6150:
2311:
5717:
1691:
4411:
6145:{\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}\operatorname {Re} \left(V_{\text{c}}\cdot e^{i\omega t}\right)&=\operatorname {Re} \left({\frac {\mathrm {d} }{\mathrm {d} t}}{\mathord {\left(V_{\text{c}}\cdot e^{i\omega t}\right)}}\right)=\operatorname {Re} \left(i\omega V_{\text{c}}\cdot e^{i\omega t}\right)\\{\frac {\mathrm {d} }{\mathrm {d} t}}\operatorname {Im} \left(V_{\text{c}}\cdot e^{i\omega t}\right)&=\operatorname {Im} \left({\frac {\mathrm {d} }{\mathrm {d} t}}{\mathord {\left(V_{\text{c}}\cdot e^{i\omega t}\right)}}\right)=\operatorname {Im} \left(i\omega V_{\text{c}}\cdot e^{i\omega t}\right).\end{aligned}}}
2306:{\displaystyle {\begin{aligned}&A_{1}\cos(\omega t+\theta _{1})+A_{2}\cos(\omega t+\theta _{2})\\={}&\operatorname {Re} \left(A_{1}e^{i\theta _{1}}e^{i\omega t}\right)+\operatorname {Re} \left(A_{2}e^{i\theta _{2}}e^{i\omega t}\right)\\={}&\operatorname {Re} \left(A_{1}e^{i\theta _{1}}e^{i\omega t}+A_{2}e^{i\theta _{2}}e^{i\omega t}\right)\\={}&\operatorname {Re} \left(\left(A_{1}e^{i\theta _{1}}+A_{2}e^{i\theta _{2}}\right)e^{i\omega t}\right)\\={}&\operatorname {Re} \left(\left(A_{3}e^{i\theta _{3}}\right)e^{i\omega t}\right)\\={}&A_{3}\cos(\omega t+\theta _{3}),\end{aligned}}}
4003:
6573:
4406:{\displaystyle {\begin{aligned}&\operatorname {Re} \left({\frac {\mathrm {d} }{\mathrm {d} t}}{\mathord {\left(Ae^{i\theta }\cdot e^{i\omega t}\right)}}\right)\\={}&\operatorname {Re} \left(Ae^{i\theta }\cdot i\omega e^{i\omega t}\right)\\={}&\operatorname {Re} \left(Ae^{i\theta }\cdot e^{i\pi /2}\omega e^{i\omega t}\right)\\={}&\operatorname {Re} \left(\omega Ae^{i(\theta +\pi /2)}\cdot e^{i\omega t}\right)\\={}&\omega A\cdot \cos \left(\omega t+\theta +{\frac {\pi }{2}}\right).\end{aligned}}}
52:
6198:
3693:
7458:
1626:
5696:
6568:{\displaystyle {\begin{aligned}i\omega V_{\text{c}}\cdot e^{i\omega t}+{\frac {1}{RC}}V_{\text{c}}\cdot e^{i\omega t}&={\frac {1}{RC}}V_{\text{s}}\cdot e^{i\omega t}\\\left(i\omega V_{\text{c}}+{\frac {1}{RC}}V_{\text{c}}\right)\!\cdot e^{i\omega t}&=\left({\frac {1}{RC}}V_{\text{s}}\right)\cdot e^{i\omega t}\\i\omega V_{\text{c}}+{\frac {1}{RC}}V_{\text{c}}&={\frac {1}{RC}}V_{\text{s}}.\end{aligned}}}
5404:
1382:
5491:
7226:
5206:
3906: radians representing one complete cycle. If the length of its moving tip is transferred at different angular intervals in time to a graph as shown above, a sinusoidal waveform would be drawn starting at the left with zero time. Each position along the horizontal axis indicates the time that has elapsed since zero time,
3968:
Sometimes when we are analysing alternating waveforms we may need to know the position of the phasor, representing the alternating quantity at some particular instant in time especially when we want to compare two different waveforms on the same axis. For example, voltage and current. We have assumed
3870:
In the example of three waves, the phase difference between the first and the last wave was 240°, while for two waves destructive interference happens at 180°. In the limit of many waves, the phasors must form a circle for destructive interference, so that the first phasor is nearly parallel with the
3704:
with each other, constructively or destructively. The static vector concept provides useful insight into questions like this: "What phase difference would be required between three identical sinusoids for perfect cancellation?" In this case, simply imagine taking three vectors of equal length and
1100:
whose real part is the original sinusoid. The benefit of the complex representation is that linear operations with other complex representations produces a complex result whose real part reflects the same linear operations with the real parts of the other complex sinusoids. Furthermore, all the
8199:
Frequency modulation is a similar representation except that the modulating phasors are not in phase with the carrier. In this case the vector sum of the modulating phasors is shifted 90° from the carrier phase. Strictly, frequency modulation representation requires additional small modulation
1671:
the shorthand notation for another phasor. Multiplying a phasor current by an impedance produces a phasor voltage. But the product of two phasors (or squaring a phasor) would represent the product of two sinusoids, which is a non-linear operation that produces new frequency components. Phasor
170:
where all signals are assumed to be sinusoidal with a common frequency. Phasor representation allows the analyst to represent the amplitude and phase of the signal using a single complex number. The only difference in their analytic representations is the complex amplitude (phasor). A linear
6732:
3502:
3200:
1621:{\displaystyle {\begin{aligned}&\operatorname {Re} \left(\left(Ae^{i\theta }\cdot Be^{i\phi }\right)\cdot e^{i\omega t}\right)\\={}&\operatorname {Re} \left(\left(ABe^{i(\theta +\phi )}\right)\cdot e^{i\omega t}\right)\\={}&AB\cos(\omega t+(\theta +\phi )).\end{aligned}}}
1681:
2962:
7052:
4757:
5691:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\operatorname {Im} \left(V_{\text{c}}\cdot e^{i\omega t}\right)+{\frac {1}{RC}}\operatorname {Im} \left(V_{\text{c}}\cdot e^{i\omega t}\right)={\frac {1}{RC}}\operatorname {Im} \left(V_{\text{s}}\cdot e^{i\omega t}\right).}
1095:
2488:
3866:
6991:
5399:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\operatorname {Re} \left(V_{\text{c}}\cdot e^{i\omega t}\right)+{\frac {1}{RC}}\operatorname {Re} (V_{\text{c}}\cdot e^{i\omega t})={\frac {1}{RC}}\operatorname {Re} \left(V_{\text{s}}\cdot e^{i\omega t}\right)}
3961:. Then the time axis of the waveform represents the angle either in degrees or radians through which the phasor has moved. So we can say that a phasor represents a scaled voltage or current value of a rotating vector which is "frozen" at some point in time, (
2703:
7702:
can then be viewed as a projection of this vector onto the real axis. A modulated waveform is represented by this phasor (the carrier) and two additional phasors (the modulation phasors). If the modulating signal is a single tone of the form
7429:, graphically represented as unit magnitudes at angles of 0, 120 and 240 degrees. By treating polyphase AC circuit quantities as phasors, balanced circuits can be simplified and unbalanced circuits can be treated as an algebraic combination of
5015:
4927:
5193:
7572:
3979:
But if a second waveform starts to the left or to the right of this zero point, or if we want to represent in phasor notation the relationship between the two waveforms, then we will need to take into account this phase difference,
6581:
3677:
3299:
1696:
3580:, which is what makes phasor notation possible. The time and frequency dependence can be suppressed and re-inserted into the outcome as long as the only operations used in between are ones that produce another phasor. In
8003:
The two modulation phasors are phased such that their vector sum is always in phase with the carrier phasor. An alternative representation is two phasors counter rotating around the end of the carrier phasor at a rate
4627:
4838:
7999:
7900:
7433:. This approach greatly simplifies the work required in electrical calculations of voltage drop, power flow, and short-circuit currents. In the context of power systems analysis, the phase angle is often given in
2316:
7448:
uses digital instruments to measure the phasors representing transmission system voltages at widespread points in a transmission network. Differences among the phasors indicate power flow and system stability.
3744:
3047:
4538:
6854:
8195:
8386:
8112:
3278:
3040:
2815:
2784:
7461:
A: phasor representation of amplitude modulation, B: alternate representation of amplitude modulation, C: phasor representation of frequency modulation, D: alternate representation of frequency modulation
945:
3554:
596:
5071:
6203:
5722:
4008:
1387:
7044:
7221:{\displaystyle v_{\text{C}}(t)=\operatorname {Re} \left(V_{\text{c}}\cdot e^{i\omega t}\right)={\frac {1}{\sqrt {1+(\omega RC)^{2}}}}\cdot V_{\text{P}}\cos(\omega t+\theta -\phi (\omega )).}
143:, which decomposes a sinusoid into the product of a complex constant and a factor depending on time and frequency. The complex constant, which depends on amplitude and phase, is known as a
4466:
930:
304:
7394:. Multiple frequency linear AC circuits and AC circuits with different waveforms can be analyzed to find voltages and currents by transforming all waveforms to sine wave components (using
536:
1344:
5481:
1272:
1219:
850:
4933:
7751:
5106:
2804:
713:
2561:
7478:
463:
6797:
3587:
488:
4574:
1135:
793:
252:
8244:
4614:
2596:
1168:
6824:
6761:
5098:
2588:
1661:
1377:
677:
248:
of an RLC circuit. However, the
Laplace transform is mathematically more difficult to apply and the effort may be unjustified if only steady state analysis is required.
6847:
3889:
636:
7668:
4845:
746:
434:
8029:
7798:
7622:
7700:
7648:
6193:
7771:
7595:
5440:
873:
411:
4764:
7904:
7805:
4616:
out of all terms of the equation, and reinserting it into the answer. For example, consider the following differential equation for the voltage across the
1688:
The sum of multiple phasors produces another phasor. That is because the sum of sinusoids with the same frequency is also a sinusoid with that frequency:
3705:
placing them head to tail such that the last head matches up with the first tail. Clearly, the shape which satisfies these conditions is an equilateral
7402:. This solution method applies only to inputs that are sinusoidal and for solutions that are in steady state, i.e., after all transients have died out.
8116:
8036:
6727:{\displaystyle V_{\text{c}}={\frac {1}{1+i\omega RC}}\cdot V_{\text{s}}={\frac {1-i\omega RC}{1+(\omega RC)^{2}}}\cdot V_{\text{P}}e^{i\theta }.}
3871:
last. This means that for many sources, destructive interference happens when the first and last wave differ by 360 degrees, a full wavelength
3497:{\displaystyle A_{3}^{2}=A_{1}^{2}+A_{2}^{2}-2A_{1}A_{2}\cos(180^{\circ }-\Delta \theta )=A_{1}^{2}+A_{2}^{2}+2A_{1}A_{2}\cos(\Delta \theta ),}
3195:{\displaystyle \pi +\arctan \left({\frac {A_{1}\sin \theta _{1}+A_{2}\sin \theta _{2}}{A_{1}\cos \theta _{1}+A_{2}\cos \theta _{2}}}\right),}
237:. Heaviside's operational calculus was modified so that the variable p becomes jω. The complex number j has simple meaning: phase shift.
4474:
2957:{\displaystyle \arctan \left({\frac {A_{1}\sin \theta _{1}+A_{2}\sin \theta _{2}}{A_{1}\cos \theta _{1}+A_{2}\cos \theta _{2}}}\right),}
3205:
2967:
8315:
4752:{\displaystyle {\frac {\mathrm {d} \,v_{\text{C}}(t)}{\mathrm {d} t}}+{\frac {1}{RC}}v_{\text{C}}(t)={\frac {1}{RC}}v_{\text{S}}(t).}
2708:
1274:
Figure 2 depicts it as a rotating vector in the complex plane. It is sometimes convenient to refer to the entire function as a
3293:
1090:{\displaystyle A\cos(\omega t+\theta )+i\cdot A\sin(\omega t+\theta )=Ae^{i(\omega t+\theta )}=Ae^{i\theta }\cdot e^{i\omega t},}
3902:
As the single vector rotates in an anti-clockwise direction, its tip at point A will rotate one complete revolution of 360° or 2
199:
71:
without axis shown), which must not be confused with the arrows in the lower diagram, which are the reference polarity for the
2483:{\displaystyle A_{3}^{2}=(A_{1}\cos \theta _{1}+A_{2}\cos \theta _{2})^{2}+(A_{1}\sin \theta _{1}+A_{2}\sin \theta _{2})^{2},}
8818:
8764:
8737:
8712:
8682:
8657:
8621:
8588:
8560:
8530:
8505:
8467:
8434:
3861:{\displaystyle \cos(\omega t)+\cos \left(\omega t+{\frac {2\pi }{3}}\right)+\cos \left(\omega t-{\frac {2\pi }{3}}\right)=0.}
3507:
544:
5020:
6986:{\displaystyle {\frac {1-i\omega RC}{1+(\omega RC)^{2}}}={\frac {1}{\sqrt {1+(\omega RC)^{2}}}}\cdot e^{-i\phi (\omega )},}
244:(limited to a single frequency), which, in contrast to phasor representation, can be used to (simultaneously) derive the
1379:, produces another phasor. That means its only effect is to change the amplitude and phase of the underlying sinusoid:
8837:. International Journal of Research in Engineering and Science (IJRES) Vol.2, N.1, Jan., pp.11-18, 2014. ISSN 2320-9364
8793:
8255:
7239:
is the ratio of two phasors, which is not a phasor, because it does not correspond to a sinusoidally varying function.
6996:
182:
The origin of the term phasor rightfully suggests that a (diagrammatic) calculus somewhat similar to that possible for
8860:
885:
258:
4418:
4415:
Therefore, in phasor representation, the time derivative of a sinusoid becomes just multiplication by the constant
496:
7800:
is the frequency of the modulating signal, then for amplitude modulation the two modulation phasors are given by,
3701:
3686:
1300:
8879:
7387:
1230:
1176:
808:
5445:
45:
8925:
4580:
240:
Glossing over some mathematical details, the phasor transform can also be seen as a particular case of the
7706:
2789:
1672:
notation can only represent systems with one frequency, such as a linear system stimulated by a sinusoid.
682:
7303:
7465:
The rotating frame picture using phasor can be a powerful tool to understand analog modulations such as
1287:
2698:{\textstyle \operatorname {sgn}(A_{1}\sin(\theta _{1})+A_{2}\sin(\theta _{2}))\cdot {\frac {\pi }{2}},}
206:
3913:. When the vector is horizontal the tip of the vector represents the angles at 0°, 180°, and at 360°.
2495:
439:
6766:
467:
390:
226:
8900:
4543:
1104:
751:
8649:
8203:
4586:
1140:
8895:
8426:
7390:
with phasors to analyze single frequency linear AC circuits containing resistors, capacitors, and
6802:
6739:
5076:
2566:
1633:
1349:
1223:
639:
382:
186:
is possible for phasors as well. An important additional feature of the phasor transform is that
38:
31:
17:
5010:{\displaystyle v_{\text{C}}(t)=\operatorname {Re} \left(V_{\text{c}}\cdot e^{i\omega t}\right),}
4922:{\displaystyle v_{\text{S}}(t)=\operatorname {Re} \left(V_{\text{s}}\cdot e^{i\omega t}\right).}
656:
7430:
386:
8772:
8459:
7380:
6829:
3874:
605:
171:
combination of such functions can be represented as a linear combination of phasors (known as
8940:
8935:
8260:
7653:
7399:
5188:{\displaystyle i\omega V_{\text{c}}+{\frac {1}{RC}}V_{\text{c}}={\frac {1}{RC}}V_{\text{s}}.}
718:
419:
378:
218:
128:
8641:
8451:
8418:
8246:
etc, but for most practical purposes these are ignored because their effect is very small.
8007:
7776:
7600:
7470:
7466:
7406:
7297:
7236:
1664:
7676:
194:
of sinusoidal signals (having constant amplitude, period and phase) corresponds to simple
8:
8642:
8419:
8389:
7627:
7567:{\displaystyle x(t)=\operatorname {Re} \left(Ae^{i\theta }\cdot e^{i2\pi f_{0}t}\right),}
7426:
3984:
of the waveform. Consider the diagram below from the previous Phase
Difference tutorial.
3916:
Likewise, when the tip of the vector is vertical it represents the positive peak value, (
203:
195:
167:
8784:
Singh, Ravish R (2009). "Section 4.5: Phasor
Representation of Alternating Quantities".
6175:
3672:{\displaystyle A_{1}\angle \theta _{1}+A_{2}\angle \theta _{2}=A_{3}\angle \theta _{3}.}
7756:
7580:
7398:) with magnitude and phase then analyzing each frequency separately, as allowed by the
5425:
3899:
from the far edge travels a full wavelength further than the light from the near edge.
936:
858:
599:
396:
245:
214:
163:
8905:
7316:) which is a representation of the average power into the circuit and reactive power (
8930:
8910:
8875:
8856:
8814:
8789:
8760:
8733:
8708:
8678:
8653:
8617:
8594:
8584:
8556:
8526:
8501:
8463:
8430:
7358:
3735:
241:
132:
8452:
8273:
7438:
7410:
802:
A real-valued sinusoid with constant amplitude, frequency, and phase has the form:
491:
234:
230:
183:
76:
8754:
8267:
3734:. So the phase difference between each wave must also be 120°, as is the case in
2807:
365:
140:
8270:, a generalization of phasors for time-variant amplitude, phase, and frequency.
7574:
where the term in brackets is viewed as a rotating vector in the complex plane.
7413:
between the voltage applied to the impedance and the current driven through it.
7267:
has no time delays and therefore doesn't change the phase of a signal therefore
7434:
7395:
7253:
3581:
3285:
876:
646:
539:
307:
104:
8919:
8598:
8393:
7445:
7321:
3289:
225:
coefficients) in the time domain. The originator of the phasor transform was
120:
68:
51:
44:"Complex amplitude" redirects here. For the quantum-mechanical concept, see
8280:
7425:
AC power systems, usually a set of phasors is defined as the three complex
3692:
217:(albeit with complex coefficients) in the phasor domain instead of solving
67:. The arrows in the upper diagram are phasors, drawn in a phasor diagram (
7422:
4833:{\displaystyle v_{\text{S}}(t)=V_{\text{P}}\cdot \cos(\omega t+\theta ),}
3892:
222:
210:
96:
88:
56:
7994:{\displaystyle {1 \over 2}Ame^{i\theta }\cdot e^{i2\pi (f_{0}-f_{m})t}.}
7895:{\displaystyle {1 \over 2}Ame^{i\theta }\cdot e^{i2\pi (f_{0}+f_{m})t},}
5103:
In the phasor shorthand notation, the differential equation reduces to:
179:) and the time/frequency dependent factor that they all have in common.
7320:) which indicates power flowing back and forth. We can also define the
4621:
3993:
187:
7457:
4617:
112:
108:
6851:
In polar coordinate form, the first term of the last expression is:
8525:(2nd ed.). Springer Science & Business Media. p. 13.
7391:
7264:
4533:{\textstyle {\frac {1}{i\omega }}={\frac {e^{-i\pi /2}}{\omega }}.}
3997:
3706:
191:
8190:{\displaystyle {1 \over 2}Ame^{i\theta }\cdot e^{-i2\pi f_{m}t}.}
4471:
Similarly, integrating a phasor corresponds to multiplication by
3696:
Phasor diagram of three waves in perfect destructive interference
3689:
to produce a resultant vector with coordinates (see animation).
348:
84:
72:
8381:{\textstyle {\frac {d}{dt}}e^{i\omega t}=i\omega e^{i\omega t},}
8107:{\displaystyle {1 \over 2}Ame^{i\theta }\cdot e^{i2\pi f_{m}t},}
3273:{\displaystyle A_{1}\cos \theta _{1}+A_{2}\cos \theta _{2}<0}
3035:{\displaystyle A_{1}\cos \theta _{1}+A_{2}\cos \theta _{2}>0}
8639:
7341:. The power law for an AC circuit expressed in phasors is then
3976:
with a corresponding phase angle in either degrees or radians.
650:
8672:
2779:{\displaystyle A_{1}\cos \theta _{1}+A_{2}\cos \theta _{2}=0,}
1680:
8835:
About the Phasor
Pathways in Analogical Amplitude Modulations
3896:
414:
8874:(1 ed.). Boca Raton, FL: CRC Press. pp. 152–155.
306:
is depicted in the complex plane, the vector formed by its
162:
A common application is in the steady-state analysis of an
139:) is fixed. It is related to a more general concept called
939:, the factoring property described in the lead paragraph:
251:
8523:
Electromagnetic Theory for
Microwaves and Optoelectronics
7452:
8732:(1st ed.). Prometheus Books Learning. p. 230.
7367:, and the magnitudes of the voltage and current phasors
3965:) and in our example above, this is at an angle of 30°.
3709:, so the angle between each phasor to the next is 120° (
3700:
In physics, this sort of addition occurs when sinusoids
1292:
8702:
4761:
When the voltage source in this circuit is sinusoidal:
3969:
in the waveform above that the waveform starts at time
3549:{\displaystyle \Delta \theta =\theta _{1}-\theta _{2}.}
591:{\displaystyle \cos \theta +i\sin \theta =e^{i\theta }}
8318:
7441:
value rather than the peak amplitude of the sinusoid.
7405:
The concept is frequently involved in representing an
5448:
5066:{\displaystyle V_{\text{s}}=V_{\text{P}}e^{i\theta },}
4477:
4421:
2599:
2498:
233:
in the late 19th century. He got his inspiration from
8870:
Dorf, Richard C.; Tallarida, Ronald J. (1993-07-15).
8520:
8416:
8206:
8119:
8039:
8010:
7907:
7808:
7779:
7759:
7709:
7679:
7656:
7630:
7603:
7583:
7481:
7256:
circuits can be applied to solve linear AC circuits.
7055:
6999:
6857:
6832:
6805:
6769:
6763:
represents differences of the amplitude and phase of
6742:
6584:
6201:
6178:
5720:
5494:
5428:
5209:
5109:
5079:
5023:
4936:
4848:
4767:
4630:
4589:
4546:
4006:
3877:
3747:
3590:
3510:
3302:
3208:
3050:
2970:
2818:
2792:
2711:
2569:
2319:
1694:
1667:, which is independent of time. In particular it is
1636:
1385:
1352:
1303:
1233:
1179:
1143:
1107:
948:
888:
861:
811:
754:
721:
685:
659:
608:
547:
499:
470:
442:
422:
399:
322:
is the angle it forms with the positive real axis at
261:
198:
on the phasors; the phasor transform thus allows the
1170:
is reinserted prior to the real part of the result.
8850:
8583:. Thaddeus Adam Roppel. Boca Raton, FL: CRC Press.
4000:of a phasor produces another phasor. For example:
8495:
8380:
8238:
8189:
8106:
8023:
7993:
7894:
7792:
7765:
7745:
7694:
7662:
7642:
7616:
7589:
7566:
7383:values of the voltage and current, respectively).
7308:Work with voltages and current as complex phasors.
7280:Ohm's law for resistors, inductors, and capacitors
7220:
7038:
6985:
6841:
6818:
6791:
6755:
6726:
6567:
6187:
6144:
5690:
5475:
5434:
5398:
5187:
5092:
5065:
5009:
4921:
4832:
4751:
4608:
4568:
4532:
4460:
4405:
3987:
3883:
3860:
3671:
3548:
3496:
3272:
3194:
3034:
2956:
2798:
2778:
2697:
2582:
2555:
2482:
2305:
1684:The sum of phasors as addition of rotating vectors
1655:
1620:
1371:
1338:
1266:
1213:
1162:
1129:
1089:
924:
867:
844:
787:
740:
707:
671:
630:
590:
530:
482:
457:
428:
405:
298:
27:Complex number representing a particular sine wave
8753:Nilsson, James William; Riedel, Susan A. (2008).
8666:
8635:
8633:
8611:
7337:and the apparent power which is the magnitude of
7039:{\displaystyle \phi (\omega )=\arctan(\omega RC)}
6397:
8917:
8616:. Morgan & Claypool Publishers. p. 51.
6578:Solving for the phasor capacitor voltage gives:
4583:with phasor arithmetic, we are merely factoring
8677:(5th ed.). Cengage Learning. p. 536.
8648:(8th ed.). John Wiley & Sons. p.
8546:
8544:
8542:
8872:Pocket Book of Electrical Engineering Formulas
8788:. Mcgraw Hill Higher Education. p. 4.13.
8721:
8698:
8696:
8694:
8630:
8614:Pragmatic Electrical Engineering: Fundamentals
8299:
8297:
4461:{\textstyle i\omega =e^{i\pi /2}\cdot \omega }
1288:Complex number § Relations and operations
1101:mathematics can be done with just the phasors
925:{\displaystyle i\cdot A\sin(\omega t+\theta )}
299:{\displaystyle A\cdot e^{i(\omega t+\theta )}}
8869:
8811:Introduction to electromagnetic compatibility
8752:
3722: radians), or one third of a wavelength
531:{\displaystyle (\cos \theta ,\,\sin \theta )}
8759:(8th ed.). Prentice Hall. p. 338.
8553:Electrical Machines & their Applications
8550:
8539:
8449:
8410:
3294:trigonometric identity for angle differences
310:rotates around the origin. Its magnitude is
8727:
8707:. John Wiley & Sons. pp. 256–261.
8691:
8421:Mathematics for Engineers and Technologists
8294:
649:with an implied conversion from degrees to
8640:Richard C. Dorf; James A. Svoboda (2010).
8514:
8484:The Fourier Transform and Its Applications
8443:
8031:relative to the carrier phasor. That is,
7386:Given this we can apply the techniques of
5100:is the unknown quantity to be determined.
1339:{\displaystyle Ae^{i\theta }e^{i\omega t}}
8673:Allan H. Robbins; Wilhelm Miller (2012).
8605:
7252:With phasors, the techniques for solving
4639:
3741:In other words, what this shows is that:
3681:Another way to view addition is that two
731:
515:
8730:The Forgotten Genius of Oliver Heaviside
8500:. Pearson Education India. p. 272.
8489:
8476:
7670:with respect to the positive real axis.
7456:
6736:As we have seen, the factor multiplying
5476:{\textstyle t-{\frac {\pi }{2\omega }},}
3691:
3584:, the operation shown above is written:
1679:
1267:{\displaystyle A\cos(\omega t+\theta ).}
1214:{\displaystyle Ae^{i(\omega t+\theta )}}
845:{\displaystyle A\cos(\omega t+\theta ),}
250:
50:
8808:
7409:. In this case, the phase angle is the
14:
8918:
8705:Circuit Systems with MATLAB and PSpice
8581:Fundamentals of electrical engineering
8555:(4th ed.). Elsevier. p. 58.
8303:Including analysis of the AC circuits.
7597:, rotates anti-clockwise at a rate of
7453:Telecommunications: analog modulations
875:is time-variant. The inclusion of an
8827:
8783:
8675:Circuit Analysis: Theory and Practice
8578:
8574:
8572:
7312:In an AC circuit we have real power (
1293:Multiplication by a constant (scalar)
8853:Physics for Scientists and Engineers
7746:{\displaystyle Am\cos {2\pi f_{m}t}}
7624:revolutions per second, and at time
7416:
5485:
5200:
2799:{\displaystyle \operatorname {sgn} }
708:{\displaystyle 1\angle 90^{\circ },}
314:, and it completes one cycle every 2
75:and the reference direction for the
7230:
24:
8844:
8833:de Oliveira, H.M. and Nunes, F.D.
8703:Won Y. Yang; Seung C. Lee (2008).
8569:
8256:In-phase and quadrature components
6023:
6017:
5941:
5935:
5816:
5810:
5734:
5728:
5504:
5498:
5219:
5213:
4662:
4635:
4032:
4026:
3653:
3627:
3601:
3511:
3482:
3402:
689:
663:
474:
446:
25:
8952:
8889:
8644:Introduction to Electric Circuits
8425:. Butterworth-Heinemann. p.
6195:and adding both equations gives:
2556:{\textstyle \theta _{3}\in \left}
8901:Visual Representation of Phasors
8521:Kequian Zhang; Dejie Li (2007).
8458:(2nd ed.). Newnes. p.
8417:Huw Fox; William Bolton (2002).
1278:, as we do in the next section.
458:{\displaystyle A\angle \theta .}
8802:
8777:
8746:
8306:
7247:
7242:
6792:{\displaystyle v_{\text{C}}(t)}
3988:Differentiation and integration
483:{\displaystyle 1\angle \theta }
8906:Polar and Rectangular Notation
8498:Electric Circuits and Networks
7980:
7954:
7881:
7855:
7689:
7683:
7491:
7485:
7388:analysis of resistive circuits
7212:
7209:
7203:
7182:
7151:
7138:
7072:
7066:
7033:
7021:
7009:
7003:
6975:
6969:
6940:
6927:
6900:
6887:
6786:
6780:
6683:
6670:
6168:
6162:
6156:
5714:It is also readily seen that:
5704:
5412:
5330:
5298:
4953:
4947:
4865:
4859:
4824:
4809:
4784:
4778:
4743:
4737:
4706:
4700:
4656:
4650:
4569:{\displaystyle e^{i\omega t},}
4306:
4286:
3938:and the negative peak value, (
3763:
3754:
3488:
3479:
3408:
3386:
2676:
2673:
2660:
2638:
2625:
2606:
2468:
2409:
2397:
2338:
2293:
2271:
1782:
1760:
1738:
1716:
1608:
1605:
1593:
1581:
1524:
1512:
1258:
1243:
1206:
1191:
1130:{\displaystyle Ae^{i\theta },}
1041:
1026:
1009:
994:
973:
958:
919:
904:
836:
821:
788:{\displaystyle e^{i\pi /2}=i.}
735:
722:
525:
500:
291:
276:
13:
1:
8403:
8239:{\displaystyle 2f_{m},3f_{m}}
5422:Since this must hold for all
4609:{\displaystyle e^{i\omega t}}
3891:. This is why in single slit
1297:Multiplication of the phasor
1281:
1163:{\displaystyle e^{i\omega t}}
935:gives it, in accordance with
797:
46:Complex probability amplitude
8851:Douglas C. Giancoli (1989).
8287:
8283:, a phasor of unit magnitude
7773:is the modulation depth and
6819:{\displaystyle V_{\text{P}}}
6756:{\displaystyle V_{\text{s}}}
5093:{\displaystyle V_{\text{c}}}
4581:linear differential equation
7:
8911:Phasor in Telecommunication
8496:K. S. Suresh Kumar (2008).
8396:of the derivative operator.
8249:
4540:The time-dependent factor,
3685:with coordinates and are
2583:{\displaystyle \theta _{3}}
1675:
1656:{\displaystyle Be^{i\phi }}
1372:{\displaystyle Be^{i\phi }}
645:The angle may be stated in
359:
10:
8957:
8612:William J. Eccles (2011).
8579:Gross, Charles A. (2012).
7235:A quantity called complex
1285:
672:{\displaystyle 1\angle 90}
363:
43:
36:
29:
8486:. McGraw-Hill, 1965. p269
490:can represent either the
227:Charles Proteus Steinmetz
7304:Kirchhoff's circuit laws
6842:{\displaystyle \theta .}
6154:Substituting these into
3895:, the minima occur when
3884:{\displaystyle \lambda }
631:{\displaystyle i^{2}=-1}
308:imaginary and real parts
37:Not to be confused with
7663:{\displaystyle \theta }
7469:(and its variants) and
7437:, and the magnitude in
7260:Ohm's law for resistors
1346:by a complex constant,
1224:analytic representation
741:{\displaystyle (0,\,1)}
679:would be assumed to be
429:{\displaystyle \theta }
383:electronics engineering
151:, and (in older texts)
141:analytic representation
32:Phasor (disambiguation)
8813:. Wiley. p. 861.
8809:Clayton, Paul (2008).
8382:
8240:
8191:
8108:
8025:
7995:
7896:
7794:
7767:
7747:
7696:
7664:
7644:
7618:
7591:
7577:The phasor has length
7568:
7462:
7431:symmetrical components
7222:
7040:
6987:
6843:
6820:
6793:
6757:
6728:
6569:
6189:
6146:
5692:
5477:
5436:
5400:
5189:
5094:
5067:
5011:
4923:
4834:
4753:
4610:
4570:
4534:
4462:
4407:
3885:
3862:
3697:
3673:
3550:
3498:
3274:
3196:
3036:
2958:
2800:
2780:
2699:
2584:
2557:
2484:
2307:
1685:
1657:
1622:
1373:
1340:
1268:
1215:
1164:
1137:and the common factor
1131:
1091:
926:
869:
846:
789:
742:
709:
673:
632:
592:
532:
484:
459:
430:
407:
387:electrical engineering
356:
300:
219:differential equations
80:
8551:J. Hindmarsh (1984).
8450:Clay Rawlins (2000).
8388:which means that the
8383:
8261:Constellation diagram
8241:
8192:
8109:
8026:
8024:{\displaystyle f_{m}}
7996:
7897:
7795:
7793:{\displaystyle f_{m}}
7768:
7748:
7697:
7665:
7645:
7619:
7617:{\displaystyle f_{0}}
7592:
7569:
7460:
7400:superposition theorem
7223:
7041:
6988:
6844:
6821:
6794:
6758:
6729:
6570:
6190:
6147:
5693:
5478:
5437:
5401:
5190:
5095:
5068:
5012:
4924:
4835:
4754:
4611:
4571:
4535:
4463:
4408:
3886:
3863:
3695:
3674:
3551:
3499:
3275:
3197:
3037:
2959:
2801:
2781:
2700:
2585:
2558:
2485:
2308:
1683:
1658:
1623:
1374:
1341:
1269:
1216:
1165:
1132:
1092:
927:
870:
855:where only parameter
847:
790:
743:
710:
674:
638:, both of which have
633:
593:
533:
485:
460:
431:
408:
379:mathematical notation
301:
255:Fig 2. When function
254:
202:(calculation) of the
55:An example of series
54:
8728:Basil Mahon (2017).
8316:
8204:
8117:
8037:
8008:
7905:
7806:
7777:
7757:
7707:
7695:{\displaystyle x(t)}
7677:
7654:
7628:
7601:
7581:
7479:
7471:frequency modulation
7467:amplitude modulation
7407:electrical impedance
7053:
6997:
6855:
6830:
6803:
6767:
6740:
6582:
6199:
6176:
5718:
5492:
5446:
5426:
5207:
5107:
5077:
5021:
4934:
4846:
4765:
4628:
4587:
4544:
4475:
4419:
4004:
3875:
3745:
3588:
3558:A key point is that
3508:
3300:
3206:
3048:
2968:
2816:
2790:
2709:
2597:
2567:
2496:
2317:
1692:
1634:
1383:
1350:
1301:
1231:
1177:
1141:
1105:
946:
886:
859:
809:
752:
719:
715:which is the vector
683:
657:
606:
545:
497:
468:
440:
420:
397:
259:
196:algebraic operations
168:time varying current
30:For other uses, see
8926:Electrical circuits
8786:Electrical Networks
8773:Chapter 9, page 338
8390:complex exponential
7643:{\displaystyle t=0}
7427:cube roots of unity
3446:
3428:
3353:
3335:
3317:
2334:
1663:would represent an
877:imaginary component
215:algebraic equations
109:sinusoidal function
8378:
8312:This results from
8236:
8187:
8104:
8021:
7991:
7892:
7790:
7763:
7743:
7692:
7660:
7650:makes an angle of
7640:
7614:
7587:
7564:
7463:
7218:
7036:
6983:
6839:
6816:
6789:
6753:
6724:
6565:
6563:
6188:{\displaystyle i,}
6185:
6142:
6140:
5688:
5473:
5432:
5396:
5199:
5185:
5090:
5063:
5007:
4919:
4842:we may substitute
4830:
4749:
4606:
4566:
4530:
4458:
4403:
4401:
3881:
3858:
3698:
3669:
3546:
3494:
3432:
3414:
3339:
3321:
3303:
3270:
3192:
3032:
2954:
2796:
2776:
2695:
2580:
2553:
2480:
2320:
2303:
2301:
1686:
1653:
1618:
1616:
1369:
1336:
1264:
1211:
1160:
1127:
1087:
922:
865:
842:
785:
738:
705:
669:
628:
588:
528:
480:
455:
426:
403:
357:
296:
246:transient response
213:by solving simple
164:electrical network
81:
8855:. Prentice Hall.
8820:978-81-265-2875-2
8766:978-0-13-198925-2
8756:Electric circuits
8739:978-1-63388-331-4
8714:978-0-470-82240-1
8684:978-1-285-40192-8
8659:978-0-470-52157-1
8623:978-1-60845-668-0
8590:978-1-4398-9807-9
8562:978-1-4832-9492-6
8532:978-3-540-74296-8
8507:978-81-317-1390-7
8469:978-0-08-049398-5
8454:Basic AC Circuits
8436:978-0-08-051119-1
8332:
8128:
8048:
7916:
7817:
7766:{\displaystyle m}
7590:{\displaystyle A}
7444:The technique of
7417:Power engineering
7359:complex conjugate
7173:
7161:
7160:
7096:
7063:
6950:
6949:
6910:
6813:
6777:
6750:
6705:
6693:
6635:
6623:
6592:
6555:
6546:
6523:
6514:
6495:
6451:
6442:
6389:
6380:
6361:
6317:
6308:
6266:
6257:
6219:
6108:
6047:
6031:
5969:
5949:
5901:
5840:
5824:
5762:
5742:
5712:
5711:
5658:
5638:
5595:
5575:
5532:
5512:
5483:it follows that:
5468:
5435:{\displaystyle t}
5420:
5419:
5369:
5349:
5308:
5290:
5247:
5227:
5197:
5179:
5170:
5151:
5142:
5123:
5087:
5044:
5031:
4977:
4944:
4889:
4856:
4797:
4775:
4734:
4725:
4697:
4688:
4670:
4647:
4525:
4491:
4389:
4040:
3845:
3802:
3736:three-phase power
3687:added vectorially
3572:do not depend on
3183:
2945:
2690:
2546:
2528:
868:{\displaystyle t}
406:{\displaystyle A}
391:polar coordinates
389:. A vector whose
242:Laplace transform
173:phasor arithmetic
149:complex amplitude
133:angular frequency
16:(Redirected from
8948:
8885:
8866:
8838:
8831:
8825:
8824:
8806:
8800:
8799:
8781:
8775:
8770:
8750:
8744:
8743:
8725:
8719:
8718:
8700:
8689:
8688:
8670:
8664:
8663:
8647:
8637:
8628:
8627:
8609:
8603:
8602:
8576:
8567:
8566:
8548:
8537:
8536:
8518:
8512:
8511:
8493:
8487:
8482:Bracewell, Ron.
8480:
8474:
8473:
8457:
8447:
8441:
8440:
8424:
8414:
8397:
8387:
8385:
8384:
8379:
8374:
8373:
8349:
8348:
8333:
8331:
8320:
8310:
8304:
8301:
8274:Complex envelope
8245:
8243:
8242:
8237:
8235:
8234:
8219:
8218:
8196:
8194:
8193:
8188:
8183:
8182:
8178:
8177:
8148:
8147:
8129:
8121:
8113:
8111:
8110:
8105:
8100:
8099:
8095:
8094:
8068:
8067:
8049:
8041:
8030:
8028:
8027:
8022:
8020:
8019:
8000:
7998:
7997:
7992:
7987:
7986:
7979:
7978:
7966:
7965:
7936:
7935:
7917:
7909:
7901:
7899:
7898:
7893:
7888:
7887:
7880:
7879:
7867:
7866:
7837:
7836:
7818:
7810:
7799:
7797:
7796:
7791:
7789:
7788:
7772:
7770:
7769:
7764:
7752:
7750:
7749:
7744:
7742:
7738:
7737:
7701:
7699:
7698:
7693:
7669:
7667:
7666:
7661:
7649:
7647:
7646:
7641:
7623:
7621:
7620:
7615:
7613:
7612:
7596:
7594:
7593:
7588:
7573:
7571:
7570:
7565:
7560:
7556:
7555:
7554:
7550:
7549:
7523:
7522:
7411:phase difference
7378:
7372:
7366:
7356:
7350:
7340:
7336:
7315:
7295:
7291:
7276:
7231:Ratio of phasors
7227:
7225:
7224:
7219:
7175:
7174:
7171:
7162:
7159:
7158:
7131:
7127:
7122:
7118:
7117:
7116:
7098:
7097:
7094:
7065:
7064:
7061:
7045:
7043:
7042:
7037:
6992:
6990:
6989:
6984:
6979:
6978:
6951:
6948:
6947:
6920:
6916:
6911:
6909:
6908:
6907:
6879:
6859:
6848:
6846:
6845:
6840:
6825:
6823:
6822:
6817:
6815:
6814:
6811:
6798:
6796:
6795:
6790:
6779:
6778:
6775:
6762:
6760:
6759:
6754:
6752:
6751:
6748:
6733:
6731:
6730:
6725:
6720:
6719:
6707:
6706:
6703:
6694:
6692:
6691:
6690:
6662:
6642:
6637:
6636:
6633:
6624:
6622:
6599:
6594:
6593:
6590:
6574:
6572:
6571:
6566:
6564:
6557:
6556:
6553:
6547:
6545:
6534:
6525:
6524:
6521:
6515:
6513:
6502:
6497:
6496:
6493:
6477:
6476:
6458:
6454:
6453:
6452:
6449:
6443:
6441:
6430:
6416:
6415:
6396:
6392:
6391:
6390:
6387:
6381:
6379:
6368:
6363:
6362:
6359:
6338:
6337:
6319:
6318:
6315:
6309:
6307:
6296:
6287:
6286:
6268:
6267:
6264:
6258:
6256:
6245:
6240:
6239:
6221:
6220:
6217:
6194:
6192:
6191:
6186:
6151:
6149:
6148:
6143:
6141:
6134:
6130:
6129:
6128:
6110:
6109:
6106:
6080:
6076:
6075:
6074:
6073:
6069:
6068:
6067:
6049:
6048:
6045:
6032:
6030:
6026:
6020:
6015:
5995:
5991:
5990:
5989:
5971:
5970:
5967:
5950:
5948:
5944:
5938:
5933:
5927:
5923:
5922:
5921:
5903:
5902:
5899:
5873:
5869:
5868:
5867:
5866:
5862:
5861:
5860:
5842:
5841:
5838:
5825:
5823:
5819:
5813:
5808:
5788:
5784:
5783:
5782:
5764:
5763:
5760:
5743:
5741:
5737:
5731:
5726:
5706:
5697:
5695:
5694:
5689:
5684:
5680:
5679:
5678:
5660:
5659:
5656:
5639:
5637:
5626:
5621:
5617:
5616:
5615:
5597:
5596:
5593:
5576:
5574:
5563:
5558:
5554:
5553:
5552:
5534:
5533:
5530:
5513:
5511:
5507:
5501:
5496:
5486:
5482:
5480:
5479:
5474:
5469:
5467:
5456:
5442:, specifically:
5441:
5439:
5438:
5433:
5414:
5405:
5403:
5402:
5397:
5395:
5391:
5390:
5389:
5371:
5370:
5367:
5350:
5348:
5337:
5329:
5328:
5310:
5309:
5306:
5291:
5289:
5278:
5273:
5269:
5268:
5267:
5249:
5248:
5245:
5228:
5226:
5222:
5216:
5211:
5201:
5194:
5192:
5191:
5186:
5181:
5180:
5177:
5171:
5169:
5158:
5153:
5152:
5149:
5143:
5141:
5130:
5125:
5124:
5121:
5099:
5097:
5096:
5091:
5089:
5088:
5085:
5072:
5070:
5069:
5064:
5059:
5058:
5046:
5045:
5042:
5033:
5032:
5029:
5016:
5014:
5013:
5008:
5003:
4999:
4998:
4997:
4979:
4978:
4975:
4946:
4945:
4942:
4928:
4926:
4925:
4920:
4915:
4911:
4910:
4909:
4891:
4890:
4887:
4858:
4857:
4854:
4839:
4837:
4836:
4831:
4799:
4798:
4795:
4777:
4776:
4773:
4758:
4756:
4755:
4750:
4736:
4735:
4732:
4726:
4724:
4713:
4699:
4698:
4695:
4689:
4687:
4676:
4671:
4669:
4665:
4659:
4649:
4648:
4645:
4638:
4632:
4615:
4613:
4612:
4607:
4605:
4604:
4579:When we solve a
4575:
4573:
4572:
4567:
4562:
4561:
4539:
4537:
4536:
4531:
4526:
4521:
4520:
4516:
4497:
4492:
4490:
4479:
4467:
4465:
4464:
4459:
4451:
4450:
4446:
4412:
4410:
4409:
4404:
4402:
4395:
4391:
4390:
4382:
4343:
4334:
4330:
4329:
4328:
4310:
4309:
4302:
4258:
4249:
4245:
4244:
4243:
4225:
4224:
4220:
4201:
4200:
4172:
4163:
4159:
4158:
4157:
4133:
4132:
4104:
4095:
4091:
4090:
4089:
4088:
4084:
4083:
4082:
4064:
4063:
4041:
4039:
4035:
4029:
4024:
4010:
3975:
3964:
3960:
3959:
3955:
3954:
3947:
3937:
3936:
3932:
3931:
3925:
3912:
3905:
3890:
3888:
3887:
3882:
3867:
3865:
3864:
3859:
3851:
3847:
3846:
3841:
3833:
3808:
3804:
3803:
3798:
3790:
3733:
3732:
3728:
3721:
3720:
3716:
3715:
3678:
3676:
3675:
3670:
3665:
3664:
3652:
3651:
3639:
3638:
3626:
3625:
3613:
3612:
3600:
3599:
3555:
3553:
3552:
3547:
3542:
3541:
3529:
3528:
3503:
3501:
3500:
3495:
3472:
3471:
3462:
3461:
3445:
3440:
3427:
3422:
3398:
3397:
3379:
3378:
3369:
3368:
3352:
3347:
3334:
3329:
3316:
3311:
3279:
3277:
3276:
3271:
3263:
3262:
3247:
3246:
3234:
3233:
3218:
3217:
3201:
3199:
3198:
3193:
3188:
3184:
3182:
3181:
3180:
3165:
3164:
3152:
3151:
3136:
3135:
3125:
3124:
3123:
3108:
3107:
3095:
3094:
3079:
3078:
3068:
3041:
3039:
3038:
3033:
3025:
3024:
3009:
3008:
2996:
2995:
2980:
2979:
2963:
2961:
2960:
2955:
2950:
2946:
2944:
2943:
2942:
2927:
2926:
2914:
2913:
2898:
2897:
2887:
2886:
2885:
2870:
2869:
2857:
2856:
2841:
2840:
2830:
2805:
2803:
2802:
2797:
2785:
2783:
2782:
2777:
2766:
2765:
2750:
2749:
2737:
2736:
2721:
2720:
2704:
2702:
2701:
2696:
2691:
2683:
2672:
2671:
2653:
2652:
2637:
2636:
2618:
2617:
2589:
2587:
2586:
2581:
2579:
2578:
2562:
2560:
2559:
2554:
2552:
2548:
2547:
2542:
2534:
2529:
2521:
2508:
2507:
2492:and, if we take
2489:
2487:
2486:
2481:
2476:
2475:
2466:
2465:
2450:
2449:
2437:
2436:
2421:
2420:
2405:
2404:
2395:
2394:
2379:
2378:
2366:
2365:
2350:
2349:
2333:
2328:
2312:
2310:
2309:
2304:
2302:
2292:
2291:
2264:
2263:
2252:
2243:
2239:
2238:
2237:
2222:
2218:
2217:
2216:
2215:
2214:
2197:
2196:
2169:
2160:
2156:
2155:
2154:
2139:
2135:
2134:
2133:
2132:
2131:
2114:
2113:
2101:
2100:
2099:
2098:
2081:
2080:
2053:
2044:
2040:
2039:
2038:
2023:
2022:
2021:
2020:
2003:
2002:
1990:
1989:
1974:
1973:
1972:
1971:
1954:
1953:
1931:
1922:
1918:
1917:
1916:
1901:
1900:
1899:
1898:
1881:
1880:
1857:
1853:
1852:
1851:
1836:
1835:
1834:
1833:
1816:
1815:
1793:
1781:
1780:
1753:
1752:
1737:
1736:
1709:
1708:
1698:
1662:
1660:
1659:
1654:
1652:
1651:
1630:In electronics,
1627:
1625:
1624:
1619:
1617:
1566:
1557:
1553:
1552:
1551:
1533:
1529:
1528:
1527:
1479:
1470:
1466:
1465:
1464:
1446:
1442:
1441:
1440:
1422:
1421:
1389:
1378:
1376:
1375:
1370:
1368:
1367:
1345:
1343:
1342:
1337:
1335:
1334:
1319:
1318:
1273:
1271:
1270:
1265:
1220:
1218:
1217:
1212:
1210:
1209:
1169:
1167:
1166:
1161:
1159:
1158:
1136:
1134:
1133:
1128:
1123:
1122:
1096:
1094:
1093:
1088:
1083:
1082:
1064:
1063:
1045:
1044:
931:
929:
928:
923:
874:
872:
871:
866:
851:
849:
848:
843:
794:
792:
791:
786:
775:
774:
770:
747:
745:
744:
739:
714:
712:
711:
706:
701:
700:
678:
676:
675:
670:
637:
635:
634:
629:
618:
617:
597:
595:
594:
589:
587:
586:
537:
535:
534:
529:
489:
487:
486:
481:
464:
462:
461:
456:
435:
433:
432:
427:
412:
410:
409:
404:
354:
346:
328:
317:
305:
303:
302:
297:
295:
294:
235:Oliver Heaviside
231:General Electric
138:
126:
118:
66:
21:
8956:
8955:
8951:
8950:
8949:
8947:
8946:
8945:
8916:
8915:
8896:Phasor Phactory
8892:
8882:
8863:
8847:
8845:Further reading
8842:
8841:
8832:
8828:
8821:
8807:
8803:
8796:
8782:
8778:
8767:
8751:
8747:
8740:
8726:
8722:
8715:
8701:
8692:
8685:
8671:
8667:
8660:
8638:
8631:
8624:
8610:
8606:
8591:
8577:
8570:
8563:
8549:
8540:
8533:
8519:
8515:
8508:
8494:
8490:
8481:
8477:
8470:
8448:
8444:
8437:
8415:
8411:
8406:
8401:
8400:
8363:
8359:
8338:
8334:
8324:
8319:
8317:
8314:
8313:
8311:
8307:
8302:
8295:
8290:
8268:Analytic signal
8252:
8230:
8226:
8214:
8210:
8205:
8202:
8201:
8173:
8169:
8156:
8152:
8140:
8136:
8120:
8118:
8115:
8114:
8090:
8086:
8076:
8072:
8060:
8056:
8040:
8038:
8035:
8034:
8015:
8011:
8009:
8006:
8005:
7974:
7970:
7961:
7957:
7944:
7940:
7928:
7924:
7908:
7906:
7903:
7902:
7875:
7871:
7862:
7858:
7845:
7841:
7829:
7825:
7809:
7807:
7804:
7803:
7784:
7780:
7778:
7775:
7774:
7758:
7755:
7754:
7733:
7729:
7722:
7708:
7705:
7704:
7678:
7675:
7674:
7655:
7652:
7651:
7629:
7626:
7625:
7608:
7604:
7602:
7599:
7598:
7582:
7579:
7578:
7545:
7541:
7531:
7527:
7515:
7511:
7507:
7503:
7480:
7477:
7476:
7455:
7421:In analysis of
7419:
7374:
7368:
7362:
7352:
7342:
7338:
7324:
7313:
7296:is the complex
7293:
7283:
7268:
7250:
7245:
7233:
7170:
7166:
7154:
7150:
7126:
7106:
7102:
7093:
7089:
7088:
7084:
7060:
7056:
7054:
7051:
7050:
6998:
6995:
6994:
6959:
6955:
6943:
6939:
6915:
6903:
6899:
6880:
6860:
6858:
6856:
6853:
6852:
6831:
6828:
6827:
6810:
6806:
6804:
6801:
6800:
6774:
6770:
6768:
6765:
6764:
6747:
6743:
6741:
6738:
6737:
6712:
6708:
6702:
6698:
6686:
6682:
6663:
6643:
6641:
6632:
6628:
6603:
6598:
6589:
6585:
6583:
6580:
6579:
6576:
6562:
6561:
6552:
6548:
6538:
6533:
6526:
6520:
6516:
6506:
6501:
6492:
6488:
6479:
6478:
6466:
6462:
6448:
6444:
6434:
6429:
6428:
6424:
6417:
6405:
6401:
6386:
6382:
6372:
6367:
6358:
6354:
6347:
6343:
6340:
6339:
6327:
6323:
6314:
6310:
6300:
6295:
6288:
6276:
6272:
6263:
6259:
6249:
6244:
6229:
6225:
6216:
6212:
6202:
6200:
6197:
6196:
6177:
6174:
6173:
6139:
6138:
6118:
6114:
6105:
6101:
6094:
6090:
6057:
6053:
6044:
6040:
6039:
6035:
6034:
6033:
6022:
6021:
6016:
6014:
6013:
6009:
5996:
5979:
5975:
5966:
5962:
5961:
5957:
5940:
5939:
5934:
5932:
5929:
5928:
5911:
5907:
5898:
5894:
5887:
5883:
5850:
5846:
5837:
5833:
5832:
5828:
5827:
5826:
5815:
5814:
5809:
5807:
5806:
5802:
5789:
5772:
5768:
5759:
5755:
5754:
5750:
5733:
5732:
5727:
5725:
5721:
5719:
5716:
5715:
5668:
5664:
5655:
5651:
5650:
5646:
5630:
5625:
5605:
5601:
5592:
5588:
5587:
5583:
5567:
5562:
5542:
5538:
5529:
5525:
5524:
5520:
5503:
5502:
5497:
5495:
5493:
5490:
5489:
5460:
5455:
5447:
5444:
5443:
5427:
5424:
5423:
5379:
5375:
5366:
5362:
5361:
5357:
5341:
5336:
5318:
5314:
5305:
5301:
5282:
5277:
5257:
5253:
5244:
5240:
5239:
5235:
5218:
5217:
5212:
5210:
5208:
5205:
5204:
5176:
5172:
5162:
5157:
5148:
5144:
5134:
5129:
5120:
5116:
5108:
5105:
5104:
5084:
5080:
5078:
5075:
5074:
5051:
5047:
5041:
5037:
5028:
5024:
5022:
5019:
5018:
4987:
4983:
4974:
4970:
4969:
4965:
4941:
4937:
4935:
4932:
4931:
4899:
4895:
4886:
4882:
4881:
4877:
4853:
4849:
4847:
4844:
4843:
4794:
4790:
4772:
4768:
4766:
4763:
4762:
4731:
4727:
4717:
4712:
4694:
4690:
4680:
4675:
4661:
4660:
4644:
4640:
4634:
4633:
4631:
4629:
4626:
4625:
4594:
4590:
4588:
4585:
4584:
4576:is unaffected.
4551:
4547:
4545:
4542:
4541:
4512:
4502:
4498:
4496:
4483:
4478:
4476:
4473:
4472:
4442:
4435:
4431:
4420:
4417:
4416:
4400:
4399:
4381:
4365:
4361:
4344:
4342:
4336:
4335:
4318:
4314:
4298:
4282:
4278:
4271:
4267:
4259:
4257:
4251:
4250:
4233:
4229:
4216:
4209:
4205:
4193:
4189:
4185:
4181:
4173:
4171:
4165:
4164:
4147:
4143:
4125:
4121:
4117:
4113:
4105:
4103:
4097:
4096:
4072:
4068:
4056:
4052:
4048:
4044:
4043:
4042:
4031:
4030:
4025:
4023:
4022:
4018:
4007:
4005:
4002:
4001:
3990:
3983:
3970:
3962:
3957:
3952:
3950:
3949:
3946:
3939:
3934:
3929:
3928:
3927:
3924:
3917:
3907:
3903:
3876:
3873:
3872:
3834:
3832:
3822:
3818:
3791:
3789:
3779:
3775:
3746:
3743:
3742:
3730:
3727:
3724:
3723:
3718:
3713:
3711:
3710:
3660:
3656:
3647:
3643:
3634:
3630:
3621:
3617:
3608:
3604:
3595:
3591:
3589:
3586:
3585:
3571:
3564:
3537:
3533:
3524:
3520:
3509:
3506:
3505:
3467:
3463:
3457:
3453:
3441:
3436:
3423:
3418:
3393:
3389:
3374:
3370:
3364:
3360:
3348:
3343:
3330:
3325:
3312:
3307:
3301:
3298:
3297:
3258:
3254:
3242:
3238:
3229:
3225:
3213:
3209:
3207:
3204:
3203:
3176:
3172:
3160:
3156:
3147:
3143:
3131:
3127:
3126:
3119:
3115:
3103:
3099:
3090:
3086:
3074:
3070:
3069:
3067:
3063:
3049:
3046:
3045:
3020:
3016:
3004:
3000:
2991:
2987:
2975:
2971:
2969:
2966:
2965:
2938:
2934:
2922:
2918:
2909:
2905:
2893:
2889:
2888:
2881:
2877:
2865:
2861:
2852:
2848:
2836:
2832:
2831:
2829:
2825:
2817:
2814:
2813:
2808:signum function
2791:
2788:
2787:
2761:
2757:
2745:
2741:
2732:
2728:
2716:
2712:
2710:
2707:
2706:
2682:
2667:
2663:
2648:
2644:
2632:
2628:
2613:
2609:
2598:
2595:
2594:
2574:
2570:
2568:
2565:
2564:
2535:
2533:
2520:
2516:
2512:
2503:
2499:
2497:
2494:
2493:
2471:
2467:
2461:
2457:
2445:
2441:
2432:
2428:
2416:
2412:
2400:
2396:
2390:
2386:
2374:
2370:
2361:
2357:
2345:
2341:
2329:
2324:
2318:
2315:
2314:
2300:
2299:
2287:
2283:
2259:
2255:
2253:
2251:
2245:
2244:
2227:
2223:
2210:
2206:
2202:
2198:
2192:
2188:
2187:
2183:
2182:
2178:
2170:
2168:
2162:
2161:
2144:
2140:
2127:
2123:
2119:
2115:
2109:
2105:
2094:
2090:
2086:
2082:
2076:
2072:
2071:
2067:
2066:
2062:
2054:
2052:
2046:
2045:
2028:
2024:
2016:
2012:
2008:
2004:
1998:
1994:
1979:
1975:
1967:
1963:
1959:
1955:
1949:
1945:
1944:
1940:
1932:
1930:
1924:
1923:
1906:
1902:
1894:
1890:
1886:
1882:
1876:
1872:
1871:
1867:
1841:
1837:
1829:
1825:
1821:
1817:
1811:
1807:
1806:
1802:
1794:
1792:
1786:
1785:
1776:
1772:
1748:
1744:
1732:
1728:
1704:
1700:
1695:
1693:
1690:
1689:
1678:
1644:
1640:
1635:
1632:
1631:
1615:
1614:
1567:
1565:
1559:
1558:
1541:
1537:
1508:
1504:
1497:
1493:
1492:
1488:
1480:
1478:
1472:
1471:
1454:
1450:
1433:
1429:
1414:
1410:
1406:
1402:
1401:
1397:
1386:
1384:
1381:
1380:
1360:
1356:
1351:
1348:
1347:
1324:
1320:
1311:
1307:
1302:
1299:
1298:
1295:
1290:
1284:
1232:
1229:
1228:
1187:
1183:
1178:
1175:
1174:
1148:
1144:
1142:
1139:
1138:
1115:
1111:
1106:
1103:
1102:
1072:
1068:
1056:
1052:
1022:
1018:
947:
944:
943:
937:Euler's formula
887:
884:
883:
860:
857:
856:
810:
807:
806:
800:
766:
759:
755:
753:
750:
749:
720:
717:
716:
696:
692:
684:
681:
680:
658:
655:
654:
613:
609:
607:
604:
603:
600:Euler's formula
598:, according to
579:
575:
546:
543:
542:
498:
495:
494:
469:
466:
465:
441:
438:
437:
421:
418:
417:
398:
395:
394:
373:(also known as
371:Phasor notation
368:
366:Vector notation
362:
352:
330:
323:
315:
272:
268:
260:
257:
256:
188:differentiation
136:
124:
116:
107:representing a
64:
63:for a specific
59:and respective
49:
42:
35:
28:
23:
22:
15:
12:
11:
5:
8954:
8944:
8943:
8938:
8933:
8928:
8914:
8913:
8908:
8903:
8898:
8891:
8890:External links
8888:
8887:
8886:
8880:
8867:
8861:
8846:
8843:
8840:
8839:
8826:
8819:
8801:
8795:978-0070260962
8794:
8776:
8765:
8745:
8738:
8720:
8713:
8690:
8683:
8665:
8658:
8629:
8622:
8604:
8589:
8568:
8561:
8538:
8531:
8513:
8506:
8488:
8475:
8468:
8442:
8435:
8408:
8407:
8405:
8402:
8399:
8398:
8377:
8372:
8369:
8366:
8362:
8358:
8355:
8352:
8347:
8344:
8341:
8337:
8330:
8327:
8323:
8305:
8292:
8291:
8289:
8286:
8285:
8284:
8278:
8277:
8276:
8265:
8264:
8263:
8251:
8248:
8233:
8229:
8225:
8222:
8217:
8213:
8209:
8186:
8181:
8176:
8172:
8168:
8165:
8162:
8159:
8155:
8151:
8146:
8143:
8139:
8135:
8132:
8127:
8124:
8103:
8098:
8093:
8089:
8085:
8082:
8079:
8075:
8071:
8066:
8063:
8059:
8055:
8052:
8047:
8044:
8018:
8014:
7990:
7985:
7982:
7977:
7973:
7969:
7964:
7960:
7956:
7953:
7950:
7947:
7943:
7939:
7934:
7931:
7927:
7923:
7920:
7915:
7912:
7891:
7886:
7883:
7878:
7874:
7870:
7865:
7861:
7857:
7854:
7851:
7848:
7844:
7840:
7835:
7832:
7828:
7824:
7821:
7816:
7813:
7787:
7783:
7762:
7741:
7736:
7732:
7728:
7725:
7721:
7718:
7715:
7712:
7691:
7688:
7685:
7682:
7659:
7639:
7636:
7633:
7611:
7607:
7586:
7563:
7559:
7553:
7548:
7544:
7540:
7537:
7534:
7530:
7526:
7521:
7518:
7514:
7510:
7506:
7502:
7499:
7496:
7493:
7490:
7487:
7484:
7454:
7451:
7446:synchrophasors
7418:
7415:
7396:Fourier series
7310:
7309:
7306:
7301:
7281:
7278:
7277:remains valid.
7261:
7249:
7246:
7244:
7241:
7232:
7229:
7217:
7214:
7211:
7208:
7205:
7202:
7199:
7196:
7193:
7190:
7187:
7184:
7181:
7178:
7169:
7165:
7157:
7153:
7149:
7146:
7143:
7140:
7137:
7134:
7130:
7125:
7121:
7115:
7112:
7109:
7105:
7101:
7092:
7087:
7083:
7080:
7077:
7074:
7071:
7068:
7059:
7035:
7032:
7029:
7026:
7023:
7020:
7017:
7014:
7011:
7008:
7005:
7002:
6982:
6977:
6974:
6971:
6968:
6965:
6962:
6958:
6954:
6946:
6942:
6938:
6935:
6932:
6929:
6926:
6923:
6919:
6914:
6906:
6902:
6898:
6895:
6892:
6889:
6886:
6883:
6878:
6875:
6872:
6869:
6866:
6863:
6838:
6835:
6809:
6788:
6785:
6782:
6773:
6746:
6723:
6718:
6715:
6711:
6701:
6697:
6689:
6685:
6681:
6678:
6675:
6672:
6669:
6666:
6661:
6658:
6655:
6652:
6649:
6646:
6640:
6631:
6627:
6621:
6618:
6615:
6612:
6609:
6606:
6602:
6597:
6588:
6560:
6551:
6544:
6541:
6537:
6532:
6529:
6527:
6519:
6512:
6509:
6505:
6500:
6491:
6487:
6484:
6481:
6480:
6475:
6472:
6469:
6465:
6461:
6457:
6447:
6440:
6437:
6433:
6427:
6423:
6420:
6418:
6414:
6411:
6408:
6404:
6400:
6395:
6385:
6378:
6375:
6371:
6366:
6357:
6353:
6350:
6346:
6342:
6341:
6336:
6333:
6330:
6326:
6322:
6313:
6306:
6303:
6299:
6294:
6291:
6289:
6285:
6282:
6279:
6275:
6271:
6262:
6255:
6252:
6248:
6243:
6238:
6235:
6232:
6228:
6224:
6215:
6211:
6208:
6205:
6204:
6184:
6181:
6166:, multiplying
6137:
6133:
6127:
6124:
6121:
6117:
6113:
6104:
6100:
6097:
6093:
6089:
6086:
6083:
6079:
6072:
6066:
6063:
6060:
6056:
6052:
6043:
6038:
6029:
6025:
6019:
6012:
6008:
6005:
6002:
5999:
5997:
5994:
5988:
5985:
5982:
5978:
5974:
5965:
5960:
5956:
5953:
5947:
5943:
5937:
5931:
5930:
5926:
5920:
5917:
5914:
5910:
5906:
5897:
5893:
5890:
5886:
5882:
5879:
5876:
5872:
5865:
5859:
5856:
5853:
5849:
5845:
5836:
5831:
5822:
5818:
5812:
5805:
5801:
5798:
5795:
5792:
5790:
5787:
5781:
5778:
5775:
5771:
5767:
5758:
5753:
5749:
5746:
5740:
5736:
5730:
5724:
5723:
5710:
5709:
5700:
5698:
5687:
5683:
5677:
5674:
5671:
5667:
5663:
5654:
5649:
5645:
5642:
5636:
5633:
5629:
5624:
5620:
5614:
5611:
5608:
5604:
5600:
5591:
5586:
5582:
5579:
5573:
5570:
5566:
5561:
5557:
5551:
5548:
5545:
5541:
5537:
5528:
5523:
5519:
5516:
5510:
5506:
5500:
5472:
5466:
5463:
5459:
5454:
5451:
5431:
5418:
5417:
5408:
5406:
5394:
5388:
5385:
5382:
5378:
5374:
5365:
5360:
5356:
5353:
5347:
5344:
5340:
5335:
5332:
5327:
5324:
5321:
5317:
5313:
5304:
5300:
5297:
5294:
5288:
5285:
5281:
5276:
5272:
5266:
5263:
5260:
5256:
5252:
5243:
5238:
5234:
5231:
5225:
5221:
5215:
5196:
5184:
5175:
5168:
5165:
5161:
5156:
5147:
5140:
5137:
5133:
5128:
5119:
5115:
5112:
5083:
5062:
5057:
5054:
5050:
5040:
5036:
5027:
5006:
5002:
4996:
4993:
4990:
4986:
4982:
4973:
4968:
4964:
4961:
4958:
4955:
4952:
4949:
4940:
4918:
4914:
4908:
4905:
4902:
4898:
4894:
4885:
4880:
4876:
4873:
4870:
4867:
4864:
4861:
4852:
4829:
4826:
4823:
4820:
4817:
4814:
4811:
4808:
4805:
4802:
4793:
4789:
4786:
4783:
4780:
4771:
4748:
4745:
4742:
4739:
4730:
4723:
4720:
4716:
4711:
4708:
4705:
4702:
4693:
4686:
4683:
4679:
4674:
4668:
4664:
4658:
4655:
4652:
4643:
4637:
4603:
4600:
4597:
4593:
4565:
4560:
4557:
4554:
4550:
4529:
4524:
4519:
4515:
4511:
4508:
4505:
4501:
4495:
4489:
4486:
4482:
4457:
4454:
4449:
4445:
4441:
4438:
4434:
4430:
4427:
4424:
4398:
4394:
4388:
4385:
4380:
4377:
4374:
4371:
4368:
4364:
4360:
4357:
4354:
4351:
4348:
4345:
4341:
4338:
4337:
4333:
4327:
4324:
4321:
4317:
4313:
4308:
4305:
4301:
4297:
4294:
4291:
4288:
4285:
4281:
4277:
4274:
4270:
4266:
4263:
4260:
4256:
4253:
4252:
4248:
4242:
4239:
4236:
4232:
4228:
4223:
4219:
4215:
4212:
4208:
4204:
4199:
4196:
4192:
4188:
4184:
4180:
4177:
4174:
4170:
4167:
4166:
4162:
4156:
4153:
4150:
4146:
4142:
4139:
4136:
4131:
4128:
4124:
4120:
4116:
4112:
4109:
4106:
4102:
4099:
4098:
4094:
4087:
4081:
4078:
4075:
4071:
4067:
4062:
4059:
4055:
4051:
4047:
4038:
4034:
4028:
4021:
4017:
4014:
4011:
4009:
3989:
3986:
3981:
3944:
3922:
3880:
3857:
3854:
3850:
3844:
3840:
3837:
3831:
3828:
3825:
3821:
3817:
3814:
3811:
3807:
3801:
3797:
3794:
3788:
3785:
3782:
3778:
3774:
3771:
3768:
3765:
3762:
3759:
3756:
3753:
3750:
3725:
3668:
3663:
3659:
3655:
3650:
3646:
3642:
3637:
3633:
3629:
3624:
3620:
3616:
3611:
3607:
3603:
3598:
3594:
3582:angle notation
3569:
3562:
3545:
3540:
3536:
3532:
3527:
3523:
3519:
3516:
3513:
3493:
3490:
3487:
3484:
3481:
3478:
3475:
3470:
3466:
3460:
3456:
3452:
3449:
3444:
3439:
3435:
3431:
3426:
3421:
3417:
3413:
3410:
3407:
3404:
3401:
3396:
3392:
3388:
3385:
3382:
3377:
3373:
3367:
3363:
3359:
3356:
3351:
3346:
3342:
3338:
3333:
3328:
3324:
3320:
3315:
3310:
3306:
3286:law of cosines
3282:
3281:
3269:
3266:
3261:
3257:
3253:
3250:
3245:
3241:
3237:
3232:
3228:
3224:
3221:
3216:
3212:
3191:
3187:
3179:
3175:
3171:
3168:
3163:
3159:
3155:
3150:
3146:
3142:
3139:
3134:
3130:
3122:
3118:
3114:
3111:
3106:
3102:
3098:
3093:
3089:
3085:
3082:
3077:
3073:
3066:
3062:
3059:
3056:
3053:
3043:
3031:
3028:
3023:
3019:
3015:
3012:
3007:
3003:
2999:
2994:
2990:
2986:
2983:
2978:
2974:
2953:
2949:
2941:
2937:
2933:
2930:
2925:
2921:
2917:
2912:
2908:
2904:
2901:
2896:
2892:
2884:
2880:
2876:
2873:
2868:
2864:
2860:
2855:
2851:
2847:
2844:
2839:
2835:
2828:
2824:
2821:
2811:
2795:
2775:
2772:
2769:
2764:
2760:
2756:
2753:
2748:
2744:
2740:
2735:
2731:
2727:
2724:
2719:
2715:
2694:
2689:
2686:
2681:
2678:
2675:
2670:
2666:
2662:
2659:
2656:
2651:
2647:
2643:
2640:
2635:
2631:
2627:
2624:
2621:
2616:
2612:
2608:
2605:
2602:
2577:
2573:
2551:
2545:
2541:
2538:
2532:
2527:
2524:
2519:
2515:
2511:
2506:
2502:
2479:
2474:
2470:
2464:
2460:
2456:
2453:
2448:
2444:
2440:
2435:
2431:
2427:
2424:
2419:
2415:
2411:
2408:
2403:
2399:
2393:
2389:
2385:
2382:
2377:
2373:
2369:
2364:
2360:
2356:
2353:
2348:
2344:
2340:
2337:
2332:
2327:
2323:
2298:
2295:
2290:
2286:
2282:
2279:
2276:
2273:
2270:
2267:
2262:
2258:
2254:
2250:
2247:
2246:
2242:
2236:
2233:
2230:
2226:
2221:
2213:
2209:
2205:
2201:
2195:
2191:
2186:
2181:
2177:
2174:
2171:
2167:
2164:
2163:
2159:
2153:
2150:
2147:
2143:
2138:
2130:
2126:
2122:
2118:
2112:
2108:
2104:
2097:
2093:
2089:
2085:
2079:
2075:
2070:
2065:
2061:
2058:
2055:
2051:
2048:
2047:
2043:
2037:
2034:
2031:
2027:
2019:
2015:
2011:
2007:
2001:
1997:
1993:
1988:
1985:
1982:
1978:
1970:
1966:
1962:
1958:
1952:
1948:
1943:
1939:
1936:
1933:
1929:
1926:
1925:
1921:
1915:
1912:
1909:
1905:
1897:
1893:
1889:
1885:
1879:
1875:
1870:
1866:
1863:
1860:
1856:
1850:
1847:
1844:
1840:
1832:
1828:
1824:
1820:
1814:
1810:
1805:
1801:
1798:
1795:
1791:
1788:
1787:
1784:
1779:
1775:
1771:
1768:
1765:
1762:
1759:
1756:
1751:
1747:
1743:
1740:
1735:
1731:
1727:
1724:
1721:
1718:
1715:
1712:
1707:
1703:
1699:
1697:
1677:
1674:
1650:
1647:
1643:
1639:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1571:
1568:
1564:
1561:
1560:
1556:
1550:
1547:
1544:
1540:
1536:
1532:
1526:
1523:
1520:
1517:
1514:
1511:
1507:
1503:
1500:
1496:
1491:
1487:
1484:
1481:
1477:
1474:
1473:
1469:
1463:
1460:
1457:
1453:
1449:
1445:
1439:
1436:
1432:
1428:
1425:
1420:
1417:
1413:
1409:
1405:
1400:
1396:
1393:
1390:
1388:
1366:
1363:
1359:
1355:
1333:
1330:
1327:
1323:
1317:
1314:
1310:
1306:
1294:
1291:
1283:
1280:
1263:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1186:
1182:
1157:
1154:
1151:
1147:
1126:
1121:
1118:
1114:
1110:
1098:
1097:
1086:
1081:
1078:
1075:
1071:
1067:
1062:
1059:
1055:
1051:
1048:
1043:
1040:
1037:
1034:
1031:
1028:
1025:
1021:
1017:
1014:
1011:
1008:
1005:
1002:
999:
996:
993:
990:
987:
984:
981:
978:
975:
972:
969:
966:
963:
960:
957:
954:
951:
933:
932:
921:
918:
915:
912:
909:
906:
903:
900:
897:
894:
891:
864:
853:
852:
841:
838:
835:
832:
829:
826:
823:
820:
817:
814:
799:
796:
784:
781:
778:
773:
769:
765:
762:
758:
748:or the number
737:
734:
730:
727:
724:
704:
699:
695:
691:
688:
668:
665:
662:
653:. For example
627:
624:
621:
616:
612:
585:
582:
578:
574:
571:
568:
565:
562:
559:
556:
553:
550:
540:complex number
527:
524:
521:
518:
514:
511:
508:
505:
502:
479:
476:
473:
454:
451:
448:
445:
425:
402:
393:are magnitude
375:angle notation
361:
358:
293:
290:
287:
284:
281:
278:
275:
271:
267:
264:
177:phasor algebra
129:time-invariant
105:complex number
61:phasor diagram
26:
9:
6:
4:
3:
2:
8953:
8942:
8939:
8937:
8934:
8932:
8929:
8927:
8924:
8923:
8921:
8912:
8909:
8907:
8904:
8902:
8899:
8897:
8894:
8893:
8883:
8877:
8873:
8868:
8864:
8862:0-13-666322-2
8858:
8854:
8849:
8848:
8836:
8830:
8822:
8816:
8812:
8805:
8797:
8791:
8787:
8780:
8774:
8768:
8762:
8758:
8757:
8749:
8741:
8735:
8731:
8724:
8716:
8710:
8706:
8699:
8697:
8695:
8686:
8680:
8676:
8669:
8661:
8655:
8651:
8646:
8645:
8636:
8634:
8625:
8619:
8615:
8608:
8600:
8596:
8592:
8586:
8582:
8575:
8573:
8564:
8558:
8554:
8547:
8545:
8543:
8534:
8528:
8524:
8517:
8509:
8503:
8499:
8492:
8485:
8479:
8471:
8465:
8461:
8456:
8455:
8446:
8438:
8432:
8428:
8423:
8422:
8413:
8409:
8395:
8394:eigenfunction
8391:
8375:
8370:
8367:
8364:
8360:
8356:
8353:
8350:
8345:
8342:
8339:
8335:
8328:
8325:
8321:
8309:
8300:
8298:
8293:
8282:
8279:
8275:
8272:
8271:
8269:
8266:
8262:
8259:
8258:
8257:
8254:
8253:
8247:
8231:
8227:
8223:
8220:
8215:
8211:
8207:
8197:
8184:
8179:
8174:
8170:
8166:
8163:
8160:
8157:
8153:
8149:
8144:
8141:
8137:
8133:
8130:
8125:
8122:
8101:
8096:
8091:
8087:
8083:
8080:
8077:
8073:
8069:
8064:
8061:
8057:
8053:
8050:
8045:
8042:
8032:
8016:
8012:
8001:
7988:
7983:
7975:
7971:
7967:
7962:
7958:
7951:
7948:
7945:
7941:
7937:
7932:
7929:
7925:
7921:
7918:
7913:
7910:
7889:
7884:
7876:
7872:
7868:
7863:
7859:
7852:
7849:
7846:
7842:
7838:
7833:
7830:
7826:
7822:
7819:
7814:
7811:
7801:
7785:
7781:
7760:
7739:
7734:
7730:
7726:
7723:
7719:
7716:
7713:
7710:
7686:
7680:
7673:The waveform
7671:
7657:
7637:
7634:
7631:
7609:
7605:
7584:
7575:
7561:
7557:
7551:
7546:
7542:
7538:
7535:
7532:
7528:
7524:
7519:
7516:
7512:
7508:
7504:
7500:
7497:
7494:
7488:
7482:
7474:
7472:
7468:
7459:
7450:
7447:
7442:
7440:
7436:
7432:
7428:
7424:
7414:
7412:
7408:
7403:
7401:
7397:
7393:
7389:
7384:
7382:
7377:
7371:
7365:
7360:
7355:
7349:
7345:
7335:
7331:
7327:
7323:
7322:complex power
7319:
7307:
7305:
7302:
7299:
7290:
7286:
7282:
7279:
7275:
7271:
7266:
7262:
7259:
7258:
7257:
7255:
7240:
7238:
7228:
7215:
7206:
7200:
7197:
7194:
7191:
7188:
7185:
7179:
7176:
7167:
7163:
7155:
7147:
7144:
7141:
7135:
7132:
7128:
7123:
7119:
7113:
7110:
7107:
7103:
7099:
7090:
7085:
7081:
7078:
7075:
7069:
7057:
7047:
7030:
7027:
7024:
7018:
7015:
7012:
7006:
7000:
6980:
6972:
6966:
6963:
6960:
6956:
6952:
6944:
6936:
6933:
6930:
6924:
6921:
6917:
6912:
6904:
6896:
6893:
6890:
6884:
6881:
6876:
6873:
6870:
6867:
6864:
6861:
6849:
6836:
6833:
6807:
6783:
6771:
6744:
6734:
6721:
6716:
6713:
6709:
6699:
6695:
6687:
6679:
6676:
6673:
6667:
6664:
6659:
6656:
6653:
6650:
6647:
6644:
6638:
6629:
6625:
6619:
6616:
6613:
6610:
6607:
6604:
6600:
6595:
6586:
6575:
6558:
6549:
6542:
6539:
6535:
6530:
6528:
6517:
6510:
6507:
6503:
6498:
6489:
6485:
6482:
6473:
6470:
6467:
6463:
6459:
6455:
6445:
6438:
6435:
6431:
6425:
6421:
6419:
6412:
6409:
6406:
6402:
6398:
6393:
6383:
6376:
6373:
6369:
6364:
6355:
6351:
6348:
6344:
6334:
6331:
6328:
6324:
6320:
6311:
6304:
6301:
6297:
6292:
6290:
6283:
6280:
6277:
6273:
6269:
6260:
6253:
6250:
6246:
6241:
6236:
6233:
6230:
6226:
6222:
6213:
6209:
6206:
6182:
6179:
6171:
6170:
6165:
6164:
6159:
6158:
6152:
6135:
6131:
6125:
6122:
6119:
6115:
6111:
6102:
6098:
6095:
6091:
6087:
6084:
6081:
6077:
6070:
6064:
6061:
6058:
6054:
6050:
6041:
6036:
6027:
6010:
6006:
6003:
6000:
5998:
5992:
5986:
5983:
5980:
5976:
5972:
5963:
5958:
5954:
5951:
5945:
5924:
5918:
5915:
5912:
5908:
5904:
5895:
5891:
5888:
5884:
5880:
5877:
5874:
5870:
5863:
5857:
5854:
5851:
5847:
5843:
5834:
5829:
5820:
5803:
5799:
5796:
5793:
5791:
5785:
5779:
5776:
5773:
5769:
5765:
5756:
5751:
5747:
5744:
5738:
5708:
5701:
5699:
5685:
5681:
5675:
5672:
5669:
5665:
5661:
5652:
5647:
5643:
5640:
5634:
5631:
5627:
5622:
5618:
5612:
5609:
5606:
5602:
5598:
5589:
5584:
5580:
5577:
5571:
5568:
5564:
5559:
5555:
5549:
5546:
5543:
5539:
5535:
5526:
5521:
5517:
5514:
5508:
5488:
5487:
5484:
5470:
5464:
5461:
5457:
5452:
5449:
5429:
5416:
5409:
5407:
5392:
5386:
5383:
5380:
5376:
5372:
5363:
5358:
5354:
5351:
5345:
5342:
5338:
5333:
5325:
5322:
5319:
5315:
5311:
5302:
5295:
5292:
5286:
5283:
5279:
5274:
5270:
5264:
5261:
5258:
5254:
5250:
5241:
5236:
5232:
5229:
5223:
5203:
5202:
5195:
5182:
5173:
5166:
5163:
5159:
5154:
5145:
5138:
5135:
5131:
5126:
5117:
5113:
5110:
5101:
5081:
5060:
5055:
5052:
5048:
5038:
5034:
5025:
5017:where phasor
5004:
5000:
4994:
4991:
4988:
4984:
4980:
4971:
4966:
4962:
4959:
4956:
4950:
4938:
4929:
4916:
4912:
4906:
4903:
4900:
4896:
4892:
4883:
4878:
4874:
4871:
4868:
4862:
4850:
4840:
4827:
4821:
4818:
4815:
4812:
4806:
4803:
4800:
4791:
4787:
4781:
4769:
4759:
4746:
4740:
4728:
4721:
4718:
4714:
4709:
4703:
4691:
4684:
4681:
4677:
4672:
4666:
4653:
4641:
4623:
4619:
4601:
4598:
4595:
4591:
4582:
4577:
4563:
4558:
4555:
4552:
4548:
4527:
4522:
4517:
4513:
4509:
4506:
4503:
4499:
4493:
4487:
4484:
4480:
4469:
4455:
4452:
4447:
4443:
4439:
4436:
4432:
4428:
4425:
4422:
4413:
4396:
4392:
4386:
4383:
4378:
4375:
4372:
4369:
4366:
4362:
4358:
4355:
4352:
4349:
4346:
4339:
4331:
4325:
4322:
4319:
4315:
4311:
4303:
4299:
4295:
4292:
4289:
4283:
4279:
4275:
4272:
4268:
4264:
4261:
4254:
4246:
4240:
4237:
4234:
4230:
4226:
4221:
4217:
4213:
4210:
4206:
4202:
4197:
4194:
4190:
4186:
4182:
4178:
4175:
4168:
4160:
4154:
4151:
4148:
4144:
4140:
4137:
4134:
4129:
4126:
4122:
4118:
4114:
4110:
4107:
4100:
4092:
4085:
4079:
4076:
4073:
4069:
4065:
4060:
4057:
4053:
4049:
4045:
4036:
4019:
4015:
4012:
3999:
3995:
3985:
3977:
3973:
3966:
3948:) at 270° or
3943:
3921:
3914:
3910:
3900:
3898:
3894:
3878:
3868:
3855:
3852:
3848:
3842:
3838:
3835:
3829:
3826:
3823:
3819:
3815:
3812:
3809:
3805:
3799:
3795:
3792:
3786:
3783:
3780:
3776:
3772:
3769:
3766:
3760:
3757:
3751:
3748:
3739:
3737:
3708:
3703:
3694:
3690:
3688:
3684:
3679:
3666:
3661:
3657:
3648:
3644:
3640:
3635:
3631:
3622:
3618:
3614:
3609:
3605:
3596:
3592:
3583:
3579:
3575:
3568:
3561:
3556:
3543:
3538:
3534:
3530:
3525:
3521:
3517:
3514:
3491:
3485:
3476:
3473:
3468:
3464:
3458:
3454:
3450:
3447:
3442:
3437:
3433:
3429:
3424:
3419:
3415:
3411:
3405:
3399:
3394:
3390:
3383:
3380:
3375:
3371:
3365:
3361:
3357:
3354:
3349:
3344:
3340:
3336:
3331:
3326:
3322:
3318:
3313:
3308:
3304:
3295:
3291:
3290:complex plane
3287:
3267:
3264:
3259:
3255:
3251:
3248:
3243:
3239:
3235:
3230:
3226:
3222:
3219:
3214:
3210:
3189:
3185:
3177:
3173:
3169:
3166:
3161:
3157:
3153:
3148:
3144:
3140:
3137:
3132:
3128:
3120:
3116:
3112:
3109:
3104:
3100:
3096:
3091:
3087:
3083:
3080:
3075:
3071:
3064:
3060:
3057:
3054:
3051:
3044:
3029:
3026:
3021:
3017:
3013:
3010:
3005:
3001:
2997:
2992:
2988:
2984:
2981:
2976:
2972:
2951:
2947:
2939:
2935:
2931:
2928:
2923:
2919:
2915:
2910:
2906:
2902:
2899:
2894:
2890:
2882:
2878:
2874:
2871:
2866:
2862:
2858:
2853:
2849:
2845:
2842:
2837:
2833:
2826:
2822:
2819:
2812:
2809:
2793:
2773:
2770:
2767:
2762:
2758:
2754:
2751:
2746:
2742:
2738:
2733:
2729:
2725:
2722:
2717:
2713:
2692:
2687:
2684:
2679:
2668:
2664:
2657:
2654:
2649:
2645:
2641:
2633:
2629:
2622:
2619:
2614:
2610:
2603:
2600:
2593:
2592:
2591:
2575:
2571:
2549:
2543:
2539:
2536:
2530:
2525:
2522:
2517:
2513:
2509:
2504:
2500:
2490:
2477:
2472:
2462:
2458:
2454:
2451:
2446:
2442:
2438:
2433:
2429:
2425:
2422:
2417:
2413:
2406:
2401:
2391:
2387:
2383:
2380:
2375:
2371:
2367:
2362:
2358:
2354:
2351:
2346:
2342:
2335:
2330:
2325:
2321:
2296:
2288:
2284:
2280:
2277:
2274:
2268:
2265:
2260:
2256:
2248:
2240:
2234:
2231:
2228:
2224:
2219:
2211:
2207:
2203:
2199:
2193:
2189:
2184:
2179:
2175:
2172:
2165:
2157:
2151:
2148:
2145:
2141:
2136:
2128:
2124:
2120:
2116:
2110:
2106:
2102:
2095:
2091:
2087:
2083:
2077:
2073:
2068:
2063:
2059:
2056:
2049:
2041:
2035:
2032:
2029:
2025:
2017:
2013:
2009:
2005:
1999:
1995:
1991:
1986:
1983:
1980:
1976:
1968:
1964:
1960:
1956:
1950:
1946:
1941:
1937:
1934:
1927:
1919:
1913:
1910:
1907:
1903:
1895:
1891:
1887:
1883:
1877:
1873:
1868:
1864:
1861:
1858:
1854:
1848:
1845:
1842:
1838:
1830:
1826:
1822:
1818:
1812:
1808:
1803:
1799:
1796:
1789:
1777:
1773:
1769:
1766:
1763:
1757:
1754:
1749:
1745:
1741:
1733:
1729:
1725:
1722:
1719:
1713:
1710:
1705:
1701:
1682:
1673:
1670:
1666:
1648:
1645:
1641:
1637:
1628:
1611:
1602:
1599:
1596:
1590:
1587:
1584:
1578:
1575:
1572:
1569:
1562:
1554:
1548:
1545:
1542:
1538:
1534:
1530:
1521:
1518:
1515:
1509:
1505:
1501:
1498:
1494:
1489:
1485:
1482:
1475:
1467:
1461:
1458:
1455:
1451:
1447:
1443:
1437:
1434:
1430:
1426:
1423:
1418:
1415:
1411:
1407:
1403:
1398:
1394:
1391:
1364:
1361:
1357:
1353:
1331:
1328:
1325:
1321:
1315:
1312:
1308:
1304:
1289:
1279:
1277:
1261:
1255:
1252:
1249:
1246:
1240:
1237:
1234:
1226:
1225:
1203:
1200:
1197:
1194:
1188:
1184:
1180:
1173:The function
1171:
1155:
1152:
1149:
1145:
1124:
1119:
1116:
1112:
1108:
1084:
1079:
1076:
1073:
1069:
1065:
1060:
1057:
1053:
1049:
1046:
1038:
1035:
1032:
1029:
1023:
1019:
1015:
1012:
1006:
1003:
1000:
997:
991:
988:
985:
982:
979:
976:
970:
967:
964:
961:
955:
952:
949:
942:
941:
940:
938:
916:
913:
910:
907:
901:
898:
895:
892:
889:
882:
881:
880:
878:
862:
839:
833:
830:
827:
824:
818:
815:
812:
805:
804:
803:
795:
782:
779:
776:
771:
767:
763:
760:
756:
732:
728:
725:
702:
697:
693:
686:
666:
660:
652:
648:
643:
641:
625:
622:
619:
614:
610:
601:
583:
580:
576:
572:
569:
566:
563:
560:
557:
554:
551:
548:
541:
522:
519:
516:
512:
509:
506:
503:
493:
477:
471:
452:
449:
443:
423:
416:
400:
392:
388:
384:
380:
376:
372:
367:
350:
345:
341:
337:
333:
326:
321:
313:
309:
288:
285:
282:
279:
273:
269:
265:
262:
253:
249:
247:
243:
238:
236:
232:
228:
224:
220:
216:
212:
208:
205:
201:
197:
193:
189:
185:
180:
178:
174:
169:
165:
160:
158:
154:
150:
146:
142:
134:
130:
122:
121:initial phase
114:
110:
106:
102:
98:
94:
90:
86:
78:
74:
70:
69:complex plane
62:
58:
53:
47:
40:
33:
19:
8941:Trigonometry
8936:Interference
8871:
8852:
8834:
8829:
8810:
8804:
8785:
8779:
8755:
8748:
8729:
8723:
8704:
8674:
8668:
8643:
8613:
8607:
8580:
8552:
8522:
8516:
8497:
8491:
8483:
8478:
8453:
8445:
8420:
8412:
8308:
8281:Phase factor
8198:
8033:
8002:
7802:
7672:
7576:
7475:
7464:
7443:
7420:
7404:
7385:
7375:
7369:
7363:
7353:
7347:
7343:
7333:
7329:
7325:
7317:
7311:
7288:
7284:
7273:
7269:
7251:
7248:Circuit laws
7243:Applications
7234:
7048:
6850:
6799:relative to
6735:
6577:
6167:
6161:
6155:
6153:
5713:
5702:
5421:
5410:
5102:
4930:
4841:
4760:
4578:
4470:
4414:
3991:
3978:
3971:
3967:
3941:
3926:) at 90° or
3919:
3915:
3908:
3901:
3869:
3740:
3699:
3682:
3680:
3577:
3573:
3566:
3559:
3557:
3284:or, via the
3283:
2491:
1687:
1668:
1629:
1296:
1275:
1222:
1172:
1099:
934:
854:
801:
644:
374:
370:
369:
343:
339:
335:
331:
324:
319:
311:
239:
211:RLC circuits
207:steady state
181:
176:
172:
161:
156:
152:
148:
144:
101:phase vector
100:
92:
82:
60:
8200:phasors at
7423:three phase
7049:Therefore:
5073:and phasor
3893:diffraction
436:is written
229:working at
192:integration
166:powered by
97:portmanteau
89:engineering
57:RLC circuit
8920:Categories
8881:0849344735
8404:References
5198:Derivation
4622:RC circuit
3994:derivative
1286:See also:
1282:Arithmetic
798:Definition
640:magnitudes
364:See also:
351:values of
131:and whose
8599:863646311
8368:ω
8357:ω
8343:ω
8288:Footnotes
8167:π
8158:−
8150:⋅
8145:θ
8084:π
8070:⋅
8065:θ
7968:−
7952:π
7938:⋅
7933:θ
7853:π
7839:⋅
7834:θ
7727:π
7720:
7658:θ
7539:π
7525:⋅
7520:θ
7501:
7392:inductors
7298:impedance
7237:impedance
7207:ω
7201:ϕ
7198:−
7195:θ
7186:ω
7180:
7164:⋅
7142:ω
7111:ω
7100:⋅
7082:
7025:ω
7019:
7007:ω
7001:ϕ
6973:ω
6967:ϕ
6961:−
6953:⋅
6931:ω
6891:ω
6871:ω
6865:−
6834:θ
6717:θ
6696:⋅
6674:ω
6654:ω
6648:−
6626:⋅
6614:ω
6486:ω
6471:ω
6460:⋅
6410:ω
6399:⋅
6352:ω
6332:ω
6321:⋅
6281:ω
6270:⋅
6234:ω
6223:⋅
6210:ω
6123:ω
6112:⋅
6099:ω
6088:
6062:ω
6051:⋅
6007:
5984:ω
5973:⋅
5955:
5916:ω
5905:⋅
5892:ω
5881:
5855:ω
5844:⋅
5800:
5777:ω
5766:⋅
5748:
5673:ω
5662:⋅
5644:
5610:ω
5599:⋅
5581:
5547:ω
5536:⋅
5518:
5465:ω
5458:π
5453:−
5384:ω
5373:⋅
5355:
5323:ω
5312:⋅
5296:
5262:ω
5251:⋅
5233:
5114:ω
5056:θ
4992:ω
4981:⋅
4963:
4904:ω
4893:⋅
4875:
4822:θ
4813:ω
4807:
4801:⋅
4618:capacitor
4599:ω
4556:ω
4523:ω
4510:π
4504:−
4488:ω
4456:ω
4453:⋅
4440:π
4426:ω
4384:π
4376:θ
4367:ω
4359:
4353:⋅
4347:ω
4323:ω
4312:⋅
4296:π
4290:θ
4273:ω
4265:
4238:ω
4227:ω
4214:π
4203:⋅
4198:θ
4179:
4152:ω
4141:ω
4135:⋅
4130:θ
4111:
4077:ω
4066:⋅
4061:θ
4016:
3992:The time
3879:λ
3839:π
3830:−
3824:ω
3816:
3796:π
3781:ω
3773:
3758:ω
3752:
3702:interfere
3658:θ
3654:∠
3632:θ
3628:∠
3606:θ
3602:∠
3535:θ
3531:−
3522:θ
3515:θ
3512:Δ
3486:θ
3483:Δ
3477:
3406:θ
3403:Δ
3400:−
3395:∘
3384:
3355:−
3256:θ
3252:
3227:θ
3223:
3174:θ
3170:
3145:θ
3141:
3117:θ
3113:
3088:θ
3084:
3061:
3052:π
3018:θ
3014:
2989:θ
2985:
2936:θ
2932:
2907:θ
2903:
2879:θ
2875:
2850:θ
2846:
2823:
2759:θ
2755:
2730:θ
2726:
2685:π
2680:⋅
2665:θ
2658:
2630:θ
2623:
2604:
2572:θ
2540:π
2523:π
2518:−
2510:∈
2501:θ
2459:θ
2455:
2430:θ
2426:
2388:θ
2384:
2359:θ
2355:
2285:θ
2275:ω
2269:
2232:ω
2208:θ
2176:
2149:ω
2125:θ
2092:θ
2060:
2033:ω
2014:θ
1984:ω
1965:θ
1938:
1911:ω
1892:θ
1865:
1846:ω
1827:θ
1800:
1774:θ
1764:ω
1758:
1730:θ
1720:ω
1714:
1665:impedance
1649:ϕ
1603:ϕ
1597:θ
1585:ω
1579:
1546:ω
1535:⋅
1522:ϕ
1516:θ
1486:
1459:ω
1448:⋅
1438:ϕ
1424:⋅
1419:θ
1395:
1365:ϕ
1329:ω
1316:θ
1256:θ
1247:ω
1241:
1204:θ
1195:ω
1153:ω
1120:θ
1077:ω
1066:⋅
1061:θ
1039:θ
1030:ω
1007:θ
998:ω
992:
983:⋅
971:θ
962:ω
956:
917:θ
908:ω
902:
893:⋅
834:θ
825:ω
819:
764:π
698:∘
690:∠
664:∠
623:−
584:θ
570:θ
567:
555:θ
552:
523:θ
520:
510:θ
507:
478:θ
475:∠
450:θ
447:∠
424:θ
289:θ
280:ω
266:⋅
157:complexor
113:amplitude
8931:AC power
8250:See also
7753:, where
7379:are the
7265:resistor
3998:integral
3707:triangle
3292:(or the
1676:Addition
381:used in
360:Notation
347:for all
329:(and at
200:analysis
155:or even
119:), and
73:voltages
8392:is the
7435:degrees
7373:and of
7357:is the
7351:(where
3956:⁄
3933:⁄
3729:⁄
3717:⁄
3683:vectors
3288:on the
2563:, then
2313:where:
651:radians
647:degrees
538:or the
377:) is a
349:integer
184:vectors
103:) is a
85:physics
77:current
18:Phasors
8878:
8859:
8817:
8792:
8763:
8736:
8711:
8681:
8656:
8620:
8597:
8587:
8559:
8529:
8504:
8466:
8433:
7292:where
7016:arctan
6993:where
4620:in an
3504:where
3058:arctan
2820:arctan
1276:phasor
1221:is an
642:of 1.
492:vector
221:(with
145:phasor
127:) are
111:whose
93:phasor
39:phaser
3897:light
2786:with
602:with
415:angle
153:sinor
147:, or
8876:ISBN
8857:ISBN
8815:ISBN
8790:ISBN
8761:ISBN
8734:ISBN
8709:ISBN
8679:ISBN
8654:ISBN
8618:ISBN
8595:OCLC
8585:ISBN
8557:ISBN
8527:ISBN
8502:ISBN
8464:ISBN
8431:ISBN
6826:and
6169:Eq.2
6163:Eq.2
6160:and
6157:Eq.1
5705:Eq.2
5413:Eq.1
3565:and
3265:<
3027:>
2806:the
2590:is:
413:and
385:and
318:/ω.
223:real
190:and
91:, a
87:and
8650:661
8460:124
7717:cos
7439:RMS
7381:RMS
7361:of
7177:cos
6172:by
4804:cos
4356:cos
3996:or
3974:= 0
3945:max
3923:max
3911:= 0
3813:cos
3770:cos
3749:cos
3576:or
3474:cos
3391:180
3381:cos
3296:):
3249:cos
3220:cos
3202:if
3167:cos
3138:cos
3110:sin
3081:sin
3011:cos
2982:cos
2964:if
2929:cos
2900:cos
2872:sin
2843:sin
2794:sgn
2752:cos
2723:cos
2705:if
2655:sin
2620:sin
2601:sgn
2452:sin
2423:sin
2381:cos
2352:cos
2266:cos
1755:cos
1711:cos
1669:not
1576:cos
1238:cos
1227:of
989:sin
953:cos
899:sin
816:cos
564:sin
549:cos
517:sin
504:cos
327:= 0
209:of
175:or
99:of
95:(a
83:In
8922::
8771:,
8693:^
8652:.
8632:^
8593:.
8571:^
8541:^
8462:.
8429:.
8427:30
8296:^
7498:Re
7473:.
7348:VI
7346:=
7334:jQ
7332:+
7328:=
7289:IZ
7287:=
7274:IR
7272:=
7263:A
7254:DC
7079:Re
7046:.
6085:Im
6004:Im
5952:Im
5878:Re
5797:Re
5745:Re
5641:Im
5578:Im
5515:Im
5352:Re
5293:Re
5230:Re
4960:Re
4872:Re
4624::
4468:.
4262:Re
4176:Re
4108:Re
4013:Re
3856:0.
3738:.
2173:Re
2057:Re
1935:Re
1862:Re
1797:Re
1483:Re
1392:Re
879::
694:90
667:90
355:).
334:=
204:AC
159:.
8884:.
8865:.
8823:.
8798:.
8769:.
8742:.
8717:.
8687:.
8662:.
8626:.
8601:.
8565:.
8535:.
8510:.
8472:.
8439:.
8376:,
8371:t
8365:i
8361:e
8354:i
8351:=
8346:t
8340:i
8336:e
8329:t
8326:d
8322:d
8232:m
8228:f
8224:3
8221:,
8216:m
8212:f
8208:2
8185:.
8180:t
8175:m
8171:f
8164:2
8161:i
8154:e
8142:i
8138:e
8134:m
8131:A
8126:2
8123:1
8102:,
8097:t
8092:m
8088:f
8081:2
8078:i
8074:e
8062:i
8058:e
8054:m
8051:A
8046:2
8043:1
8017:m
8013:f
7989:.
7984:t
7981:)
7976:m
7972:f
7963:0
7959:f
7955:(
7949:2
7946:i
7942:e
7930:i
7926:e
7922:m
7919:A
7914:2
7911:1
7890:,
7885:t
7882:)
7877:m
7873:f
7869:+
7864:0
7860:f
7856:(
7850:2
7847:i
7843:e
7831:i
7827:e
7823:m
7820:A
7815:2
7812:1
7786:m
7782:f
7761:m
7740:t
7735:m
7731:f
7724:2
7714:m
7711:A
7690:)
7687:t
7684:(
7681:x
7638:0
7635:=
7632:t
7610:0
7606:f
7585:A
7562:,
7558:)
7552:t
7547:0
7543:f
7536:2
7533:i
7529:e
7517:i
7513:e
7509:A
7505:(
7495:=
7492:)
7489:t
7486:(
7483:x
7376:I
7370:V
7364:I
7354:I
7344:S
7339:S
7330:P
7326:S
7318:Q
7314:P
7300:.
7294:Z
7285:V
7270:V
7216:.
7213:)
7210:)
7204:(
7192:+
7189:t
7183:(
7172:P
7168:V
7156:2
7152:)
7148:C
7145:R
7139:(
7136:+
7133:1
7129:1
7124:=
7120:)
7114:t
7108:i
7104:e
7095:c
7091:V
7086:(
7076:=
7073:)
7070:t
7067:(
7062:C
7058:v
7034:)
7031:C
7028:R
7022:(
7013:=
7010:)
7004:(
6981:,
6976:)
6970:(
6964:i
6957:e
6945:2
6941:)
6937:C
6934:R
6928:(
6925:+
6922:1
6918:1
6913:=
6905:2
6901:)
6897:C
6894:R
6888:(
6885:+
6882:1
6877:C
6874:R
6868:i
6862:1
6837:.
6812:P
6808:V
6787:)
6784:t
6781:(
6776:C
6772:v
6749:s
6745:V
6722:.
6714:i
6710:e
6704:P
6700:V
6688:2
6684:)
6680:C
6677:R
6671:(
6668:+
6665:1
6660:C
6657:R
6651:i
6645:1
6639:=
6634:s
6630:V
6620:C
6617:R
6611:i
6608:+
6605:1
6601:1
6596:=
6591:c
6587:V
6559:.
6554:s
6550:V
6543:C
6540:R
6536:1
6531:=
6522:c
6518:V
6511:C
6508:R
6504:1
6499:+
6494:c
6490:V
6483:i
6474:t
6468:i
6464:e
6456:)
6450:s
6446:V
6439:C
6436:R
6432:1
6426:(
6422:=
6413:t
6407:i
6403:e
6394:)
6388:c
6384:V
6377:C
6374:R
6370:1
6365:+
6360:c
6356:V
6349:i
6345:(
6335:t
6329:i
6325:e
6316:s
6312:V
6305:C
6302:R
6298:1
6293:=
6284:t
6278:i
6274:e
6265:c
6261:V
6254:C
6251:R
6247:1
6242:+
6237:t
6231:i
6227:e
6218:c
6214:V
6207:i
6183:,
6180:i
6136:.
6132:)
6126:t
6120:i
6116:e
6107:c
6103:V
6096:i
6092:(
6082:=
6078:)
6071:)
6065:t
6059:i
6055:e
6046:c
6042:V
6037:(
6028:t
6024:d
6018:d
6011:(
6001:=
5993:)
5987:t
5981:i
5977:e
5968:c
5964:V
5959:(
5946:t
5942:d
5936:d
5925:)
5919:t
5913:i
5909:e
5900:c
5896:V
5889:i
5885:(
5875:=
5871:)
5864:)
5858:t
5852:i
5848:e
5839:c
5835:V
5830:(
5821:t
5817:d
5811:d
5804:(
5794:=
5786:)
5780:t
5774:i
5770:e
5761:c
5757:V
5752:(
5739:t
5735:d
5729:d
5707:)
5703:(
5686:.
5682:)
5676:t
5670:i
5666:e
5657:s
5653:V
5648:(
5635:C
5632:R
5628:1
5623:=
5619:)
5613:t
5607:i
5603:e
5594:c
5590:V
5585:(
5572:C
5569:R
5565:1
5560:+
5556:)
5550:t
5544:i
5540:e
5531:c
5527:V
5522:(
5509:t
5505:d
5499:d
5471:,
5462:2
5450:t
5430:t
5415:)
5411:(
5393:)
5387:t
5381:i
5377:e
5368:s
5364:V
5359:(
5346:C
5343:R
5339:1
5334:=
5331:)
5326:t
5320:i
5316:e
5307:c
5303:V
5299:(
5287:C
5284:R
5280:1
5275:+
5271:)
5265:t
5259:i
5255:e
5246:c
5242:V
5237:(
5224:t
5220:d
5214:d
5183:.
5178:s
5174:V
5167:C
5164:R
5160:1
5155:=
5150:c
5146:V
5139:C
5136:R
5132:1
5127:+
5122:c
5118:V
5111:i
5086:c
5082:V
5061:,
5053:i
5049:e
5043:P
5039:V
5035:=
5030:s
5026:V
5005:,
5001:)
4995:t
4989:i
4985:e
4976:c
4972:V
4967:(
4957:=
4954:)
4951:t
4948:(
4943:C
4939:v
4917:.
4913:)
4907:t
4901:i
4897:e
4888:s
4884:V
4879:(
4869:=
4866:)
4863:t
4860:(
4855:S
4851:v
4828:,
4825:)
4819:+
4816:t
4810:(
4796:P
4792:V
4788:=
4785:)
4782:t
4779:(
4774:S
4770:v
4747:.
4744:)
4741:t
4738:(
4733:S
4729:v
4722:C
4719:R
4715:1
4710:=
4707:)
4704:t
4701:(
4696:C
4692:v
4685:C
4682:R
4678:1
4673:+
4667:t
4663:d
4657:)
4654:t
4651:(
4646:C
4642:v
4636:d
4602:t
4596:i
4592:e
4564:,
4559:t
4553:i
4549:e
4528:.
4518:2
4514:/
4507:i
4500:e
4494:=
4485:i
4481:1
4448:2
4444:/
4437:i
4433:e
4429:=
4423:i
4397:.
4393:)
4387:2
4379:+
4373:+
4370:t
4363:(
4350:A
4340:=
4332:)
4326:t
4320:i
4316:e
4307:)
4304:2
4300:/
4293:+
4287:(
4284:i
4280:e
4276:A
4269:(
4255:=
4247:)
4241:t
4235:i
4231:e
4222:2
4218:/
4211:i
4207:e
4195:i
4191:e
4187:A
4183:(
4169:=
4161:)
4155:t
4149:i
4145:e
4138:i
4127:i
4123:e
4119:A
4115:(
4101:=
4093:)
4086:)
4080:t
4074:i
4070:e
4058:i
4054:e
4050:A
4046:(
4037:t
4033:d
4027:d
4020:(
3982:Φ
3972:t
3963:t
3958:2
3953:π
3951:3
3942:A
3940:−
3935:2
3930:π
3920:A
3918:+
3909:t
3904:π
3853:=
3849:)
3843:3
3836:2
3827:t
3820:(
3810:+
3806:)
3800:3
3793:2
3787:+
3784:t
3777:(
3767:+
3764:)
3761:t
3755:(
3731:3
3726:λ
3719:3
3714:π
3712:2
3667:.
3662:3
3649:3
3645:A
3641:=
3636:2
3623:2
3619:A
3615:+
3610:1
3597:1
3593:A
3578:t
3574:ω
3570:3
3567:θ
3563:3
3560:A
3544:.
3539:2
3526:1
3518:=
3492:,
3489:)
3480:(
3469:2
3465:A
3459:1
3455:A
3451:2
3448:+
3443:2
3438:2
3434:A
3430:+
3425:2
3420:1
3416:A
3412:=
3409:)
3387:(
3376:2
3372:A
3366:1
3362:A
3358:2
3350:2
3345:2
3341:A
3337:+
3332:2
3327:1
3323:A
3319:=
3314:2
3309:3
3305:A
3280:.
3268:0
3260:2
3244:2
3240:A
3236:+
3231:1
3215:1
3211:A
3190:,
3186:)
3178:2
3162:2
3158:A
3154:+
3149:1
3133:1
3129:A
3121:2
3105:2
3101:A
3097:+
3092:1
3076:1
3072:A
3065:(
3055:+
3042:;
3030:0
3022:2
3006:2
3002:A
2998:+
2993:1
2977:1
2973:A
2952:,
2948:)
2940:2
2924:2
2920:A
2916:+
2911:1
2895:1
2891:A
2883:2
2867:2
2863:A
2859:+
2854:1
2838:1
2834:A
2827:(
2810:;
2774:,
2771:0
2768:=
2763:2
2747:2
2743:A
2739:+
2734:1
2718:1
2714:A
2693:,
2688:2
2677:)
2674:)
2669:2
2661:(
2650:2
2646:A
2642:+
2639:)
2634:1
2626:(
2615:1
2611:A
2607:(
2576:3
2550:]
2544:2
2537:3
2531:,
2526:2
2514:[
2505:3
2478:,
2473:2
2469:)
2463:2
2447:2
2443:A
2439:+
2434:1
2418:1
2414:A
2410:(
2407:+
2402:2
2398:)
2392:2
2376:2
2372:A
2368:+
2363:1
2347:1
2343:A
2339:(
2336:=
2331:2
2326:3
2322:A
2297:,
2294:)
2289:3
2281:+
2278:t
2272:(
2261:3
2257:A
2249:=
2241:)
2235:t
2229:i
2225:e
2220:)
2212:3
2204:i
2200:e
2194:3
2190:A
2185:(
2180:(
2166:=
2158:)
2152:t
2146:i
2142:e
2137:)
2129:2
2121:i
2117:e
2111:2
2107:A
2103:+
2096:1
2088:i
2084:e
2078:1
2074:A
2069:(
2064:(
2050:=
2042:)
2036:t
2030:i
2026:e
2018:2
2010:i
2006:e
2000:2
1996:A
1992:+
1987:t
1981:i
1977:e
1969:1
1961:i
1957:e
1951:1
1947:A
1942:(
1928:=
1920:)
1914:t
1908:i
1904:e
1896:2
1888:i
1884:e
1878:2
1874:A
1869:(
1859:+
1855:)
1849:t
1843:i
1839:e
1831:1
1823:i
1819:e
1813:1
1809:A
1804:(
1790:=
1783:)
1778:2
1770:+
1767:t
1761:(
1750:2
1746:A
1742:+
1739:)
1734:1
1726:+
1723:t
1717:(
1706:1
1702:A
1646:i
1642:e
1638:B
1612:.
1609:)
1606:)
1600:+
1594:(
1591:+
1588:t
1582:(
1573:B
1570:A
1563:=
1555:)
1549:t
1543:i
1539:e
1531:)
1525:)
1519:+
1513:(
1510:i
1506:e
1502:B
1499:A
1495:(
1490:(
1476:=
1468:)
1462:t
1456:i
1452:e
1444:)
1435:i
1431:e
1427:B
1416:i
1412:e
1408:A
1404:(
1399:(
1362:i
1358:e
1354:B
1332:t
1326:i
1322:e
1313:i
1309:e
1305:A
1262:.
1259:)
1253:+
1250:t
1244:(
1235:A
1207:)
1201:+
1198:t
1192:(
1189:i
1185:e
1181:A
1156:t
1150:i
1146:e
1125:,
1117:i
1113:e
1109:A
1085:,
1080:t
1074:i
1070:e
1058:i
1054:e
1050:A
1047:=
1042:)
1036:+
1033:t
1027:(
1024:i
1020:e
1016:A
1013:=
1010:)
1004:+
1001:t
995:(
986:A
980:i
977:+
974:)
968:+
965:t
959:(
950:A
920:)
914:+
911:t
905:(
896:A
890:i
863:t
840:,
837:)
831:+
828:t
822:(
813:A
783:.
780:i
777:=
772:2
768:/
761:i
757:e
736:)
733:1
729:,
726:0
723:(
703:,
687:1
661:1
626:1
620:=
615:2
611:i
581:i
577:e
573:=
561:i
558:+
526:)
513:,
501:(
472:1
453:.
444:A
401:A
353:n
344:ω
342:/
340:π
338:2
336:n
332:t
325:t
320:θ
316:π
312:A
292:)
286:+
283:t
277:(
274:i
270:e
263:A
137:ω
135:(
125:θ
123:(
117:A
115:(
79:.
65:ω
48:.
41:.
34:.
20:)
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