colin55,
Sorry I did not answer sooner, but I had places to go and things to do.
Now, if you looked at my analysis, you would have seen immediately from my node equations that I was considering the sources as current sources, not voltage sources like you did. Since the sources were marked with current direction, and were missing voltage polarity signs, I believe that was a good assumption.
But OK, let's assume they are voltage sources.
The circuit is very complex until you realise one thing.
Actually, I don't think it is complex at all. What thing should I realize?
Firstly, remove the 6R, 18v and 12R.
Maximum power in Rx is achieved when Rx is equal to 3R.
To see if this is true, simply change Rx to 2R or 4R and work out the watts dissipated.
Another way to look at it is this: When Rx is zero, the watts dissipated is zero, when Rx is very high, the watts dissipated is very small.
For the partial circuit you specify, that is true.
Make Rx = 3R and see what happens:
The voltage across Rx will be 18v.
Now put-back the 6R, 18v supply and 12R load resistor.
The voltage across Rx is 18v. This means no current will flow though the 6R resistor because it has 18v on each end.
Current through Rx = 36v/6R = 6 amp
Wattage dissipated by 3R = V2/3 = 108watts or I2xR = 62 x 3 = 108watts
We have turned a seemingly complex question into a very simple answer.
By fallacious reasoning, you have obtained the wrong answer to a slightly difficult problem. The goal is to find the resistance for maximum dissipation, not the dissipation at Rx=3. Since you have not performed any differentiation of the power with respect to Rx, you have no idea what the resistance for max power dissipation is. As you can see in the attachment, it occurs at Rx=2 for a dissipation of 112.5 watts, not 108 watts. The voltage increases across Rx as its resistance increases from zero, but after Rx=2, the power drops even though the voltage across the resistor still increases. The reasoning you expound above does not take that into effect.
See this review on Spot The Mistake:
**broken link removed** .
See the attachment which proves your mistake. I am surprised no one caught your error after all this time.
Ratch
[MODNOTE]Removed unnecessary comments.[/MODNOTE]