p33_065

# p33_065 - 65. The ampliﬁer is connected across the...

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Unformatted text preview: 65. The ampliﬁer is connected across the primary windings of a transformer and the resistor R is connected across the secondary windings. If Is is the rms current in the secondary coil then the average power 2 delivered to R is Pavg = Is R. Using Is = (Np /Ns )Ip , we obtain Pavg = Ip Np Ns 2 R. Next, we ﬁnd the current in the primary circuit. This is eﬀectively a circuit consisting of a generator and two resistors in series. One resistance is that of the ampliﬁer (r), and the other is the equivalent resistance Req of the secondary circuit. Therefore, Ip = Erms Erms = r + Req r + (Np /Ns )2 R where Eq. 33-82 is used for Req . Consequently, Pavg = E 2 (Np /Ns )2 R 2 [r + (Np /Ns )2 R] . Now, we wish to ﬁnd the value of Np /Ns such that Pavg is a maximum. For brevity, let x = (Np /Ns )2 . Then E 2 Rx , Pavg = (r + xR)2 and the derivative with respect to x is dPavg E 2 R(r − xR) . = dx (r + xR)3 This is zero for x = r/R = (1000 Ω)/(10 Ω) = 100. We note that for small x, Pavg increases linearly with x, and for large x it decreases in proportion to 1/x. Thus x = r/R is indeed a maximum, not a minimum. √ Recalling x = (Np /Ns )2 , we conclude that the maximum power is achieved for Np /Ns = x = 10. The diagram below is a schematic of a transformer with a ten to one turns ratio. An actual transformer would have many more turns in both the primary and secondary coils. ................... ................... . ... .. . .. ..... ......... ..... . ... .. .. . ... . ......... ........ .. . . ... .. . . .. ..... ......... ..... . ... .. .. . . .. ..... ......... ..... . ... .. .. . ......... ......... . ... . ... .. .. . . .. ..... ......... ..... . ... .. .. . ... ......... ........ ... .. .. .. . . . .. ..... ......... ..... . ... .. .. . .. .. .... ......... ..... . ... .. .. . ........................... .......................... to ampliﬁer .. ...................... ........................... ..... ... ................... ................... to load resistance R ...
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## This note was uploaded on 11/12/2011 for the course PHYS 2001 taught by Professor Sprunger during the Fall '08 term at LSU.

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