2551:
divider with an output ratio of 3.3/5 might be used to reduce the 5 V signal to 3.3 V, to allow the circuits to interoperate without damaging the 3.3 V circuit. For this to be feasible, the 5 V source impedance and 3.3 V input impedance must be negligible, or they must be constant and the divider resistor values must account for their impedances. If the input impedance is capacitive, a purely resistive divider will limit the data rate. This can be roughly overcome by adding a capacitor in series with the top resistor, to make both legs of the divider capacitive as well as resistive.
2533:. The high voltage is applied across the divider, and the divider output—which outputs a lower voltage that is within the meter's input range—is measured by the meter. High voltage resistor divider probes designed specifically for this purpose can be used to measure voltages up to 100 kV. Special high-voltage resistors are used in such probes as they must be able to tolerate high input voltages and, to produce accurate results, must have matched
100:
2500:
722:
1800:
1451:
2329:
1612:
1293:
2445:
To obtain a sufficiently stable output voltage, the output current must either be stable (and so be made part of the calculation of the potential divider values) or limited to an appropriately small percentage of the divider's input current. Load sensitivity can be decreased by reducing the impedance
2550:
to interface two circuits that use different operating voltages. For example, some logic circuits operate at 5 V whereas others operate at 3.3 V. Directly interfacing a 5 V logic output to a 3.3 V input may cause permanent damage to the 3.3 V circuit. In this case, a voltage
2490:
Another example that is commonly used involves a potentiometer (variable resistor) as one of the resistive elements. When the shaft of the potentiometer is rotated the resistance it produces either increases or decreases, the change in resistance corresponds to the angular change of the shaft. If
2478:
Voltage dividers can be used to allow a microcontroller to measure the resistance of a sensor. The sensor is wired in series with a known resistance to form a voltage divider and a known voltage is applied across the divider. The microcontroller's analog-to-digital converter is connected to the
2082:
1095:
1943:
2479:
center tap of the divider so that it can measure the tap voltage and, by using the measured voltage and the known resistance and voltage, compute the sensor resistance. This technique is commonly used to measure the resistance of temperature sensors such as
1409:
1795:{\displaystyle {\frac {V_{\mathrm {out} }}{V_{\mathrm {in} }}}={\frac {Z_{\mathrm {2} }}{Z_{\mathrm {1} }+Z_{\mathrm {2} }}}={\frac {\frac {1}{\mathrm {j} \omega C}}{{\frac {1}{\mathrm {j} \omega C}}+R}}={\frac {1}{1+\mathrm {j} \omega RC}}\ .}
2639:
691:
2604:
1134:
1563:
2048:
867:
575:
275:
2537:
and very low voltage coefficients. Capacitive divider probes are typically used for voltages above 100 kV, as the heat caused by power losses in resistor divider probes at such high voltages could be excessive.
953:
86:
at low frequencies. For direct current and relatively low frequencies, a voltage divider may be sufficiently accurate if made only of resistors; where frequency response over a wide range is required (such as in an
479:
2632:
2491:
coupled with a stable voltage reference, the output voltage can be fed into an analog-to-digital converter and a display can show the angle. Such circuits are commonly used in reading control knobs.
2324:{\displaystyle V_{\mathrm {out} }={\frac {Xc_{2}}{Xc_{1}+Xc_{2}}}\cdot V_{\mathrm {in} }={\frac {1/C_{2}}{1/C_{1}+1/C_{2}}}\cdot V_{\mathrm {in} }={\frac {C_{1}}{C_{1}+C_{2}}}\cdot V_{\mathrm {in} }}
356:
91:
probe), a voltage divider may have capacitive elements added to compensate load capacitance. In electric power transmission, a capacitive voltage divider is used for measurement of high voltage.
978:
2591:
411:
1840:
1311:
2633:"A very quick and dirty introduction to Sensors, Microcontrollers, and Electronics; Part Three: how sensors and actuators work and how to hook them up to a microcontroller"
82:
Resistor voltage dividers are commonly used to create reference voltages, or to reduce the magnitude of a voltage so it can be measured, and may also be used as signal
2446:
of both halves of the divider, though this increases the divider's quiescent input current and results in higher power consumption (and wasted heat) in the divider.
594:
1288:{\displaystyle R_{1}={\frac {R_{2}\cdot V_{\mathrm {in} }}{V_{\mathrm {out} }}}-R_{2}=R_{2}\cdot \left({{\frac {V_{\mathrm {in} }}{V_{\mathrm {out} }}}-1}\right)}
2344:
elements in the proper proportions, the same division ratio can be maintained over a useful range of frequencies. This is the principle applied in compensated
1489:
1958:
777:
485:
185:
892:
2458:
Voltage dividers are used for adjusting the level of a signal, for bias of active devices in amplifiers, and for measurement of voltages. A
2336:
Any leakage current in the capactive elements requires use of the generalized expression with two impedances. By selection of parallel
2666:
1822:
The ratio then depends on frequency, in this case decreasing as frequency increases. This circuit is, in fact, a basic (first-order)
71:
is the result of distributing the input voltage among the components of the divider. A simple example of a voltage divider is two
417:
2484:
2619:
It is required for the R1 resistor to be at least ten times larger than R2 to avoid a divider circuit (R2 ≤ 10 R1).
2600:
291:
79:, with the input voltage applied across the resistor pair and the output voltage emerging from the connection between them.
1428:
greater than 1 is not possible. That is, using resistors alone it is not possible to either invert the voltage or increase
2450:
are often used in lieu of passive voltage dividers when it is necessary to accommodate high or fluctuating load currents.
1090:{\displaystyle {\frac {V_{\mathrm {out} }}{V_{\mathrm {in} }}}={\frac {R_{2}}{R_{1}+R_{2}}}={\frac {6}{9}}={\frac {2}{3}}}
76:
1938:{\displaystyle \left|{\frac {V_{\mathrm {out} }}{V_{\mathrm {in} }}}\right|={\frac {1}{\sqrt {1+(\omega RC)^{2}}}}\ .}
362:
2356:
The output voltage of a voltage divider will vary according to the electric current it is supplying to its external
2066:
Inductive dividers split AC input according to the reactance of the elements as for the resistive divider above.
2681:
83:
1404:{\displaystyle R_{2}=R_{1}\cdot {\frac {1}{\left({{\frac {V_{\mathrm {in} }}{V_{\mathrm {out} }}}-1}\right)}}}
2534:
1831:
1576:
2514:
probe tip and ground is connected to the other end of the divider via the black cable. The divider output (
112:
108:
2593:
SN74HCS72 Schmitt-Trigger Input Dual D-Type
Negative-Edge-Triggered Flip-Flops With Clear and Preset
2547:
2399:
2565:
115:
in series, as shown in Figure 1. The input voltage is applied across the series impedances
2570:
2459:
2447:
2382:
2056:
701:
686:{\displaystyle H={\frac {V_{\mathrm {out} }}{V_{\mathrm {in} }}}={\frac {Z_{2}}{Z_{1}+Z_{2}}}}
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1600:
697:
165:
If the current in the output wire is zero then the relationship between the input voltage,
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2060:
1823:
43:
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1558:{\displaystyle Z_{2}=-\mathrm {j} X_{\mathrm {C} }={\frac {1}{\mathrm {j} \omega C}}\ ,}
1826:. The ratio contains an imaginary number, and actually contains both the amplitude and
1592:
46:
2043:{\displaystyle V_{\mathrm {out} }={\frac {L_{2}}{L_{1}+L_{2}}}\cdot V_{\mathrm {in} }}
862:{\displaystyle V_{\mathrm {out} }={\frac {R_{2}}{R_{1}+R_{2}}}\cdot V_{\mathrm {in} }}
570:{\displaystyle V_{\mathrm {out} }=V_{\mathrm {in} }\cdot {\frac {Z_{2}}{Z_{1}+Z_{2}}}}
270:{\displaystyle V_{\mathrm {out} }={\frac {Z_{2}}{Z_{1}+Z_{2}}}\cdot V_{\mathrm {in} }}
27:
Linear circuit that produces an output voltage that is a fraction of its input voltage
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1827:
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17:
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information of the filter. To extract just the amplitude ratio, calculate the
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is used as a variable voltage divider in the volume control of many radios.
1459:
705:
159:
155:
151:
72:
948:{\displaystyle V_{\mathrm {out} }={\frac {1}{2}}\cdot V_{\mathrm {in} }}
1450:
1101:
50:
2498:
720:
98:
2503:
High voltage (HV) resistor divider probe. The HV to be measured (
1953:
Inductive dividers split AC input according to inductance:
2360:. The effective source impedance coming from a divider of
2333:(with components in the same positions as Figure 2.)
2052:(with components in the same positions as Figure 2.)
474:{\displaystyle I={\frac {V_{\mathrm {in} }}{Z_{1}+Z_{2}}}}
729:
A resistive divider is the case where both impedances,
150:
may be composed of any combination of elements such as
972: = 9 V (both commonly used voltages), then:
351:{\displaystyle V_{\mathrm {in} }=I\cdot (Z_{1}+Z_{2})}
2085:
2055:
The above equation is for non-interacting inductors;
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1843:
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2525:A voltage divider can be used to scale down a very
2323:
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1937:
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2521:) appears on the connector adjacent to the cable.
2077:For an AC input a simple capacitive equation is:
1454:Figure 3: Resistor/capacitor voltage divider
1458:Consider a divider consisting of a resistor and
1606:This divider will then have the voltage ratio:
725:Figure 2: Simple resistive voltage divider
406:{\displaystyle V_{\mathrm {out} }=I\cdot Z_{2}}
8:
1483:is the impedance of the capacitor, given by
60:) that is a fraction of its input voltage (
2348:probes to increase measurement bandwidth.
2074:Capacitive dividers do not pass DC input.
2546:A voltage divider can be used as a crude
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1465:Comparing with the general case, we see
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743:, are purely resistive (Figure 2).
2582:
696:In general this transfer function is a
103:Figure 1: A simple voltage divider
129:and the output is the voltage across
7:
2603:. June 2020 . p. 11. SCLS801A.
771:into the previous expression gives:
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25:
2466:both include voltage dividers. A
2529:so that it can be measured by a
107:A voltage divider referenced to
2645:from the original on 2023-07-20
2610:from the original on 2023-07-20
2601:Texas Instruments Incorporated
1917:
1904:
345:
319:
1:
2590:"9.2.1 Design Requirements".
584:(also known as the divider's
111:is created by connecting two
1462:as shown in Figure 3.
1111:must be twice the value of
2703:
2667:Voltage Divider Calculator
2063:) will alter the results.
172:, and the output voltage,
2535:temperature coefficients
2495:High voltage measurement
49:that produces an output
1834:of the ratio, that is:
2522:
2325:
2044:
1939:
1796:
1603:of the input voltage.
1559:
1455:
1405:
1289:
1091:
949:
863:
726:
687:
588:) of this circuit is:
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407:
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2502:
2326:
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1406:
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1100:and by solving using
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965: = 6 V and
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864:
724:
688:
572:
476:
408:
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272:
113:electrical impedances
102:
2542:Logic level shifting
2510:) is applied to the
2374:, as above, will be
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2548:logic level shifter
2392:(sometimes written
2566:DC-to-DC converter
2523:
2474:Sensor measurement
2448:Voltage regulators
2321:
2070:Capacitive divider
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1935:
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1587:of the capacitor,
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1446:Low-pass RC filter
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1949:Inductive divider
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717:Resistive divider
702:rational function
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40:potential divider
38:(also known as a
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69:Voltage division
36:voltage divider
21:
18:Voltage division
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2682:Analog circuits
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2561:Current divider
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2061:autotransformer
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2661:External links
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2638:. 2014-07-02.
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2617:. p. 11:
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2518:
2507:
2496:
2493:
2475:
2472:
2455:
2452:
2439:
2432:
2425:
2418:
2412:
2405:
2396:
2389:
2378:
2371:
2364:
2353:
2352:Loading effect
2350:
2317:
2314:
2309:
2305:
2297:
2293:
2289:
2284:
2280:
2273:
2269:
2263:
2257:
2254:
2249:
2245:
2237:
2233:
2228:
2224:
2221:
2216:
2212:
2207:
2203:
2196:
2192:
2187:
2183:
2177:
2171:
2168:
2163:
2159:
2151:
2147:
2143:
2140:
2135:
2131:
2127:
2120:
2116:
2112:
2106:
2100:
2097:
2094:
2089:
2071:
2068:
2036:
2033:
2028:
2024:
2016:
2012:
2008:
2003:
1999:
1992:
1988:
1982:
1976:
1973:
1970:
1965:
1950:
1947:
1946:
1945:
1934:
1923:
1919:
1915:
1912:
1909:
1906:
1903:
1900:
1896:
1891:
1887:
1879:
1876:
1871:
1864:
1861:
1858:
1853:
1847:
1813:is called the
1803:
1802:
1791:
1782:
1779:
1776:
1772:
1768:
1765:
1761:
1756:
1750:
1747:
1741:
1738:
1734:
1729:
1721:
1718:
1714:
1709:
1703:
1694:
1689:
1685:
1679:
1674:
1666:
1661:
1655:
1647:
1644:
1639:
1632:
1629:
1626:
1621:
1593:imaginary unit
1572:
1566:
1565:
1554:
1545:
1542:
1538:
1533:
1528:
1522:
1517:
1512:
1508:
1505:
1500:
1496:
1480:
1469:
1447:
1444:
1439:
1432:
1425:
1418:
1412:
1411:
1397:
1393:
1390:
1382:
1379:
1376:
1371:
1364:
1361:
1356:
1349:
1345:
1340:
1335:
1331:
1327:
1322:
1318:
1302:
1296:
1295:
1283:
1279:
1276:
1268:
1265:
1262:
1257:
1250:
1247:
1242:
1235:
1231:
1226:
1222:
1218:
1213:
1209:
1205:
1197:
1194:
1191:
1186:
1178:
1175:
1170:
1166:
1161:
1157:
1150:
1145:
1141:
1125:
1115:
1108:
1098:
1097:
1084:
1081:
1076:
1071:
1068:
1063:
1055:
1051:
1047:
1042:
1038:
1031:
1027:
1021:
1013:
1010:
1005:
998:
995:
992:
987:
969:
962:
956:
955:
941:
938:
933:
929:
924:
921:
916:
910:
907:
904:
899:
883:
876:
870:
869:
855:
852:
847:
843:
835:
831:
827:
822:
818:
811:
807:
801:
795:
792:
789:
784:
768:
764:
757:
750:
740:
733:
718:
715:
713:
710:
694:
693:
677:
673:
669:
664:
660:
653:
649:
643:
635:
632:
627:
620:
617:
614:
609:
603:
600:
578:
577:
561:
557:
553:
548:
544:
537:
533:
527:
521:
518:
513:
509:
503:
500:
497:
492:
481:
465:
461:
457:
452:
448:
440:
437:
432:
426:
423:
413:
400:
396:
392:
389:
386:
380:
377:
374:
369:
358:
347:
342:
338:
334:
329:
325:
321:
318:
315:
312:
306:
303:
298:
278:
277:
263:
260:
255:
251:
243:
239:
235:
230:
226:
219:
215:
209:
203:
200:
197:
192:
176:
169:
147:
140:
133:
126:
119:
96:
93:
64:
57:
47:linear circuit
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2699:
2688:
2685:
2683:
2680:
2679:
2677:
2668:
2665:
2664:
2660:
2641:
2634:
2628:
2625:
2620:
2606:
2602:
2595:
2594:
2586:
2583:
2576:
2572:
2569:
2567:
2564:
2562:
2559:
2558:
2554:
2552:
2549:
2541:
2539:
2536:
2532:
2528:
2517:
2513:
2506:
2501:
2494:
2492:
2488:
2486:
2482:
2473:
2471:
2469:
2468:potentiometer
2465:
2461:
2453:
2451:
2449:
2443:
2438:
2431:
2424:
2417:
2411:
2408:), that is: (
2404:
2401:
2395:
2388:
2384:
2377:
2370:
2363:
2359:
2351:
2349:
2347:
2343:
2339:
2334:
2331:
2307:
2303:
2295:
2291:
2287:
2282:
2278:
2271:
2267:
2261:
2247:
2243:
2235:
2231:
2226:
2222:
2219:
2214:
2210:
2205:
2201:
2194:
2190:
2185:
2181:
2175:
2161:
2157:
2149:
2145:
2141:
2138:
2133:
2129:
2125:
2118:
2114:
2110:
2104:
2087:
2078:
2075:
2069:
2067:
2064:
2062:
2058:
2053:
2050:
2026:
2022:
2014:
2010:
2006:
2001:
1997:
1990:
1986:
1980:
1963:
1954:
1948:
1932:
1921:
1913:
1910:
1907:
1901:
1898:
1894:
1889:
1885:
1869:
1851:
1845:
1837:
1836:
1835:
1833:
1829:
1825:
1820:
1818:
1817:
1816:time constant
1812:
1809:(tau) =
1808:
1789:
1780:
1777:
1774:
1766:
1763:
1759:
1754:
1748:
1745:
1739:
1736:
1727:
1719:
1716:
1707:
1701:
1692:
1687:
1683:
1677:
1672:
1664:
1659:
1653:
1637:
1619:
1609:
1608:
1607:
1604:
1602:
1598:
1594:
1590:
1586:
1582:
1578:
1571:
1552:
1543:
1540:
1531:
1526:
1515:
1506:
1503:
1498:
1494:
1486:
1485:
1484:
1479:
1475:
1468:
1463:
1461:
1452:
1445:
1443:
1438:
1431:
1424:
1417:
1395:
1391:
1388:
1369:
1354:
1347:
1343:
1338:
1333:
1329:
1325:
1320:
1316:
1308:
1307:
1306:
1301:
1298:To solve for
1281:
1277:
1274:
1255:
1240:
1233:
1229:
1224:
1220:
1216:
1211:
1207:
1203:
1184:
1168:
1164:
1159:
1155:
1148:
1143:
1139:
1131:
1130:
1129:
1124:
1121:To solve for
1119:
1114:
1107:
1103:
1082:
1079:
1074:
1069:
1066:
1061:
1053:
1049:
1045:
1040:
1036:
1029:
1025:
1019:
1003:
985:
975:
974:
973:
968:
961:
931:
927:
922:
919:
914:
897:
889:
888:
887:
882:
875:
845:
841:
833:
829:
825:
820:
816:
809:
805:
799:
782:
774:
773:
772:
763:
756:
749:
746:Substituting
744:
739:
732:
723:
716:
711:
709:
707:
703:
699:
675:
671:
667:
662:
658:
651:
647:
641:
625:
607:
601:
598:
591:
590:
589:
587:
586:voltage ratio
583:
559:
555:
551:
546:
542:
535:
531:
525:
511:
507:
490:
482:
463:
459:
455:
450:
446:
430:
424:
421:
414:
398:
394:
390:
387:
384:
367:
359:
340:
336:
332:
327:
323:
316:
313:
310:
296:
288:
287:
286:
283:
280:Proof (using
253:
249:
241:
237:
233:
228:
224:
217:
213:
207:
190:
182:
181:
180:
175:
168:
163:
161:
157:
153:
146:
139:
132:
125:
118:
114:
110:
101:
94:
92:
90:
85:
80:
78:
75:connected in
74:
70:
63:
56:
52:
48:
45:
41:
37:
33:
19:
2647:. Retrieved
2627:
2618:
2612:. Retrieved
2592:
2585:
2545:
2527:high voltage
2524:
2515:
2504:
2489:
2477:
2457:
2454:Applications
2444:
2436:
2429:
2422:
2415:
2409:
2402:
2393:
2386:
2375:
2368:
2361:
2355:
2346:oscilloscope
2341:
2337:
2335:
2332:
2079:
2076:
2073:
2065:
2054:
2051:
1955:
1952:
1821:
1814:
1810:
1806:
1805:The product
1804:
1605:
1596:
1588:
1580:
1569:
1567:
1477:
1473:
1466:
1464:
1457:
1436:
1429:
1422:
1415:
1413:
1299:
1297:
1122:
1120:
1112:
1105:
1099:
966:
959:
957:
880:
873:
871:
761:
754:
747:
745:
737:
730:
728:
695:
585:
579:
279:
173:
166:
164:
144:
137:
130:
123:
116:
106:
95:General case
89:oscilloscope
81:
68:
61:
54:
39:
35:
29:
2512:corona ball
2481:thermistors
1828:phase shift
1585:capacitance
84:attenuators
32:electronics
2676:Categories
2649:2015-11-02
2622:(23 pages)
2614:2023-07-20
2577:References
2531:volt meter
2464:multimeter
2421:) / (
2059:(as in an
1414:Any ratio
160:capacitors
2435:) =
2304:⋅
2244:⋅
2158:⋅
2023:⋅
1908:ω
1832:magnitude
1775:ω
1737:ω
1717:ω
1577:reactance
1541:ω
1507:−
1460:capacitor
1389:−
1339:⋅
1275:−
1230:⋅
1204:−
1165:⋅
928:⋅
842:⋅
767: = R
706:frequency
526:⋅
391:⋅
317:⋅
282:Ohm's law
250:⋅
156:inductors
152:resistors
73:resistors
2640:Archived
2605:Archived
2555:See also
2428: +
2383:parallel
1472: =
1421: /
879: =
753: =
712:Examples
2687:Voltage
1591:is the
1583:is the
1575:is the
1102:algebra
698:complex
51:voltage
44:passive
42:) is a
2462:and a
1930:
1787:
1595:, and
1568:where
1550:
1435:above
179:, is:
109:ground
77:series
2643:(PDF)
2636:(PDF)
2608:(PDF)
2597:(PDF)
2385:with
886:then
2485:RTDs
2483:and
2367:and
2340:and
1476:and
760:and
736:and
580:The
158:and
143:and
122:and
34:, a
2519:out
2381:in
1433:out
1419:out
963:out
958:If
872:If
704:of
177:out
67:).
58:out
30:In
2678::
2599:.
2508:in
2487:.
2442:.
2437:HZ
2400://
1811:RC
1442:.
1440:in
1426:in
1305::
1128::
1118:.
1104:,
970:in
708:.
700:,
284:):
170:in
162:.
154:,
136:.
65:in
2652:.
2516:V
2505:V
2440:1
2433:2
2430:Z
2426:1
2423:Z
2419:2
2416:Z
2413:1
2410:Z
2406:2
2403:Z
2397:1
2394:Z
2390:2
2387:Z
2379:1
2376:Z
2372:2
2369:Z
2365:1
2362:Z
2342:C
2338:R
2316:n
2313:i
2308:V
2296:2
2292:C
2288:+
2283:1
2279:C
2272:1
2268:C
2262:=
2256:n
2253:i
2248:V
2236:2
2232:C
2227:/
2223:1
2220:+
2215:1
2211:C
2206:/
2202:1
2195:2
2191:C
2186:/
2182:1
2176:=
2170:n
2167:i
2162:V
2150:2
2146:c
2142:X
2139:+
2134:1
2130:c
2126:X
2119:2
2115:c
2111:X
2105:=
2099:t
2096:u
2093:o
2088:V
2035:n
2032:i
2027:V
2015:2
2011:L
2007:+
2002:1
1998:L
1991:2
1987:L
1981:=
1975:t
1972:u
1969:o
1964:V
1933:.
1922:2
1918:)
1914:C
1911:R
1905:(
1902:+
1899:1
1895:1
1890:=
1886:|
1878:n
1875:i
1870:V
1863:t
1860:u
1857:o
1852:V
1846:|
1807:τ
1790:.
1781:C
1778:R
1771:j
1767:+
1764:1
1760:1
1755:=
1749:R
1746:+
1740:C
1733:j
1728:1
1720:C
1713:j
1708:1
1702:=
1693:2
1688:Z
1684:+
1678:1
1673:Z
1665:2
1660:Z
1654:=
1646:n
1643:i
1638:V
1631:t
1628:u
1625:o
1620:V
1597:ω
1589:j
1581:C
1573:C
1570:X
1553:,
1544:C
1537:j
1532:1
1527:=
1521:C
1516:X
1511:j
1504:=
1499:2
1495:Z
1481:2
1478:Z
1474:R
1470:1
1467:Z
1437:V
1430:V
1423:V
1416:V
1396:)
1392:1
1381:t
1378:u
1375:o
1370:V
1363:n
1360:i
1355:V
1348:(
1344:1
1334:1
1330:R
1326:=
1321:2
1317:R
1303:2
1300:R
1282:)
1278:1
1267:t
1264:u
1261:o
1256:V
1249:n
1246:i
1241:V
1234:(
1225:2
1221:R
1217:=
1212:2
1208:R
1196:t
1193:u
1190:o
1185:V
1177:n
1174:i
1169:V
1160:2
1156:R
1149:=
1144:1
1140:R
1126:1
1123:R
1116:1
1113:R
1109:2
1106:R
1083:3
1080:2
1075:=
1070:9
1067:6
1062:=
1054:2
1050:R
1046:+
1041:1
1037:R
1030:2
1026:R
1020:=
1012:n
1009:i
1004:V
997:t
994:u
991:o
986:V
967:V
960:V
940:n
937:i
932:V
923:2
920:1
915:=
909:t
906:u
903:o
898:V
884:2
881:R
877:1
874:R
854:n
851:i
846:V
834:2
830:R
826:+
821:1
817:R
810:2
806:R
800:=
794:t
791:u
788:o
783:V
769:2
765:2
762:Z
758:1
755:R
751:1
748:Z
741:2
738:Z
734:1
731:Z
676:2
672:Z
668:+
663:1
659:Z
652:2
648:Z
642:=
634:n
631:i
626:V
619:t
616:u
613:o
608:V
602:=
599:H
560:2
556:Z
552:+
547:1
543:Z
536:2
532:Z
520:n
517:i
512:V
508:=
502:t
499:u
496:o
491:V
464:2
460:Z
456:+
451:1
447:Z
439:n
436:i
431:V
425:=
422:I
399:2
395:Z
388:I
385:=
379:t
376:u
373:o
368:V
346:)
341:2
337:Z
333:+
328:1
324:Z
320:(
314:I
311:=
305:n
302:i
297:V
262:n
259:i
254:V
242:2
238:Z
234:+
229:1
225:Z
218:2
214:Z
208:=
202:t
199:u
196:o
191:V
174:V
167:V
148:2
145:Z
141:1
138:Z
134:2
131:Z
127:2
124:Z
120:1
117:Z
62:V
55:V
53:(
20:)
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