Page 312 - Schaum's Outline of Theory and Problems of Applied Physics

CHAP. 25] DIRECT-CURRENTCIRCUITS 297

IMPEDANCE MATCHING
When a source of electric energy is connected to a load, the power transfer is a maximum when both source and
load have the same resistance. This is an example of impedance matching: When energy is being transferred
from one system to another (here from an emf source to a load), the efficiency is greatest when both systems have
the same impedance, which is a general term for resistance to the flow of energy. Another example of impedance
matching occurs when a moving body strikes a stationary one. When both bodies have the same mass (which
is a measure of the resistance of a body to a change in its state of motion or of rest), the energy transfer to the
stationary body is a maximum.
Figure 25-7 shows an emf source of internal resistance r connected to a load resistance R. If the current that
flows is I,

V emf
I =
R + r
The power in the load is

V 2 R
2 emf
P = I R =
(R + r) 2
The maximum value of P corresponds to R = r, when

2
V emf
P max =
4R
Figure 25-8 shows how P/P max varies with the resistance ratio R/r.

1.00
Power dissipated in R relative to maximum 0.75

0.50

0.25

0
0 1 2 3 4 5
Resistance ratio, R/r
Fig. 25-8. (From Modern Technical Physics, 6th Ed., Arthur Beiser,
Fig. 25-7 c 1992. Reprinted by permission of Pearson Education, Inc.)

SOLVED PROBLEM 25.20
Verify that P max occurs when R = r by calculating the power delivered by a battery of emf 10 V and
internal resistance 0.5 when it is connected to (a) 0.25- ,(b) 0.5- , and (c)1- resistors.
V e 10 V
(a) I 1 = = = 13.3A
R 1 + r 0.25 + 0.5

2
2
P 1 = I R 1 = (13.3A) (0.25 ) = 44 W
1