Impedance matching with lossy components

Impedance matching networks couple a generator to a load. When lossless matching elements are used, the load power
P
and the power
P'
extracted from the generator terminals are the same. Then the familiar conjugate impedance match condition maximizes
P'
, and hence
P
. But if the matching elements are lossy
P \neq P'
; the power difference
P' - P
is wasted in the matching network. Maximizing
P'
does not maximize
P
. Suppose the load is purely resistive and that the quality factors of all elements used in the matching network must not exceed a prescribed value
Q
. If the load resistance value
R
is not prescribed, but can be adjusted to facilitate matching, then the maximum obtainable load power is
P_m
, given by Theorem 1. When
R
is prescribed, the maximum power may be less than
P_m
. However, there is a range of load resistance values, wide in many cases, for which Theorem 2 applies to deliver the same maximum power
P_m
.
P_m
depends on
Q
and on another quality factor
q
of the generator.
P_m
is small if
Q \ll q
, as may be the case if the generator is a short receiving antenna.