Millman’s Theorem
Network Analysis Techniques
Don’t just sit there! Build something!! 
Learning to mathematically analyze circuits requires much study and practice. Typically, students practice by working through lots of sample problems and checking their answers against those provided by the textbook or the instructor. While this is good, there is a much better way.
You will learn much more by actually building and analyzing real circuits, letting your test equipment provide the “answers” instead of a book or another person. For successful circuitbuilding exercises, follow these steps:
 Carefully measure and record all component values prior to circuit construction.
 Draw the schematic diagram for the circuit to be analyzed.
 Carefully build this circuit on a breadboard or other convenient medium.
 Check the accuracy of the circuit’s construction, following each wire to each connection point, and verifying these elements onebyone on the diagram.
 Mathematically analyze the circuit, solving for all values of voltage, current, etc.
 Carefully measure those quantities, to verify the accuracy of your analysis.
 If there are any substantial errors (greater than a few percent), carefully check your circuit’s construction against the diagram, then carefully recalculate the values and remeasure.
Avoid very high and very low resistor values, to avoid measurement errors caused by meter “loading”. I recommend resistors between 1 kΩ and 100 kΩ, unless, of course, the purpose of the circuit is to illustrate the effects of meter loading!
One way you can save time and reduce the possibility of error is to begin with a very simple circuit and incrementally add components to increase its complexity after each analysis, rather than building a whole new circuit for each practice problem. Another timesaving technique is to reuse the same components in a variety of different circuit configurations. This way, you won’t have to measure any component’s value more than once.
A set of batteries are connected in parallel to form a battery bank. Ideally, their individual voltages would be exactly equal, and there would be no stray resistance anywhere in the circuit, but in reality what we have is something like this:

Use Millman’s theorem to calculate the total voltage between the two busses for the battery bank, given these specifications for the four batteries:
Battery Voltage R_{connection +} R_{connection −} R_{internal}
1 11.9 1.2 Ω 1.1 Ω 5.5 Ω
2 12.2 1.0 Ω 1.3 Ω 5.1 Ω
3 12.0 1.4 Ω 0.9 Ω 4.7 Ω
4 12.1 1.1 Ω 1.2 Ω 5.5 Ω
Calculate the voltage across the starter motor terminals of the “dead” car, and the current through the starter motor, while a second car is giving it a jumpstart:

Regard the starter motor itself as a 0.15 Ω resistor, and disregard any resistance of the jumper cables connecting the two cars’ electrical systems together.
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