421:
analogy is a weight suspended on a spring which will oscillate up and down when released. This is no passing metaphor; a weight on a spring is described by exactly the same second order differential equation as an RLC circuit and for all the properties of the one system there will be found an analogous property of the other. The mechanical property answering to the resistor in the circuit is friction in the spring–weight system. Friction will slowly bring any oscillation to a halt if there is no external force driving it. Likewise, the resistance in an RLC circuit will "damp" the oscillation, diminishing it with time if there is no driving AC power source in the circuit.
211:
5691:
442:
the way that a tuning fork will carry on ringing after it has been struck, and the effect is often called ringing. This effect is the peak natural resonance frequency of the circuit and in general is not exactly the same as the driven resonance frequency, although the two will usually be quite close to each other. Various terms are used by different authors to distinguish the two, but resonance frequency unqualified usually means the driven resonance frequency. The driven frequency may be called the
3075:
6352:
8506:
8478:
1407:
8493:
8465:
8015:
6981:
314:. All three elements in series or all three elements in parallel are the simplest in concept and the most straightforward to analyse. There are, however, other arrangements, some with practical importance in real circuits. One issue often encountered is the need to take into account inductor resistance. Inductors are typically constructed from coils of wire, the resistance of which is not usually desirable, but it often has a significant effect on the circuit.
2640:
6671:
8450:
8437:
2758:
203:
7886:
6904:
3070:{\displaystyle {\begin{aligned}s_{1}&=-\alpha +{\sqrt {\alpha ^{2}-\omega _{0}^{2}\,}}=-\omega _{0}\left(\zeta -{\sqrt {\zeta ^{2}-1\,}}\right)={\frac {-\omega _{0}}{\ \zeta +{\sqrt {\zeta ^{2}-1\ }}\ }}\ ,\\s_{2}&=-\alpha -{\sqrt {\alpha ^{2}-\omega _{0}^{2}\,}}=-\omega _{0}\left(\zeta +{\sqrt {\zeta ^{2}-1\,}}\right)\;.\end{aligned}}}
1228:
8329:
was discharged through a wire wound around an iron needle, sometimes the needle was left magnetized in one direction and sometimes in the opposite direction. He correctly deduced that this was caused by a damped oscillating discharge current in the wire, which reversed the magnetization of the needle
428:
of the circuit is at a minimum. Equivalently, it can be defined as the frequency at which the impedance is purely real (that is, purely resistive). This occurs because the impedances of the inductor and capacitor at resonant are equal but of opposite sign and cancel out. Circuits where L and C are in
8520:
In the filtering application, the resistor becomes the load that the filter is working into. The value of the damping factor is chosen based on the desired bandwidth of the filter. For a wider bandwidth, a larger value of the damping factor is required (and vice versa). The three components give the
8370:
One of the first demonstrations of resonance between tuned circuits was Lodge's "syntonic jars" experiment around 1889 He placed two resonant circuits next to each other, each consisting of a Leyden jar connected to an adjustable one-turn coil with a spark gap. When a high voltage from an induction
3207:
of the specific problem being analysed. That is, they are set by the values of the currents and voltages in the circuit at the onset of the transient and the presumed value they will settle to after infinite time. The differential equation for the circuit solves in three different ways depending on
441:
The resonance frequency is defined in terms of the impedance presented to a driving source. It is still possible for the circuit to carry on oscillating (for a time) after the driving source has been removed or it is subjected to a step in voltage (including a step down to zero). This is similar to
8913:
Figure 10 shows a band-stop filter formed by a series LC circuit in shunt across the load. Figure 11 is a band-stop filter formed by a parallel LC circuit in series with the load. The first case requires a high impedance source so that the current is diverted into the resonator when it becomes low
2154:
1393:
The article next gives the analysis for the series RLC circuit in detail. Other configurations are not described in such detail, but the key differences from the series case are given. The general form of the differential equations given in the series circuit section are applicable to all second
8739:
A band-pass filter can be formed with an RLC circuit by either placing a series LC circuit in series with the load resistor or else by placing a parallel LC circuit in parallel with the load resistor. These arrangements are shown in
Figures 8 and 9 respectively. The centre frequency is given by
3426:
705:
The resonance effect can be used for filtering, the rapid change in impedance near resonance can be used to pass or block signals close to the resonance frequency. Both band-pass and band-stop filters can be constructed and some filter circuits are shown later in the article. A key parameter in
420:
occurs because energy for this situation is stored in two different ways: in an electric field as the capacitor is charged and in a magnetic field as current flows through the inductor. Energy can be transferred from one to the other within the circuit and this can be oscillatory. A mechanical
8025:
In the same vein, a resistor in parallel with the capacitor in a series LC circuit can be used to represent a capacitor with a lossy dielectric. This configuration is shown in Figure 5. The resonant frequency (frequency at which the impedance has zero imaginary part) in this case is given by
6657:
7599:
7434:
8289:
2302:
1991:
602:
is caused by the resistance in the circuit. It determines whether or not the circuit will resonate naturally (that is, without a driving source). Circuits that will resonate in this way are described as underdamped and those that will not are overdamped. Damping attenuation (symbol
6727:
3995:
The critically damped response represents the circuit response that decays in the fastest possible time without going into oscillation. This consideration is important in control systems where it is required to reach the desired state as quickly as possible without overshooting.
446:
resonance frequency or undamped natural frequency and the peak frequency may be called the damped resonance frequency or the damped natural frequency. The reason for this terminology is that the driven resonance frequency in a series or parallel resonant circuit has the value.
2460:
9820:
5857:
4584:
6991:
A series resistor with the inductor in a parallel LC circuit as shown in Figure 4 is a topology commonly encountered where there is a need to take into account the resistance of the coil winding and its self-capacitance. Parallel LC circuits are frequently used for
1071:
1038:
3592:
3866:
4743:
7588:
5407:
513:
RLC circuit is the same as a circuit in which there is no damping, hence undamped resonant frequency. The resonant frequency peak amplitude, on the other hand, does depend on the value of the resistor and is described as the damped resonant frequency. A
7109:
This is the resonant frequency of the circuit defined as the frequency at which the admittance has zero imaginary part. The frequency that appears in the generalised form of the characteristic equation (which is the same for this circuit as previously)
5981:
4408:
9576:
6283:
1828:
7104:
5638:
6173:
9246:
A similar effect is observed with currents in the parallel circuit. Even though the circuit appears as high impedance to the external source, there is a large current circulating in the internal loop of the parallel inductor and capacitor.
5235:
8371:
coil was applied to one tuned circuit, creating sparks and thus oscillating currents, sparks were excited in the other tuned circuit only when the inductors were adjusted to resonance. Lodge and some
English scientists preferred the term "
714:, most frequently defined as the frequencies at which the power passed through the circuit has fallen to half the value passed at resonance. There are two of these half-power frequencies, one above, and one below the resonance frequency
8348:, by discharging a large battery of Leyden jars through a long wire, created a tuned circuit with its resonant frequency in the audio range, which produced a musical tone from the spark when it was discharged. In 1857, German physicist
2626:
8131:
6073:
From the frequency response of the current, the frequency response of the voltages across the various circuit elements can also be determined (see figure). Moreover, the maximum voltage across the capacitor happens at a frequency
8386:
to allow the receiver to be tuned to the transmitter. The first patent for a radio system that allowed tuning was filed by Lodge in 1897, although the first practical systems were invented in 1900 by Anglo
Italian radio pioneer
2006:
3697:
3254:
1729:
844:
per second respectively. Other units may require a conversion factor. A more general measure of bandwidth is the fractional bandwidth, which expresses the bandwidth as a fraction of the resonance frequency and is given by
7881:{\displaystyle \ |Z|_{\mathrm {max} }={\frac {RQ_{L}^{2}}{\ {\sqrt {2Q_{L}{\sqrt {Q_{L}^{2}+2\ }}-\left(2Q_{L}^{2}+1\right)~}}\ }}={\frac {RQ_{L}}{\ {\sqrt {2{\sqrt {1+2/Q_{L}^{2}\ }}-\left(2+1/Q_{L}^{2}\right)~}}\ }}\ .}
8922:
For applications in oscillator circuits, it is generally desirable to make the attenuation (or equivalently, the damping factor) as small as possible. In practice, this objective requires making the circuit's resistance
6546:
7267:
6005:
at this peak is, in this particular case, equal to the undamped natural resonance frequency. This means that the maximum voltage across the resistor, and thus maximum heat dissipation, occurs at the natural frequency.
9350:
1554:
245:(C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components of this circuit, where the sequence of the components may vary from RLC.
6519:
4276:
904:
4951:
6971:
when the circuit is driven by a constant voltage. On the other hand, if driven by a constant current, there would be a maximum in the voltage which would follow the same curve as the current in the series circuit.
1355:
9355:
Such a circuit could consist of an energy storage capacitor, a load in the form of a resistance, some circuit inductance and a switch – all in series. The initial conditions are that the capacitor is at voltage,
6899:{\displaystyle {\begin{aligned}{\frac {1}{\,Z\,}}&={\frac {1}{\,Z_{L}\,}}+{\frac {1}{\,Z_{C}\,}}+{\frac {1}{\,Z_{R}\,}}\\&={\frac {1}{\,j\,\omega \,L\,}}+j\,\omega \,C+{\frac {1}{\,R\,}}\,.\end{aligned}}}
4155:
4842:
9647:
8664:
2165:
1836:
7287:
2534:
8792:
8726:
8597:
8151:
6068:
3990:
681:
7187:
9068:
2365:
8009:
9713:
3871:
This is called the damped resonance frequency or the damped natural frequency. It is the frequency the circuit will naturally oscillate at if not driven by an external source. The resonance frequency,
504:
9918:
6339:
768:
9424:
9255:
An overdamped series RLC circuit can be used as a pulse discharge circuit. Often it is useful to know the values of components that could be used to produce a waveform. This is described by the form
5739:
4438:
3180:
1223:{\displaystyle Q={\frac {X}{\,R\;}}={\frac {1}{\,\omega _{\text{o}}R\,C\,}}={\frac {\,\omega _{\text{o}}L\,}{R}}={\frac {1}{\,R\;}}{\sqrt {{\frac {L}{\,C\,}}\,}}={\frac {\,Z_{\text{o}}\,}{R\;}}\;,}
8903:
8845:
2743:
9702:
9183:
6732:
6551:
2763:
2370:
6969:
589:
6448:
961:
3452:
931:
is a widespread measure used to characterise resonators. It is defined as the peak energy stored in the circuit divided by the average energy dissipated in it per radian at resonance. Low-
9478:
6909:
The change from a series arrangement to a parallel arrangement results in the circuit having a peak in impedance at resonance rather than a minimum, so the circuit is an anti-resonator.
3769:
835:
9924:
may be present. The driven resonance does not occur at the damped free oscillation frequency, with a more complicated formula (see below) that produces a reduced value due to damping (
8980:
2643:
Plot showing underdamped and overdamped responses of a series RLC circuit to a voltage input step of 1 V. The critical damping plot is the bold red curve. The plots are normalised for
412:
4608:
9241:
5259:
3880:, which is the frequency at which the circuit will resonate when driven by an external oscillation, may often be referred to as the undamped resonance frequency to distinguish it.
5872:
9116:
8545:
An RLC circuit can be used as a low-pass filter. The circuit configuration is shown in Figure 6. The corner frequency, that is, the frequency of the 3 dB point, is given by
4287:
533:
Circuits with topologies more complex than straightforward series or parallel (some examples described later in the article) have a driven resonance frequency that deviates from
7484:
6395:
of a series RLC. Considering this, it becomes clear that the differential equations describing this circuit are identical to the general form of those describing a series RLC.
1660:
909:
The fractional bandwidth is also often stated as a percentage. The damping of filter circuits is adjusted to result in the required bandwidth. A narrow band filter, such as a
9489:
7931:
9016:
8914:
impedance at resonance. The second case requires a low impedance source so that the voltage is dropped across the antiresonator when it becomes high impedance at resonance.
6184:
1734:
8521:
designer three degrees of freedom. Two of these are required to set the bandwidth and resonant frequency. The designer is still left with one which can be used to scale
7006:
5431:
6080:
8356:
calculated the effect of applying an alternating current to a circuit with inductance and capacitance, showing that the response is maximum at the resonant frequency.
5037:
8367:
in his pioneering paper on the discovery of radio waves, showing the length of spark obtainable from his spark-gap LC resonator detectors as a function of frequency.
8424:
material that has the effect of increasing inductance) is threaded so that it can be screwed further in, or screwed further out of the inductor winding as required.
189:
1301:
1279:
1253:
8352:
photographed the spark produced by a resonant Leyden jar circuit in a rotating mirror, providing visible evidence of the oscillations. In 1868, Scottish physicist
2545:
1390:. This means that circuits which have similar parameters share similar characteristics regardless of whether or not they are operating in the same frequency band.
8850:
The shunt version of the circuit is intended to be driven by a high impedance source, that is, a constant current source. Under those conditions the bandwidth is
8341:(Lord Kelvin) in 1853 showed mathematically that the discharge of a Leyden jar through an inductance should be oscillatory, and derived its resonant frequency.
8032:
283:
to select a narrow frequency range from ambient radio waves. In this role, the circuit is often referred to as a tuned circuit. An RLC circuit can be used as a
2149:{\displaystyle {\frac {\mathrm {d} ^{2}}{\mathrm {d} t^{2}}}I(t)+{\frac {R}{\,L\,}}{\frac {\mathrm {d} }{\mathrm {d} t}}I(t)+{\frac {1}{\,L\,C\,}}I(t)=0\;.}
697:
and represents the case of a circuit that is just on the border of oscillation. It is the minimum damping that can be applied without causing oscillation.
3421:{\displaystyle I(t)=A_{1}e^{-\omega _{0}\left(\zeta +{\sqrt {\zeta ^{2}-1}}\right)t}+A_{2}e^{-\omega _{0}\left(\zeta -{\sqrt {\zeta ^{2}-1}}\right)t}\;.}
264:. The resistor also reduces the peak resonant frequency. Some resistance is unavoidable even if a resistor is not specifically included as a component.
6652:{\displaystyle {\begin{aligned}B_{\mathrm {f} }&={\frac {1}{\,R\,}}{\sqrt {{\frac {L}{C}}~}}\\Q&=R{\sqrt {{\frac {C}{L}}~}}\,~.\end{aligned}}}
3607:
1664:
518:
circuit will fail to resonate at all, when not driven. A circuit with a value of resistor that causes it to be just on the edge of ringing is called
509:
This is exactly the same as the resonance frequency of a lossless LC circuit – that is, one with no resistor present. The resonant frequency for a
9125:
is small, consisting only of the inductor winding resistance say, then this current will be large. It will drop a voltage across the inductor of
8404:
A very frequent use of these circuits is in the tuning circuits of analogue radios. Adjustable tuning is commonly achieved with a parallel plate
9261:
1481:
221:. This circuit does not have a resistor like the above, but all tuned circuits have some resistance, causing them to function as an RLC circuit.
6459:
4188:
851:
4887:
182:
7202:
2297:{\displaystyle {\frac {\mathrm {d} ^{2}}{\mathrm {d} t^{2}}}I(t)+2\alpha {\frac {\mathrm {d} }{\mathrm {d} t}}I(t)+\omega _{0}^{2}I(t)=0\;.}
1986:{\displaystyle RI(t)+L{\frac {\,\mathrm {d} I(t)\,}{\mathrm {d} t}}+V(0)+{\frac {1}{\,C\,}}\int _{0}^{t}I(\tau )\,\mathrm {d} \tau =V(t)\;.}
1306:
7429:{\displaystyle \ \omega _{\mathrm {max} }=\omega '_{0}\ {\sqrt {-{\frac {1}{\ Q_{L}^{2}\ ~}}+{\sqrt {1+{\frac {2}{\ Q_{L}^{2}\ }}~}}~}}\ ,}
4068:
429:
parallel rather than series actually have a maximum impedance rather than a minimum impedance. For this reason they are often described as
8284:{\displaystyle \ \omega _{\mathrm {m} }=\omega '_{0}\ {\sqrt {-{\frac {1}{\ Q_{C}^{2}\ }}+{\sqrt {1+{\frac {2}{\ Q_{C}^{2}\ }}\ }}\ }}\ ,}
4789:
2339:
of the circuit will die away after the stimulus has been removed. Neper occurs in the name because the units can also be considered to be
9591:
8608:
2491:
8746:
8680:
8551:
3598:
6012:
2455:{\displaystyle {\begin{aligned}\alpha &={\frac {R}{\,2L\,}}\\\omega _{0}&={\frac {1}{\,{\sqrt {L\,C\,}}\,}}\;.\end{aligned}}}
3901:
629:
9815:{\displaystyle V_{0}={\frac {\,-I_{0}R\,\alpha \,\beta ~}{\alpha +\beta }}\left({\frac {1}{\beta }}-{\frac {1}{\alpha }}\right)\,.}
2672:
175:
9027:
591:, and for those the undamped resonance frequency, damped resonance frequency and driven resonance frequency can all be different.
7939:
5852:{\displaystyle {\big |}Y(j\omega ){\big |}={\frac {1}{{\sqrt {R^{2}+\left(\omega L-{\frac {1}{\,\omega C\,}}\right)^{2}}}\,}}\;.}
4579:{\displaystyle Y(s)={\frac {I(s)}{\,V(s)\,}}={\frac {s}{\,L\ \left(s^{2}+{\frac {R}{\,L\,}}s+{\frac {1}{\,L\,C\,}}\right)\,}}\,.}
8416:
in the radio where the tuning is preset in the factory, the more usual solution is an adjustable core in the inductor to adjust
453:
9978:
9863:
6291:
720:
9375:
7116:
10494:
10131:
3086:
6987:
Series RL, parallel C circuit with resistance in series with the inductor is the standard model for a self-resonant inductor
8856:
9192:
can be made sufficiently small, these voltages can be several times the input voltage. The voltage ratio is, in fact, the
8803:
2681:
10518:
9658:
9131:
8931:
to as much as possible for a parallel circuit. In either case, the RLC circuit becomes a good approximation to an ideal
1033:{\displaystyle Q={\frac {1}{\,B_{\mathrm {f} }\,}}={\frac {\omega _{\text{o}}}{\,\operatorname {\Delta } \omega \,}}\;.}
6915:
3587:{\displaystyle I(t)=B_{1}e^{-\alpha t}\cos(\omega _{\mathrm {d} }t)+B_{2}e^{-\alpha t}\sin(\omega _{\mathrm {d} }t)\,.}
536:
6408:
3080:
The general solution of the differential equation is an exponential in either root or a linear superposition of both,
10475:
10456:
10437:
10418:
10351:
8338:
8321:
The first evidence that a capacitor could produce electrical oscillations was discovered in 1826 by French scientist
6912:
The graph opposite shows that there is a minimum in the frequency response of the current at the resonance frequency
10196:
6536:
circuit in the other topology when constructed from components with identical values. The fractional bandwidth and
3861:{\displaystyle \omega _{\mathrm {d} }={\sqrt {\omega _{0}^{2}-\alpha ^{2}}}=\omega _{0}{\sqrt {1-\zeta ^{2}\,}}\;.}
10197:"Finding the exact maximum impedance resonant frequency of a practical parallel resonant circuit without calculus"
9435:
8337:
repeated Savary's experiment in 1842 and came to the same conclusion, apparently independently. British scientist
8421:
4738:{\displaystyle Y(s)={\frac {I(s)}{\,V(s)\,}}={\frac {s}{\,L\ \left(s^{2}+2\alpha s+\omega _{0}^{2}\right)\,}}\;.}
5402:{\displaystyle \ I(t)={\frac {1}{\ L\ }}\ \int _{0}^{t}V(t-\tau )e^{-\alpha \tau }\ \left\ \mathrm {d} \tau \ ,}
804:
9963:
8948:
367:
299:. The tuning application, for instance, is an example of band-pass filtering. The RLC filter is described as a
8330:
back and forth until it was too small to have an effect, leaving the needle magnetized in a random direction.
9202:
6388:
5976:{\displaystyle {\big |}I(j\omega ){\big |}={\big |}Y(j\omega ){\big |}\cdot {\bigl |}V(j\omega ){\bigr |}\;.}
707:
4403:{\displaystyle I(s)={\frac {s}{\,L\ \left(s^{2}+{\frac {R}{\,L\,}}s+{\frac {1}{\,L\,C\,}}\right)\,}}V(s)\;.}
7583:{\displaystyle \ \omega _{\mathrm {max} }\approx \omega '_{0}\ {\sqrt {1-{\frac {1}{\ 2Q_{L}^{4}\ }}~}}\ .}
100:
9084:
210:
9571:{\displaystyle V_{0}=-I_{0}L\,\alpha \,\beta \,\left({\frac {1}{\beta }}-{\frac {1}{\alpha }}\right)\,.}
9188:
An equal magnitude voltage will also be seen across the capacitor but in antiphase to the inductor. If
6662:
Notice that the formulas here are the reciprocals of the formulas for the series circuit, given above.
1468:
1613:
7894:
6278:{\displaystyle \omega _{L}={\frac {1}{\ C{\sqrt {{\tfrac {L}{\ C\ }}-{\tfrac {1}{2}}R^{2}\ }}\ }}\;.}
5694:
Bode magnitude plot for the voltages across the elements of an RLC series circuit. Natural frequency
117:
70:
65:
33:
8991:
1823:{\displaystyle \,V_{\mathrm {C} }=V(0)+{\frac {1}{\,C\,}}\int _{0}^{t}I(\tau )\,\mathrm {d} \tau \,}
433:; it is still usual, however, to name the frequency at which this occurs as the resonant frequency.
10513:
9078:
In a series RLC circuit at resonance, the current is limited only by the resistance of the circuit
8349:
7099:{\displaystyle \ \omega _{0}={\sqrt {{\frac {1}{\ LC\ }}-\left({\frac {R}{\ L\ }}\right)^{2}~}}\ .}
5633:{\displaystyle \ I(t)={\frac {1}{\ L\ }}\ \int _{0}^{t}V(t-\tau )e^{-\alpha \tau }\left\,d\tau \ ,}
1996:
For the case where the source is an unchanging voltage, taking the time derivative and dividing by
8674:
A high-pass filter is shown in Figure 7. The corner frequency is the same as the low-pass filter:
6168:{\displaystyle \omega _{C}={\frac {\ {\sqrt {{\tfrac {L}{\ C\ }}-{\tfrac {1}{2}}R^{2}}}\ }{L}}\;,}
5230:{\displaystyle I(t)={\frac {1}{\,L\,}}\ \int _{0}^{t}V(t-\tau )e^{-\alpha \tau }\left\,d\tau \,,}
4848:
4754:
111:
6721:
The complex admittance of this circuit is given by adding up the admittances of the components:
2631:
The value of the damping factor determines the type of transient that the circuit will exhibit.
8413:
311:
303:
circuit, meaning that any voltage or current in the circuit can be described by a second-order
123:
83:
10341:
10319:
939:
circuits are underdamped and prone to amplitude extremes if driven at the resonant frequency.
327:
An important property of this circuit is its ability to resonate at a specific frequency, the
8383:
3754:
in the second form) are arbitrary constants determined by boundary conditions. The frequency
2621:{\displaystyle \zeta ={\frac {\,R\,}{2}}{\sqrt {{\frac {C}{\,L\,}}\,}}={\frac {1}{\ 2Q\ }}~.}
1472:
1464:
304:
268:
260:. Introducing the resistor increases the decay of these oscillations, which is also known as
234:
10246:
1284:
1262:
1236:
425:
328:
8537:
may be predetermined by the external circuitry which will use the last degree of freedom.
8126:{\displaystyle \ \omega _{0}={\sqrt {{\frac {1}{\ LC\ }}-{\frac {1}{\ (RC)^{2}\ }}\ }}\ ,}
8:
8939:-factor), issues such as dielectric losses of coils and capacitors can become important.
8353:
5690:
5679:
4022:
The series RLC can be analyzed for both transient and steady AC state behavior using the
1047:
280:
249:
3601:
the two trigonometric functions may be expressed as a single sinusoid with phase shift,
10266:
10262:
8405:
4878:
3204:
2336:
1395:
230:
128:
40:
10490:
10471:
10452:
10433:
10414:
10347:
10127:
8388:
4042:
4023:
2320:
519:
345:
218:
90:
22:
10270:
4985:
of the characteristic polynomial of the differential equation in the section above.
10258:
3720:
2344:
711:
296:
288:
284:
272:
95:
45:
9860:
RLC response to driving voltage occurs at the frequency for lossless oscillation,
9369:
is known, then the remaining parameters are given by the following – capacitance:
8927:
as small as physically possible for a series circuit, or alternatively increasing
7000:
is largely governed by this resistance. The resonant frequency of this circuit is
6993:
3431:
The overdamped response is a decay of the transient current without oscillation.
310:
The three circuit elements, R, L and C, can be combined in a number of different
292:
60:
50:
3692:{\displaystyle I(t)=B_{3}e^{-\alpha t}\sin(\omega _{\mathrm {d} }t+\varphi )\;.}
9841:
8364:
6392:
5730:
1724:{\displaystyle \,V_{\mathrm {L} }=L{\frac {\mathrm {d} I(t)}{\mathrm {d} t}}\,}
1460:
276:
798:
is the upper half-power frequency. The bandwidth is related to attenuation by
10507:
430:
105:
55:
8021:
Parallel RC, series L circuit with resistance in parallel with the capacitor
357:, is used because it is more mathematically convenient. This is measured in
8345:
8334:
8322:
6703:
4026:. If the voltage source above produces a waveform with Laplace-transformed
910:
9977:(course notes). Electrical & Computer Engineering. Santa Barbara, CA:
9345:{\displaystyle I(t)=I_{0}\left(\,e^{-\alpha \,t}-e^{-\beta \,t}\right)\,.}
6351:
2485:
instead; hence requiring careful specification of one's use of that term.
2359:
For the case of the series RLC circuit these two parameters are given by:
1549:{\displaystyle V_{\mathrm {R} }+V_{\mathrm {L} }+V_{\mathrm {C} }=V(t)\,,}
8602:
This is also the bandwidth of the filter. The damping factor is given by
8505:
8477:
6514:{\displaystyle \zeta ={\frac {1}{\,2\,R\,}}{\sqrt {{\frac {L}{C}}~}}\,~.}
4271:{\displaystyle I(s)={\frac {1}{\,R+L\,s+{\frac {1}{\,C\,s\,}}\,}}V(s)\;.}
1394:
order circuits and can be used to describe the voltage or current in any
899:{\displaystyle B_{\mathrm {f} }={\frac {\Delta \omega }{\omega _{0}}}\,.}
523:
206:
A series RLC network (in order): a resistor, an inductor, and a capacitor
6373:– the equivalent resistance of the combined source, load, and components
4946:{\displaystyle s=-\alpha \pm {\sqrt {\alpha ^{2}-\omega _{0}^{2}\,}}\;.}
1406:
9836:
9831:
8932:
8492:
8464:
8326:
8014:
6980:
4419:
4171:
is the
Laplace-transformed current through all components. Solving for
1432:, the effective resistance of the combined load, source, and components
527:
257:
156:
151:
146:
7262:{\displaystyle \ \omega '_{0}\equiv {\frac {1}{\ {\sqrt {LC~}}\ }}\ .}
2539:
In the case of the series RLC circuit, the damping factor is given by
1350:{\displaystyle \,Z_{\text{o}}\equiv {\sqrt {{\frac {L}{\,C\,}}\,}}\;.}
9365:, and there is no current flowing in the inductor. If the inductance
8412:
to be changed and tune to stations on different frequencies. For the
8360:
6528:
are also reciprocals of each other. This means that a wide-band, low-
5733:. Taking the magnitude of the above equation with this substitution:
4150:{\displaystyle V(s)=I(s)\left(R+L\,s+{\frac {1}{\,C\,s\,}}\right)\,,}
2639:
1449:
417:
253:
242:
215:
6670:
6391:
of electrical circuits and considering that the parallel RLC is the
6387:
The properties of the parallel RLC circuit can be obtained from the
4837:{\displaystyle s=0\quad {\mbox{and}}\quad |s|\rightarrow \infty \;.}
2159:
This can usefully be expressed in a more generally applicable form:
10138:(Some notations have been changed to fit the rest of this article.)
9642:{\displaystyle C={\frac {~\alpha +\beta ~}{R\,\alpha \,\beta }}\,,}
8659:{\displaystyle \zeta ={\frac {1}{2R_{L}}}{\sqrt {\frac {L}{C}}}\,.}
8484:
RLC circuit as a parallel band-pass filter in shunt across the line
7475:
1439:
925:
443:
361:
per second. They are related to each other by a simple proportion,
238:
8512:
RLC circuit as a parallel band-stop filter in series with the line
2529:{\displaystyle \zeta \equiv {\frac {\alpha }{\,\omega _{0}\,}}\;.}
2473:, which is defined as the ratio of these two; although, sometimes
913:, requires low damping. A wide band filter requires high damping.
8787:{\displaystyle \omega _{\mathrm {c} }={\frac {1}{\sqrt {LC}}}\,,}
8721:{\displaystyle \omega _{\mathrm {c} }={\frac {1}{\sqrt {LC}}}\,.}
8592:{\displaystyle \omega _{\mathrm {c} }={\frac {1}{\sqrt {LC}}}\,.}
8499:
RLC circuit as a series band-stop filter in shunt across the line
3711:. The oscillation decays at a rate determined by the attenuation
1459:
In this circuit, the three components are all in series with the
612:
599:
261:
8471:
RLC circuit as a series band-pass filter in series with the line
8449:
6524:
Likewise, the other scaled parameters, fractional bandwidth and
6063:{\displaystyle \omega _{0}={\frac {1}{\,{\sqrt {L\ C\ }}\,}}\;.}
3702:
The underdamped response is a decaying oscillation at frequency
424:
The resonant frequency is defined as the frequency at which the
8436:
7593:
Furthermore, the exact maximum impedance magnitude is given by
358:
9928:) which only applies to free oscillations (no driving signal).
3985:{\displaystyle I(t)=D_{1}te^{-\alpha t}+D_{2}e^{-\alpha t}\;.}
676:{\displaystyle \ \zeta ={\frac {\alpha }{\ \omega _{0}\ }}\ .}
8382:
The first practical use for RLC circuits was in the 1890s in
6693:
2340:
841:
608:
341:
202:
9063:{\displaystyle \omega _{\mathrm {d} }\approx \omega _{0}\,.}
619:, zeta) is often a more useful measure, which is related to
8004:{\displaystyle \ |Z|_{\mathrm {max} }\approx RQ_{L}^{2}\ .}
7281:, at which the impedance magnitude is maximum, is given by
7192:
is not the same frequency. In this case it is the natural,
6713:
4014:
are arbitrary constants determined by boundary conditions.
2000:
leads to the following second order differential equation:
10126:(2nd ed.). Chapman & Hall/CRC. pp. 198–202.
6178:
whereas the maximum voltage across the inductor occurs at
499:{\displaystyle \omega _{0}={\frac {1}{\sqrt {L\,C~}}}\,~.}
9913:{\displaystyle \ {\frac {1}{\ 2\pi {\sqrt {LC\ }}\ }}\ ,}
6683:
6334:{\displaystyle \omega _{C}<\omega _{0}<\omega _{L}}
4059:
763:{\displaystyle \Delta \omega =\omega _{2}-\omega _{1}\,,}
10383:
10381:
10194:
9419:{\displaystyle C={\frac {1}{~L\,\alpha \,\beta \,~}}\,,}
8145:
at which the impedance magnitude is minimum is given by
7182:{\displaystyle \ s^{2}+2\alpha s+{\omega '_{0}}^{2}=0\ }
6532:
circuit in one topology will become a narrow-band, high-
10324:(2nd ed.). New York: Henry Hold. pp. 516–517.
3175:{\displaystyle I(t)=A_{1}e^{s_{1}t}+A_{2}e^{s_{2}t}\;.}
6239:
6216:
6129:
6106:
4804:
10411:
Foundations of Analog and
Digital Electronic Circuits
10378:
10195:
Cartwright, K.V.; Joseph, E.; Kaminsky, E.J. (2010).
9866:
9716:
9661:
9594:
9492:
9438:
9378:
9264:
9205:
9134:
9087:
9030:
8994:
8951:
8898:{\displaystyle \Delta \omega ={\frac {1}{CR_{L}}}\,.}
8859:
8806:
8749:
8683:
8611:
8554:
8154:
8035:
7942:
7897:
7602:
7487:
7290:
7205:
7119:
7009:
6918:
6730:
6549:
6462:
6411:
6294:
6187:
6083:
6015:
5875:
5742:
5434:
5262:
5040:
4890:
4792:
4611:
4441:
4290:
4191:
4071:
3904:
3772:
3610:
3455:
3257:
3089:
2761:
2684:
2548:
2494:
2368:
2168:
2009:
1839:
1737:
1667:
1616:
1484:
1309:
1287:
1265:
1239:
1074:
964:
854:
807:
723:
632:
539:
456:
370:
10190:
10188:
10186:
10184:
8840:{\displaystyle \Delta \omega ={\frac {R_{L}}{L}}\,.}
2738:{\displaystyle s^{2}+2\alpha s+\omega _{0}^{2}=0\,.}
522:. Either side of critically damped are described as
10290:Savary, Felix (1827). "Memoirs sur l'Aimentation".
9697:{\displaystyle L={\frac {R}{\,\alpha +\beta ~}}\,,}
9178:{\displaystyle V_{L}={\frac {V}{R}}\omega _{0}L\,.}
8935:. However, for very low-attenuation circuits (high
951:circuits are narrow-band. In fact, it happens that
9912:
9814:
9696:
9641:
9570:
9472:
9418:
9344:
9235:
9177:
9110:
9062:
9010:
8974:
8897:
8839:
8786:
8720:
8658:
8591:
8283:
8125:
8003:
7925:
7880:
7582:
7428:
7261:
7181:
7098:
6963:
6898:
6651:
6513:
6442:
6333:
6277:
6167:
6062:
5975:
5851:
5715:Sinusoidal steady state is represented by letting
5632:
5401:
5229:
4945:
4836:
4737:
4578:
4402:
4270:
4149:
3984:
3860:
3691:
3586:
3420:
3174:
3069:
2737:
2620:
2528:
2454:
2296:
2148:
1985:
1822:
1723:
1654:
1548:
1349:
1295:
1273:
1247:
1222:
1032:
898:
829:
762:
675:
583:
498:
406:
10485:Nilsson, James William; Riedel, Susan A. (2008).
10181:
9957:
9955:
9953:
6964:{\displaystyle ~\omega _{0}=1/{\sqrt {\,L\,C~}}~}
935:circuits are therefore damped and lossy and high-
584:{\displaystyle \ \omega _{0}=1/{\sqrt {L\,C~}}\ }
10505:
9962:Long, Steve (2004-04-15) . Rodwell, Mark (ed.).
8363:curve was published in 1887 by German physicist
6677:Sinusoidal steady-state analysis. Normalised to
6443:{\displaystyle \alpha ={\frac {1}{\,2\,R\,C\,}}}
5004:, the solution obtained by inverse transform of
8533:to convenient practical values. Alternatively,
10204:The Technology Interface International Journal
9950:
7478:of the coil. This can be well approximated by
1475:for each of the three elements. From the KVL,
10484:
10121:
10120:This section is based on Example 4.2.13 from
10071:
10069:
5964:
5942:
5932:
5910:
5900:
5878:
5767:
5745:
3883:
1607:is the time-varying voltage from the source.
183:
10335:
10333:
10331:
10240:
10238:
10236:
10234:
10232:
10230:
10228:
10226:
10032:
10030:
9473:{\displaystyle R=L\,(\,\alpha +\beta \,)\,,}
8797:and the bandwidth for the series circuit is
6383:– the capacitance of the capacitor component
6345:
10408:
10343:The Worldwide History of Telecommunications
10122:Debnath, Lokenath; Bhatta, Dambaru (2007).
9961:
8420:. In this design, the core (made of a high
840:where the units are radians per second and
10313:
10311:
10309:
10307:
10305:
10124:Integral Transforms and Their Applications
10066:
9250:
8731:The filter has a stop-band of this width.
8399:
6398:For the parallel circuit, the attenuation
6378:– the inductance of the inductor component
6368:– the current admitted through the circuit
6271:
6161:
6056:
5969:
5845:
5685:
4939:
4830:
4731:
4396:
4264:
3978:
3854:
3685:
3414:
3168:
3059:
2522:
2444:
2290:
2142:
1979:
1426:, the current admitted through the circuit
1343:
1216:
1212:
1166:
1091:
1026:
830:{\displaystyle \Delta \omega =2\alpha \,,}
190:
176:
10409:Agarwal, Anant; Lang, Jeffrey H. (2005).
10339:
10328:
10244:
10223:
10027:
9808:
9754:
9750:
9733:
9690:
9674:
9635:
9628:
9624:
9564:
9530:
9526:
9522:
9466:
9462:
9452:
9448:
9412:
9405:
9401:
9397:
9338:
9327:
9307:
9295:
9229:
9171:
9104:
9056:
9004:
8975:{\displaystyle \alpha \ll \omega _{0}\,,}
8968:
8891:
8833:
8780:
8714:
8652:
8585:
6949:
6945:
6888:
6884:
6880:
6867:
6863:
6853:
6849:
6845:
6841:
6821:
6810:
6797:
6786:
6773:
6762:
6745:
6741:
6638:
6583:
6579:
6504:
6483:
6479:
6475:
6436:
6432:
6428:
6424:
6363:– the voltage source powering the circuit
6052:
6035:
5841:
5824:
5817:
5617:
5223:
5216:
5066:
5062:
4936:
4727:
4670:
4657:
4644:
4602:defined in the previous section, we have
4572:
4568:
4559:
4555:
4551:
4535:
4531:
4500:
4487:
4474:
4380:
4371:
4367:
4363:
4347:
4343:
4312:
4248:
4244:
4240:
4236:
4223:
4213:
4143:
4134:
4130:
4126:
4113:
3851:
3580:
3051:
3003:
2870:
2822:
2731:
2585:
2581:
2577:
2562:
2558:
2518:
2507:
2440:
2437:
2433:
2427:
2396:
2389:
2120:
2116:
2112:
2068:
2064:
1955:
1924:
1920:
1882:
1864:
1819:
1810:
1779:
1775:
1738:
1720:
1668:
1648:
1542:
1420:, the voltage source powering the circuit
1340:
1336:
1332:
1310:
1292:
1288:
1270:
1266:
1244:
1240:
1206:
1195:
1186:
1182:
1178:
1162:
1146:
1132:
1122:
1118:
1104:
1087:
1022:
1010:
990:
977:
892:
823:
756:
569:
489:
479:
407:{\displaystyle \omega _{0}=2\pi f_{0}\,.}
400:
9652:inductance (total of circuit and load):
9429:resistance (total of circuit and load):
8504:
8491:
8476:
8463:
8448:
8435:
8013:
6979:
6669:
6350:
5689:
2638:
1405:
710:. The bandwidth is measured between the
209:
201:
10317:
10302:
9707:initial terminal voltage of capacitor:
9483:initial terminal voltage of capacitor:
9236:{\displaystyle {\frac {V_{L}}{V}}=Q\,.}
6975:
6453:and the damping factor is consequently
3434:
1054:factor depends inversely on bandwidth.
955:is the inverse of fractional bandwidth
340:. Frequencies are measured in units of
267:RLC circuits have many applications as
10506:
10468:Electromagnetic Compatibility Handbook
10465:
10289:
4062:can be applied in the Laplace domain:
3236:
789:is the lower half-power frequency and
10446:
10427:
10346:. USA: Wiley-IEEE. pp. 199–200.
10247:"The History of Electrical Resonance"
9073:
6540:of the parallel circuit are given by
4413:
2634:
1065:factor can be calculated as follows:
9964:"Resonant circuits – resonators and
2356:is the angular resonance frequency.
1359:
436:
9111:{\displaystyle I={\frac {V}{R}}\,.}
8908:
8734:
8669:
8359:The first example of an electrical
6665:
6344:
4988:
2335:, and is a measure of how fast the
1058:
1046:factor is directly proportional to
16:Resistor Inductor Capacitor Circuit
13:
10449:Basic Engineering Circuit Analysis
10263:10.1002/j.1538-7305.1941.tb03608.x
10245:Blanchard, Julian (October 1941).
9037:
8860:
8807:
8756:
8690:
8561:
8540:
8164:
7968:
7965:
7962:
7628:
7625:
7622:
7503:
7500:
7497:
7306:
7303:
7300:
6560:
5600:
5573:
5543:
5386:
5199:
5172:
5142:
4827:
4748:
3779:
3667:
3568:
3512:
2671:The differential equation has the
2232:
2226:
2185:
2173:
2081:
2075:
2026:
2014:
1957:
1886:
1866:
1812:
1745:
1710:
1691:
1675:
1521:
1506:
1491:
1467:can be found by substituting into
1012:
984:
873:
861:
808:
724:
611:per second. However, the unitless
14:
10530:
8456:RLC circuit as a high-pass filter
8375:" for this effect, but the term "
7933:this can be well approximated by
5862:and the current as a function of
4017:
1401:
317:
10318:Kimball, Arthur Lalanne (1917).
10292:Annales de Chimie et de Physique
10149:Electric Circuits & Networks
10045:Nilsson and Riedel, pp. 287–288.
8443:RLC circuit as a low-pass filter
3888:The critically damped response (
1830:into the equation above yields:
1655:{\displaystyle V_{R}=R\ I(t)\,,}
947:circuits are wide-band and high-
10402:
10390:
10369:
10360:
10283:
10214:
10172:
10163:
10154:
10141:
10114:
10105:
10096:
10087:
10078:
10057:
10048:
9581:Rearranging for the case where
8394:
7926:{\displaystyle \ Q_{L}\gg 1\ ,}
5241:In the critically damped case,
4810:
4802:
1057:For a series resonant circuit (
10321:A College Text-book of Physics
10039:
10018:
10009:
10000:
9991:
9941:
9854:
9463:
9449:
9274:
9268:
9011:{\displaystyle \zeta \ll 1\,.}
8917:
8097:
8087:
7956:
7947:
7616:
7607:
5959:
5950:
5927:
5918:
5895:
5886:
5762:
5753:
5609:
5591:
5552:
5534:
5504:
5492:
5447:
5441:
5332:
5320:
5275:
5269:
5208:
5190:
5151:
5133:
5103:
5091:
5050:
5044:
4824:
4820:
4812:
4654:
4648:
4639:
4633:
4621:
4615:
4484:
4478:
4469:
4463:
4451:
4445:
4393:
4387:
4300:
4294:
4261:
4255:
4201:
4195:
4096:
4090:
4081:
4075:
3914:
3908:
3682:
3658:
3620:
3614:
3577:
3559:
3521:
3503:
3465:
3459:
3267:
3261:
3099:
3093:
2281:
2275:
2251:
2245:
2211:
2205:
2133:
2127:
2100:
2094:
2052:
2046:
1976:
1970:
1952:
1946:
1908:
1902:
1879:
1873:
1852:
1846:
1807:
1801:
1763:
1757:
1704:
1698:
1645:
1639:
1539:
1533:
1:
10251:Bell System Technical Journal
9935:
4589:Simplifying using parameters
2748:The roots of the equation in
943:is related to bandwidth; low-
66:Optimum "L" (Legendre) filter
10340:Huurdeman, Anton A. (2003).
9920:even though loss resistance
9847:
8384:spark-gap radio transmitters
700:
322:
7:
10466:Kaiser, Kenneth L. (2004).
10178:Agarwal & Lang, p. 805.
10160:Nilsson and Riedel, p. 286.
10102:Nilsson and Riedel, p. 303.
10075:Nilsson and Riedel, p. 295.
9997:Nilsson and Riedel, p. 308.
9825:
4967:are identical to the roots
916:
10:
10535:
10519:Electronic filter topology
8427:
8408:which allows the value of
8316:
3884:Critically damped response
3439:The underdamped response (
3226:), and critically damped (
2465:A useful parameter is the
2343:per second, neper being a
594:
256:in a manner similar to an
10257:(4). USA: AT&T: 415.
10093:Agarwal and Lang, p. 692.
10063:Agarwal and Lang, p. 648.
10036:Agarwal and Lang, p. 656.
10015:Agarwal and Lang, p. 646.
10006:Agarwal and Lang, p. 641.
8942:In an oscillator circuit
8344:British radio researcher
5986:There is a peak value of
5019:In the underdamped case,
4281:And rearranging, we have
3241:The overdamped response (
1448:, the capacitance of the
530:(ringing is suppressed).
118:Bridged T delay equaliser
34:Network synthesis filters
10447:Irwin, J. David (2006).
10432:. Taylor & Francis.
9585:is known – capacitance:
8350:Berend Wilhelm Feddersen
5413:In the overdamped case,
4418:Solving for the Laplace
3599:trigonometric identities
3212:. These are overdamped (
1584:are the voltages across
1438:, the inductance of the
9251:Pulse discharge circuit
8400:Variable tuned circuits
8325:. He found that when a
5686:Sinusoidal steady state
2673:characteristic equation
1469:Kirchhoff's voltage law
84:Image impedance filters
51:Elliptic (Cauer) filter
10430:Dynamics of Structures
10396:Kaiser, pp. 7.30–7.34.
10387:Kaiser, pp. 7.21–7.27.
10366:Kaiser, pp. 7.14–7.16.
10298:. Paris: Masson: 5–37.
10220:Kaiser, pp. 5.25–5.26.
10169:Kaiser, pp. 5.26–5.27
9973:. ECE145B / ECE 218B.
9914:
9816:
9698:
9643:
9572:
9474:
9420:
9346:
9237:
9179:
9112:
9064:
9012:
8976:
8899:
8841:
8788:
8722:
8660:
8593:
8513:
8500:
8485:
8472:
8457:
8444:
8285:
8127:
8022:
8005:
7927:
7882:
7584:
7430:
7263:
7183:
7100:
6988:
6965:
6900:
6718:
6653:
6515:
6444:
6384:
6335:
6279:
6169:
6064:
5977:
5853:
5712:
5634:
5403:
5231:
4947:
4838:
4739:
4580:
4404:
4272:
4151:
3986:
3862:
3693:
3588:
3422:
3203:are determined by the
3176:
3071:
2739:
2668:
2622:
2530:
2456:
2298:
2150:
1987:
1824:
1725:
1656:
1550:
1456:
1351:
1297:
1275:
1249:
1224:
1034:
900:
831:
764:
677:
585:
526:(ringing happens) and
500:
408:
222:
207:
124:Composite image filter
10428:Humar, J. L. (2002).
9947:Kaiser, pp. 7.71–7.72
9915:
9817:
9699:
9644:
9573:
9475:
9421:
9347:
9238:
9180:
9113:
9065:
9013:
8977:
8900:
8842:
8789:
8723:
8661:
8594:
8508:
8495:
8480:
8467:
8452:
8439:
8286:
8128:
8017:
8006:
7928:
7883:
7585:
7431:
7264:
7184:
7101:
6983:
6966:
6901:
6673:
6654:
6516:
6445:
6354:
6336:
6280:
6170:
6065:
5978:
5854:
5693:
5635:
5404:
5232:
4948:
4839:
4740:
4581:
4405:
4273:
4152:
3987:
3863:
3715:. The exponential in
3694:
3597:By applying standard
3589:
3423:
3177:
3072:
2740:
2642:
2623:
2531:
2457:
2319:are both in units of
2299:
2151:
1988:
1825:
1726:
1657:
1551:
1473:constitutive equation
1465:differential equation
1409:
1352:
1298:
1296:{\displaystyle \,C\,}
1276:
1274:{\displaystyle \,L\,}
1250:
1248:{\displaystyle \,X\,}
1225:
1035:
901:
832:
765:
678:
586:
501:
409:
307:in circuit analysis.
305:differential equation
213:
205:
101:General image filters
71:Linkwitz–Riley filter
9864:
9714:
9659:
9592:
9490:
9436:
9376:
9262:
9203:
9132:
9085:
9028:
8992:
8949:
8857:
8804:
8747:
8681:
8609:
8552:
8379:" eventually stuck.
8152:
8136:while the frequency
8033:
7940:
7895:
7600:
7485:
7288:
7203:
7196:resonant frequency:
7117:
7007:
6976:Other configurations
6916:
6728:
6547:
6460:
6409:
6389:duality relationship
6358:RLC parallel circuit
6292:
6185:
6081:
6013:
5873:
5740:
5680:hyperbolic functions
5432:
5260:
5038:
4888:
4862:are those values of
4790:
4768:are those values of
4609:
4439:
4288:
4189:
4069:
3902:
3770:
3750:and the phase shift
3723:of the oscillation.
3608:
3453:
3435:Underdamped response
3255:
3087:
2759:
2682:
2546:
2492:
2366:
2166:
2007:
1837:
1735:
1665:
1614:
1596:, respectively, and
1482:
1307:
1285:
1263:
1237:
1072:
962:
852:
805:
721:
686:The special case of
630:
537:
454:
368:
248:The circuit forms a
10413:. Morgan Kaufmann.
10084:Humar, pp. 223–224.
10024:Irwin, pp. 217–220.
8354:James Clerk Maxwell
8333:American physicist
8258:
8217:
8185:
7994:
7855:
7813:
7731:
7694:
7657:
7562:
7524:
7403:
7359:
7327:
7221:
7162:
5488:
5316:
5087:
4935:
4721:
3804:
3237:Overdamped response
3205:boundary conditions
3002:
2821:
2724:
2271:
1942:
1797:
1413:RLC series circuit
344:. In this article,
329:resonance frequency
269:oscillator circuits
250:harmonic oscillator
214:Tuned circuit of a
108:(constant R) filter
9979:U.C. Santa Barbara
9910:
9812:
9694:
9639:
9568:
9470:
9416:
9342:
9233:
9175:
9108:
9074:Voltage multiplier
9060:
9008:
8972:
8895:
8837:
8784:
8718:
8656:
8589:
8514:
8501:
8486:
8473:
8458:
8445:
8406:variable capacitor
8281:
8244:
8203:
8173:
8123:
8023:
8001:
7980:
7923:
7878:
7841:
7799:
7717:
7680:
7643:
7580:
7548:
7512:
7426:
7389:
7345:
7315:
7259:
7209:
7179:
7150:
7096:
6994:bandpass filtering
6989:
6961:
6896:
6894:
6719:
6649:
6647:
6511:
6440:
6385:
6331:
6275:
6248:
6233:
6165:
6138:
6123:
6060:
5973:
5866:can be found from
5849:
5713:
5630:
5474:
5399:
5302:
5227:
5073:
4943:
4921:
4834:
4808:
4735:
4707:
4576:
4414:Laplace admittance
4400:
4268:
4147:
3982:
3858:
3790:
3689:
3584:
3418:
3172:
3067:
3065:
2988:
2807:
2735:
2710:
2669:
2635:Transient response
2618:
2526:
2481:is referred to as
2452:
2450:
2337:transient response
2294:
2257:
2146:
1983:
1928:
1820:
1783:
1721:
1652:
1546:
1457:
1381:are all scaled to
1347:
1303:at resonance, and
1293:
1271:
1245:
1220:
1030:
896:
827:
780:is the bandwidth,
760:
712:cutoff frequencies
673:
581:
496:
404:
231:electrical circuit
223:
208:
41:Butterworth filter
25:electronic filters
10496:978-0-13-198925-2
10489:. Prentice Hall.
10487:Electric Circuits
10147:Kumar and Kumar,
10133:978-1-58488-575-7
9906:
9902:
9900:
9896:
9895:
9878:
9869:
9801:
9788:
9773:
9760:
9688:
9686:
9633:
9618:
9606:
9557:
9544:
9410:
9408:
9393:
9221:
9156:
9102:
8889:
8831:
8778:
8777:
8712:
8711:
8650:
8649:
8638:
8583:
8582:
8518:
8517:
8389:Guglielmo Marconi
8277:
8273:
8272:
8268:
8267:
8263:
8261:
8243:
8222:
8220:
8202:
8188:
8157:
8119:
8115:
8114:
8110:
8108:
8086:
8073:
8071:
8062:
8038:
7997:
7945:
7919:
7900:
7874:
7870:
7868:
7864:
7863:
7817:
7816:
7777:
7752:
7750:
7746:
7745:
7704:
7703:
7662:
7605:
7576:
7572:
7571:
7567:
7565:
7544:
7527:
7490:
7422:
7418:
7417:
7413:
7412:
7408:
7406:
7388:
7367:
7365:
7362:
7344:
7330:
7293:
7255:
7251:
7249:
7245:
7244:
7233:
7208:
7178:
7122:
7092:
7088:
7087:
7073:
7071:
7065:
7047:
7045:
7036:
7012:
6960:
6956:
6955:
6921:
6886:
6855:
6823:
6799:
6775:
6747:
6641:
6636:
6635:
6631:
6602:
6601:
6597:
6585:
6507:
6502:
6501:
6497:
6485:
6438:
6269:
6267:
6263:
6262:
6247:
6232:
6230:
6224:
6209:
6159:
6154:
6150:
6137:
6122:
6120:
6114:
6102:
6054:
6050:
6049:
6043:
5843:
5839:
5826:
5626:
5583:
5581:
5566:
5473:
5469:
5467:
5461:
5437:
5395:
5384:
5376:
5361:
5353:
5301:
5297:
5295:
5289:
5265:
5182:
5180:
5165:
5072:
5068:
4993:For an arbitrary
4937:
4879:quadratic formula
4807:
4729:
4676:
4659:
4570:
4561:
4537:
4506:
4489:
4382:
4373:
4349:
4318:
4250:
4246:
4136:
4043:complex frequency
4024:Laplace transform
3852:
3818:
3402:
3330:
3185:The coefficients
3052:
3004:
2939:
2935:
2933:
2929:
2928:
2901:
2871:
2823:
2614:
2610:
2608:
2599:
2586:
2583:
2567:
2520:
2477:is not used, and
2442:
2438:
2398:
2321:angular frequency
2240:
2200:
2122:
2089:
2070:
2041:
1926:
1894:
1781:
1718:
1635:
1398:of each circuit.
1360:Scaled parameters
1341:
1338:
1318:
1255:is the reactance
1214:
1203:
1187:
1184:
1168:
1151:
1140:
1124:
1112:
1093:
1024:
1006:
992:
890:
706:filter design is
669:
665:
663:
650:
635:
607:) is measured in
580:
576:
575:
542:
520:critically damped
492:
487:
486:
485:
437:Natural frequency
346:angular frequency
252:for current, and
219:radio transmitter
200:
199:
91:Constant k filter
10526:
10500:
10481:
10462:
10443:
10424:
10397:
10394:
10388:
10385:
10376:
10375:Kaiser, p. 7.21.
10373:
10367:
10364:
10358:
10357:
10337:
10326:
10325:
10315:
10300:
10299:
10287:
10281:
10280:
10278:
10277:
10242:
10221:
10218:
10212:
10211:
10201:
10192:
10179:
10176:
10170:
10167:
10161:
10158:
10152:
10145:
10139:
10137:
10118:
10112:
10109:
10103:
10100:
10094:
10091:
10085:
10082:
10076:
10073:
10064:
10061:
10055:
10052:
10046:
10043:
10037:
10034:
10025:
10022:
10016:
10013:
10007:
10004:
9998:
9995:
9989:
9988:
9986:
9985:
9972:
9967:
9959:
9948:
9945:
9929:
9927:
9923:
9919:
9917:
9916:
9911:
9904:
9903:
9901:
9898:
9897:
9893:
9886:
9876:
9871:
9867:
9858:
9821:
9819:
9818:
9813:
9807:
9803:
9802:
9794:
9789:
9781:
9774:
9772:
9761:
9758:
9746:
9745:
9731:
9726:
9725:
9703:
9701:
9700:
9695:
9689:
9687:
9684:
9669:
9648:
9646:
9645:
9640:
9634:
9632:
9619:
9616:
9604:
9602:
9584:
9577:
9575:
9574:
9569:
9563:
9559:
9558:
9550:
9545:
9537:
9518:
9517:
9502:
9501:
9479:
9477:
9476:
9471:
9425:
9423:
9422:
9417:
9411:
9409:
9406:
9391:
9386:
9368:
9364:
9351:
9349:
9348:
9343:
9337:
9333:
9332:
9331:
9312:
9311:
9289:
9288:
9242:
9240:
9239:
9234:
9222:
9217:
9216:
9207:
9196:of the circuit,
9195:
9191:
9184:
9182:
9181:
9176:
9167:
9166:
9157:
9149:
9144:
9143:
9124:
9117:
9115:
9114:
9109:
9103:
9095:
9069:
9067:
9066:
9061:
9055:
9054:
9042:
9041:
9040:
9017:
9015:
9014:
9009:
8985:or equivalently
8981:
8979:
8978:
8973:
8967:
8966:
8938:
8930:
8926:
8909:Band-stop filter
8904:
8902:
8901:
8896:
8890:
8888:
8887:
8886:
8870:
8846:
8844:
8843:
8838:
8832:
8827:
8826:
8817:
8793:
8791:
8790:
8785:
8779:
8770:
8766:
8761:
8760:
8759:
8735:Band-pass filter
8727:
8725:
8724:
8719:
8713:
8704:
8700:
8695:
8694:
8693:
8670:High-pass filter
8665:
8663:
8662:
8657:
8651:
8642:
8641:
8639:
8637:
8636:
8635:
8619:
8598:
8596:
8595:
8590:
8584:
8575:
8571:
8566:
8565:
8564:
8536:
8532:
8528:
8524:
8432:
8431:
8419:
8411:
8312:
8290:
8288:
8287:
8282:
8275:
8274:
8270:
8269:
8265:
8264:
8262:
8259:
8257:
8252:
8241:
8236:
8228:
8223:
8221:
8218:
8216:
8211:
8200:
8195:
8190:
8186:
8181:
8169:
8168:
8167:
8155:
8144:
8132:
8130:
8129:
8124:
8117:
8116:
8112:
8111:
8109:
8106:
8105:
8104:
8084:
8079:
8074:
8072:
8069:
8060:
8055:
8053:
8048:
8047:
8036:
8010:
8008:
8007:
8002:
7995:
7993:
7988:
7973:
7972:
7971:
7959:
7950:
7943:
7932:
7930:
7929:
7924:
7917:
7910:
7909:
7898:
7887:
7885:
7884:
7879:
7872:
7871:
7869:
7866:
7865:
7861:
7860:
7856:
7854:
7849:
7840:
7818:
7814:
7812:
7807:
7798:
7784:
7779:
7775:
7773:
7772:
7771:
7758:
7753:
7751:
7748:
7747:
7743:
7742:
7738:
7730:
7725:
7705:
7701:
7693:
7688:
7679:
7677:
7676:
7664:
7660:
7658:
7656:
7651:
7638:
7633:
7632:
7631:
7619:
7610:
7603:
7589:
7587:
7586:
7581:
7574:
7573:
7569:
7568:
7566:
7563:
7561:
7556:
7542:
7537:
7529:
7525:
7520:
7508:
7507:
7506:
7488:
7473:
7472:
7470:
7469:
7464:
7461:
7435:
7433:
7432:
7427:
7420:
7419:
7415:
7414:
7410:
7409:
7407:
7404:
7402:
7397:
7386:
7381:
7373:
7368:
7366:
7363:
7360:
7358:
7353:
7342:
7337:
7332:
7328:
7323:
7311:
7310:
7309:
7291:
7280:
7268:
7266:
7265:
7260:
7253:
7252:
7250:
7247:
7246:
7242:
7235:
7231:
7226:
7217:
7206:
7188:
7186:
7185:
7180:
7176:
7169:
7168:
7163:
7158:
7132:
7131:
7120:
7105:
7103:
7102:
7097:
7090:
7089:
7085:
7084:
7083:
7078:
7074:
7072:
7069:
7063:
7058:
7048:
7046:
7043:
7034:
7029:
7027:
7022:
7021:
7010:
6999:
6970:
6968:
6967:
6962:
6958:
6957:
6953:
6944:
6942:
6931:
6930:
6919:
6905:
6903:
6902:
6897:
6895:
6887:
6885:
6875:
6856:
6854:
6836:
6828:
6824:
6822:
6820:
6819:
6805:
6800:
6798:
6796:
6795:
6781:
6776:
6774:
6772:
6771:
6757:
6748:
6746:
6736:
6716:
6706:
6696:
6686:
6666:Frequency domain
6658:
6656:
6655:
6650:
6648:
6639:
6637:
6633:
6632:
6624:
6622:
6603:
6599:
6598:
6590:
6588:
6586:
6584:
6574:
6565:
6564:
6563:
6539:
6535:
6531:
6527:
6520:
6518:
6517:
6512:
6505:
6503:
6499:
6498:
6490:
6488:
6486:
6484:
6470:
6449:
6447:
6446:
6441:
6439:
6437:
6419:
6401:
6382:
6377:
6372:
6367:
6362:
6347:
6346:Parallel circuit
6340:
6338:
6337:
6332:
6330:
6329:
6317:
6316:
6304:
6303:
6284:
6282:
6281:
6276:
6270:
6268:
6265:
6264:
6260:
6259:
6258:
6249:
6240:
6234:
6231:
6228:
6222:
6217:
6214:
6207:
6202:
6197:
6196:
6174:
6172:
6171:
6166:
6160:
6155:
6152:
6151:
6149:
6148:
6139:
6130:
6124:
6121:
6118:
6112:
6107:
6104:
6100:
6098:
6093:
6092:
6069:
6067:
6066:
6061:
6055:
6053:
6051:
6047:
6041:
6037:
6030:
6025:
6024:
6004:
6000:
5998:
5982:
5980:
5979:
5974:
5968:
5967:
5946:
5945:
5936:
5935:
5914:
5913:
5904:
5903:
5882:
5881:
5865:
5858:
5856:
5855:
5850:
5844:
5842:
5840:
5838:
5837:
5832:
5828:
5827:
5825:
5812:
5792:
5791:
5782:
5776:
5771:
5770:
5749:
5748:
5728:
5724:
5710:
5704:, damping ratio
5703:
5677:
5673:
5669:
5668:
5667:
5639:
5637:
5636:
5631:
5624:
5616:
5612:
5605:
5604:
5603:
5584:
5582:
5579:
5578:
5577:
5576:
5564:
5559:
5548:
5547:
5546:
5522:
5521:
5487:
5482:
5471:
5470:
5468:
5465:
5459:
5454:
5435:
5425:
5408:
5406:
5405:
5400:
5393:
5389:
5382:
5381:
5377:
5374:
5359:
5351:
5350:
5349:
5315:
5310:
5299:
5298:
5296:
5293:
5287:
5282:
5263:
5253:
5236:
5234:
5233:
5228:
5215:
5211:
5204:
5203:
5202:
5183:
5181:
5178:
5177:
5176:
5175:
5163:
5158:
5147:
5146:
5145:
5121:
5120:
5086:
5081:
5070:
5069:
5067:
5057:
5031:
5014:
5003:
4989:General solution
4984:
4975:
4966:
4952:
4950:
4949:
4944:
4938:
4934:
4929:
4917:
4916:
4907:
4876:
4865:
4861:
4843:
4841:
4840:
4835:
4823:
4815:
4809:
4805:
4782:
4771:
4767:
4744:
4742:
4741:
4736:
4730:
4728:
4726:
4722:
4720:
4715:
4691:
4690:
4674:
4665:
4660:
4658:
4642:
4628:
4601:
4592:
4585:
4583:
4582:
4577:
4571:
4569:
4567:
4563:
4562:
4560:
4546:
4538:
4536:
4526:
4521:
4520:
4504:
4495:
4490:
4488:
4472:
4458:
4431:
4409:
4407:
4406:
4401:
4383:
4381:
4379:
4375:
4374:
4372:
4358:
4350:
4348:
4338:
4333:
4332:
4316:
4307:
4277:
4275:
4274:
4269:
4251:
4249:
4247:
4245:
4231:
4208:
4181:
4170:
4156:
4154:
4153:
4148:
4142:
4138:
4137:
4135:
4121:
4057:
4040:
4036:
4013:
4004:
3991:
3989:
3988:
3983:
3977:
3976:
3961:
3960:
3948:
3947:
3929:
3928:
3894:
3879:
3867:
3865:
3864:
3859:
3853:
3850:
3849:
3834:
3832:
3831:
3819:
3817:
3816:
3803:
3798:
3789:
3784:
3783:
3782:
3762:
3753:
3749:
3740:
3731:
3718:
3714:
3710:
3698:
3696:
3695:
3690:
3672:
3671:
3670:
3651:
3650:
3635:
3634:
3593:
3591:
3590:
3585:
3573:
3572:
3571:
3552:
3551:
3536:
3535:
3517:
3516:
3515:
3496:
3495:
3480:
3479:
3445:
3427:
3425:
3424:
3419:
3413:
3412:
3408:
3404:
3403:
3395:
3394:
3385:
3372:
3371:
3354:
3353:
3341:
3340:
3336:
3332:
3331:
3323:
3322:
3313:
3300:
3299:
3282:
3281:
3247:
3232:
3225:
3219:), underdamped (
3218:
3211:
3202:
3193:
3181:
3179:
3178:
3173:
3167:
3166:
3162:
3161:
3147:
3146:
3134:
3133:
3129:
3128:
3114:
3113:
3076:
3074:
3073:
3068:
3066:
3058:
3054:
3053:
3044:
3043:
3034:
3021:
3020:
3005:
3001:
2996:
2984:
2983:
2974:
2956:
2955:
2937:
2936:
2934:
2931:
2930:
2926:
2919:
2918:
2909:
2899:
2897:
2896:
2895:
2882:
2877:
2873:
2872:
2863:
2862:
2853:
2840:
2839:
2824:
2820:
2815:
2803:
2802:
2793:
2775:
2774:
2751:
2744:
2742:
2741:
2736:
2723:
2718:
2694:
2693:
2666:
2656:
2649:
2627:
2625:
2624:
2619:
2612:
2611:
2609:
2606:
2597:
2592:
2587:
2584:
2582:
2572:
2570:
2568:
2563:
2556:
2535:
2533:
2532:
2527:
2521:
2519:
2517:
2516:
2502:
2480:
2476:
2472:
2461:
2459:
2458:
2453:
2451:
2443:
2441:
2439:
2429:
2422:
2413:
2412:
2399:
2397:
2384:
2355:
2347:of attenuation.
2345:logarithmic unit
2326:
2318:
2309:
2303:
2301:
2300:
2295:
2270:
2265:
2241:
2239:
2235:
2229:
2224:
2201:
2199:
2198:
2197:
2188:
2182:
2181:
2176:
2170:
2155:
2153:
2152:
2147:
2123:
2121:
2107:
2090:
2088:
2084:
2078:
2073:
2071:
2069:
2059:
2042:
2040:
2039:
2038:
2029:
2023:
2022:
2017:
2011:
1999:
1992:
1990:
1989:
1984:
1960:
1941:
1936:
1927:
1925:
1915:
1895:
1893:
1889:
1883:
1869:
1862:
1829:
1827:
1826:
1821:
1815:
1796:
1791:
1782:
1780:
1770:
1750:
1749:
1748:
1730:
1728:
1727:
1722:
1719:
1717:
1713:
1707:
1694:
1688:
1680:
1679:
1678:
1661:
1659:
1658:
1653:
1633:
1626:
1625:
1606:
1595:
1591:
1587:
1583:
1574:
1565:
1555:
1553:
1552:
1547:
1526:
1525:
1524:
1511:
1510:
1509:
1496:
1495:
1494:
1463:. The governing
1447:
1437:
1431:
1425:
1419:
1389:
1380:
1376:
1367:
1356:
1354:
1353:
1348:
1342:
1339:
1337:
1327:
1325:
1320:
1319:
1316:
1302:
1300:
1299:
1294:
1280:
1278:
1277:
1272:
1254:
1252:
1251:
1246:
1229:
1227:
1226:
1221:
1215:
1213:
1207:
1205:
1204:
1201:
1193:
1188:
1185:
1183:
1173:
1171:
1169:
1167:
1157:
1152:
1147:
1142:
1141:
1138:
1130:
1125:
1123:
1114:
1113:
1110:
1099:
1094:
1092:
1082:
1064:
1053:
1045:
1039:
1037:
1036:
1031:
1025:
1023:
1015:
1008:
1007:
1004:
998:
993:
991:
989:
988:
987:
972:
954:
950:
946:
942:
938:
934:
928:
919:
905:
903:
902:
897:
891:
889:
888:
879:
871:
866:
865:
864:
836:
834:
833:
828:
797:
788:
779:
769:
767:
766:
761:
755:
754:
742:
741:
695:critical damping
692:
682:
680:
679:
674:
667:
666:
664:
661:
660:
659:
648:
643:
633:
622:
618:
606:
590:
588:
587:
582:
578:
577:
573:
565:
563:
552:
551:
540:
505:
503:
502:
497:
490:
488:
483:
475:
471:
466:
465:
413:
411:
410:
405:
399:
398:
380:
379:
356:
339:
297:high-pass filter
289:band-stop filter
285:band-pass filter
233:consisting of a
192:
185:
178:
96:m-derived filter
46:Chebyshev filter
19:
18:
10534:
10533:
10529:
10528:
10527:
10525:
10524:
10523:
10514:Analog circuits
10504:
10503:
10497:
10478:
10459:
10440:
10421:
10405:
10400:
10395:
10391:
10386:
10379:
10374:
10370:
10365:
10361:
10354:
10338:
10329:
10316:
10303:
10288:
10284:
10275:
10273:
10243:
10224:
10219:
10215:
10199:
10193:
10182:
10177:
10173:
10168:
10164:
10159:
10155:
10146:
10142:
10134:
10119:
10115:
10110:
10106:
10101:
10097:
10092:
10088:
10083:
10079:
10074:
10067:
10062:
10058:
10053:
10049:
10044:
10040:
10035:
10028:
10023:
10019:
10014:
10010:
10005:
10001:
9996:
9992:
9983:
9981:
9970:
9965:
9960:
9951:
9946:
9942:
9938:
9933:
9932:
9925:
9921:
9885:
9875:
9870:
9865:
9862:
9861:
9859:
9855:
9850:
9828:
9793:
9780:
9779:
9775:
9762:
9741:
9737:
9732:
9730:
9721:
9717:
9715:
9712:
9711:
9673:
9668:
9660:
9657:
9656:
9620:
9603:
9601:
9593:
9590:
9589:
9582:
9549:
9536:
9535:
9531:
9513:
9509:
9497:
9493:
9491:
9488:
9487:
9437:
9434:
9433:
9390:
9385:
9377:
9374:
9373:
9366:
9363:
9357:
9320:
9316:
9300:
9296:
9294:
9290:
9284:
9280:
9263:
9260:
9259:
9253:
9212:
9208:
9206:
9204:
9201:
9200:
9193:
9189:
9162:
9158:
9148:
9139:
9135:
9133:
9130:
9129:
9122:
9094:
9086:
9083:
9082:
9076:
9050:
9046:
9036:
9035:
9031:
9029:
9026:
9025:
8993:
8990:
8989:
8962:
8958:
8950:
8947:
8946:
8936:
8928:
8924:
8920:
8911:
8882:
8878:
8874:
8869:
8858:
8855:
8854:
8822:
8818:
8816:
8805:
8802:
8801:
8765:
8755:
8754:
8750:
8748:
8745:
8744:
8737:
8699:
8689:
8688:
8684:
8682:
8679:
8678:
8672:
8640:
8631:
8627:
8623:
8618:
8610:
8607:
8606:
8570:
8560:
8559:
8555:
8553:
8550:
8549:
8543:
8541:Low-pass filter
8534:
8530:
8526:
8522:
8430:
8417:
8409:
8402:
8397:
8339:William Thomson
8319:
8308:
8300:
8295:
8253:
8248:
8240:
8235:
8227:
8212:
8207:
8199:
8194:
8189:
8177:
8163:
8162:
8158:
8153:
8150:
8149:
8143:
8137:
8100:
8096:
8083:
8078:
8059:
8054:
8052:
8043:
8039:
8034:
8031:
8030:
7989:
7984:
7961:
7960:
7955:
7954:
7946:
7941:
7938:
7937:
7905:
7901:
7896:
7893:
7892:
7850:
7845:
7836:
7826:
7822:
7808:
7803:
7794:
7783:
7778:
7774:
7767:
7763:
7759:
7757:
7726:
7721:
7713:
7709:
7689:
7684:
7678:
7672:
7668:
7663:
7659:
7652:
7647:
7639:
7637:
7621:
7620:
7615:
7614:
7606:
7601:
7598:
7597:
7557:
7552:
7541:
7536:
7528:
7516:
7496:
7495:
7491:
7486:
7483:
7482:
7465:
7462:
7457:
7451:
7450:
7448:
7445:
7440:
7398:
7393:
7385:
7380:
7372:
7354:
7349:
7341:
7336:
7331:
7319:
7299:
7298:
7294:
7289:
7286:
7285:
7279:
7273:
7234:
7230:
7225:
7213:
7204:
7201:
7200:
7164:
7154:
7149:
7148:
7127:
7123:
7118:
7115:
7114:
7079:
7062:
7057:
7053:
7052:
7033:
7028:
7026:
7017:
7013:
7008:
7005:
7004:
6997:
6978:
6943:
6938:
6926:
6922:
6917:
6914:
6913:
6893:
6892:
6879:
6874:
6840:
6835:
6826:
6825:
6815:
6811:
6809:
6804:
6791:
6787:
6785:
6780:
6767:
6763:
6761:
6756:
6749:
6740:
6735:
6731:
6729:
6726:
6725:
6708:
6698:
6688:
6678:
6668:
6646:
6645:
6623:
6621:
6611:
6605:
6604:
6589:
6587:
6578:
6573:
6566:
6559:
6558:
6554:
6550:
6548:
6545:
6544:
6537:
6533:
6529:
6525:
6489:
6487:
6474:
6469:
6461:
6458:
6457:
6423:
6418:
6410:
6407:
6406:
6399:
6380:
6379:
6375:
6374:
6370:
6369:
6365:
6364:
6360:
6359:
6349:
6325:
6321:
6312:
6308:
6299:
6295:
6293:
6290:
6289:
6254:
6250:
6238:
6221:
6215:
6213:
6206:
6201:
6192:
6188:
6186:
6183:
6182:
6144:
6140:
6128:
6111:
6105:
6103:
6099:
6097:
6088:
6084:
6082:
6079:
6078:
6036:
6034:
6029:
6020:
6016:
6014:
6011:
6010:
6002:
6001:. The value of
5989:
5987:
5963:
5962:
5941:
5940:
5931:
5930:
5909:
5908:
5899:
5898:
5877:
5876:
5874:
5871:
5870:
5863:
5833:
5816:
5811:
5801:
5797:
5796:
5787:
5783:
5781:
5780:
5775:
5766:
5765:
5744:
5743:
5741:
5738:
5737:
5726:
5716:
5705:
5701:
5695:
5688:
5675:
5671:
5666:
5656:
5654:
5652:
5646:
5599:
5598:
5594:
5572:
5571:
5567:
5563:
5558:
5542:
5541:
5537:
5527:
5523:
5511:
5507:
5483:
5478:
5458:
5453:
5433:
5430:
5429:
5420:
5414:
5385:
5358:
5354:
5339:
5335:
5311:
5306:
5286:
5281:
5261:
5258:
5257:
5248:
5242:
5198:
5197:
5193:
5171:
5170:
5166:
5162:
5157:
5141:
5140:
5136:
5126:
5122:
5110:
5106:
5082:
5077:
5061:
5056:
5039:
5036:
5035:
5026:
5020:
5005:
4994:
4991:
4983:
4977:
4974:
4968:
4957:
4930:
4925:
4912:
4908:
4906:
4889:
4886:
4885:
4867:
4863:
4852:
4819:
4811:
4803:
4791:
4788:
4787:
4773:
4769:
4758:
4751:
4749:Poles and zeros
4716:
4711:
4686:
4682:
4681:
4677:
4669:
4664:
4643:
4629:
4627:
4610:
4607:
4606:
4600:
4594:
4590:
4550:
4545:
4530:
4525:
4516:
4512:
4511:
4507:
4499:
4494:
4473:
4459:
4457:
4440:
4437:
4436:
4422:
4416:
4362:
4357:
4342:
4337:
4328:
4324:
4323:
4319:
4311:
4306:
4289:
4286:
4285:
4235:
4230:
4212:
4207:
4190:
4187:
4186:
4172:
4161:
4125:
4120:
4103:
4099:
4070:
4067:
4066:
4045:
4038:
4027:
4020:
4012:
4006:
4003:
3997:
3966:
3962:
3956:
3952:
3937:
3933:
3924:
3920:
3903:
3900:
3899:
3889:
3886:
3878:
3872:
3845:
3841:
3833:
3827:
3823:
3812:
3808:
3799:
3794:
3788:
3778:
3777:
3773:
3771:
3768:
3767:
3761:
3755:
3751:
3748:
3742:
3739:
3733:
3730:
3724:
3716:
3712:
3709:
3703:
3666:
3665:
3661:
3640:
3636:
3630:
3626:
3609:
3606:
3605:
3567:
3566:
3562:
3541:
3537:
3531:
3527:
3511:
3510:
3506:
3485:
3481:
3475:
3471:
3454:
3451:
3450:
3440:
3437:
3390:
3386:
3384:
3377:
3373:
3367:
3363:
3359:
3355:
3349:
3345:
3318:
3314:
3312:
3305:
3301:
3295:
3291:
3287:
3283:
3277:
3273:
3256:
3253:
3252:
3242:
3239:
3227:
3220:
3213:
3209:
3201:
3195:
3192:
3186:
3157:
3153:
3152:
3148:
3142:
3138:
3124:
3120:
3119:
3115:
3109:
3105:
3088:
3085:
3084:
3064:
3063:
3039:
3035:
3033:
3026:
3022:
3016:
3012:
2997:
2992:
2979:
2975:
2973:
2957:
2951:
2947:
2944:
2943:
2914:
2910:
2908:
2898:
2891:
2887:
2883:
2881:
2858:
2854:
2852:
2845:
2841:
2835:
2831:
2816:
2811:
2798:
2794:
2792:
2776:
2770:
2766:
2762:
2760:
2757:
2756:
2749:
2719:
2714:
2689:
2685:
2683:
2680:
2679:
2664:
2658:
2651:
2644:
2637:
2596:
2591:
2576:
2571:
2569:
2557:
2555:
2547:
2544:
2543:
2512:
2508:
2506:
2501:
2493:
2490:
2489:
2478:
2474:
2470:
2449:
2448:
2428:
2426:
2421:
2414:
2408:
2404:
2401:
2400:
2388:
2383:
2376:
2369:
2367:
2364:
2363:
2354:
2348:
2329:neper frequency
2324:
2317:
2311:
2307:
2266:
2261:
2231:
2230:
2225:
2223:
2193:
2189:
2184:
2183:
2177:
2172:
2171:
2169:
2167:
2164:
2163:
2111:
2106:
2080:
2079:
2074:
2072:
2063:
2058:
2034:
2030:
2025:
2024:
2018:
2013:
2012:
2010:
2008:
2005:
2004:
1997:
1956:
1937:
1932:
1919:
1914:
1885:
1884:
1865:
1863:
1861:
1838:
1835:
1834:
1811:
1792:
1787:
1774:
1769:
1744:
1743:
1739:
1736:
1733:
1732:
1709:
1708:
1690:
1689:
1687:
1674:
1673:
1669:
1666:
1663:
1662:
1621:
1617:
1615:
1612:
1611:
1597:
1593:
1589:
1585:
1582:
1576:
1573:
1567:
1564:
1560:
1520:
1519:
1515:
1505:
1504:
1500:
1490:
1489:
1485:
1483:
1480:
1479:
1455:
1445:
1435:
1429:
1423:
1417:
1404:
1388:
1382:
1378:
1375:
1369:
1365:
1364:The parameters
1362:
1331:
1326:
1324:
1315:
1311:
1308:
1305:
1304:
1286:
1283:
1282:
1264:
1261:
1260:
1238:
1235:
1234:
1208:
1200:
1196:
1194:
1192:
1177:
1172:
1170:
1161:
1156:
1137:
1133:
1131:
1129:
1109:
1105:
1103:
1098:
1086:
1081:
1073:
1070:
1069:
1062:
1051:
1043:
1011:
1009:
1003:
999:
997:
983:
982:
978:
976:
971:
963:
960:
959:
952:
948:
944:
940:
936:
932:
926:
922:
917:
884:
880:
872:
870:
860:
859:
855:
853:
850:
849:
806:
803:
802:
796:
790:
787:
781:
774:
750:
746:
737:
733:
722:
719:
718:
703:
687:
655:
651:
647:
642:
631:
628:
627:
620:
616:
604:
597:
564:
559:
547:
543:
538:
535:
534:
470:
461:
457:
455:
452:
451:
439:
394:
390:
375:
371:
369:
366:
365:
355:
349:
338:
332:
325:
320:
293:low-pass filter
277:television sets
273:Radio receivers
196:
167:
166:
142:
134:
133:
129:mm'-type filter
86:
76:
75:
61:Gaussian filter
36:
24:
17:
12:
11:
5:
10532:
10522:
10521:
10516:
10502:
10501:
10495:
10482:
10476:
10463:
10457:
10444:
10438:
10425:
10419:
10404:
10401:
10399:
10398:
10389:
10377:
10368:
10359:
10352:
10327:
10301:
10282:
10222:
10213:
10180:
10171:
10162:
10153:
10140:
10132:
10113:
10111:Irwin, p. 220.
10104:
10095:
10086:
10077:
10065:
10056:
10054:Irwin, p. 532.
10047:
10038:
10026:
10017:
10008:
9999:
9990:
9949:
9939:
9937:
9934:
9931:
9930:
9909:
9892:
9889:
9884:
9881:
9874:
9852:
9851:
9849:
9846:
9845:
9844:
9842:Linear circuit
9839:
9834:
9827:
9824:
9823:
9822:
9811:
9806:
9800:
9797:
9792:
9787:
9784:
9778:
9771:
9768:
9765:
9757:
9753:
9749:
9744:
9740:
9736:
9729:
9724:
9720:
9705:
9704:
9693:
9683:
9680:
9677:
9672:
9667:
9664:
9650:
9649:
9638:
9631:
9627:
9623:
9615:
9612:
9609:
9600:
9597:
9579:
9578:
9567:
9562:
9556:
9553:
9548:
9543:
9540:
9534:
9529:
9525:
9521:
9516:
9512:
9508:
9505:
9500:
9496:
9481:
9480:
9469:
9465:
9461:
9458:
9455:
9451:
9447:
9444:
9441:
9427:
9426:
9415:
9404:
9400:
9396:
9389:
9384:
9381:
9361:
9353:
9352:
9341:
9336:
9330:
9326:
9323:
9319:
9315:
9310:
9306:
9303:
9299:
9293:
9287:
9283:
9279:
9276:
9273:
9270:
9267:
9252:
9249:
9244:
9243:
9232:
9228:
9225:
9220:
9215:
9211:
9186:
9185:
9174:
9170:
9165:
9161:
9155:
9152:
9147:
9142:
9138:
9119:
9118:
9107:
9101:
9098:
9093:
9090:
9075:
9072:
9071:
9070:
9059:
9053:
9049:
9045:
9039:
9034:
9019:
9018:
9007:
9003:
9000:
8997:
8983:
8982:
8971:
8965:
8961:
8957:
8954:
8919:
8916:
8910:
8907:
8906:
8905:
8894:
8885:
8881:
8877:
8873:
8868:
8865:
8862:
8848:
8847:
8836:
8830:
8825:
8821:
8815:
8812:
8809:
8795:
8794:
8783:
8776:
8773:
8769:
8764:
8758:
8753:
8736:
8733:
8729:
8728:
8717:
8710:
8707:
8703:
8698:
8692:
8687:
8671:
8668:
8667:
8666:
8655:
8648:
8645:
8634:
8630:
8626:
8622:
8617:
8614:
8600:
8599:
8588:
8581:
8578:
8574:
8569:
8563:
8558:
8542:
8539:
8516:
8515:
8502:
8488:
8487:
8474:
8460:
8459:
8446:
8429:
8426:
8401:
8398:
8396:
8393:
8365:Heinrich Hertz
8318:
8315:
8306:
8298:
8292:
8291:
8280:
8256:
8251:
8247:
8239:
8234:
8231:
8226:
8215:
8210:
8206:
8198:
8193:
8184:
8180:
8176:
8172:
8166:
8161:
8141:
8134:
8133:
8122:
8103:
8099:
8095:
8092:
8089:
8082:
8077:
8068:
8065:
8058:
8051:
8046:
8042:
8012:
8011:
8000:
7992:
7987:
7983:
7979:
7976:
7970:
7967:
7964:
7958:
7953:
7949:
7922:
7916:
7913:
7908:
7904:
7891:For values of
7889:
7888:
7877:
7859:
7853:
7848:
7844:
7839:
7835:
7832:
7829:
7825:
7821:
7811:
7806:
7802:
7797:
7793:
7790:
7787:
7782:
7770:
7766:
7762:
7756:
7741:
7737:
7734:
7729:
7724:
7720:
7716:
7712:
7708:
7700:
7697:
7692:
7687:
7683:
7675:
7671:
7667:
7655:
7650:
7646:
7642:
7636:
7630:
7627:
7624:
7618:
7613:
7609:
7591:
7590:
7579:
7560:
7555:
7551:
7547:
7540:
7535:
7532:
7523:
7519:
7515:
7511:
7505:
7502:
7499:
7494:
7476:quality factor
7455:
7443:
7437:
7436:
7425:
7401:
7396:
7392:
7384:
7379:
7376:
7371:
7357:
7352:
7348:
7340:
7335:
7326:
7322:
7318:
7314:
7308:
7305:
7302:
7297:
7277:
7272:The frequency
7270:
7269:
7258:
7241:
7238:
7229:
7224:
7220:
7216:
7212:
7190:
7189:
7175:
7172:
7167:
7161:
7157:
7153:
7147:
7144:
7141:
7138:
7135:
7130:
7126:
7107:
7106:
7095:
7082:
7077:
7068:
7061:
7056:
7051:
7042:
7039:
7032:
7025:
7020:
7016:
6977:
6974:
6952:
6948:
6941:
6937:
6934:
6929:
6925:
6907:
6906:
6891:
6883:
6878:
6873:
6870:
6866:
6862:
6859:
6852:
6848:
6844:
6839:
6834:
6831:
6829:
6827:
6818:
6814:
6808:
6803:
6794:
6790:
6784:
6779:
6770:
6766:
6760:
6755:
6752:
6750:
6744:
6739:
6734:
6733:
6667:
6664:
6660:
6659:
6644:
6630:
6627:
6620:
6617:
6614:
6612:
6610:
6607:
6606:
6596:
6593:
6582:
6577:
6572:
6569:
6567:
6562:
6557:
6553:
6552:
6522:
6521:
6510:
6496:
6493:
6482:
6478:
6473:
6468:
6465:
6451:
6450:
6435:
6431:
6427:
6422:
6417:
6414:
6393:dual impedance
6348:
6343:
6328:
6324:
6320:
6315:
6311:
6307:
6302:
6298:
6286:
6285:
6274:
6257:
6253:
6246:
6243:
6237:
6227:
6220:
6212:
6205:
6200:
6195:
6191:
6176:
6175:
6164:
6158:
6147:
6143:
6136:
6133:
6127:
6117:
6110:
6096:
6091:
6087:
6071:
6070:
6059:
6046:
6040:
6033:
6028:
6023:
6019:
5984:
5983:
5972:
5966:
5961:
5958:
5955:
5952:
5949:
5944:
5939:
5934:
5929:
5926:
5923:
5920:
5917:
5912:
5907:
5902:
5897:
5894:
5891:
5888:
5885:
5880:
5860:
5859:
5848:
5836:
5831:
5823:
5820:
5815:
5810:
5807:
5804:
5800:
5795:
5790:
5786:
5779:
5774:
5769:
5764:
5761:
5758:
5755:
5752:
5747:
5731:imaginary unit
5699:
5687:
5684:
5678:are the usual
5664:
5650:
5643:
5642:
5641:
5640:
5629:
5623:
5620:
5615:
5611:
5608:
5602:
5597:
5593:
5590:
5587:
5575:
5570:
5562:
5557:
5554:
5551:
5545:
5540:
5536:
5533:
5530:
5526:
5520:
5517:
5514:
5510:
5506:
5503:
5500:
5497:
5494:
5491:
5486:
5481:
5477:
5464:
5457:
5452:
5449:
5446:
5443:
5440:
5418:
5411:
5410:
5409:
5398:
5392:
5388:
5380:
5373:
5370:
5367:
5364:
5357:
5348:
5345:
5342:
5338:
5334:
5331:
5328:
5325:
5322:
5319:
5314:
5309:
5305:
5292:
5285:
5280:
5277:
5274:
5271:
5268:
5246:
5239:
5238:
5237:
5226:
5222:
5219:
5214:
5210:
5207:
5201:
5196:
5192:
5189:
5186:
5174:
5169:
5161:
5156:
5153:
5150:
5144:
5139:
5135:
5132:
5129:
5125:
5119:
5116:
5113:
5109:
5105:
5102:
5099:
5096:
5093:
5090:
5085:
5080:
5076:
5065:
5060:
5055:
5052:
5049:
5046:
5043:
5024:
4990:
4987:
4981:
4972:
4954:
4953:
4942:
4933:
4928:
4924:
4920:
4915:
4911:
4905:
4902:
4899:
4896:
4893:
4845:
4844:
4833:
4829:
4826:
4822:
4818:
4814:
4801:
4798:
4795:
4750:
4747:
4746:
4745:
4734:
4725:
4719:
4714:
4710:
4706:
4703:
4700:
4697:
4694:
4689:
4685:
4680:
4673:
4668:
4663:
4656:
4653:
4650:
4647:
4641:
4638:
4635:
4632:
4626:
4623:
4620:
4617:
4614:
4598:
4587:
4586:
4575:
4566:
4558:
4554:
4549:
4544:
4541:
4534:
4529:
4524:
4519:
4515:
4510:
4503:
4498:
4493:
4486:
4483:
4480:
4477:
4471:
4468:
4465:
4462:
4456:
4453:
4450:
4447:
4444:
4415:
4412:
4411:
4410:
4399:
4395:
4392:
4389:
4386:
4378:
4370:
4366:
4361:
4356:
4353:
4346:
4341:
4336:
4331:
4327:
4322:
4315:
4310:
4305:
4302:
4299:
4296:
4293:
4279:
4278:
4267:
4263:
4260:
4257:
4254:
4243:
4239:
4234:
4229:
4226:
4222:
4219:
4216:
4211:
4206:
4203:
4200:
4197:
4194:
4158:
4157:
4146:
4141:
4133:
4129:
4124:
4119:
4116:
4112:
4109:
4106:
4102:
4098:
4095:
4092:
4089:
4086:
4083:
4080:
4077:
4074:
4019:
4018:Laplace domain
4016:
4010:
4001:
3993:
3992:
3981:
3975:
3972:
3969:
3965:
3959:
3955:
3951:
3946:
3943:
3940:
3936:
3932:
3927:
3923:
3919:
3916:
3913:
3910:
3907:
3885:
3882:
3876:
3869:
3868:
3857:
3848:
3844:
3840:
3837:
3830:
3826:
3822:
3815:
3811:
3807:
3802:
3797:
3793:
3787:
3781:
3776:
3759:
3746:
3737:
3728:
3719:describes the
3707:
3700:
3699:
3688:
3684:
3681:
3678:
3675:
3669:
3664:
3660:
3657:
3654:
3649:
3646:
3643:
3639:
3633:
3629:
3625:
3622:
3619:
3616:
3613:
3595:
3594:
3583:
3579:
3576:
3570:
3565:
3561:
3558:
3555:
3550:
3547:
3544:
3540:
3534:
3530:
3526:
3523:
3520:
3514:
3509:
3505:
3502:
3499:
3494:
3491:
3488:
3484:
3478:
3474:
3470:
3467:
3464:
3461:
3458:
3436:
3433:
3429:
3428:
3417:
3411:
3407:
3401:
3398:
3393:
3389:
3383:
3380:
3376:
3370:
3366:
3362:
3358:
3352:
3348:
3344:
3339:
3335:
3329:
3326:
3321:
3317:
3311:
3308:
3304:
3298:
3294:
3290:
3286:
3280:
3276:
3272:
3269:
3266:
3263:
3260:
3238:
3235:
3199:
3190:
3183:
3182:
3171:
3165:
3160:
3156:
3151:
3145:
3141:
3137:
3132:
3127:
3123:
3118:
3112:
3108:
3104:
3101:
3098:
3095:
3092:
3078:
3077:
3062:
3057:
3050:
3047:
3042:
3038:
3032:
3029:
3025:
3019:
3015:
3011:
3008:
3000:
2995:
2991:
2987:
2982:
2978:
2972:
2969:
2966:
2963:
2960:
2958:
2954:
2950:
2946:
2945:
2942:
2925:
2922:
2917:
2913:
2907:
2904:
2894:
2890:
2886:
2880:
2876:
2869:
2866:
2861:
2857:
2851:
2848:
2844:
2838:
2834:
2830:
2827:
2819:
2814:
2810:
2806:
2801:
2797:
2791:
2788:
2785:
2782:
2779:
2777:
2773:
2769:
2765:
2764:
2746:
2745:
2734:
2730:
2727:
2722:
2717:
2713:
2709:
2706:
2703:
2700:
2697:
2692:
2688:
2662:
2636:
2633:
2629:
2628:
2617:
2605:
2602:
2595:
2590:
2580:
2575:
2566:
2561:
2554:
2551:
2537:
2536:
2525:
2515:
2511:
2505:
2500:
2497:
2483:damping factor
2467:damping factor
2463:
2462:
2447:
2436:
2432:
2425:
2420:
2417:
2415:
2411:
2407:
2403:
2402:
2395:
2392:
2387:
2382:
2379:
2377:
2375:
2372:
2371:
2352:
2327:is called the
2315:
2305:
2304:
2293:
2289:
2286:
2283:
2280:
2277:
2274:
2269:
2264:
2260:
2256:
2253:
2250:
2247:
2244:
2238:
2234:
2228:
2222:
2219:
2216:
2213:
2210:
2207:
2204:
2196:
2192:
2187:
2180:
2175:
2157:
2156:
2145:
2141:
2138:
2135:
2132:
2129:
2126:
2119:
2115:
2110:
2105:
2102:
2099:
2096:
2093:
2087:
2083:
2077:
2067:
2062:
2057:
2054:
2051:
2048:
2045:
2037:
2033:
2028:
2021:
2016:
1994:
1993:
1982:
1978:
1975:
1972:
1969:
1966:
1963:
1959:
1954:
1951:
1948:
1945:
1940:
1935:
1931:
1923:
1918:
1913:
1910:
1907:
1904:
1901:
1898:
1892:
1888:
1881:
1878:
1875:
1872:
1868:
1860:
1857:
1854:
1851:
1848:
1845:
1842:
1818:
1814:
1809:
1806:
1803:
1800:
1795:
1790:
1786:
1778:
1773:
1768:
1765:
1762:
1759:
1756:
1753:
1747:
1742:
1716:
1712:
1706:
1703:
1700:
1697:
1693:
1686:
1683:
1677:
1672:
1651:
1647:
1644:
1641:
1638:
1632:
1629:
1624:
1620:
1580:
1571:
1562:
1557:
1556:
1545:
1541:
1538:
1535:
1532:
1529:
1523:
1518:
1514:
1508:
1503:
1499:
1493:
1488:
1461:voltage source
1454:
1453:
1443:
1433:
1427:
1421:
1414:
1403:
1402:Series circuit
1400:
1386:
1373:
1361:
1358:
1346:
1335:
1330:
1323:
1314:
1291:
1269:
1243:
1231:
1230:
1219:
1211:
1199:
1191:
1181:
1176:
1165:
1160:
1155:
1150:
1145:
1136:
1128:
1121:
1117:
1108:
1102:
1097:
1090:
1085:
1080:
1077:
1059:as shown below
1041:
1040:
1029:
1021:
1018:
1014:
1002:
996:
986:
981:
975:
970:
967:
921:
915:
907:
906:
895:
887:
883:
878:
875:
869:
863:
858:
838:
837:
826:
822:
819:
816:
813:
810:
794:
785:
771:
770:
759:
753:
749:
745:
740:
736:
732:
729:
726:
702:
699:
684:
683:
672:
658:
654:
646:
641:
638:
613:damping factor
596:
593:
572:
568:
562:
558:
555:
550:
546:
507:
506:
495:
482:
478:
474:
469:
464:
460:
438:
435:
431:antiresonators
415:
414:
403:
397:
393:
389:
386:
383:
378:
374:
353:
336:
324:
321:
319:
318:Basic concepts
316:
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:
10531:
10520:
10517:
10515:
10512:
10511:
10509:
10498:
10492:
10488:
10483:
10479:
10477:0-8493-2087-9
10473:
10470:. CRC Press.
10469:
10464:
10460:
10458:7-302-13021-3
10454:
10450:
10445:
10441:
10439:90-5809-245-3
10435:
10431:
10426:
10422:
10420:1-55860-735-8
10416:
10412:
10407:
10406:
10393:
10384:
10382:
10372:
10363:
10355:
10353:0-471-20505-2
10349:
10345:
10344:
10336:
10334:
10332:
10323:
10322:
10314:
10312:
10310:
10308:
10306:
10297:
10293:
10286:
10272:
10268:
10264:
10260:
10256:
10252:
10248:
10241:
10239:
10237:
10235:
10233:
10231:
10229:
10227:
10217:
10209:
10205:
10198:
10191:
10189:
10187:
10185:
10175:
10166:
10157:
10150:
10144:
10135:
10129:
10125:
10117:
10108:
10099:
10090:
10081:
10072:
10070:
10060:
10051:
10042:
10033:
10031:
10021:
10012:
10003:
9994:
9980:
9976:
9969:
9958:
9956:
9954:
9944:
9940:
9907:
9890:
9887:
9882:
9879:
9872:
9857:
9853:
9843:
9840:
9838:
9835:
9833:
9830:
9829:
9809:
9804:
9798:
9795:
9790:
9785:
9782:
9776:
9769:
9766:
9763:
9755:
9751:
9747:
9742:
9738:
9734:
9727:
9722:
9718:
9710:
9709:
9708:
9691:
9681:
9678:
9675:
9670:
9665:
9662:
9655:
9654:
9653:
9636:
9629:
9625:
9621:
9613:
9610:
9607:
9598:
9595:
9588:
9587:
9586:
9565:
9560:
9554:
9551:
9546:
9541:
9538:
9532:
9527:
9523:
9519:
9514:
9510:
9506:
9503:
9498:
9494:
9486:
9485:
9484:
9467:
9459:
9456:
9453:
9445:
9442:
9439:
9432:
9431:
9430:
9413:
9402:
9398:
9394:
9387:
9382:
9379:
9372:
9371:
9370:
9360:
9339:
9334:
9328:
9324:
9321:
9317:
9313:
9308:
9304:
9301:
9297:
9291:
9285:
9281:
9277:
9271:
9265:
9258:
9257:
9256:
9248:
9230:
9226:
9223:
9218:
9213:
9209:
9199:
9198:
9197:
9172:
9168:
9163:
9159:
9153:
9150:
9145:
9140:
9136:
9128:
9127:
9126:
9105:
9099:
9096:
9091:
9088:
9081:
9080:
9079:
9057:
9051:
9047:
9043:
9032:
9024:
9023:
9022:
9021:As a result,
9005:
9001:
8998:
8995:
8988:
8987:
8986:
8969:
8963:
8959:
8955:
8952:
8945:
8944:
8943:
8940:
8934:
8915:
8892:
8883:
8879:
8875:
8871:
8866:
8863:
8853:
8852:
8851:
8834:
8828:
8823:
8819:
8813:
8810:
8800:
8799:
8798:
8781:
8774:
8771:
8767:
8762:
8751:
8743:
8742:
8741:
8732:
8715:
8708:
8705:
8701:
8696:
8685:
8677:
8676:
8675:
8653:
8646:
8643:
8632:
8628:
8624:
8620:
8615:
8612:
8605:
8604:
8603:
8586:
8579:
8576:
8572:
8567:
8556:
8548:
8547:
8546:
8538:
8511:
8507:
8503:
8498:
8494:
8490:
8489:
8483:
8479:
8475:
8470:
8466:
8462:
8461:
8455:
8451:
8447:
8442:
8438:
8434:
8433:
8425:
8423:
8415:
8407:
8392:
8390:
8385:
8380:
8378:
8374:
8368:
8366:
8362:
8357:
8355:
8351:
8347:
8342:
8340:
8336:
8331:
8328:
8324:
8314:
8311:
8305:
8301:
8278:
8254:
8249:
8245:
8237:
8232:
8229:
8224:
8213:
8208:
8204:
8196:
8191:
8182:
8178:
8174:
8170:
8159:
8148:
8147:
8146:
8140:
8120:
8101:
8093:
8090:
8080:
8075:
8066:
8063:
8056:
8049:
8044:
8040:
8029:
8028:
8027:
8020:
8016:
7998:
7990:
7985:
7981:
7977:
7974:
7951:
7936:
7935:
7934:
7920:
7914:
7911:
7906:
7902:
7875:
7857:
7851:
7846:
7842:
7837:
7833:
7830:
7827:
7823:
7819:
7809:
7804:
7800:
7795:
7791:
7788:
7785:
7780:
7768:
7764:
7760:
7754:
7739:
7735:
7732:
7727:
7722:
7718:
7714:
7710:
7706:
7698:
7695:
7690:
7685:
7681:
7673:
7669:
7665:
7653:
7648:
7644:
7640:
7634:
7611:
7596:
7595:
7594:
7577:
7558:
7553:
7549:
7545:
7538:
7533:
7530:
7521:
7517:
7513:
7509:
7492:
7481:
7480:
7479:
7477:
7468:
7460:
7454:
7446:
7423:
7399:
7394:
7390:
7382:
7377:
7374:
7369:
7355:
7350:
7346:
7338:
7333:
7324:
7320:
7316:
7312:
7295:
7284:
7283:
7282:
7276:
7256:
7239:
7236:
7227:
7222:
7218:
7214:
7210:
7199:
7198:
7197:
7195:
7173:
7170:
7165:
7159:
7155:
7151:
7145:
7142:
7139:
7136:
7133:
7128:
7124:
7113:
7112:
7111:
7093:
7080:
7075:
7066:
7059:
7054:
7049:
7040:
7037:
7030:
7023:
7018:
7014:
7003:
7002:
7001:
6995:
6986:
6982:
6973:
6950:
6946:
6939:
6935:
6932:
6927:
6923:
6910:
6889:
6881:
6876:
6871:
6868:
6864:
6860:
6857:
6850:
6846:
6842:
6837:
6832:
6830:
6816:
6812:
6806:
6801:
6792:
6788:
6782:
6777:
6768:
6764:
6758:
6753:
6751:
6742:
6737:
6724:
6723:
6722:
6715:
6711:
6705:
6701:
6695:
6691:
6685:
6681:
6676:
6672:
6663:
6642:
6628:
6625:
6618:
6615:
6613:
6608:
6594:
6591:
6580:
6575:
6570:
6568:
6555:
6543:
6542:
6541:
6508:
6494:
6491:
6480:
6476:
6471:
6466:
6463:
6456:
6455:
6454:
6433:
6429:
6425:
6420:
6415:
6412:
6405:
6404:
6403:
6396:
6394:
6390:
6357:
6353:
6342:
6326:
6322:
6318:
6313:
6309:
6305:
6300:
6296:
6272:
6255:
6251:
6244:
6241:
6235:
6225:
6218:
6210:
6203:
6198:
6193:
6189:
6181:
6180:
6179:
6162:
6156:
6145:
6141:
6134:
6131:
6125:
6115:
6108:
6094:
6089:
6085:
6077:
6076:
6075:
6057:
6044:
6038:
6031:
6026:
6021:
6017:
6009:
6008:
6007:
5996:
5992:
5970:
5956:
5953:
5947:
5937:
5924:
5921:
5915:
5905:
5892:
5889:
5883:
5869:
5868:
5867:
5846:
5834:
5829:
5821:
5818:
5813:
5808:
5805:
5802:
5798:
5793:
5788:
5784:
5777:
5772:
5759:
5756:
5750:
5736:
5735:
5734:
5732:
5723:
5719:
5708:
5698:
5692:
5683:
5681:
5663:
5659:
5649:
5627:
5621:
5618:
5613:
5606:
5595:
5588:
5585:
5568:
5560:
5555:
5549:
5538:
5531:
5528:
5524:
5518:
5515:
5512:
5508:
5501:
5498:
5495:
5489:
5484:
5479:
5475:
5462:
5455:
5450:
5444:
5438:
5428:
5427:
5424:
5417:
5412:
5396:
5390:
5378:
5371:
5368:
5365:
5362:
5355:
5346:
5343:
5340:
5336:
5329:
5326:
5323:
5317:
5312:
5307:
5303:
5290:
5283:
5278:
5272:
5266:
5256:
5255:
5252:
5245:
5240:
5224:
5220:
5217:
5212:
5205:
5194:
5187:
5184:
5167:
5159:
5154:
5148:
5137:
5130:
5127:
5123:
5117:
5114:
5111:
5107:
5100:
5097:
5094:
5088:
5083:
5078:
5074:
5063:
5058:
5053:
5047:
5041:
5034:
5033:
5030:
5023:
5018:
5017:
5016:
5012:
5008:
5001:
4997:
4986:
4980:
4971:
4964:
4960:
4956:The poles of
4940:
4931:
4926:
4922:
4918:
4913:
4909:
4903:
4900:
4897:
4894:
4891:
4884:
4883:
4882:
4880:
4874:
4870:
4859:
4855:
4850:
4831:
4816:
4799:
4796:
4793:
4786:
4785:
4784:
4780:
4776:
4765:
4761:
4756:
4732:
4723:
4717:
4712:
4708:
4704:
4701:
4698:
4695:
4692:
4687:
4683:
4678:
4671:
4666:
4661:
4651:
4645:
4636:
4630:
4624:
4618:
4612:
4605:
4604:
4603:
4597:
4573:
4564:
4556:
4552:
4547:
4542:
4539:
4532:
4527:
4522:
4517:
4513:
4508:
4501:
4496:
4491:
4481:
4475:
4466:
4460:
4454:
4448:
4442:
4435:
4434:
4433:
4429:
4425:
4421:
4397:
4390:
4384:
4376:
4368:
4364:
4359:
4354:
4351:
4344:
4339:
4334:
4329:
4325:
4320:
4313:
4308:
4303:
4297:
4291:
4284:
4283:
4282:
4265:
4258:
4252:
4241:
4237:
4232:
4227:
4224:
4220:
4217:
4214:
4209:
4204:
4198:
4192:
4185:
4184:
4183:
4179:
4175:
4168:
4164:
4144:
4139:
4131:
4127:
4122:
4117:
4114:
4110:
4107:
4104:
4100:
4093:
4087:
4084:
4078:
4072:
4065:
4064:
4063:
4061:
4056:
4052:
4048:
4044:
4034:
4030:
4025:
4015:
4009:
4000:
3979:
3973:
3970:
3967:
3963:
3957:
3953:
3949:
3944:
3941:
3938:
3934:
3930:
3925:
3921:
3917:
3911:
3905:
3898:
3897:
3896:
3892:
3881:
3875:
3855:
3846:
3842:
3838:
3835:
3828:
3824:
3820:
3813:
3809:
3805:
3800:
3795:
3791:
3785:
3774:
3766:
3765:
3764:
3758:
3745:
3736:
3727:
3722:
3706:
3686:
3679:
3676:
3673:
3662:
3655:
3652:
3647:
3644:
3641:
3637:
3631:
3627:
3623:
3617:
3611:
3604:
3603:
3602:
3600:
3581:
3574:
3563:
3556:
3553:
3548:
3545:
3542:
3538:
3532:
3528:
3524:
3518:
3507:
3500:
3497:
3492:
3489:
3486:
3482:
3476:
3472:
3468:
3462:
3456:
3449:
3448:
3447:
3443:
3432:
3415:
3409:
3405:
3399:
3396:
3391:
3387:
3381:
3378:
3374:
3368:
3364:
3360:
3356:
3350:
3346:
3342:
3337:
3333:
3327:
3324:
3319:
3315:
3309:
3306:
3302:
3296:
3292:
3288:
3284:
3278:
3274:
3270:
3264:
3258:
3251:
3250:
3249:
3245:
3234:
3230:
3223:
3216:
3208:the value of
3206:
3198:
3189:
3169:
3163:
3158:
3154:
3149:
3143:
3139:
3135:
3130:
3125:
3121:
3116:
3110:
3106:
3102:
3096:
3090:
3083:
3082:
3081:
3060:
3055:
3048:
3045:
3040:
3036:
3030:
3027:
3023:
3017:
3013:
3009:
3006:
2998:
2993:
2989:
2985:
2980:
2976:
2970:
2967:
2964:
2961:
2959:
2952:
2948:
2940:
2923:
2920:
2915:
2911:
2905:
2902:
2892:
2888:
2884:
2878:
2874:
2867:
2864:
2859:
2855:
2849:
2846:
2842:
2836:
2832:
2828:
2825:
2817:
2812:
2808:
2804:
2799:
2795:
2789:
2786:
2783:
2780:
2778:
2771:
2767:
2755:
2754:
2753:
2752:-domain are,
2732:
2728:
2725:
2720:
2715:
2711:
2707:
2704:
2701:
2698:
2695:
2690:
2686:
2678:
2677:
2676:
2674:
2661:
2654:
2647:
2641:
2632:
2615:
2603:
2600:
2593:
2588:
2578:
2573:
2564:
2559:
2552:
2549:
2542:
2541:
2540:
2523:
2513:
2509:
2503:
2498:
2495:
2488:
2487:
2486:
2484:
2468:
2445:
2434:
2430:
2423:
2418:
2416:
2409:
2405:
2393:
2390:
2385:
2380:
2378:
2373:
2362:
2361:
2360:
2357:
2351:
2346:
2342:
2338:
2334:
2330:
2322:
2314:
2291:
2287:
2284:
2278:
2272:
2267:
2262:
2258:
2254:
2248:
2242:
2236:
2220:
2217:
2214:
2208:
2202:
2194:
2190:
2178:
2162:
2161:
2160:
2143:
2139:
2136:
2130:
2124:
2117:
2113:
2108:
2103:
2097:
2091:
2085:
2065:
2060:
2055:
2049:
2043:
2035:
2031:
2019:
2003:
2002:
2001:
1980:
1973:
1967:
1964:
1961:
1949:
1943:
1938:
1933:
1929:
1921:
1916:
1911:
1905:
1899:
1896:
1890:
1876:
1870:
1858:
1855:
1849:
1843:
1840:
1833:
1832:
1831:
1816:
1804:
1798:
1793:
1788:
1784:
1776:
1771:
1766:
1760:
1754:
1751:
1740:
1714:
1701:
1695:
1684:
1681:
1670:
1649:
1642:
1636:
1630:
1627:
1622:
1618:
1610:Substituting
1608:
1604:
1600:
1579:
1570:
1543:
1536:
1530:
1527:
1516:
1512:
1501:
1497:
1486:
1478:
1477:
1476:
1474:
1470:
1466:
1462:
1451:
1444:
1441:
1434:
1428:
1422:
1416:
1415:
1412:
1408:
1399:
1397:
1391:
1385:
1372:
1357:
1344:
1333:
1328:
1321:
1312:
1289:
1267:
1258:
1241:
1217:
1209:
1197:
1189:
1179:
1174:
1163:
1158:
1153:
1148:
1143:
1134:
1126:
1119:
1115:
1106:
1100:
1095:
1088:
1083:
1078:
1075:
1068:
1067:
1066:
1060:
1055:
1049:
1027:
1019:
1016:
1000:
994:
979:
973:
968:
965:
958:
957:
956:
930:
914:
912:
893:
885:
881:
876:
867:
856:
848:
847:
846:
843:
824:
820:
817:
814:
811:
801:
800:
799:
793:
784:
778:
757:
751:
747:
743:
738:
734:
730:
727:
717:
716:
715:
713:
709:
698:
696:
690:
670:
656:
652:
644:
639:
636:
626:
625:
624:
614:
610:
601:
592:
570:
566:
560:
556:
553:
548:
544:
531:
529:
525:
521:
517:
516:highly damped
512:
493:
480:
476:
472:
467:
462:
458:
450:
449:
448:
445:
434:
432:
427:
422:
419:
401:
395:
391:
387:
384:
381:
376:
372:
364:
363:
362:
360:
352:
347:
343:
335:
330:
315:
313:
308:
306:
302:
298:
294:
290:
286:
282:
279:use them for
278:
274:
270:
265:
263:
259:
255:
251:
246:
244:
240:
236:
232:
228:
220:
217:
212:
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:
10486:
10467:
10448:
10429:
10410:
10403:Bibliography
10392:
10371:
10362:
10342:
10320:
10295:
10291:
10285:
10274:. Retrieved
10254:
10250:
10216:
10207:
10203:
10174:
10165:
10156:
10148:
10143:
10123:
10116:
10107:
10098:
10089:
10080:
10059:
10050:
10041:
10020:
10011:
10002:
9993:
9982:. Retrieved
9975:ece.ucsb.edu
9974:
9943:
9856:
9706:
9651:
9580:
9482:
9428:
9358:
9354:
9254:
9245:
9187:
9120:
9077:
9020:
8984:
8941:
8921:
8912:
8849:
8796:
8738:
8730:
8673:
8601:
8544:
8519:
8509:
8496:
8481:
8468:
8453:
8440:
8422:permeability
8403:
8395:Applications
8381:
8376:
8372:
8369:
8358:
8346:Oliver Lodge
8343:
8335:Joseph Henry
8332:
8323:Felix Savary
8320:
8309:
8303:
8296:
8293:
8138:
8135:
8024:
8018:
7890:
7592:
7466:
7458:
7452:
7441:
7438:
7274:
7271:
7193:
7191:
7108:
6990:
6984:
6911:
6908:
6720:
6709:
6699:
6689:
6679:
6674:
6661:
6523:
6452:
6402:is given by
6397:
6386:
6355:
6287:
6177:
6072:
5994:
5990:
5985:
5861:
5721:
5717:
5714:
5706:
5696:
5661:
5657:
5647:
5644:
5422:
5415:
5250:
5243:
5028:
5021:
5010:
5006:
4999:
4995:
4992:
4978:
4969:
4962:
4958:
4955:
4872:
4868:
4857:
4853:
4846:
4778:
4774:
4763:
4759:
4752:
4595:
4588:
4427:
4423:
4417:
4280:
4177:
4173:
4166:
4162:
4159:
4054:
4050:
4046:
4032:
4028:
4021:
4007:
3998:
3994:
3890:
3887:
3873:
3870:
3763:is given by
3756:
3743:
3734:
3725:
3704:
3701:
3596:
3441:
3438:
3430:
3243:
3240:
3228:
3221:
3214:
3196:
3187:
3184:
3079:
2747:
2670:
2659:
2652:
2645:
2630:
2538:
2482:
2466:
2464:
2358:
2349:
2332:
2328:
2312:
2306:
2158:
1995:
1609:
1602:
1598:
1577:
1568:
1558:
1458:
1410:
1392:
1383:
1370:
1363:
1256:
1232:
1056:
1042:
923:
911:notch filter
908:
839:
791:
782:
776:
772:
704:
694:
688:
685:
598:
532:
515:
510:
508:
440:
423:
416:
350:
333:
326:
309:
301:second-order
300:
266:
247:
226:
224:
161:
10210:(1): 26–34.
8918:Oscillators
2333:attenuation
1048:selectivity
524:underdamped
241:(L), and a
227:RLC circuit
10508:Categories
10276:2013-02-25
9984:2016-10-21
9936:References
9837:RL circuit
9832:RC circuit
8933:LC circuit
8510:Figure 11.
8497:Figure 10.
8327:Leyden jar
6288:It holds:
4881:, we find
4420:admittance
1471:(KVL) the
693:is called
528:overdamped
312:topologies
258:LC circuit
162:RLC filter
120:(all-pass)
114:(all-pass)
10451:. Wiley.
10151:, p. 464.
9883:π
9848:Footnotes
9799:α
9791:−
9786:β
9770:β
9764:α
9756:β
9752:α
9735:−
9682:β
9676:α
9630:β
9626:α
9614:β
9608:α
9555:α
9547:−
9542:β
9528:β
9524:α
9507:−
9460:β
9454:α
9403:β
9399:α
9325:β
9322:−
9314:−
9305:α
9302:−
9160:ω
9048:ω
9044:≈
9033:ω
8999:≪
8996:ζ
8960:ω
8956:≪
8953:α
8864:ω
8861:Δ
8811:ω
8808:Δ
8752:ω
8686:ω
8613:ζ
8557:ω
8482:Figure 9.
8469:Figure 8.
8454:Figure 7.
8441:Figure 6.
8377:resonance
8361:resonance
8192:−
8175:ω
8160:ω
8076:−
8041:ω
8019:Figure 5.
7975:≈
7912:≫
7820:−
7707:−
7534:−
7514:ω
7510:≈
7493:ω
7334:−
7317:ω
7296:ω
7223:≡
7211:ω
7152:ω
7140:α
7050:−
7015:ω
6985:Figure 4.
6924:ω
6865:ω
6847:ω
6675:Figure 3.
6464:ζ
6413:α
6356:Figure 2.
6323:ω
6310:ω
6297:ω
6236:−
6190:ω
6126:−
6086:ω
6018:ω
5957:ω
5938:⋅
5925:ω
5893:ω
5819:ω
5809:−
5803:ω
5760:ω
5702:= 1 rad/s
5622:τ
5607:τ
5596:ω
5589:
5569:ω
5561:α
5556:−
5550:τ
5539:ω
5532:
5519:τ
5516:α
5513:−
5502:τ
5499:−
5476:∫
5391:τ
5372:τ
5369:α
5366:−
5347:τ
5344:α
5341:−
5330:τ
5327:−
5304:∫
5221:τ
5206:τ
5195:ω
5188:
5168:ω
5160:α
5155:−
5149:τ
5138:ω
5131:
5118:τ
5115:α
5112:−
5101:τ
5098:−
5075:∫
4923:ω
4919:−
4910:α
4904:±
4901:α
4898:−
4877:. By the
4828:∞
4825:→
4709:ω
4699:α
3971:α
3968:−
3942:α
3939:−
3843:ζ
3839:−
3825:ω
3810:α
3806:−
3792:ω
3775:ω
3680:φ
3663:ω
3656:
3645:α
3642:−
3564:ω
3557:
3546:α
3543:−
3508:ω
3501:
3490:α
3487:−
3397:−
3388:ζ
3382:−
3379:ζ
3365:ω
3361:−
3325:−
3316:ζ
3307:ζ
3293:ω
3289:−
3046:−
3037:ζ
3028:ζ
3014:ω
3010:−
2990:ω
2986:−
2977:α
2971:−
2968:α
2965:−
2921:−
2912:ζ
2903:ζ
2889:ω
2885:−
2865:−
2856:ζ
2850:−
2847:ζ
2833:ω
2829:−
2809:ω
2805:−
2796:α
2787:α
2784:−
2712:ω
2702:α
2550:ζ
2510:ω
2504:α
2499:≡
2496:ζ
2406:ω
2374:α
2259:ω
2221:α
1962:τ
1950:τ
1930:∫
1817:τ
1805:τ
1785:∫
1452:component
1450:capacitor
1442:component
1411:Figure 1:
1322:≡
1135:ω
1107:ω
1050:, as the
1020:ω
1017:
1013:Δ
1001:ω
882:ω
877:ω
874:Δ
821:α
812:ω
809:Δ
748:ω
744:−
735:ω
728:ω
725:Δ
708:bandwidth
701:Bandwidth
653:ω
645:α
637:ζ
545:ω
459:ω
426:impedance
418:Resonance
388:π
373:ω
323:Resonance
254:resonates
243:capacitor
216:shortwave
157:LC filter
152:RL filter
147:RC filter
10271:51669988
9826:See also
8414:IF stage
8183:′
7522:′
7325:′
7219:′
7194:undamped
7160:′
6996:and the
5725:, where
3721:envelope
1440:inductor
615:(symbol
444:undamped
239:inductor
237:(R), an
235:resistor
8428:Filters
8373:syntony
8317:History
7474:is the
7471:
7449:
5729:is the
5655:√
4058:), the
4041:is the
4037:(where
1396:element
1061:), the
600:Damping
595:Damping
359:radians
262:damping
10493:
10474:
10455:
10436:
10417:
10350:
10269:
10130:
9905:
9899:
9894:
9877:
9868:
9759:
9685:
9617:
9605:
9407:
9392:
8294:where
8276:
8271:
8266:
8260:
8242:
8219:
8201:
8187:
8156:
8118:
8113:
8107:
8085:
8070:
8061:
8037:
7996:
7944:
7918:
7899:
7873:
7867:
7862:
7815:
7776:
7749:
7744:
7702:
7661:
7604:
7575:
7570:
7564:
7543:
7526:
7489:
7439:where
7421:
7416:
7411:
7405:
7387:
7364:
7361:
7343:
7329:
7292:
7254:
7248:
7243:
7232:
7207:
7177:
7121:
7091:
7086:
7070:
7064:
7044:
7035:
7011:
6959:
6954:
6920:
6707:, and
6640:
6634:
6600:
6506:
6500:
6266:
6261:
6229:
6223:
6208:
6153:
6119:
6113:
6101:
6048:
6042:
5999:|
5988:|
5670:, and
5645:where
5625:
5580:
5565:
5472:
5466:
5460:
5436:
5394:
5383:
5375:
5360:
5352:
5300:
5294:
5288:
5264:
5179:
5164:
5071:
4866:where
4772:where
4675:
4505:
4317:
4160:where
3444:< 1
3246:> 1
3224:< 1
3217:> 1
2938:
2932:
2927:
2900:
2613:
2607:
2598:
2341:nepers
1634:
1592:, and
1559:where
1377:, and
1281:or of
1257:either
1233:where
929:factor
920:factor
842:nepers
773:where
668:
662:
649:
634:
609:nepers
579:
574:
541:
511:driven
491:
484:
281:tuning
229:is an
10267:S2CID
10200:(PDF)
9971:(PDF)
5709:= 0.4
5421:<
5027:>
4875:) → ∞
4849:poles
4781:) = 0
4755:zeros
3895:) is
3446:) is
3248:) is
2331:, or
342:hertz
10491:ISBN
10472:ISBN
10453:ISBN
10434:ISBN
10415:ISBN
10348:ISBN
10128:ISBN
8529:and
6712:= 1
6702:= 1
6692:= 1
6682:= 1
6319:<
6306:<
5676:sinh
5674:and
5672:cosh
5586:sinh
5529:cosh
5015:is:
4976:and
4847:The
4753:The
4593:and
4005:and
3741:(or
3732:and
3194:and
2657:and
2310:and
1731:and
1575:and
924:The
275:and
10259:doi
9121:If
7278:max
5185:sin
5128:cos
4851:of
4806:and
4757:of
4060:KVL
3893:= 1
3653:sin
3554:sin
3498:cos
3233:).
3231:= 1
2665:= 1
2655:= 1
2648:= 1
1259:of
691:= 1
623:by
295:or
225:An
10510::
10380:^
10330:^
10304:^
10296:34
10294:.
10265:.
10255:20
10253:.
10249:.
10225:^
10208:11
10206:.
10202:.
10183:^
10068:^
10029:^
9952:^
8525:,
8391:.
8313:.
8310:RC
8304:ω′
8302:=
7453:ω′
7447:≡
6697:,
6687:,
6341:.
5995:jω
5722:jω
5720:=
5682:.
5660:−
5653:=
5426::
5254::
5249:=
5032::
4783::
4432::
4182::
4055:jω
4053:+
4049:=
2675:,
2650:,
2469:,
2323:.
1588:,
1566:,
1368:,
348:,
331:,
291:,
287:,
271:.
10499:.
10480:.
10461:.
10442:.
10423:.
10356:.
10279:.
10261::
10136:.
9987:.
9968:"
9966:Q
9926:R
9922:R
9908:,
9891:C
9888:L
9880:2
9873:1
9810:.
9805:)
9796:1
9783:1
9777:(
9767:+
9748:R
9743:0
9739:I
9728:=
9723:0
9719:V
9692:,
9679:+
9671:R
9666:=
9663:L
9637:,
9622:R
9611:+
9599:=
9596:C
9583:R
9566:.
9561:)
9552:1
9539:1
9533:(
9520:L
9515:0
9511:I
9504:=
9499:0
9495:V
9468:,
9464:)
9457:+
9450:(
9446:L
9443:=
9440:R
9414:,
9395:L
9388:1
9383:=
9380:C
9367:L
9362:0
9359:V
9340:.
9335:)
9329:t
9318:e
9309:t
9298:e
9292:(
9286:0
9282:I
9278:=
9275:)
9272:t
9269:(
9266:I
9231:.
9227:Q
9224:=
9219:V
9214:L
9210:V
9194:Q
9190:R
9173:.
9169:L
9164:0
9154:R
9151:V
9146:=
9141:L
9137:V
9123:R
9106:.
9100:R
9097:V
9092:=
9089:I
9058:.
9052:0
9038:d
9006:.
9002:1
8970:,
8964:0
8937:Q
8929:R
8925:R
8893:.
8884:L
8880:R
8876:C
8872:1
8867:=
8835:.
8829:L
8824:L
8820:R
8814:=
8782:,
8775:C
8772:L
8768:1
8763:=
8757:c
8716:.
8709:C
8706:L
8702:1
8697:=
8691:c
8654:.
8647:C
8644:L
8633:L
8629:R
8625:2
8621:1
8616:=
8587:.
8580:C
8577:L
8573:1
8568:=
8562:c
8535:R
8531:C
8527:L
8523:R
8418:L
8410:C
8307:0
8299:C
8297:Q
8279:,
8255:2
8250:C
8246:Q
8238:2
8233:+
8230:1
8225:+
8214:2
8209:C
8205:Q
8197:1
8179:0
8171:=
8165:m
8142:m
8139:ω
8121:,
8102:2
8098:)
8094:C
8091:R
8088:(
8081:1
8067:C
8064:L
8057:1
8050:=
8045:0
7999:.
7991:2
7986:L
7982:Q
7978:R
7969:x
7966:a
7963:m
7957:|
7952:Z
7948:|
7921:,
7915:1
7907:L
7903:Q
7876:.
7858:)
7852:2
7847:L
7843:Q
7838:/
7834:1
7831:+
7828:2
7824:(
7810:2
7805:L
7801:Q
7796:/
7792:2
7789:+
7786:1
7781:2
7769:L
7765:Q
7761:R
7755:=
7740:)
7736:1
7733:+
7728:2
7723:L
7719:Q
7715:2
7711:(
7699:2
7696:+
7691:2
7686:L
7682:Q
7674:L
7670:Q
7666:2
7654:2
7649:L
7645:Q
7641:R
7635:=
7629:x
7626:a
7623:m
7617:|
7612:Z
7608:|
7578:.
7559:4
7554:L
7550:Q
7546:2
7539:1
7531:1
7518:0
7504:x
7501:a
7498:m
7467:R
7463:/
7459:L
7456:0
7444:L
7442:Q
7424:,
7400:2
7395:L
7391:Q
7383:2
7378:+
7375:1
7370:+
7356:2
7351:L
7347:Q
7339:1
7321:0
7313:=
7307:x
7304:a
7301:m
7275:ω
7257:.
7240:C
7237:L
7228:1
7215:0
7174:0
7171:=
7166:2
7156:0
7146:+
7143:s
7137:2
7134:+
7129:2
7125:s
7094:.
7081:2
7076:)
7067:L
7060:R
7055:(
7041:C
7038:L
7031:1
7024:=
7019:0
6998:Q
6951:C
6947:L
6940:/
6936:1
6933:=
6928:0
6890:.
6882:R
6877:1
6872:+
6869:C
6861:j
6858:+
6851:L
6843:j
6838:1
6833:=
6817:R
6813:Z
6807:1
6802:+
6793:C
6789:Z
6783:1
6778:+
6769:L
6765:Z
6759:1
6754:=
6743:Z
6738:1
6717:.
6714:V
6710:V
6704:H
6700:L
6694:F
6690:C
6684:Ω
6680:R
6643:.
6629:L
6626:C
6619:R
6616:=
6609:Q
6595:C
6592:L
6581:R
6576:1
6571:=
6561:f
6556:B
6538:Q
6534:Q
6530:Q
6526:Q
6509:.
6495:C
6492:L
6481:R
6477:2
6472:1
6467:=
6434:C
6430:R
6426:2
6421:1
6416:=
6400:α
6381:C
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6314:0
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6003:ω
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5993:(
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5971:.
5965:|
5960:)
5954:j
5951:(
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5943:|
5933:|
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5864:ω
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2325:α
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2308:α
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2020:2
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1981:.
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1974:t
1971:(
1968:V
1965:=
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1947:(
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1628:=
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1190:=
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1127:=
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1079:=
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177:v
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