qr code vb.net library * In cycles/second eq. 404 would be d f f0 =f0 . in .NET

Drawer Code-128 in .NET * In cycles/second eq. 404 would be d f f0 =f0 .

* In cycles/second eq. 404 would be d f f0 =f0 .
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406
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CHAPTER 10 Magnetic Coupling. Transformers
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Now, going back to eq. (400), replace (XL XC ) with the right-hand side of the above equation. Next, in eq. (400), set Xm !M k!L (by eq. (371)) and also set XC 1=!C. Doing this, eq. (400) becomes " G r2 k2 !2 L2 jgm kL=!C2 4!2 L2 d 2 j4r!0 Ld 0 407
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Next, what is called the Q factor of an inductor coil is universally de ned as the ratio of the REACTANCE of the coil to its RESISTANCE. If the coil appears in a resonant circuit, then we ll de ne the Q in terms of the resonant frequency of the circuit; thus Q !0 L r 408
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where r is the total resistance in series with the coil, including any resistance the coil windings may have. One reason it s convenient to work in terms of Q is because it s easy to nd both the inductance of a coil and its Q by use of a standard piece of laboratory equipment called a Q-meter. Let us therefore write our equations in terms of Q; thus, noting that r !0 L=Q, eq. (407) becomes " G jgm kL=!C2 !2 L2 j4!2 L2 d 0 0 k2 !2 L2 4!2 L2 d 2 0 Q Q2 409
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We next might note that the second term in the denominator of the above equation can be written as  2 ! 2 2 2 2 2 2 2 2 2 2 2 k ! L k ! !0 L =!0 k !0 L !0 and, upon making this change, note that !2 L2 factors from the denominator. Doing this, 0 then multiplying the numerator and denominator by Q2 , eq. (409) becomes " G jgm LkQ2 =!C2 !2 L2 1 k2 Q2 !=!0 2 4Q2 d 2 j4Qd 0 410
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Now multiply the denominator by 1=!2 L2 and the numerator by !2 C 2 (which is per0 0 missible, because, by eq. (401), 1=!0 L !0 C). Doing this, eq. (410) becomes " G jgm !0 LQ2 !0 =! k 1 k2 Q2 !=!0 2 4Q2 d 2 j4Qd 411
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At this point let s pause to consider some relationships that are known to exist in practical applications of Fig. 232. To make the rst point, note that, by eq. (404), ! 1 d !0 and therefore (see discussion following eq. (405)) it follows that, for almost all practical applications, it s perfectly permissible to write that (!=!0 !0 =! 1; hence, for practical purposes eq. (411) becomes " G jgm !0 LkQ2 1 k2 Q2 4d 2 Q2 j4dQ 412
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CHAPTER 10 Magnetic Coupling. Transformers
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In the above it might seem, at rst glance, that the entire denominator could be reduced to 1, since, in practical work, d and k are both very much less than 1. This, however, will not be the case because in practical work Q will generally be much greater than 1 (a value of Q 100 is entirely possible). Thus products of the forms dQ and kQ can have signi cant values, and hence must not be dropped from the denominator. " In the above equation the voltage gain G is expressed in complex form. However, in " most work it will be su cient to know only how the magnitude of G varies with frequency. With this in mind, let us write eq. (412) in the form kQ2 " jGj G gm !0 L q 1 k2 Q2 4d 2 Q2 2 16d 2 Q2 413
In the above, remember that d is the fractional deviation of any frequency ! from the resonant frequency !0 (thus d 0 for ! !0 ). What we now wish to investigate is how the value of k a ects the way in which G varies relative to the value of d. That is, how the value of k a ects the shape of the curve of G versus d. As an illustration of how G varies with d, let s rst consider a speci c value, after which we ll point out some practical, general conclusions regarding Fig. 232. Let us take, as our example, the case for Q 50 (a reasonable value). Then, for this value of Q (and omitting the constant multiplier gm !0 L), eq. (413) becomes 2500k G q 1 2500k2 10,000d 2 2 40,000d 2 414
in which we ll take d as the independent variable and k, the coe cient of coupling, as a parameter whose e ect we wish to investigate. In regard to k, it should be noted that what is called critical coupling is de ned as being equal to 1=Q. Thus, if kc denotes critical coupling, we have that kc 1=Q. Hence, in the present example we have that kc 1=50 0:02 2%. In the above example it will be interesting to plot the curves of G versus d for several di erent values of the parameter k; let us select the values k 0:01, k kc 0:02, k 0:03. To do this, we successively substitute into eq. (414) the chosen values of k, with the results shown in the table below, with nal calculator values rounded to one decimal place.
Value of G for k 0:01 20.0 17.7 12.4 5.2 2.6 1.5 Value of G for k 0:02 25.0 24.8 22.4 11.2 5.4 3.1 Value of G for k 0:03 23.1 23.7 24.9 18.4 9.0 5.0
d 0.000 0.005 0.010 0.020 0.030 0.040
The above results are plotted in Figs. 239 and 240, in which d, on the horizontal axis, is understood to be multiplied by 10 2 . In regard to the above, let s return to Fig. 232 and rst suppose that the two coils are physically far apart, so that only very loose coupling exists between the coils. As we would expect, the voltage gain would be quite low in such a case. This would, for example, be the condition for the case of k 0:01 in Fig. 239.
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