Polarity of voltage drops:  We can trace the direction that electrons will flow in the same circuit by starting at the negative (-) terminal and following through to the positive ( ) terminal of the battery, the only source of voltage in the circuit. From this we can see that the electrons are moving counter-clockwise, from point 6 to 5 to 4 to 3 to 2 to 1 and back to 6 again.

As the current encounters the 5 Ω resistance, voltage is dropped across the resistor's ends. The polarity of this voltage drop is negative (-) at point 4 with respect to positive ( ) at point 3. We can mark the polarity of the resistor's voltage drop with these negative and positive symbols, in accordance with the direction of current (whichever end of the resistor the current is entering is negative with respect to the end of the resistor it is exiting:

We could make our table of voltages a little more complete by marking the polarity of the voltage for each pair of points in this circuit:

Between points 1 ( ) and 4 (-) = 10 volts, Between points 2 ( ) and 4 (-) = 10 volts
Between points 3 ( ) and 4 (-) = 10 volts, Between points 1 ( ) and 5 (-) = 10 volts
Between points 2 ( ) and 5 (-) = 10 volts, Between points 3 ( ) and 5 (-) = 10 volts
Between points 1 ( ) and 6 (-) = 10 volts,Between points 2 ( ) and 6 (-) = 10 volts
Between points 3 ( ) and 6 (-) = 10 volts

While it might seem a little silly to document polarity of voltage drop in this circuit, it is an important concept to master. It will be critically important in the analysis of more complex circuits involving multiple resistors and/or batteries. It should be understood that polarity has nothing to do with Ohm's Law: there will never be negative voltages, currents, or resistance entered into any Ohm's Law equations! There are other mathematical principles of electricity that do take polarity into account through the use of signs ( or -), but not Ohm's Law.

Dc Network Analysis:

Generally speaking, network analysis is any structured technique used to mathematically analyze a circuit (a "network" of interconnected components). Quite often the technician or engineer will encounter circuits containing multiple sources of power or component configurations which defy simplification by series/parallel analysis techniques. In those cases, he or she will be forced to use other means. This chapter presents a few techniques useful in analyzing such complex circuits.

To illustrate how even a simple circuit can defy analysis by breakdown into series and parallel portions, take start with this series-parallel circuit:

To analyze the above circuit, one would first find the equivalent of R₂ and R₃ in parallel, then add R₁ in series to arrive at a total resistance. Then, taking the voltage of battery B₁ with that total circuit resistance, the total current could be calculated through the use of Ohm's Law (I=E/R), then that current figure used to calculate voltage drops in the circuit. All in all, a fairly simple procedure.

However, the addition of just one more battery could change all of that:

Resistors R₂ and R₃ are no longer in parallel with each other, because B₂ has been inserted into R₃'s branch of the circuit. Upon closer inspection, it appears there are no two resistors in this circuit directly in series or parallel with each other. This is the crux of our problem: in series-parallel analysis, we started off by identifying sets of resistors that were directly in series or parallel with each other, and then reduce them to single, equivalent resistances.