
1893:
1884:
1875:
1859:
1766:frequency transformations
1717:Prototype transformations
701:is the limiting value of
454:representation of signals
221:of identical sections of
118:Bridged T delay equaliser
34:Network synthesis filters
2436:Ghosh, Smarajit (2005),
2080:
1752:. This prototype can be
718:characteristic impedance
450:and on knowledge of the
365:of the input and output
2468:Image impedance filters
2142:10.1023/A:1026506828275
1432:transmission parameters
1400:Transmission parameters
548:{\displaystyle \omega }
259:single element circuits
84:Image impedance filters
51:Elliptic (Cauer) filter
2409:(1923), pp. 1–46.
2373:Matthaei et al., p.68.
2328:Matthaei et al., p.61.
2297:
2249:"Section 22-7. Filter"
2075:Composite image filter
2015:
2006:
1987:
1971:
1962:
1953:
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1881:
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350:
288:telephone channels on
124:Composite image filter
2298:
2177:U.S. patent 1,509,184
2130:BT Technology Journal
2014:
2005:
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101:General image filters
71:Linkwitz–Riley filter
2314:Zobel, 1923, pp.3-4.
2264:
2095:Bell System Tech. J.
1630:
1544:
1500:
1441:
1416:and phase change in
1327:
1251:
1196:
1107:
985:
912:
836:
742:
634:
586:
559:
539:
411:section is called a
2241:Leighton, Robert B.
1489:and for a chain of
1383:
1291:
298:Tchebyscheff filter
235:other methodologies
213:designed using the
108:(constant R) filter
2293:
2016:
2007:
1988:
1972:
1963:
1954:
1916:
1900:
1891:
1882:
1790:
1772:Cascading sections
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1079:
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599:
572:
545:
520:
478:
442:representation of
371:mid-series section
351:
290:transmission lines
203:Constant k filters
41:Butterworth filter
25:electronic filters
2398:McGraw-Hill 1964.
2346:Zobel, 1923, p.3.
2291:
2290:
2189:Zobel, 1923, p.6.
2051:
2050:
2047:
2046:
2043:
2020:
2019:
1920:
1919:
1784:) for a chain of
1780:Gain response, H(
1701:
1685:
1608:
1474:
1410:transfer function
1384:
1305:
1225:
1224:
1187:cut-off frequency
1170:
1077:
1017:
839:
792:
791:
777:
776:
658:
388:mid-shunt section
326:two-port sections
231:rarely considered
211:electronic filter
200:
199:
91:Constant k filter
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2371:
2365:
2362:
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2302:
2300:
2299:
2294:
2292:
2289:
2288:
2279:
2278:
2269:
2268:
2237:Feynman, Richard
2233:
2227:
2226:
2214:
2208:
2207:
2196:
2190:
2187:
2181:
2179:
2168:
2162:
2159:
2153:
2152:
2125:
2119:
2116:
2110:
2109:
2090:
2065:m-derived filter
2039:
2026:
2025:
1929:
1857:
1840:
1831:
1822:
1814:
1809:
1800:
1758:frequency scaled
1754:impedance scaled
1744:and capacitance
1723:prototype filter
1712:
1710:
1709:
1704:
1702:
1694:
1686:
1684:
1683:
1671:
1666:
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1355:
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1303:
1290:
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1234:
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1125:
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1080:
1078:
1076:
1075:
1063:
1058:
1057:
1045:
1044:
1039:
1038:
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1036:
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1016:
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1008:
1007:
989:
974:
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938:
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803:
801:
800:
795:
793:
784:
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764:
753:
752:
685:to be constant,
669:
667:
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645:
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605:
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597:
581:
579:
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570:
554:
552:
551:
546:
526:Image impedance
452:frequency domain
432:
423:
410:
406:
402:
398:
385:
381:
264:m-derived filter
209:, are a type of
192:
185:
178:
96:m-derived filter
46:Chebyshev filter
19:
18:
2493:
2492:
2488:
2487:
2486:
2484:
2483:
2482:
2473:Analog circuits
2458:
2457:
2450:
2427:
2425:Further reading
2382:
2377:
2372:
2368:
2363:
2359:
2354:
2350:
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2332:
2327:
2318:
2313:
2309:
2284:
2280:
2274:
2270:
2267:
2265:
2262:
2261:
2257:. Vol. 2.
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2175:
2169:
2165:
2160:
2156:
2126:
2122:
2117:
2113:
2091:
2087:
2083:
2070:mm'-type filter
2060:Image impedance
2056:
1977:Ladder network
1937:C Half-section
1905:Ladder network
1871:Π Section
1865:L Half section
1848:
1847:filter sections
1832:
1829:
1823:
1820:
1815:section. Where
1812:
1810:
1807:
1801:
1798:
1787:
1783:
1774:
1731:
1719:
1693:
1679:
1675:
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897:
895:Image impedance
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739:
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404:
400:
397:
391:
383:
380:
374:
363:image impedance
322:
294:ladder topology
247:George Campbell
243:
196:
167:
166:
142:
134:
133:
129:mm'-type filter
86:
76:
75:
61:Gaussian filter
36:
24:
17:
12:
11:
5:
2491:
2481:
2480:
2475:
2470:
2454:
2453:
2448:
2432:
2431:
2426:
2423:
2422:
2421:
2410:
2399:
2392:
2381:
2378:
2376:
2375:
2366:
2357:
2348:
2339:
2330:
2316:
2307:
2287:
2283:
2277:
2273:
2245:Sands, Matthew
2228:
2209:
2200:Lee, Thomas H.
2191:
2182:
2163:
2154:
2136:(1): 107–132,
2120:
2111:
2084:
2082:
2079:
2078:
2077:
2072:
2067:
2062:
2055:
2052:
2049:
2048:
2045:
2044:
2037:
2030:
2022:
2021:
2018:
2017:
2008:
1998:
1997:
1994:
1990:
1989:
1979:
1978:
1974:
1973:
1964:
1955:
1945:
1944:
1941:
1938:
1934:
1933:
1925:
1924:
1921:
1918:
1917:
1907:
1906:
1902:
1901:
1892:
1883:
1873:
1872:
1869:
1866:
1862:
1861:
1854:
1850:
1849:
1843:
1828:
1819:
1806:
1802:faces another
1797:
1785:
1781:
1773:
1770:
1729:
1718:
1715:
1714:
1713:
1700:
1697:
1692:
1689:
1682:
1678:
1674:
1669:
1664:
1661:
1657:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1621:
1620:
1605:
1601:
1597:
1592:
1587:
1584:
1580:
1576:
1573:
1570:
1567:
1564:
1561:
1558:
1555:
1552:
1549:
1535:
1534:
1521:
1518:
1515:
1510:
1506:
1493:half-sections
1487:
1486:
1473:
1470:
1465:
1460:
1457:
1453:
1449:
1446:
1401:
1398:
1397:
1396:
1381:
1376:
1372:
1368:
1363:
1359:
1353:
1350:
1347:
1341:
1338:
1333:
1318:
1317:
1302:
1298:
1294:
1289:
1284:
1280:
1274:
1271:
1265:
1262:
1257:
1238:
1237:
1223:
1220:
1216:
1211:
1206:
1202:
1183:
1182:
1169:
1166:
1161:
1156:
1152:
1148:
1145:
1142:
1139:
1136:
1131:
1123:
1120:
1115:
1090:
1089:
1074:
1070:
1066:
1061:
1056:
1052:
1048:
1043:
1035:
1032:
1027:
1021:
1014:
1006:
1003:
998:
992:
976:
975:
962:
958:
954:
949:
945:
941:
936:
928:
925:
920:
896:
893:
891:
890:
878:
875:
869:
864:
858:
855:
852:
849:
846:
842:
830:
806:
805:
790:
787:
781:
774:
771:
768:
763:
760:
757:
750:
747:
724:
705:
691:dual impedance
671:
670:
657:
654:
649:
644:
640:
620:, and a shunt
596:
592:
569:
565:
544:
530:
513:
506:
496:
489:
462:
459:
458:
457:
429:
420:
395:
390:is designated
378:
330:ladder network
321:
318:
242:
239:
219:ladder network
207:k-type filters
198:
197:
195:
194:
187:
180:
172:
169:
168:
165:
164:
159:
154:
149:
143:
141:Simple filters
140:
139:
136:
135:
132:
131:
126:
121:
115:
112:Lattice filter
109:
103:
98:
93:
87:
82:
81:
78:
77:
74:
73:
68:
63:
58:
53:
48:
43:
37:
32:
31:
28:
27:
15:
9:
6:
4:
3:
2:
2490:
2479:
2476:
2474:
2471:
2469:
2466:
2465:
2463:
2456:
2451:
2449:81-203-2638-5
2445:
2441:
2440:
2434:
2433:
2429:
2428:
2419:
2415:
2411:
2408:
2404:
2401:Zobel, O. J.,
2400:
2397:
2393:
2390:
2389:
2384:
2383:
2370:
2361:
2352:
2343:
2334:
2325:
2323:
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2311:
2303:
2285:
2281:
2275:
2271:
2256:
2255:
2250:
2246:
2242:
2238:
2232:
2224:
2220:
2213:
2205:
2201:
2195:
2186:
2178:
2173:
2167:
2158:
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2147:
2143:
2139:
2135:
2131:
2124:
2115:
2108:
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2100:
2096:
2089:
2085:
2076:
2073:
2071:
2068:
2066:
2063:
2061:
2058:
2057:
2042:
2035:
2027:
2023:
2013:
2004:
1999:
1991:
1985:
1980:
1975:
1969:
1960:
1951:
1946:
1935:
1930:
1927:
1926:
1913:
1908:
1903:
1897:
1888:
1879:
1874:
1863:
1858:
1855:
1851:
1846:
1841:
1838:
1834:
1827:
1818:
1805:
1796:
1778:
1769:
1767:
1763:
1759:
1755:
1751:
1747:
1743:
1739:
1735:
1728:
1724:
1698:
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1676:
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1659:
1655:
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1642:
1639:
1636:
1633:
1626:
1625:
1624:
1603:
1599:
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1590:
1585:
1582:
1578:
1574:
1571:
1568:
1565:
1562:
1559:
1556:
1553:
1550:
1547:
1540:
1539:
1538:
1519:
1516:
1513:
1508:
1504:
1496:
1495:
1494:
1492:
1471:
1468:
1463:
1458:
1455:
1451:
1447:
1444:
1437:
1436:
1435:
1433:
1427:
1419:
1415:
1411:
1406:
1379:
1374:
1370:
1366:
1361:
1357:
1351:
1348:
1345:
1331:
1323:
1322:
1321:
1300:
1296:
1292:
1287:
1282:
1278:
1272:
1269:
1255:
1247:
1246:
1245:
1243:
1221:
1218:
1214:
1209:
1204:
1200:
1192:
1191:
1190:
1188:
1167:
1164:
1159:
1154:
1146:
1143:
1137:
1134:
1129:
1113:
1103:
1102:
1101:
1099:
1095:
1072:
1068:
1064:
1059:
1054:
1050:
1046:
1041:
1025:
1019:
1012:
996:
990:
981:
980:
979:
960:
956:
952:
947:
943:
939:
934:
918:
908:
907:
906:
902:
876:
873:
862:
856:
850:
847:
844:
832:
831:
829:
827:
823:
819:
815:
811:
788:
785:
779:
772:
769:
766:
761:
758:
755:
748:
745:
738:
737:
736:
734:
730:
723:
719:
715:
711:
704:
700:
696:
692:
688:
684:
680:
676:
655:
652:
647:
642:
638:
630:
629:
628:
626:
623:
619:
616:
594:
590:
567:
563:
542:
533:
529:
524:
518:
512:
505:
501:
495:
488:
482:
476:
472:
467:
455:
453:
449:
445:
441:
436:
435:
434:
428:
419:
414:
394:
389:
377:
372:
368:
364:
360:
356:
348:
344:
342:
338:
335:
331:
327:
317:
315:
311:
307:
303:
299:
295:
291:
287:
282:
280:
276:
272:
267:
265:
260:
256:
252:
248:
238:
236:
232:
228:
224:
220:
216:
212:
208:
204:
193:
188:
186:
181:
179:
174:
173:
171:
170:
163:
160:
158:
155:
153:
150:
148:
145:
144:
138:
137:
130:
127:
125:
122:
119:
116:
113:
110:
107:
106:Zobel network
104:
102:
99:
97:
94:
92:
89:
88:
85:
80:
79:
72:
69:
67:
64:
62:
59:
57:
56:Bessel filter
54:
52:
49:
47:
44:
42:
39:
38:
35:
30:
29:
26:
23:Linear analog
21:
20:
2455:
2438:
2413:
2406:
2402:
2395:
2386:
2369:
2360:
2351:
2342:
2333:
2310:
2258:
2253:
2231:
2222:
2212:
2203:
2194:
2185:
2171:
2170:Zobel, O J,
2166:
2157:
2133:
2129:
2123:
2114:
2098:
2094:
2088:
1943:Box Section
1835:
1825:
1816:
1803:
1794:
1791:
1745:
1737:
1733:
1726:
1720:
1622:
1536:
1490:
1488:
1429:
1319:
1239:
1184:
1091:
977:
904:
825:
821:
817:
813:
809:
807:
732:
721:
713:
709:
702:
698:
694:
689:must be the
686:
682:
674:
672:
624:
617:
612:
531:
527:
516:
510:
503:
499:
493:
486:
474:
470:
437:
426:
417:
413:half-section
412:
399:due to the "
392:
387:
382:due to the "
375:
370:
358:
354:
352:
336:
323:
283:
268:
244:
227:ideal filter
206:
202:
201:
90:
2161:Bray, p.62.
2101:(2): 1–32,
1860:Unbalanced
1762:transformed
1240:Below this
320:Terminology
286:multiplexed
251:AT&T Co
2462:Categories
2385:Bray, J.,
2380:References
1940:H Section
1868:T Section
1424:See also:
899:See also:
622:admittance
461:Derivation
444:capacitors
334:inductance
271:RL circuit
255:Otto Zobel
162:RLC filter
120:(all-pass)
114:(all-pass)
1932:Balanced
1696:π
1677:ω
1673:ω
1668:
1660:−
1649:β
1640:α
1634:γ
1600:ω
1596:ω
1591:
1583:−
1563:β
1554:α
1548:γ
1520:γ
1505:γ
1464:
1456:−
1445:γ
1371:ω
1367:−
1358:ω
1297:ω
1293:−
1279:ω
1242:frequency
1201:ω
1189:given by
1144:ω
1138:−
1098:imaginary
1034:Π
1005:Π
854:→
808:Elements
770:ω
759:ω
729:resonance
615:impedance
591:ω
564:ω
543:ω
473:is equal
448:inductors
440:impedance
314:Band-stop
312:filters.
310:band-pass
306:high-pass
279:pass band
275:stop band
157:LC filter
152:RL filter
147:RC filter
2150:62360033
2054:See also
2034:balanced
302:low-pass
2036:lines.
1923:
1853:
1418:radians
716:is the
241:History
223:passive
205:, also
2446:
2407:Vol. 2
2148:
1824:faces
1414:nepers
673:Thus,
2146:S2CID
2081:Notes
2029:N.B.
1845:Image
1750:farad
1742:henry
339:– in
215:image
2444:ISBN
2041:edit
1756:and
1748:= 1
1740:= 1
1656:cosh
1452:sinh
1430:The
1408:The
1094:real
978:and
824:and
812:and
679:ohms
502:and
446:and
367:port
308:and
2138:doi
2103:doi
1579:sin
841:lim
820:.
693:of
409:"Π"
407:or
405:"T"
2464::
2319:^
2251:.
2247:.
2243:;
2239:;
2221:.
2174:,
2144:,
2134:18
2132:,
2097:,
1830:iΠ
1821:iΠ
1808:iT
1799:iT
1768:.
720:,
532:iT
509:=
492:=
475:Ck
430:iΠ
421:iT
396:iΠ
379:iT
304:,
253:,
2305:.
2286:0
2282:C
2276:0
2272:L
2225:.
2140::
2105::
2099:1
1826:Z
1817:Z
1813:Π
1804:Z
1795:Z
1786:n
1782:ω
1746:C
1738:L
1734:k
1730:c
1727:ω
1699:2
1691:i
1688:+
1681:c
1663:1
1652:=
1646:i
1643:+
1637:=
1604:c
1586:1
1575:i
1572:+
1569:0
1566:=
1560:i
1557:+
1551:=
1517:n
1514:=
1509:n
1491:n
1472:k
1469:Z
1459:1
1448:=
1420:.
1380:2
1375:c
1362:2
1352:L
1349:i
1346:=
1340:T
1337:i
1332:Z
1301:2
1288:2
1283:c
1273:L
1270:=
1264:T
1261:i
1256:Z
1222:C
1219:L
1215:1
1210:=
1205:c
1168:C
1165:L
1160:+
1155:2
1151:)
1147:L
1141:(
1135:=
1130:2
1122:T
1119:i
1114:Z
1073:2
1069:k
1065:1
1060:+
1055:2
1051:Y
1047:=
1042:2
1031:i
1026:Y
1020:=
1013:2
1002:i
997:Z
991:1
961:2
957:k
953:+
948:2
944:Z
940:=
935:2
927:T
924:i
919:Z
889:.
877:k
874:=
868:i
863:Z
857:0
851:Y
848:,
845:Z
826:Y
822:Z
818:k
814:C
810:L
804:.
789:C
786:L
780:=
773:C
767:i
762:L
756:i
749:=
746:k
733:ω
725:0
722:Z
714:k
710:k
706:i
703:Z
699:k
695:Z
687:Y
683:k
675:k
656:Y
653:Z
648:=
643:2
639:k
625:Y
618:Z
609:.
595:c
568:c
528:Z
517:k
514:1
511:C
507:2
504:L
500:k
497:2
494:C
490:1
487:L
471:L
456:.
427:Z
418:Z
401:Π
393:Z
384:T
376:Z
359:Y
355:Z
337:L
191:e
184:t
177:v
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