Parallel Circuits vs. Series Circuits*

Claimed by Yu Fu FALL 2016

Parallel Circuits vs. Series Circuits

In a circuit containing a power source and different electrical elements such as resistors, capacitors, or bulb, the elements of the circuit can be connected either in parallel or in series, relative to the power source. Each type of connection affects the resistivity and overall current in the circuit.

Suppose we have three bulbs and a battery to connect together in a circuit.

One way to connect the bulbs is connect them in line with the battery, in such a way that a charge traveling from the high potential end of the battery to its low potential end would have to travel through all three bulbs to get to there. These bulbs are connected in series.

Another way to connect the bulbs is to make them branch off from a common point connected to the high potential of the battery, creating three paths to go from one end of the battery to the other. A charge traveling from the high potential end of the battery to the low potential end would only travel through one of the three branches, and therefore through one of the three bulbs. These bulbs are connected in parallel.

A Mathematical Model

A ohmic circuit is a set of resistors linked together in an unique combination. It can also includes components such as bulbs, heaters and meters as long as they can be considered ohmic. In other words, if all components in a circuit only release heat when connected to a power source like battery, then the circuit is ohmic, and such a circuit is easy to analyze when components are connected in parallel or in series by using Ohm's law. We may need to use other laws when analyzing non-ohmic circuits. Any resistors may become non-ohmic under very high voltage, and some may be so under low voltage.

A Computational Model

For a purely in-series circuit, (or purely in-series combination of resistors)

[math]\displaystyle{ U_{total} }[/math] = [math]\displaystyle{ U_{1} }[/math] + [math]\displaystyle{ U_{2} }[/math] + [math]\displaystyle{ U_{3} }[/math] + ... + [math]\displaystyle{ U_{n} }[/math]

[math]\displaystyle{ I_{total} }[/math] = [math]\displaystyle{ I_{1} }[/math] = [math]\displaystyle{ I_{2} }[/math] = [math]\displaystyle{ I_{3} }[/math] = ... = [math]\displaystyle{ I_{n} }[/math]

[math]\displaystyle{ R_{total} }[/math] = [math]\displaystyle{ R_{1} }[/math] + [math]\displaystyle{ R_{2} }[/math] + [math]\displaystyle{ R_{3} }[/math] + ... + [math]\displaystyle{ R_{n} }[/math]

if there are n ohmic components in the circuit connected in series.

For a purely in-parallel circuit, (or purely in-parallel combination of resistors)

[math]\displaystyle{ U_{total} }[/math] = [math]\displaystyle{ U_{1} }[/math] = [math]\displaystyle{ U_{2} }[/math] = [math]\displaystyle{ U_{3} }[/math] = ... = [math]\displaystyle{ U_{n} }[/math]

[math]\displaystyle{ I_{total} }[/math] = [math]\displaystyle{ I_{1} }[/math] + [math]\displaystyle{ I_{2} }[/math] + [math]\displaystyle{ I_{3} }[/math] + ... + [math]\displaystyle{ I_{n} }[/math]

[math]\displaystyle{ {\frac{1}{R}_{total}} }[/math] = [math]\displaystyle{ {\frac{1}{R}_{1}} }[/math] +[math]\displaystyle{ {\frac{1}{R}_{2}} }[/math] + [math]\displaystyle{ {\frac{1}{R}_{3}} }[/math] + ... + [math]\displaystyle{ {\frac{1}{R}_{n}} }[/math]

For ohmic circuits,

When a circuit is analyzed, cut up circuits into purely in-series combination of components or purely in-parallel combination of components, calculate resistance of each combination of components. Reduce the circuit until it is purely in-series or in-parallel and calculate the current, total resistance and potential difference between two ends of each component.

Examples

Be sure to show all steps in your solution and include diagrams whenever possible

Simple

1. Purely in-series:
2. Purely in-parallel:

Middling

1. simple combination of one in-series component or one in-parallel component: File:Middling.jpg

Difficult

1. complex combination of in-series and in-parallel components:

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