1325:
1313:
2631:
2639:
2189:
1777:
25:
1464:
1304:= approximately 37% of its initial value. In case 4, after five time constants the function reaches a value less than 1% of its original. In most cases this 1% threshold is considered sufficient to assume that the function has decayed to zero – as a rule of thumb, in control engineering a stable system is one that exhibits such an overall damped behavior.
2056:
174:
Physically, the time constant represents the elapsed time required for the system response to decay to zero if the system had continued to decay at the initial rate, because of the progressive change in the rate of decay the response will have actually decreased in value to
1772:{\displaystyle {\begin{aligned}V(t)&=V_{0}e^{-t/\tau }+{\frac {Ae^{-t/\tau }}{\tau }}\int _{0}^{t}\,dt'\ e^{t'/\tau }e^{j\omega t'}\\&=V_{0}e^{-t/\tau }+{\frac {\frac {1}{\tau }}{j\omega +{\frac {1}{\tau }}}}A\left(e^{j\omega t}-e^{-t/\tau }\right).\end{aligned}}}
2529:
The time constant remains the same for the same system regardless of the starting conditions. Simply stated, a system approaches its final, steady-state situation at a constant rate, regardless of how close it is to that value at any arbitrary starting point.
2826:
is present, a system may exhibit unstable, increasing oscillations. In addition, physical electrical circuits are seldom truly linear systems except for very low amplitude excitations; however, the approximation of linearity is widely used.
1892:
1885:
3298:
482:
3215:
203:), and it represents both the mean lifetime of a decaying system (such as an atom) before it decays, or the time it takes for all but 36.8% of the atoms to decay. For this reason, the time constant is longer than the
2474:
3489:
3789:
2319:
1469:
1427:
2149:
4249:
is the amount of time it takes for a meteorological sensor to respond to a rapid change in a measure, and until it is measuring values within the accuracy tolerance usually expected of the sensor.
3578:
3365:
3856:
4195:
2955:
1293:
3074:
1078:
280:
3940:
1149:
1200:
842:
708:
2705:
610:
2863:
2525:
556:
4234:
3674:
955:
758:
2787:
1790:
1231:
1016:
988:
335:
4132:
4029:
3980:
3882:
3631:
2758:
2668:
2565:
of a second, the motor has sped up an additional 23 RPM, which equals 63% of that 37 RPM difference. This brings it to 86 RPM—still 14 RPM low. After a third
2224:
3224:
912:
402:
3110:
2080:
convention, defined as the frequency range where power drops by less than half (at most −3 dB). Using the frequency in hertz, rather than radians/s (
2881:. In this case, the heat transfer from the body to the ambient at a given time is proportional to the temperature difference between the body and the ambient:
2367:
3429:
2533:
For example, consider an electric motor whose startup is well modelled by a first-order LTI system. Suppose that when started from rest, the motor takes
3713:
2235:
171:(lumped capacity analysis method) for thermal systems, used when objects cool or warm uniformly under the influence of convective cooling or warming.
133:
of a first-order time-invariant system, that is, the frequency at which the output signal power drops to half the value it has at low frequencies.
43:
1346:
2095:
2051:{\displaystyle |V_{\infty }(t)|=A{\frac {1}{\tau \left(\omega ^{2}+(1/\tau )^{2}\right)^{1/2}}}=A{\frac {1}{\sqrt {1+(\omega \tau )^{2}}}}.}
3497:. The solution to this equation suggests that, in such systems, the difference between the temperature of the system and its surroundings Δ
2549:
of a second to reach 63% of its nominal speed of 100 RPM, or 63 RPM—a shortfall of 37 RPM. Then it will be found that after the next
3307:
3796:
4137:
2886:
850:(tau) is referred to as the "time constant" and can be used (as in this case) to indicate how rapidly an exponential function decays.
3014:
1431:(Response to a real cosine or sine wave input can be obtained by taking the real or imaginary part of the final result by virtue of
3594:= 0. In words, the body assumes the same temperature as the ambient at an exponentially slow rate determined by the time constant.
224:
3900:
4534:
4442:
4413:
4380:
4061:
3009:
is an outward flux). As heat is lost to the ambient, this heat transfer leads to a drop in temperature of the body given by:
781:
4056:
The larger a time constant is, the slower the rise or fall of the potential of a neuron. A long time constant can result in
650:
129:
of the step response, or using an input that is a simple sinusoidal function of time) the time constant also determines the
3511:
2677:
561:
2491:
502:
2877:(lumped capacity analysis method) for thermal systems, used when objects cool or warm uniformly under the influence of
156:– which can be modelled or approximated by first-order LTI systems. Other examples include time constant used in
3636:
1236:
4496:
195:
of its final (asymptotic) value (say from a step increase). In radioactive decay the time constant is related to the
61:
1021:
2581:
of a second, the motor will have gained an additional 9 RPM (63% of that 14 RPM difference), putting it at 95 RPM.
715:
183:
in this time (say from a step decrease). In an increasing system, the time constant is the time for the system's
4337:
2763:
1085:
2814:
Electrical circuits are often more complex than these examples, and may exhibit multiple time constants (See
2077:
302:
130:
3710:
The time constant is used to describe the rise and fall of membrane voltage, where the rise is described by
1154:
216:
4252:
This most often applies to measurements of temperature, dew-point temperature, humidity and air pressure.
4578:
2837:
2823:
4112:
3494:
2878:
4205:
2964:
1316:
An example response of system to sine wave forcing function. Time axis in units of the time constant
1337:. The response is normalized to a zero frequency value of unity, and drops to 1/√2 at the bandwidth.
426:
4465:
4281:
917:
385:
114:
4405:
3097:
is the body volume. The negative sign indicates the temperature drops when the heat transfer is
2874:
168:
4524:
4370:
1207:
4583:
4437:(NATO Advanced Study Institute on Physical-Based Modelling ed.). Springer. p. 627.
4430:
4060:, or the algebraic summation of repeated potentials. A short time constant rather produces a
2714:
995:
960:
4553:
4117:
4014:
3965:
3867:
3616:
2743:
2653:
2209:
485:
122:
8:
891:
103:
3413:. However, retaining the simple constant ambient example, by substituting the variable Δ
4588:
4558:
4502:
4452:
4398:
4301:
4057:
3611:
3493:
Systems for which cooling satisfies the above exponential equation are said to satisfy
2335:
1432:
39:
4345:
4341:
4530:
4492:
4438:
4409:
4376:
4276:
4087:
4083:
4077:
4065:
3894:
2997:
is the constant ambient temperature. The positive sign indicates the convention that
351:
describing an external driving function of time, which can be regarded as the system
289:
145:
137:
126:
99:
4522:
4506:
4484:
4306:
4271:
4266:
4046:, meaning that the time constant is the time elapsed after it has fallen to 37% of
118:
2488:
The long-time solution is time independent and independent of initial conditions:
4336:
Concretely, a first-order LTI system is a system that can be modeled by a single
4286:
1880:{\displaystyle V_{\infty }(t)={\frac {1/\tau }{j\omega +1/\tau }}Ae^{j\omega t}.}
1324:
1312:
136:
The time constant is also used to characterize the frequency response of various
109:
In the time domain, the usual choice to explore the time response is through the
4481:
The Thirty-Seventh
Asilomar Conference on Signals, Systems & Computers, 2003
4488:
4479:
Harris, D.; Sutherland, I. (2003). "Logical effort of carry propagate adders".
4198:
2819:
2199:, one above the final value and one at zero. Long-time response is a constant,
196:
157:
153:
149:
87:
2630:
4572:
4316:
4311:
4091:
3704:
3293:{\displaystyle {\frac {dT}{dt}}+{\frac {1}{\tau }}T={\frac {1}{\tau }}T_{a},}
3090:
2834:
is often used. This can be converted to time constant units via the equation
2815:
2638:
184:
141:
110:
2188:
846:
This behavior is referred to as a "decaying" exponential function. The time
477:{\displaystyle u(t)={\begin{cases}0,&t<0\\1,&t\geq 0\end{cases}}}
3700:
3210:{\displaystyle \rho c_{p}V{\frac {dT}{dt}}=-hA_{s}\left(T(t)-T_{a}\right).}
2730:
1782:
3406:
suggests the possible generalization to time-varying ambient temperatures
3219:
Evidently, this is a first-order LTI system that can be cast in the form:
2166:
and the observation that half-power corresponds to a drop in the value of
3693:
3005:
the body because its temperature is higher than the ambient temperature (
2831:
2804:
2469:{\displaystyle V(t)=V_{0}e^{-t/\tau }+A\tau \left(1-e^{-t/\tau }\right).}
79:
4435:
Physically-based
Modelling and Simulation of Climate and Climatic Change
4253:
3699:
The resistance across the membrane is a function of the number of open
2737:
2726:
2647:
2610:
of a second later this particular motor will have gained an additional
221:
First order LTI systems are characterized by the differential equation
4296:
4291:
4095:
3997:. This means that the time constant is the time elapsed after 63% of
3484:{\displaystyle {\frac {d\Delta T}{dt}}+{\frac {1}{\tau }}\Delta T=0.}
1328:
Frequency response of system vs. frequency in units of the bandwidth
204:
161:
95:
638:
An example solution to the differential equation with initial value
4563:
4554:
Conversion of time constant τ to cutoff frequency fc and vice versa
2163:
2058:
By convention, the bandwidth of this system is the frequency where
1781:
For long times the decaying exponentials become negligible and the
2183:
4526:
Fundamentals of the finite element method for heat and fluid flow
4523:
Roland Wynne Lewis; Perumal
Nithiarasu; K. N. Seetharamu (2004).
4256:
are especially affected due to their rapid increase in altitude.
3861:
3784:{\displaystyle V(t)=V_{\textrm {max}}\left(1-e^{-t/\tau }\right)}
3603:
2180:
Thus, the time constant determines the bandwidth of this system.
75:
4372:
Instrument
Engineers' Handbook: Process control and optimization
2314:{\displaystyle {\frac {dV}{dt}}+{\frac {1}{\tau }}V=f(t)=Au(t),}
766:. Thus, the response is an exponential decay with time constant
160:
for integral and derivative action controllers, which are often
4340:
in time. Examples include the simplest single-stage electrical
3607:
3105:> 0). Equating these two expressions for the heat transfer,
2740:
composed of a single resistor and capacitor, the time constant
2671:
4404:. American Institute of Aeronautics and Astronautics. p.
2650:
composed of a single resistor and inductor, the time constant
98:
characterizing the response to a step input of a first-order,
3395:
lead to more rapid temperature change (shorter time constant
2808:
106:
of a first-order LTI system. It gives speed of the response.
3380:
lead to slower changes in temperature (longer time constant
1422:{\displaystyle \tau {\frac {dV}{dt}}+V=f(t)=Ae^{j\omega t}.}
125:
input. In the frequency domain (for example, looking at the
3703:
and the capacitance is a function of the properties of the
2230:
Suppose the forcing function is chosen as a step input so:
2144:{\displaystyle f_{\mathrm {3dB} }={\frac {1}{2\pi \tau }}.}
470:
2796:
2718:
2192:
Step response of system for two different initial values
1341:
Suppose the forcing function is chosen as sinusoidal so:
1307:
1298:
After a period of one time constant the function reaches
2625:
207:, which is the time for only 50% of the atoms to decay.
1320:. The response damps out to become a simple sine wave.
1157:
4559:
All about circuits – Voltage and current calculations
4208:
4140:
4120:
4090:
isotope, the time constant can be interpreted as the
4017:
3968:
3903:
3870:
3799:
3716:
3639:
3619:
3573:{\displaystyle \Delta T(t)=\Delta T_{0}e^{-t/\tau },}
3514:
3432:
3310:
3227:
3113:
3017:
2889:
2840:
2766:
2746:
2680:
2656:
2494:
2370:
2238:
2212:
2098:
1895:
1793:
1467:
1349:
1239:
1210:
1088:
1024:
998:
963:
920:
894:
784:
718:
653:
564:
505:
405:
305:
227:
3360:{\displaystyle \tau ={\frac {\rho c_{p}V}{hA_{s}}}.}
2830:
In digital electronic circuits another measure, the
2485:
limit of the above response to a sinusoidal input.)
1435:.) The general solution to this equation for times
34:
may be too technical for most readers to understand
4397:
4228:
4197:The reciprocal of the time constant is called the
4189:
4126:
4023:
3974:
3934:
3893:is defined as the maximum voltage change from the
3876:
3851:{\displaystyle V(t)=V_{\textrm {max}}e^{-t/\tau }}
3850:
3783:
3668:
3625:
3572:
3483:
3359:
3292:
3209:
3068:
2949:
2857:
2781:
2752:
2699:
2662:
2519:
2468:
2338:. The general solution to this equation for times
2313:
2218:
2143:
2050:
1879:
1771:
1421:
1287:
1225:
1194:
1143:
1072:
1010:
982:
949:
906:
878:is the initial value (see "specific cases" below).
836:
752:
702:
604:
550:
476:
329:
274:
4190:{\displaystyle T_{1/2}=T_{\text{HL}}=\tau \ln 2.}
4570:
4478:
2950:{\displaystyle F=hA_{s}\left(T(t)-T_{a}\right),}
1288:{\displaystyle V=V_{0}e^{-5}\approx 0.0067V_{0}}
1159:
3597:
3590:is the initial temperature difference, at time
3069:{\displaystyle \rho c_{p}V{\frac {dT}{dt}}=-F,}
2184:Step response with arbitrary initial conditions
4368:
2478:(It may be observed that this response is the
1073:{\displaystyle V=V_{0}e^{-1}\approx 0.37V_{0}}
4564:Energy and Thermal Time Constant of Buildings
275:{\displaystyle \tau {\frac {dV}{dt}}+V=f(t)}
102:(LTI) system. The time constant is the main
370:, or system output. Classical examples for
152:, record cutting and replay equipment, and
3951:is the resistance across the membrane and
3864:is in millivolts, time is in seconds, and
3685:is the resistance across the membrane and
3403:
2206:. Time axis in units of the time constant
616:is the amplitude of the forcing function,
499:, and also the sinusoidal input function:
4240:
1573:
62:Learn how and when to remove this message
46:, without removing the technical details.
4428:
4395:
3935:{\displaystyle V_{\textrm {max}}=r_{m}I}
2868:
2637:
2629:
2187:
1323:
1311:
1144:{\displaystyle V=f(t)=V_{0}e^{-t/\tau }}
630:is the frequency in radians per second.
210:
4518:
4516:
1195:{\textstyle \lim _{t\to \infty }f(t)=0}
837:{\displaystyle V(t)=V_{0}e^{-t/\tau }.}
4571:
4375:(4 ed.). CRC Press. p. 100.
2822:for some examples.) In the case where
2162:stems from the expression of power in
1308:Relation of time constant to bandwidth
703:{\displaystyle V(t)=V_{0}e^{-t/\tau }}
4362:
2626:Time constants in electrical circuits
2177:by a factor of 1/2 or by 3 decibels.
44:make it understandable to non-experts
4513:
4431:"Lessons from energy balance models"
4071:
2873:Time constants are a feature of the
2700:{\displaystyle \tau ={\frac {L}{R}}}
605:{\displaystyle f(t)=Ae^{j\omega t},}
167:Time constants are a feature of the
18:
4433:. In Michael E. Schlesinger (ed.).
4422:
2978:is the temperature function, i.e.,
2858:{\displaystyle 5\tau ={\text{FO4}}}
2520:{\displaystyle V_{\infty }=A\tau .}
1889:The magnitude of this response is:
1785:solution or long-time solution is:
633:
551:{\displaystyle f(t)=A\sin(2\pi ft)}
16:Constant in physics and engineering
13:
4400:Space vehicle dynamics and control
4389:
3533:
3515:
3469:
3439:
2986:) is the body temperature at time
2500:
2111:
2108:
1906:
1799:
1169:
14:
4600:
4547:
4338:first order differential equation
3402:Comparison with the introductory
882:
4229:{\displaystyle \lambda =1/\tau }
3669:{\displaystyle \tau =r_{m}c_{m}}
2634:Capacitor voltage step-response.
23:
3602:In an excitable cell such as a
2642:Inductor voltage step-response.
620:is the frequency in Hertz, and
4472:
4330:
3809:
3803:
3726:
3720:
3527:
3521:
3384:), while larger surface areas
3369:In other words, larger masses
3183:
3177:
2923:
2917:
2380:
2374:
2305:
2299:
2287:
2281:
2069:drops to half-value, or where
2033:
2023:
1974:
1959:
1921:
1917:
1911:
1897:
1810:
1804:
1481:
1475:
1391:
1385:
1183:
1177:
1166:
1104:
1098:
794:
788:
747:
735:
663:
657:
574:
568:
545:
530:
515:
509:
415:
409:
321:
315:
269:
263:
1:
4355:
3793:and the fall is described by
3101:from the body (that is, when
2879:convective cooling or warming
773:
3598:Time constants in biophysics
3373:with higher heat capacities
950:{\displaystyle V=V_{0}e^{0}}
753:{\displaystyle V_{0}=V(t=0)}
217:Linear time-invariant system
7:
4259:
3610:, the time constant of the
2620:
647:and no forcing function is
337:The right-hand side is the
10:
4605:
4489:10.1109/ACSSC.2003.1292037
4113:exponential decay constant
4075:
3391:with higher heat transfer
214:
164:, rather than electrical.
3955:is the membrane current.
3001:is positive when heat is
2965:heat transfer coefficient
86:, usually denoted by the
4323:
2782:{\displaystyle \tau =RC}
1226:{\displaystyle t=5\tau }
762:is the initial value of
4369:Béla G. Lipták (2003).
3495:Newton's law of cooling
1011:{\displaystyle t=\tau }
983:{\displaystyle V=V_{0}}
867:in control engineering)
386:Heaviside step function
330:{\displaystyle V=V(t).}
4529:. Wiley. p. 151.
4241:Meteorological sensors
4230:
4191:
4128:
4025:
3976:
3936:
3878:
3852:
3785:
3670:
3627:
3574:
3501:as a function of time
3485:
3361:
3294:
3211:
3070:
2951:
2875:lumped system analysis
2859:
2795:is the resistance (in
2783:
2754:
2701:
2664:
2643:
2635:
2521:
2470:
2315:
2227:
2220:
2145:
2052:
1881:
1773:
1423:
1338:
1321:
1289:
1227:
1196:
1145:
1074:
1012:
984:
951:
908:
838:
754:
704:
606:
552:
478:
331:
296:is a function of time
276:
169:lumped system analysis
4231:
4192:
4129:
4127:{\displaystyle \tau }
4076:Further information:
4026:
4024:{\displaystyle \tau }
3977:
3975:{\displaystyle \tau }
3937:
3879:
3877:{\displaystyle \tau }
3853:
3786:
3671:
3628:
3626:{\displaystyle \tau }
3575:
3486:
3404:differential equation
3362:
3295:
3212:
3071:
2974:is the surface area,
2952:
2869:Thermal time constant
2860:
2784:
2755:
2753:{\displaystyle \tau }
2702:
2665:
2663:{\displaystyle \tau }
2641:
2633:
2522:
2471:
2316:
2221:
2219:{\displaystyle \tau }
2191:
2146:
2053:
1882:
1774:
1424:
1327:
1315:
1290:
1228:
1197:
1146:
1075:
1013:
985:
952:
909:
839:
755:
705:
607:
553:
479:
332:
277:
211:Differential equation
100:linear time-invariant
4483:. pp. 873–878.
4282:Lead–lag compensator
4206:
4138:
4118:
4062:coincidence detector
4015:
3966:
3901:
3868:
3797:
3714:
3637:
3617:
3512:
3430:
3308:
3225:
3111:
3015:
2887:
2838:
2764:
2744:
2678:
2654:
2492:
2368:
2236:
2210:
2096:
2076:. This is the usual
1893:
1791:
1465:
1347:
1237:
1208:
1155:
1086:
1022:
996:
961:
918:
892:
782:
716:
651:
562:
503:
403:
303:
225:
189:1 − 1 /
123:Dirac delta function
1572:
907:{\displaystyle t=0}
860:is time (generally
488:, often denoted by
388:, often denoted by
104:characteristic unit
4579:Physical constants
4460:Unknown parameter
4302:Frequency response
4226:
4187:
4124:
4111:is related to the
4058:temporal summation
4031:for the fall sets
4021:
3982:for the rise sets
3972:
3932:
3874:
3848:
3781:
3666:
3623:
3612:membrane potential
3570:
3481:
3357:
3290:
3207:
3066:
2947:
2855:
2779:
2750:
2697:
2660:
2644:
2636:
2517:
2466:
2336:unit step function
2311:
2228:
2216:
2141:
2048:
1877:
1769:
1767:
1558:
1419:
1339:
1322:
1285:
1223:
1192:
1173:
1141:
1070:
1008:
980:
947:
904:
834:
750:
700:
602:
548:
474:
469:
327:
272:
146:radio transmitters
4536:978-0-470-84789-3
4444:978-90-277-2789-3
4429:GR North (1988).
4415:978-1-56347-261-9
4396:Bong Wie (1998).
4382:978-0-8493-1081-2
4277:Exponential decay
4201:, and is denoted
4169:
4084:exponential decay
4078:Exponential decay
4072:Exponential decay
4066:spatial summation
4004:has been reached
3912:
3895:resting potential
3823:
3740:
3696:of the membrane.
3467:
3454:
3352:
3275:
3259:
3246:
3148:
3052:
2853:
2760:(in seconds) is:
2736:Similarly, in an
2695:
2270:
2257:
2136:
2043:
2042:
2003:
1853:
1707:
1704:
1685:
1587:
1556:
1371:
1158:
290:exponential decay
249:
138:signal processing
127:Fourier transform
72:
71:
64:
4596:
4541:
4540:
4520:
4511:
4510:
4476:
4470:
4469:
4463:
4458:
4456:
4448:
4426:
4420:
4419:
4403:
4393:
4387:
4386:
4366:
4349:
4334:
4307:Impulse response
4272:Cutoff frequency
4267:RC time constant
4237:
4235:
4233:
4232:
4227:
4222:
4196:
4194:
4193:
4188:
4171:
4170:
4167:
4158:
4157:
4153:
4133:
4131:
4130:
4125:
4030:
4028:
4027:
4022:
3981:
3979:
3978:
3973:
3941:
3939:
3938:
3933:
3928:
3927:
3915:
3914:
3913:
3910:
3883:
3881:
3880:
3875:
3857:
3855:
3854:
3849:
3847:
3846:
3842:
3826:
3825:
3824:
3821:
3790:
3788:
3787:
3782:
3780:
3776:
3775:
3774:
3770:
3743:
3742:
3741:
3738:
3675:
3673:
3672:
3667:
3665:
3664:
3655:
3654:
3632:
3630:
3629:
3624:
3579:
3577:
3576:
3571:
3566:
3565:
3561:
3545:
3544:
3490:
3488:
3487:
3482:
3468:
3460:
3455:
3453:
3445:
3434:
3366:
3364:
3363:
3358:
3353:
3351:
3350:
3349:
3336:
3332:
3331:
3318:
3299:
3297:
3296:
3291:
3286:
3285:
3276:
3268:
3260:
3252:
3247:
3245:
3237:
3229:
3216:
3214:
3213:
3208:
3203:
3199:
3198:
3197:
3168:
3167:
3149:
3147:
3139:
3131:
3126:
3125:
3075:
3073:
3072:
3067:
3053:
3051:
3043:
3035:
3030:
3029:
2956:
2954:
2953:
2948:
2943:
2939:
2938:
2937:
2908:
2907:
2864:
2862:
2861:
2856:
2854:
2851:
2788:
2786:
2785:
2780:
2759:
2757:
2756:
2751:
2706:
2704:
2703:
2698:
2696:
2688:
2669:
2667:
2666:
2661:
2617:
2609:
2607:
2606:
2603:
2600:
2594:
2580:
2578:
2577:
2574:
2571:
2564:
2562:
2561:
2558:
2555:
2548:
2546:
2545:
2542:
2539:
2526:
2524:
2523:
2518:
2504:
2503:
2484:
2475:
2473:
2472:
2467:
2462:
2458:
2457:
2456:
2452:
2416:
2415:
2411:
2395:
2394:
2361:
2344:
2333:
2320:
2318:
2317:
2312:
2271:
2263:
2258:
2256:
2248:
2240:
2225:
2223:
2222:
2217:
2176:
2161:
2150:
2148:
2147:
2142:
2137:
2135:
2121:
2116:
2115:
2114:
2089:
2075:
2068:
2057:
2055:
2054:
2049:
2044:
2041:
2040:
2016:
2012:
2004:
2002:
2001:
2000:
1996:
1987:
1983:
1982:
1981:
1969:
1955:
1954:
1932:
1924:
1910:
1909:
1900:
1886:
1884:
1883:
1878:
1873:
1872:
1854:
1852:
1848:
1830:
1826:
1817:
1803:
1802:
1778:
1776:
1775:
1770:
1768:
1761:
1757:
1756:
1755:
1751:
1732:
1731:
1708:
1706:
1705:
1697:
1678:
1677:
1672:
1671:
1667:
1651:
1650:
1635:
1631:
1630:
1629:
1610:
1609:
1605:
1600:
1585:
1584:
1571:
1566:
1557:
1552:
1551:
1550:
1546:
1526:
1521:
1520:
1516:
1500:
1499:
1458:
1441:
1428:
1426:
1425:
1420:
1415:
1414:
1372:
1370:
1362:
1354:
1336:
1319:
1303:
1294:
1292:
1291:
1286:
1284:
1283:
1268:
1267:
1255:
1254:
1232:
1230:
1229:
1224:
1201:
1199:
1198:
1193:
1172:
1150:
1148:
1147:
1142:
1140:
1139:
1135:
1119:
1118:
1079:
1077:
1076:
1071:
1069:
1068:
1053:
1052:
1040:
1039:
1017:
1015:
1014:
1009:
989:
987:
986:
981:
979:
978:
956:
954:
953:
948:
946:
945:
936:
935:
913:
911:
910:
905:
877:
866:
859:
849:
843:
841:
840:
835:
830:
829:
825:
809:
808:
769:
765:
759:
757:
756:
751:
728:
727:
709:
707:
706:
701:
699:
698:
694:
678:
677:
646:
634:Example solution
629:
619:
615:
611:
609:
608:
603:
598:
597:
557:
555:
554:
549:
498:
486:impulse function
483:
481:
480:
475:
473:
472:
398:
380:
365:
350:
339:forcing function
336:
334:
333:
328:
299:
295:
287:
281:
279:
278:
273:
250:
248:
240:
232:
194:
182:
177:1 /
140:systems –
119:impulse response
93:
67:
60:
56:
53:
47:
27:
26:
19:
4604:
4603:
4599:
4598:
4597:
4595:
4594:
4593:
4569:
4568:
4550:
4545:
4544:
4537:
4521:
4514:
4499:
4477:
4473:
4461:
4459:
4450:
4449:
4445:
4427:
4423:
4416:
4394:
4390:
4383:
4367:
4363:
4358:
4353:
4352:
4335:
4331:
4326:
4321:
4287:Length constant
4262:
4243:
4218:
4207:
4204:
4203:
4202:
4166:
4162:
4149:
4145:
4141:
4139:
4136:
4135:
4119:
4116:
4115:
4110:
4103:
4086:, such as of a
4080:
4074:
4052:
4045:
4039:) equal to 0.37
4016:
4013:
4012:
4003:
3996:
3990:) equal to 0.63
3967:
3964:
3963:
3950:
3923:
3919:
3909:
3908:
3904:
3902:
3899:
3898:
3892:
3884:is in seconds.
3869:
3866:
3865:
3838:
3831:
3827:
3820:
3819:
3815:
3798:
3795:
3794:
3766:
3759:
3755:
3748:
3744:
3737:
3736:
3732:
3715:
3712:
3711:
3691:
3684:
3660:
3656:
3650:
3646:
3638:
3635:
3634:
3618:
3615:
3614:
3600:
3589:
3557:
3550:
3546:
3540:
3536:
3513:
3510:
3509:
3505:, is given by:
3459:
3446:
3435:
3433:
3431:
3428:
3427:
3423:
3412:
3390:
3379:
3345:
3341:
3337:
3327:
3323:
3319:
3317:
3309:
3306:
3305:
3281:
3277:
3267:
3251:
3238:
3230:
3228:
3226:
3223:
3222:
3193:
3189:
3173:
3169:
3163:
3159:
3140:
3132:
3130:
3121:
3117:
3112:
3109:
3108:
3088:
3044:
3036:
3034:
3025:
3021:
3016:
3013:
3012:
2996:
2973:
2933:
2929:
2913:
2909:
2903:
2899:
2888:
2885:
2884:
2871:
2850:
2839:
2836:
2835:
2765:
2762:
2761:
2745:
2742:
2741:
2687:
2679:
2676:
2675:
2655:
2652:
2651:
2628:
2623:
2611:
2604:
2601:
2598:
2597:
2595:
2589:
2584:In fact, given
2575:
2572:
2569:
2568:
2566:
2559:
2556:
2553:
2552:
2550:
2543:
2540:
2537:
2536:
2534:
2499:
2495:
2493:
2490:
2489:
2479:
2448:
2441:
2437:
2430:
2426:
2407:
2400:
2396:
2390:
2386:
2369:
2366:
2365:
2360:
2346:
2339:
2324:
2262:
2249:
2241:
2239:
2237:
2234:
2233:
2211:
2208:
2207:
2205:
2198:
2186:
2174:
2167:
2160:
2154:
2125:
2120:
2104:
2103:
2099:
2097:
2094:
2093:
2081:
2070:
2066:
2059:
2036:
2032:
2011:
1992:
1988:
1977:
1973:
1965:
1950:
1946:
1945:
1941:
1940:
1936:
1931:
1920:
1905:
1901:
1896:
1894:
1891:
1890:
1862:
1858:
1844:
1831:
1822:
1818:
1816:
1798:
1794:
1792:
1789:
1788:
1766:
1765:
1747:
1740:
1736:
1721:
1717:
1716:
1712:
1696:
1686:
1676:
1663:
1656:
1652:
1646:
1642:
1633:
1632:
1622:
1615:
1611:
1601:
1593:
1592:
1588:
1577:
1567:
1562:
1542:
1535:
1531:
1527:
1525:
1512:
1505:
1501:
1495:
1491:
1484:
1468:
1466:
1463:
1462:
1457:
1443:
1436:
1433:Euler's formula
1404:
1400:
1363:
1355:
1353:
1348:
1345:
1344:
1335:
1329:
1317:
1310:
1299:
1279:
1275:
1260:
1256:
1250:
1246:
1238:
1235:
1234:
1209:
1206:
1205:
1162:
1156:
1153:
1152:
1131:
1124:
1120:
1114:
1110:
1087:
1084:
1083:
1064:
1060:
1045:
1041:
1035:
1031:
1023:
1020:
1019:
997:
994:
993:
974:
970:
962:
959:
958:
941:
937:
931:
927:
919:
916:
915:
893:
890:
889:
885:
876:
870:
861:
857:
847:
821:
814:
810:
804:
800:
783:
780:
779:
776:
767:
763:
723:
719:
717:
714:
713:
690:
683:
679:
673:
669:
652:
649:
648:
645:
639:
636:
621:
617:
613:
587:
583:
563:
560:
559:
504:
501:
500:
489:
468:
467:
456:
447:
446:
435:
422:
421:
404:
401:
400:
389:
371:
356:
341:
304:
301:
300:
297:
293:
288:represents the
285:
241:
233:
231:
226:
223:
222:
219:
213:
188:
176:
158:control systems
154:digital filters
91:
68:
57:
51:
48:
40:help improve it
37:
28:
24:
17:
12:
11:
5:
4602:
4592:
4591:
4586:
4581:
4567:
4566:
4561:
4556:
4549:
4548:External links
4546:
4543:
4542:
4535:
4512:
4497:
4471:
4443:
4421:
4414:
4388:
4381:
4360:
4359:
4357:
4354:
4351:
4350:
4328:
4327:
4325:
4322:
4320:
4319:
4314:
4309:
4304:
4299:
4294:
4289:
4284:
4279:
4274:
4269:
4263:
4261:
4258:
4242:
4239:
4225:
4221:
4217:
4214:
4211:
4199:decay constant
4186:
4183:
4180:
4177:
4174:
4165:
4161:
4156:
4152:
4148:
4144:
4123:
4108:
4101:
4073:
4070:
4050:
4043:
4020:
4001:
3994:
3971:
3948:
3931:
3926:
3922:
3918:
3907:
3890:
3873:
3845:
3841:
3837:
3834:
3830:
3818:
3814:
3811:
3808:
3805:
3802:
3779:
3773:
3769:
3765:
3762:
3758:
3754:
3751:
3747:
3735:
3731:
3728:
3725:
3722:
3719:
3689:
3682:
3663:
3659:
3653:
3649:
3645:
3642:
3622:
3599:
3596:
3587:
3581:
3580:
3569:
3564:
3560:
3556:
3553:
3549:
3543:
3539:
3535:
3532:
3529:
3526:
3523:
3520:
3517:
3480:
3477:
3474:
3471:
3466:
3463:
3458:
3452:
3449:
3444:
3441:
3438:
3424:), one finds:
3421:
3410:
3388:
3377:
3356:
3348:
3344:
3340:
3335:
3330:
3326:
3322:
3316:
3313:
3289:
3284:
3280:
3274:
3271:
3266:
3263:
3258:
3255:
3250:
3244:
3241:
3236:
3233:
3206:
3202:
3196:
3192:
3188:
3185:
3182:
3179:
3176:
3172:
3166:
3162:
3158:
3155:
3152:
3146:
3143:
3138:
3135:
3129:
3124:
3120:
3116:
3086:
3065:
3062:
3059:
3056:
3050:
3047:
3042:
3039:
3033:
3028:
3024:
3020:
2994:
2971:
2946:
2942:
2936:
2932:
2928:
2925:
2922:
2919:
2916:
2912:
2906:
2902:
2898:
2895:
2892:
2870:
2867:
2849:
2846:
2843:
2820:Pole splitting
2778:
2775:
2772:
2769:
2749:
2694:
2691:
2686:
2683:
2659:
2627:
2624:
2622:
2619:
2612:0.63 × (100 −
2588:initial speed
2516:
2513:
2510:
2507:
2502:
2498:
2465:
2461:
2455:
2451:
2447:
2444:
2440:
2436:
2433:
2429:
2425:
2422:
2419:
2414:
2410:
2406:
2403:
2399:
2393:
2389:
2385:
2382:
2379:
2376:
2373:
2358:
2310:
2307:
2304:
2301:
2298:
2295:
2292:
2289:
2286:
2283:
2280:
2277:
2274:
2269:
2266:
2261:
2255:
2252:
2247:
2244:
2215:
2203:
2196:
2185:
2182:
2172:
2158:
2140:
2134:
2131:
2128:
2124:
2119:
2113:
2110:
2107:
2102:
2064:
2047:
2039:
2035:
2031:
2028:
2025:
2022:
2019:
2015:
2010:
2007:
1999:
1995:
1991:
1986:
1980:
1976:
1972:
1968:
1964:
1961:
1958:
1953:
1949:
1944:
1939:
1935:
1930:
1927:
1923:
1919:
1916:
1913:
1908:
1904:
1899:
1876:
1871:
1868:
1865:
1861:
1857:
1851:
1847:
1843:
1840:
1837:
1834:
1829:
1825:
1821:
1815:
1812:
1809:
1806:
1801:
1797:
1764:
1760:
1754:
1750:
1746:
1743:
1739:
1735:
1730:
1727:
1724:
1720:
1715:
1711:
1703:
1700:
1695:
1692:
1689:
1684:
1681:
1675:
1670:
1666:
1662:
1659:
1655:
1649:
1645:
1641:
1638:
1636:
1634:
1628:
1625:
1621:
1618:
1614:
1608:
1604:
1599:
1596:
1591:
1583:
1580:
1576:
1570:
1565:
1561:
1555:
1549:
1545:
1541:
1538:
1534:
1530:
1524:
1519:
1515:
1511:
1508:
1504:
1498:
1494:
1490:
1487:
1485:
1483:
1480:
1477:
1474:
1471:
1470:
1455:
1418:
1413:
1410:
1407:
1403:
1399:
1396:
1393:
1390:
1387:
1384:
1381:
1378:
1375:
1369:
1366:
1361:
1358:
1352:
1333:
1309:
1306:
1296:
1295:
1282:
1278:
1274:
1271:
1266:
1263:
1259:
1253:
1249:
1245:
1242:
1222:
1219:
1216:
1213:
1202:
1191:
1188:
1185:
1182:
1179:
1176:
1171:
1168:
1165:
1161:
1138:
1134:
1130:
1127:
1123:
1117:
1113:
1109:
1106:
1103:
1100:
1097:
1094:
1091:
1080:
1067:
1063:
1059:
1056:
1051:
1048:
1044:
1038:
1034:
1030:
1027:
1007:
1004:
1001:
990:
977:
973:
969:
966:
944:
940:
934:
930:
926:
923:
903:
900:
897:
884:
883:Specific cases
881:
880:
879:
874:
868:
833:
828:
824:
820:
817:
813:
807:
803:
799:
796:
793:
790:
787:
775:
772:
749:
746:
743:
740:
737:
734:
731:
726:
722:
697:
693:
689:
686:
682:
676:
672:
668:
665:
662:
659:
656:
643:
635:
632:
601:
596:
593:
590:
586:
582:
579:
576:
573:
570:
567:
547:
544:
541:
538:
535:
532:
529:
526:
523:
520:
517:
514:
511:
508:
471:
466:
463:
460:
457:
455:
452:
449:
448:
445:
442:
439:
436:
434:
431:
428:
427:
425:
420:
417:
414:
411:
408:
326:
323:
320:
317:
314:
311:
308:
271:
268:
265:
262:
259:
256:
253:
247:
244:
239:
236:
230:
215:Main article:
212:
209:
197:decay constant
142:magnetic tapes
94:(tau), is the
70:
69:
31:
29:
22:
15:
9:
6:
4:
3:
2:
4601:
4590:
4587:
4585:
4582:
4580:
4577:
4576:
4574:
4565:
4562:
4560:
4557:
4555:
4552:
4551:
4538:
4532:
4528:
4527:
4519:
4517:
4508:
4504:
4500:
4498:0-7803-8104-1
4494:
4490:
4486:
4482:
4475:
4467:
4462:|agency=
4454:
4446:
4440:
4436:
4432:
4425:
4417:
4411:
4407:
4402:
4401:
4392:
4384:
4378:
4374:
4373:
4365:
4361:
4347:
4343:
4339:
4333:
4329:
4318:
4317:Settling time
4315:
4313:
4312:Step response
4310:
4308:
4305:
4303:
4300:
4298:
4295:
4293:
4290:
4288:
4285:
4283:
4280:
4278:
4275:
4273:
4270:
4268:
4265:
4264:
4257:
4255:
4250:
4248:
4247:time constant
4238:
4223:
4219:
4215:
4212:
4209:
4200:
4184:
4181:
4178:
4175:
4172:
4163:
4159:
4154:
4150:
4146:
4142:
4121:
4114:
4107:
4100:
4097:
4093:
4092:mean lifetime
4089:
4085:
4079:
4069:
4067:
4063:
4059:
4054:
4049:
4042:
4038:
4034:
4018:
4010:
4005:
4000:
3993:
3989:
3985:
3969:
3961:
3956:
3954:
3947:
3942:
3929:
3924:
3920:
3916:
3905:
3896:
3889:
3885:
3871:
3863:
3858:
3843:
3839:
3835:
3832:
3828:
3816:
3812:
3806:
3800:
3791:
3777:
3771:
3767:
3763:
3760:
3756:
3752:
3749:
3745:
3733:
3729:
3723:
3717:
3708:
3706:
3705:lipid bilayer
3702:
3697:
3695:
3688:
3681:
3676:
3661:
3657:
3651:
3647:
3643:
3640:
3620:
3613:
3609:
3605:
3595:
3593:
3586:
3567:
3562:
3558:
3554:
3551:
3547:
3541:
3537:
3530:
3524:
3518:
3508:
3507:
3506:
3504:
3500:
3496:
3491:
3478:
3475:
3472:
3464:
3461:
3456:
3450:
3447:
3442:
3436:
3425:
3420:
3416:
3409:
3405:
3400:
3398:
3394:
3387:
3383:
3376:
3372:
3367:
3354:
3346:
3342:
3338:
3333:
3328:
3324:
3320:
3314:
3311:
3303:
3300:
3287:
3282:
3278:
3272:
3269:
3264:
3261:
3256:
3253:
3248:
3242:
3239:
3234:
3231:
3220:
3217:
3204:
3200:
3194:
3190:
3186:
3180:
3174:
3170:
3164:
3160:
3156:
3153:
3150:
3144:
3141:
3136:
3133:
3127:
3122:
3118:
3114:
3106:
3104:
3100:
3096:
3092:
3091:specific heat
3085:
3081:
3076:
3063:
3060:
3057:
3054:
3048:
3045:
3040:
3037:
3031:
3026:
3022:
3018:
3010:
3008:
3004:
3000:
2993:
2989:
2985:
2981:
2977:
2970:
2966:
2962:
2957:
2944:
2940:
2934:
2930:
2926:
2920:
2914:
2910:
2904:
2900:
2896:
2893:
2890:
2882:
2880:
2876:
2866:
2847:
2844:
2841:
2833:
2828:
2825:
2821:
2817:
2816:Step response
2812:
2810:
2806:
2802:
2798:
2794:
2789:
2776:
2773:
2770:
2767:
2747:
2739:
2734:
2732:
2728:
2724:
2720:
2716:
2712:
2707:
2692:
2689:
2684:
2681:
2673:
2657:
2649:
2640:
2632:
2618:
2615:
2592:
2587:
2582:
2531:
2527:
2514:
2511:
2508:
2505:
2496:
2486:
2482:
2476:
2463:
2459:
2453:
2449:
2445:
2442:
2438:
2434:
2431:
2427:
2423:
2420:
2417:
2412:
2408:
2404:
2401:
2397:
2391:
2387:
2383:
2377:
2371:
2363:
2357:
2353:
2349:
2342:
2337:
2331:
2327:
2321:
2308:
2302:
2296:
2293:
2290:
2284:
2278:
2275:
2272:
2267:
2264:
2259:
2253:
2250:
2245:
2242:
2231:
2213:
2202:
2195:
2190:
2181:
2178:
2171:
2165:
2157:
2153:The notation
2151:
2138:
2132:
2129:
2126:
2122:
2117:
2105:
2100:
2091:
2088:
2084:
2079:
2073:
2063:
2045:
2037:
2029:
2026:
2020:
2017:
2013:
2008:
2005:
1997:
1993:
1989:
1984:
1978:
1970:
1966:
1962:
1956:
1951:
1947:
1942:
1937:
1933:
1928:
1925:
1914:
1902:
1887:
1874:
1869:
1866:
1863:
1859:
1855:
1849:
1845:
1841:
1838:
1835:
1832:
1827:
1823:
1819:
1813:
1807:
1795:
1786:
1784:
1779:
1762:
1758:
1752:
1748:
1744:
1741:
1737:
1733:
1728:
1725:
1722:
1718:
1713:
1709:
1701:
1698:
1693:
1690:
1687:
1682:
1679:
1673:
1668:
1664:
1660:
1657:
1653:
1647:
1643:
1639:
1637:
1626:
1623:
1619:
1616:
1612:
1606:
1602:
1597:
1594:
1589:
1581:
1578:
1574:
1568:
1563:
1559:
1553:
1547:
1543:
1539:
1536:
1532:
1528:
1522:
1517:
1513:
1509:
1506:
1502:
1496:
1492:
1488:
1486:
1478:
1472:
1460:
1454:
1450:
1446:
1439:
1434:
1429:
1416:
1411:
1408:
1405:
1401:
1397:
1394:
1388:
1382:
1379:
1376:
1373:
1367:
1364:
1359:
1356:
1350:
1342:
1332:
1326:
1314:
1305:
1302:
1280:
1276:
1272:
1269:
1264:
1261:
1257:
1251:
1247:
1243:
1240:
1220:
1217:
1214:
1211:
1203:
1189:
1186:
1180:
1174:
1163:
1136:
1132:
1128:
1125:
1121:
1115:
1111:
1107:
1101:
1095:
1092:
1089:
1081:
1065:
1061:
1057:
1054:
1049:
1046:
1042:
1036:
1032:
1028:
1025:
1005:
1002:
999:
991:
975:
971:
967:
964:
942:
938:
932:
928:
924:
921:
901:
898:
895:
887:
886:
873:
869:
864:
856:
855:
854:
851:
844:
831:
826:
822:
818:
815:
811:
805:
801:
797:
791:
785:
771:
760:
744:
741:
738:
732:
729:
724:
720:
710:
695:
691:
687:
684:
680:
674:
670:
666:
660:
654:
642:
631:
628:
624:
599:
594:
591:
588:
584:
580:
577:
571:
565:
542:
539:
536:
533:
527:
524:
521:
518:
512:
506:
496:
492:
487:
464:
461:
458:
453:
450:
443:
440:
437:
432:
429:
423:
418:
412:
406:
396:
392:
387:
382:
378:
374:
369:
363:
359:
354:
348:
344:
340:
324:
318:
312:
309:
306:
292:constant and
291:
282:
266:
260:
257:
254:
251:
245:
242:
237:
234:
228:
218:
208:
206:
202:
198:
192:
186:
185:step response
180:
172:
170:
165:
163:
159:
155:
151:
147:
143:
139:
134:
132:
128:
124:
120:
116:
112:
111:step response
107:
105:
101:
97:
89:
85:
84:time constant
81:
77:
66:
63:
55:
52:February 2023
45:
41:
35:
32:This article
30:
21:
20:
4584:Neuroscience
4525:
4480:
4474:
4434:
4424:
4399:
4391:
4371:
4364:
4332:
4251:
4246:
4244:
4105:
4098:
4081:
4055:
4047:
4040:
4036:
4032:
4008:
4007:Setting for
4006:
3998:
3991:
3987:
3983:
3959:
3958:Setting for
3957:
3952:
3945:
3943:
3887:
3886:
3859:
3792:
3709:
3701:ion channels
3698:
3686:
3679:
3677:
3601:
3591:
3584:
3582:
3502:
3498:
3492:
3426:
3418:
3414:
3407:
3401:
3396:
3392:
3385:
3381:
3374:
3370:
3368:
3304:
3301:
3221:
3218:
3107:
3102:
3098:
3094:
3083:
3079:
3077:
3011:
3006:
3002:
2998:
2991:
2987:
2983:
2979:
2975:
2968:
2960:
2958:
2883:
2872:
2829:
2813:
2800:
2792:
2790:
2735:
2722:
2710:
2708:
2645:
2613:
2590:
2585:
2583:
2532:
2528:
2487:
2480:
2477:
2364:
2355:
2351:
2347:
2340:
2329:
2325:
2322:
2232:
2229:
2200:
2193:
2179:
2169:
2155:
2152:
2092:
2086:
2082:
2071:
2061:
1888:
1787:
1783:steady-state
1780:
1461:
1452:
1448:
1444:
1437:
1430:
1343:
1340:
1330:
1300:
1297:
871:
862:
852:
845:
777:
761:
711:
640:
637:
626:
622:
494:
490:
394:
390:
383:
376:
372:
367:
361:
357:
352:
346:
342:
338:
283:
220:
200:
190:
178:
173:
166:
135:
108:
83:
73:
58:
49:
33:
4346:RL circuits
4342:RC circuits
4254:Radiosondes
4088:radioactive
3694:capacitance
3082:= density,
2805:capacitance
2345:, assuming
1442:, assuming
355:, to which
80:engineering
4573:Categories
4356:References
2738:RC circuit
2727:inductance
2715:resistance
2648:RL circuit
2593:≤ 100 RPM,
774:Discussion
115:step input
4589:Durations
4464:ignored (
4453:cite book
4297:Fall time
4292:Rise time
4224:τ
4210:λ
4182:
4176:τ
4122:τ
4096:half-life
4019:τ
3970:τ
3897:, where
3872:τ
3844:τ
3833:−
3772:τ
3761:−
3753:−
3641:τ
3621:τ
3563:τ
3552:−
3534:Δ
3516:Δ
3470:Δ
3465:τ
3440:Δ
3321:ρ
3312:τ
3273:τ
3257:τ
3187:−
3154:−
3115:ρ
3058:−
3019:ρ
2927:−
2845:τ
2768:τ
2748:τ
2682:τ
2658:τ
2512:τ
2501:∞
2454:τ
2443:−
2435:−
2424:τ
2413:τ
2402:−
2268:τ
2214:τ
2133:τ
2130:π
2078:bandwidth
2030:τ
2027:ω
1971:τ
1948:ω
1938:τ
1907:∞
1867:ω
1850:τ
1836:ω
1828:τ
1800:∞
1753:τ
1742:−
1734:−
1726:ω
1702:τ
1691:ω
1683:τ
1669:τ
1658:−
1620:ω
1607:τ
1560:∫
1554:τ
1548:τ
1537:−
1518:τ
1507:−
1409:ω
1351:τ
1270:≈
1262:−
1221:τ
1170:∞
1167:→
1151:, and so
1137:τ
1126:−
1055:≈
1047:−
1006:τ
957:, and so
827:τ
816:−
696:τ
685:−
592:ω
537:π
528:
462:≥
229:τ
205:half-life
187:to reach
162:pneumatic
150:receivers
131:bandwidth
117:, or the
96:parameter
4260:See also
4064:through
2824:feedback
2621:Examples
2164:decibels
1627:′
1598:′
1582:′
778:Suppose
368:response
4507:7880203
3862:voltage
3692:is the
3583:where Δ
3099:outward
3003:leaving
2963:is the
2803:is the
2725:is the
2713:is the
2672:seconds
2608:
2596:
2579:
2567:
2563:
2551:
2547:
2535:
2354:= 0) =
1451:= 0) =
1233:; then
1018:; then
914:; then
853:Here:
366:is the
193:≈ 63.2%
181:≈ 36.8%
90:letter
76:physics
38:Please
4533:
4505:
4495:
4441:
4412:
4379:
4094:. The
3944:where
3860:where
3678:where
3608:neuron
3604:muscle
3078:where
2990:, and
2967:, and
2959:where
2809:farads
2799:) and
2791:where
2731:henrys
2721:) and
2709:where
2674:) is
2646:In an
2616:) RPM.
1586:
1273:0.0067
865:> 0
712:where
612:where
284:where
82:, the
4503:S2CID
4324:Notes
3419:T − T
3302:with
2343:≥ 0 s
2323:with
1440:≥ 0 s
381:are:
353:input
121:to a
113:to a
88:Greek
4531:ISBN
4493:ISBN
4466:help
4439:ISBN
4410:ISBN
4377:ISBN
4344:and
3093:and
2818:and
2807:(in
2797:ohms
2729:(in
2719:ohms
2717:(in
2670:(in
2362:is:
2334:the
1459:is:
1204:Let
1082:Let
1058:0.37
992:Let
888:Let
484:the
441:<
384:The
148:and
78:and
4485:doi
4406:100
4134:by
4109:1/2
4104:or
4082:In
4051:max
4044:max
4002:max
3995:max
3911:max
3891:max
3822:max
3739:max
3633:is
3606:or
3417:≡ (
3399:).
2852:FO4
2832:FO4
2811:).
2733:).
2586:any
2483:→ 0
2159:3dB
2090:):
2085:= 2
2074:= 1
1334:3dB
1160:lim
627:π f
625:= 2
558:or
525:sin
74:In
42:to
4575::
4515:^
4501:.
4491:.
4457::
4455:}}
4451:{{
4408:.
4245:A
4185:2.
4179:ln
4168:HL
4102:HL
4068:.
4053:.
4011:=
3962:=
3707:.
3479:0.
3371:ρV
3089:=
2865:.
2087:πf
2072:ωτ
770:.
399::
144:,
4539:.
4509:.
4487::
4468:)
4447:.
4418:.
4385:.
4348:.
4236:.
4220:/
4216:1
4213:=
4173:=
4164:T
4160:=
4155:2
4151:/
4147:1
4143:T
4106:T
4099:T
4048:V
4041:V
4037:t
4035:(
4033:V
4009:t
3999:V
3992:V
3988:t
3986:(
3984:V
3960:t
3953:I
3949:m
3946:r
3930:I
3925:m
3921:r
3917:=
3906:V
3888:V
3840:/
3836:t
3829:e
3817:V
3813:=
3810:)
3807:t
3804:(
3801:V
3778:)
3768:/
3764:t
3757:e
3750:1
3746:(
3734:V
3730:=
3727:)
3724:t
3721:(
3718:V
3690:m
3687:c
3683:m
3680:r
3662:m
3658:c
3652:m
3648:r
3644:=
3592:t
3588:0
3585:T
3568:,
3559:/
3555:t
3548:e
3542:0
3538:T
3531:=
3528:)
3525:t
3522:(
3519:T
3503:t
3499:T
3476:=
3473:T
3462:1
3457:+
3451:t
3448:d
3443:T
3437:d
3422:a
3415:T
3411:a
3408:T
3397:τ
3393:h
3389:s
3386:A
3382:τ
3378:p
3375:c
3355:.
3347:s
3343:A
3339:h
3334:V
3329:p
3325:c
3315:=
3288:,
3283:a
3279:T
3270:1
3265:=
3262:T
3254:1
3249:+
3243:t
3240:d
3235:T
3232:d
3205:.
3201:)
3195:a
3191:T
3184:)
3181:t
3178:(
3175:T
3171:(
3165:s
3161:A
3157:h
3151:=
3145:t
3142:d
3137:T
3134:d
3128:V
3123:p
3119:c
3103:F
3095:V
3087:p
3084:c
3080:ρ
3064:,
3061:F
3055:=
3049:t
3046:d
3041:T
3038:d
3032:V
3027:p
3023:c
3007:F
2999:F
2995:a
2992:T
2988:t
2984:t
2982:(
2980:T
2976:T
2972:s
2969:A
2961:h
2945:,
2941:)
2935:a
2931:T
2924:)
2921:t
2918:(
2915:T
2911:(
2905:s
2901:A
2897:h
2894:=
2891:F
2848:=
2842:5
2801:C
2793:R
2777:C
2774:R
2771:=
2723:L
2711:R
2693:R
2690:L
2685:=
2614:s
2605:8
2602:/
2599:1
2591:s
2576:8
2573:/
2570:1
2560:8
2557:/
2554:1
2544:8
2541:/
2538:1
2515:.
2509:A
2506:=
2497:V
2481:ω
2464:.
2460:)
2450:/
2446:t
2439:e
2432:1
2428:(
2421:A
2418:+
2409:/
2405:t
2398:e
2392:0
2388:V
2384:=
2381:)
2378:t
2375:(
2372:V
2359:0
2356:V
2352:t
2350:(
2348:V
2341:t
2332:)
2330:t
2328:(
2326:u
2309:,
2306:)
2303:t
2300:(
2297:u
2294:A
2291:=
2288:)
2285:t
2282:(
2279:f
2276:=
2273:V
2265:1
2260:+
2254:t
2251:d
2246:V
2243:d
2226:.
2204:∞
2201:V
2197:0
2194:V
2175:|
2173:∞
2170:V
2168:|
2156:f
2139:.
2127:2
2123:1
2118:=
2112:B
2109:d
2106:3
2101:f
2083:ω
2067:|
2065:∞
2062:V
2060:|
2046:.
2038:2
2034:)
2024:(
2021:+
2018:1
2014:1
2009:A
2006:=
1998:2
1994:/
1990:1
1985:)
1979:2
1975:)
1967:/
1963:1
1960:(
1957:+
1952:2
1943:(
1934:1
1929:A
1926:=
1922:|
1918:)
1915:t
1912:(
1903:V
1898:|
1875:.
1870:t
1864:j
1860:e
1856:A
1846:/
1842:1
1839:+
1833:j
1824:/
1820:1
1814:=
1811:)
1808:t
1805:(
1796:V
1763:.
1759:)
1749:/
1745:t
1738:e
1729:t
1723:j
1719:e
1714:(
1710:A
1699:1
1694:+
1688:j
1680:1
1674:+
1665:/
1661:t
1654:e
1648:0
1644:V
1640:=
1624:t
1617:j
1613:e
1603:/
1595:t
1590:e
1579:t
1575:d
1569:t
1564:0
1544:/
1540:t
1533:e
1529:A
1523:+
1514:/
1510:t
1503:e
1497:0
1493:V
1489:=
1482:)
1479:t
1476:(
1473:V
1456:0
1453:V
1449:t
1447:(
1445:V
1438:t
1417:.
1412:t
1406:j
1402:e
1398:A
1395:=
1392:)
1389:t
1386:(
1383:f
1380:=
1377:V
1374:+
1368:t
1365:d
1360:V
1357:d
1331:f
1318:τ
1301:e
1281:0
1277:V
1265:5
1258:e
1252:0
1248:V
1244:=
1241:V
1218:5
1215:=
1212:t
1190:0
1187:=
1184:)
1181:t
1178:(
1175:f
1164:t
1133:/
1129:t
1122:e
1116:0
1112:V
1108:=
1105:)
1102:t
1099:(
1096:f
1093:=
1090:V
1066:0
1062:V
1050:1
1043:e
1037:0
1033:V
1029:=
1026:V
1003:=
1000:t
976:0
972:V
968:=
965:V
943:0
939:e
933:0
929:V
925:=
922:V
902:0
899:=
896:t
875:0
872:V
863:t
858:t
848:τ
832:.
823:/
819:t
812:e
806:0
802:V
798:=
795:)
792:t
789:(
786:V
768:τ
764:V
748:)
745:0
742:=
739:t
736:(
733:V
730:=
725:0
721:V
692:/
688:t
681:e
675:0
671:V
667:=
664:)
661:t
658:(
655:V
644:0
641:V
623:ω
618:f
614:A
600:,
595:t
589:j
585:e
581:A
578:=
575:)
572:t
569:(
566:f
546:)
543:t
540:f
534:2
531:(
522:A
519:=
516:)
513:t
510:(
507:f
497:)
495:t
493:(
491:δ
465:0
459:t
454:,
451:1
444:0
438:t
433:,
430:0
424:{
419:=
416:)
413:t
410:(
407:u
397:)
395:t
393:(
391:u
379:)
377:t
375:(
373:f
364:)
362:t
360:(
358:V
349:)
347:t
345:(
343:f
325:.
322:)
319:t
316:(
313:V
310:=
307:V
298:t
294:V
286:τ
270:)
267:t
264:(
261:f
258:=
255:V
252:+
246:t
243:d
238:V
235:d
201:λ
199:(
191:e
179:e
92:τ
65:)
59:(
54:)
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36:.
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