Thus far, we have used a single-loop series circuit to demonstrate the principles of Kirchhoff’s Loop Rule. What if, however, a circuit has multiple loops through which current ( I ) travels? To further complicate things, what if each loop contains a voltage ( V ) source? How will it be possible to determine the magnitude and direction of current flow under relatively complex and widely varying conditions? What voltage drop occurs as current moves through each resistor ( R ) within such circuitry? Even though a discussion about multi-loop electrical circuitry has been avoided thus far, we may still use Kirchhoff’s Rules to evaluate such systems.
The inevitable solution within the aforementioned dilemma is to analyze circuit loops individually. We oftentimes cannot initially determine the correct direction of current flow within multi-loop circuits, but the Law of Conservation of Energy still applies; thus, we arbitrarily choose current directions within each loop. As we have seen beforehand, a wrong direction around a loop will yield negative current values. For this reason, negative determinations of current will allow us to make any and all changes that are necessary to create diagrams that accurately reflect how a given circuit operates.
For the time being, we should become very familiar with Kirchhoff’s Loop Rule prior to solving multi-loop problems. The Loop Rule is an extension of the Principle of Conservation of Charge, which states that the net quantity of charge flowing into a steady-state system equals the net quantity of charge flowing out of the same system. Let’s first consider the following circuit comprised to two loops that each contain a voltage source:
It is useful to envision the smaller loop as being a series component that rests within the perimeter of the larger loop. The points at which the larger loop becomes enjoined with the smaller loop are called “ nodes “. A node is a place within an electrical circuit where a current either spreads out or where multiple currents combine. Although two nodes secure a smaller loop to the larger loop, there are in fact three unique paths over which current may travel:
- The outermost perimeter of the circuit can be regarded as a possible route over which current may flow. This joins the larger loop with the upper segment of the smaller loop.
- Another pathway connects the larger loop with the lower segment of the smaller loop.
- The smaller loop is a pathway unto itself.
To make things a bit easier, the letters A, B, C, D, and E are strategically placed along any segment of wire that connects two or more circuit elements. Beginning at the lower left corner of the larger loop, we place the letter “ A “. As we move upward along the circuit and cross the 10 Ω resistor, we place a “ B “ in the upper-left corner of the larger loop. Prior to reaching the smaller loop, we pass through the 20 Ω resistor and then place a “ C “ upon the node that connects the larger loop to the smaller one. Notice how the upper branch of the smaller loop contains two circuit elements while the lower branch contains only one.
Current along the upper branch of the smaller loop will pass the 20 V source and the 10 Ω resistor, while a current that travels along the lower branch of the smaller loop will pass through a single 30 Ω resistor. For this reason, the C-segment of the circuit contains the wire that connects the aforementioned 20 Ω resistor to the node and connects the two lengths of wire that leave the node and connect it to the upper and lower branches of the smaller loop. A “ D “ is then placed between the 20 V source and the 10 Ω resistor along the uppermost path. Finally, we place an “ E “ at the node that reconnects the upper and lower branches of the smaller loop. When we leave the node and travel back towards the starting point, we cross over a 40 V source; however, there is no need to label anything else. The path traveled prior to arriving at the 40 V source is part of our E-section, and the length of wire that emerges from this voltage source is part of our original A-section. Without a doubt, it is of crucial importance to correctly identify and label the various nodes and segments of a multi-loop circuit prior to using the Node Rule to solve problems.
RENAISSANCE PHYSICS 