I am currently learning how ideal sources may be replaced with linear sources to make the application of other circuit theory possible. Unfortunately, I don't understand how the following conversion is possible. This is the original circuit.
simulate this circuit – Schematic created using CircuitLab
And my prof. converts it into the following, unfortunately very quickly and with little commentary:
with: $$ V_1^* = V \frac{R_3}{R_1+R_3} $$ $$ R_1^* = \frac{R_1R_3}{R_1+R_3} $$ $$ V_2^* = V \frac{R_4}{R_2+R_4} $$ $$ R_2^* = \frac{R_2R_4}{R_2+R_$} $$
Ultimately, the question in this exercise is what voltage-drop occurs over \$R_5\$, but my question relates more to the conversion above, and less to the accomplishment of said goal. I only mention it, since it may allow some simplifications which would not be possible, would we seek information about other specific nodes or currents, for example.
Let's only consider \$V_1*\$ and \$R_1*\$ since the conversion for the other voltage-source is equivalent.
The voltage itself of course stems from a voltage-division between \$R_1\$ and \$R_3\$. Yet I don't quite see how this comes about, since, going from the cathode and through \$R_3\$, our remaining voltage should at first glance be smaller, the larger \$R_3\$ is. A second glance gives me even more trouble, since I don't see a way for current to travel from the upper cathode to the upper anode while going through the rest of the circuit (since it would ultimately have to be at node A again before going through \$R_1\$ and into the anode.
When calculating the inner resistance, the resistors \$R_1\$ and \$R_3\$ are clearly considered to be parallel. But, considering one of the voltage-sources, I see no way that current could travel from cathode to anode without going through both resistors.
I assume that in both cases my problem is that I consider only one of the sources at a time, while I should consider them both at once. They supply the same voltage, and are oriented so that no potential-difference occurs between their cathodes, which means that there is no potential-difference between their anodes as well. In this exercise they actually stem from a single source, which got split into these two to make a simplification of the circuit possible.
So, why can we make the above replacements, and why are the relations the way they are? Am I right that I should consider both sources at the same time? Some explanation and/or pointing to relevant theory would be highly welcome.