Output power and efficiency from a generator with inner resistansce (two pool)

Figure 1 describes how a generator with inner reistance Ri is loaded with an outer resistance RL. In tube amplifiers the problem is to select  RL not only so that maximum output power is achieved, which is achieved with RL=Ri, but also to look at the efficiency for the cicuit is affected for different choices of RL.

Figure 1:

Developed power in RL is calculated as:


1.1
The maximum available power from the generator is acheived when the derivate = 0 namely:


1.2
From this one realizes that maximum power in RL is developed when RL = Ri.
The generators maximum power can the be calculated as:
1.3
Now form the kvote between output power P and maximum generator power Pgenmax:


and set
1.4
We then gets:


1.5
The circuits total efficiency can be calculated as:
Maximum power is
1.6
We then gets:


1.7

Summary we kan sketch the following graphs that describes how matching works for different values of k.

Blue
 curve shows kvote between power in RL and maximum available generator power
Green
 curve corresponds to the circuits efficiency
Red
 line shows when the power has reached