barcode reader code in asp.net Figure 426 Waveforms for the AC circuit of Figure 425 in Software

Paint Denso QR Bar Code in Software Figure 426 Waveforms for the AC circuit of Figure 425

Figure 426 Waveforms for the AC circuit of Figure 425
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In a sinusoidally excited linear circuit, all branch voltages and currents are sinusoids at the same frequency as the excitation signal The amplitudes of these voltages and currents are a scaled version of the excitation amplitude, and the voltages and currents may be shifted in phase with respect to the excitation signal
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These observations indicate that three parameters uniquely de ne a sinusoid: frequency, amplitude, and phase But if this is the case, is it necessary to carry the excess luggage, that is, the sinusoidal functions Might it be possible to simply keep track of the three parameters just mentioned Fortunately, the answers to these two questions are no and yes, respectively The next section will describe the use of a notation that, with the aid of complex algebra, eliminates the need for the sinusoidal functions of time, and for the formulation and solution of differential equations, permitting the use of simpler algebraic methods
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Check Your Understanding
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411 Show that the solution to either equation 428 or equation 430 is suf cient to compute all of the currents and voltages in the circuit of Figure 425 412 Show that the equality
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A sin t + B cos t = C cos( t + ) holds if A = C sin B = C cos or, conversely, if C= A2 + B 2 A B
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= tan 1 C in equation 439
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413 Use the result of Exercise 412 to compute C and as functions of V , , R, and
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PHASORS AND IMPEDANCE
In this section, we introduce an ef cient notation to make it possible to represent sinusoidal signals as complex numbers, and to eliminate the need for solving differential equations The student who needs a brief review of complex algebra will nd a reasonably complete treatment in Appendix A, including solved examples and Check Your Understanding exercises For the remainder of the chapter, it will be assumed that you are familiar with both the rectangular and the polar forms of complex number coordinates, with the conversion between these two forms, and with the basic operations of addition, subtraction, multiplication, and division of complex numbers Euler s Identity Named after the Swiss mathematician Leonhard Euler (the last name is pronounced Oiler ), Euler s identity forms the basis of phasor notation Simply stated, the identity de nes the complex exponential ej as a point in the complex plane, which may be represented by real and imaginary components: ej = cos + j sin (440)
Leonhard Euler (1707 1783) Photo courtesy of Deutsches Museum, Munich
Im j
Figure 427 illustrates how the complex exponential may be visualized as a point (or vector, if referenced to the origin) in the complex plane Note immediately that the magnitude of ej is equal to 1:
1 1 Re
sin _1
|ej | = 1 since | cos + j sin | = cos2 + sin2 = 1
(441)
cos
(442)
_j e j = cos + j sin
Figure 427 Euler s identity
and note also that writing Euler s identity corresponds to equating the polar form of a complex number to its rectangular form For example, consider a vector of length A making an angle with the real axis The following equation illustrates the relationship between the rectangular and polar forms: Aej = A cos + j A sin = A (443)
Part I
Circuits
In effect, Euler s identity is simply a trigonometric relationship in the complex plane Phasors To see how complex numbers can be used to represent sinusoidal signals, rewrite the expression for a generalized sinusoid in light of Euler s equation: A cos( t + ) = Re [Aej ( t+ ) ] (444)
This equality is easily veri ed by expanding the right-hand side, as follows: Re [Aej ( t+ ) ] = Re [A cos( t + ) + j A sin( t + )] = A cos( t + ) We see, then, that it is possible to express a generalized sinusoid as the real part of a complex vector whose argument, or angle, is given by ( t + ) and whose length, or magnitude, is equal to the peak amplitude of the sinusoid The complex phasor corresponding to the sinusoidal signal A cos( t + ) is therefore de ned to be the complex number Aej : Aej = complex phasor notation for A cos( t + ) = A (445)
It is important to explicitly point out that this is a de nition Phasor notation arises from equation 444; however, this expression is simpli ed (for convenience, as will be promptly shown) by removing the real part of operator (Re) and factoring out and deleting the term ej t The next equation illustrates the simpli cation: A cos( t + ) = Re [Aej ( t+ ) ] = Re [Aej ej t ] (446)
The reason for this simpli cation is simply mathematical convenience, as will become apparent in the following examples; you will have to remember that the ej t term that was removed from the complex form of the sinusoid is really still present, indicating the speci c frequency of the sinusoidal signal, With these caveats, you should now be prepared to use the newly found phasor to analyze AC circuits The following comments summarize the important points developed thus far in the section
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