This preview shows page 1. Sign up to view the full content.
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. 3382 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 ...
View
Full
Document
This note was uploaded on 11/12/2011 for the course PHYS 2001 taught by Professor Sprunger during the Fall '08 term at LSU.
 Fall '08
 SPRUNGER
 Physics, Current, Power

Click to edit the document details