There's a common misconception that, for a linear circuit, the maximum
efficiency and/or power available from a voltage source occurs when the
source resistance equals the load resistance (or, more generally, when
they're complex conjugates). But this isn't universally true, as I'll
show with a simple example.

Suppose we have a 100 volt perfect voltage source in series with a
variable source resistance, and a fixed load resistance of 100 ohms. If
we make the source resistance 100 ohms, the source delivers 50 watts, 25
of which are dissipated in the source resistance and 25 watts in the
load. The efficiency, if you consider the source resistance dissipation
as wasted, is 50%. But what happens if we reduce the source resistance
to 50 ohms? Now the source delivers 66.7 watts, of which 22.2 is
dissipated in the source resistance and 44.4 in the load resistance. The
power to the load has increased, and the efficiency has increased from
50 to 66.7%. The efficiency and load power continue to increase as the
source resistance is made smaller and smaller, reaching a maximum when
the source resistance is zero. At that point, the source will deliver
100 watts, all of which is dissipated in the load, for an efficiency of
100%.

The well known and often misapplied rule about maximizing power transfer
by matching the source and load impedances applies only when you're
stuck with a fixed source resistance and can only modify the load.

Roy Lewallen, W7EL