Parallel Circuits: Difference between revisions

Latest revision as of 17:26, 7 November 2019

Claimed by Alex Oshin (Fall 2017)

The Main Idea

A simple parallel circuit. R1 and R2 are in parallel because they lie on different branches of the circuit.

Components of a circuit can be connected in two main ways, in series or in parallel. This page focuses on components connected in parallel, which means that the components are connected such that the same voltage is applied to each component. A parallel circuit is a circuit that is connected entirely in parallel. Each component in a parallel circuit occupies its own branch, or path in the circuit, and these branches are connected at locations called nodes. Thus a charge passing through the circuit has multiple possible paths that it can take when it reaches a node. This property is the reason why adding an additional resistor in a parallel circuit results in a decreased total resistance of the circuit (since there is an additional path that the charges can follow). Likewise, if a path is removed, or if the circuit is broken in a single path, more charge is forced to flow through the remaining paths, and the total resistance increases.

A Mathematical Model

Ohm's Law is essential for understanding circuits, and in particular, the relationship between current, voltage, and resistance.

[math]\displaystyle{ I=\frac{V}{R} }[/math]

The voltage is the same for all elements of a parallel circuit.

[math]\displaystyle{ V_{total}=V_1=V_2=...=V_n }[/math]

The total resistance of a parallel circuit is the reciprocal of the sum of the reciprocals of the individual resistances.

[math]\displaystyle{ \frac{1}{R_{total}}=\frac{1}{R_1}+\frac{1}{R_2}+...+\frac{1}{R_n} }[/math]

Applying Ohm's Law to these two equations, we can find an equation for the total current of a parallel circuit.

[math]\displaystyle{ I_{total}=V(\frac{1}{R_{1}}+\frac{1}{R_{2}}+...+\frac{1}{R_{n}}) }[/math]

And the current flowing through a component with resistance [math]\displaystyle{ R_{i} }[/math] will be:

[math]\displaystyle{ I_{i}=\frac{V}{R_{i}} }[/math]

These results are consistent with Kirchhoff's circuit laws, and can be similarly derived.

A Computational Model

A schematic is the easiest way to visualize a parallel circuit, and circuits in general.

The circuit above contains a battery and three resistors. Notice R1, R2, and R3 occupy different branches of the circuit. They are connected by only two nodes, one at the top of the circuit and one at the bottom, even though it may seem like there are four nodes in the schematic.

The following image shows a real-world example.

Each lightbulb acts like a resistor in the previous circuit schematic.

This website is a simple circuit builder that can be used to simulate circuits in parallel(or in series).

This website is a more complicated circuit builder that can also be used to model parallel or series circuits and test the differences in current, voltage, and resistance with with various setups.

Examples

Simple

Question

Calculate the total resistance of this parallel circuit.

Solution

Start by applying the formula for the resistance of a parallel circuit.

[math]\displaystyle{ \frac{1}{R}_{total}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3} }[/math]

In this case, we simply need to substitute the values of [math]\displaystyle{ R_1 }[/math], [math]\displaystyle{ R_2 }[/math], and [math]\displaystyle{ R_3 }[/math] into the formula, and solve for [math]\displaystyle{ R_{total} }[/math].

[math]\displaystyle{ \frac{1}{R}_{total}=\frac{1}{50Ω}+\frac{1}{70Ω}+\frac{1}{100Ω} }[/math]

[math]\displaystyle{ {R}_{total}=22.581Ω }[/math]

Middling

Question

Calculate the total current of the following circuit.

Solution

Start by finding the total resistance of the circuit.

[math]\displaystyle{ \frac{1}{R}_{total}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}+\frac{1}{R_4} }[/math]

[math]\displaystyle{ \frac{1}{R}_{total}=\frac{1}{75Ω}+\frac{1}{150Ω}+\frac{1}{25Ω}+\frac{1}{100Ω} }[/math]

[math]\displaystyle{ {R}_{total}=14.286Ω }[/math]

From here, we can use Ohm's Law to find the total current [math]\displaystyle{ I_{total} }[/math]

[math]\displaystyle{ I=\frac{V}{R} }[/math]

[math]\displaystyle{ I_{total}=\frac{V}{R_{total}} }[/math]

[math]\displaystyle{ I_{total}=\frac{9V}{14.286Ω} }[/math]

[math]\displaystyle{ I_{total}=0.63A }[/math]

This problem can be solved in one step by combining Ohm's Law with what we know about the resistance of a parallel circuit (see A Mathematical Model above)

[math]\displaystyle{ I_{total}=V(\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}+\frac{1}{R_4}) }[/math]

[math]\displaystyle{ I_{total}=(9V)(\frac{1}{75Ω}+\frac{1}{150Ω}+\frac{1}{25Ω}+\frac{1}{100Ω}) }[/math]

[math]\displaystyle{ I_{total}=0.63A }[/math]

Difficult

Question

Calculate the total current of the following circuit.

Solution

This problem relies on understanding both series and parallel circuits. See Series Circuits if any of the following assumptions are unclear.

This easiest way to solve this problem is to calculate the total resistance of the circuit, and then use Ohm's Law to calculate the total current. Since resistors in series can be simplified into a single resistor with resistance equal to the sum of all the resistors in series, we can simplify the circuit by combining R1 and R5, as well as R3 and R4.

Since R2 and R3+R4 are connected in parallel, we can use what we know about parallel circuits to further simplify the circuit.

[math]\displaystyle{ \frac{1}{R}_{total}=\frac{1}{R2}+\frac{1}{R3+R4} }[/math]

[math]\displaystyle{ R_{total}=35.714Ω }[/math]

Finally, these two simplified resistors are connected in series, so we can find the total resistance.

[math]\displaystyle{ R_{total}=(R1+R5)+(R2+R3+R4) }[/math]

[math]\displaystyle{ R_{total}=160.714Ω }[/math]

Now we can use Ohm's Law to calculate the total current.

[math]\displaystyle{ I=\frac{V}{R} }[/math]

[math]\displaystyle{ I_{total}=\frac{V}{R_{total}} }[/math]

[math]\displaystyle{ I_{total}=\frac{9V}{160.714Ω} }[/math]

[math]\displaystyle{ I_{total}=0.056A }[/math]

Connectedness

Circuits can be applied to many areas. However, it is rare that we can find a true parallel circuit in the real world. Many times, real world examples combine both series and parallel circuits. But with some generalizations, we can find examples of parallel circuits in our everyday lives. For example, your phone's battery supplies voltage to the various components and devices within your phone, such as the processor, memory, display, etc., and these are all connected in parallel. Electricity to houses is typically set up in parallel so that one appliance or light can be used without having to use all of them(which would be necessary in series to complete the circuit). On a larger scale, electric or power grids distribute electricity in parallel to the consumers that need it.

History

Electric circuits would not be possible without a constant source of current. Therefore, the battery was a necessary invention for the circuit to be developed. The invention of the battery is generally attributed to Alessandro Volta, who stacked pairs of zinc and copper soaked in brine, and realized that it produced current. The SI unit for electric potential, the volt, is named after him in his honor.

These, and all later batteries, produce direct current, or DC for short, which supplies a constant current in one direction. Circuits created using these batteries are very similar to the circuits discussed on this page. In fact, these circuits formed the basis for Thomas Edison's invention of the incandescent light bulb, and later his attempt at distributing electricity across a power grid.

However, DC has an inherent disadvantage in that there is a large power loss over long distances due the resistance of the wires used to transport the electricity. As such, Edison's rival and contemporary Nikola Tesla used alternating current, or AC, to efficiently distribute electricity over long distances. AC periodically changes direction, and this property allows it to travel long distances with little power loss, and it also generates relatively little heat during the process (another problem of DC). Using electric devices known as transformers, it is possible to change voltage levels in an AC circuit, and this forms the foundation for the electricity that is sent to your home. You will learn later about how a transformer works when you study magnetic induction.

After the invention of lighting, and power grids that could effectively distribute electricity, the further development of home appliances and other electric devices meant that the influence of electric circuits would continue to spread. Nowadays, it is hard to imagine how different our lives would be if we did not have electricity.

Parallel Circuits: Difference between revisions

Latest revision as of 17:26, 7 November 2019

Contents

The Main Idea

A Mathematical Model

A Computational Model

Examples

Simple

Middling

Difficult

Connectedness

History

See also

Further reading

External links

References

Navigation menu

@@ Line 1: / Line 1: @@
+'''Claimed by Alex Oshin (Fall 2017)'''
 ==The Main Idea==
+[[Image:Pc1.png|right|thumb|A simple parallel circuit. R1 and R2 are in parallel because they lie on different branches of the circuit.|330px]]
+Components of a circuit can be connected in two main ways, in series or in parallel. This page focuses on components connected in ''parallel'', which means that the components are connected such that the same voltage is applied to each component. A '''parallel circuit''' is a circuit that is connected entirely in parallel. Each component in a parallel circuit occupies its own ''branch'', or path in the circuit, and these branches are connected at locations called ''nodes''. Thus a charge passing through the circuit has multiple possible paths that it can take when it reaches a node. This property is the reason why adding an additional resistor in a parallel circuit results in a decreased total resistance of the circuit (since there is an additional path that the charges can follow). Likewise, if a path is removed, or if the circuit is broken in a single path, more charge is forced to flow through the remaining paths, and the total resistance increases.
+===A Mathematical Model===
+Ohm's Law is essential for understanding circuits, and in particular, the relationship between current, voltage, and resistance.
+::<math>I=\frac{V}{R}</math>
+The voltage is the same for all elements of a parallel circuit.
+::<math>V_{total}=V_1=V_2=...=V_n</math>
+The total resistance of a parallel circuit is the reciprocal of the sum of the reciprocals of the individual resistances.
+::<math>\frac{1}{R_{total}}=\frac{1}{R_1}+\frac{1}{R_2}+...+\frac{1}{R_n}</math>
+Applying Ohm's Law to these two equations, we can find an equation for the total current of a parallel circuit.
+::<math>I_{total}=V(\frac{1}{R_{1}}+\frac{1}{R_{2}}+...+\frac{1}{R_{n}})</math>
+And the current flowing through a component with resistance <math>R_{i}</math> will be:
+::<math>I_{i}=\frac{V}{R_{i}}</math>
+These results are consistent with Kirchhoff's circuit laws, and can be similarly derived.
+===A Computational Model===
+A schematic is the easiest way to visualize a parallel circuit, and circuits in general.
-Parallel Circuit is a circuit that is connected in parallel. All components in parallel circuit are linked to the same set of electric points, and they can create several branches(individual paths) within a circuit. These individual paths provide multiple pathways to the charge, so whenever the charge encounters a branch it would travel to the lower potential. This means adding an additional resistor in a parallel circuit would result in a decreased resistance.
+[[Image:pc3.png]]
-In a parallel circuit, the potential difference is identical with each resistor positioned in different branches.
-If a single parallel circuit is opened(broke), no charge would flow to that path, but other paths will have charges going through them.
+The circuit above contains a battery and three resistors. Notice R1, R2, and R3 occupy different branches of the circuit. They are connected by only two nodes, one at the top of the circuit and one at the bottom, even though it may seem like there are four nodes in the schematic.
-In order to understand the parallel circuit, it's critical to know the main difference between series circuit and parallel circuits.
+The following image shows a real-world example.
-In a series circuit the current is identical through out the circuit, whereas parallel circuit has current depending on the resistor running through the individual branch. Moreover, in a series circuit the voltage differs,whereas in parallel circuit the voltage is same everywhere.
-===A Mathematical Model===
+[[File:paralleldiagram.jpeg]]
-::<math>V=IR</math>
+Each lightbulb acts like a resistor in the previous circuit schematic.
-Using the basic Ohm's law we could determine the voltage, resistance, and current throughout a circuit.
-:::<math>\frac{1}{R}_{Total}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}+...\frac{1}{R_N}</math>
+This [https://www.physicsclassroom.com/Physics-Interactives/Electric-Circuits/Circuit-Builder/Circuit-Builder-Interactive website] is a simple circuit builder that can be used to simulate circuits in parallel(or in series).
-When calculating the total Reisistance in a parallel circuit, we need to know the basic principle:
+[[File:Parallel circuit2.png|500px]]
+This [https://www.circuitlab.com/circuit/g2qzxs/resistors-in-series-and-parallel/ website] is a more complicated circuit builder that can also be used to model parallel or series circuits and test the differences in current, voltage, and resistance with with various setups.
-'''More resistor in parallel circuit, less resistance.'''
+[[File:Parallel circuit1.png|400px]]
-So, we need to find the sum of reciprocals of individual resistors to derive the total resistor.
+==Examples==
+===Simple===
-:::<math>V_{Total}=V_1=V_2=V_3...=V_N</math>
+'''Question'''
-In a parallel circuit, potential difference through out the circuit is equal everywhere.
+Calculate the total resistance of this parallel circuit.
+[[Image:pc3.png]]
-:::<math>I_{Total}=I_1+I_2+I_3+...I_N</math>
-In a parallel circuit, the total amount of current outside the individual branch equals to the sum of individual branches in the circuit. Thus, the individual current in each branch depends on the resistor in the branch.
+'''Solution'''
-[[File:Current.gif]]
+Start by applying the formula for the resistance of a parallel circuit.
+:::<math>\frac{1}{R}_{total}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}</math>
-:::<math>I_n=I_{Total}\frac{R_{Total}}{R_n}</math>
-'''Current Divider Law'''
+In this case, we simply need to substitute the values of <math>R_1</math>, <math>R_2</math>, and <math>R_3</math> into the formula, and solve for <math>R_{total}</math>.
-Since the current running through each branch of the circuit is dependent on the impedance of that branch, the current divider law can be used to determine the magnitude the current through each individual branch.
-So if we want to find out the current in <math>R_1 </math>
+:::<math>\frac{1}{R}_{total}=\frac{1}{50&Omega;}+\frac{1}{70&Omega;}+\frac{1}{100&Omega;}</math>
-:::<math>I_1=I_{Total}\frac{R_{Total}}{R_1}</math>
-===A Computational Model===
-[[File:ParallelCircuit.jpeg]]
+:::<math>{R}_{total}=22.581&Omega;</math>
-This is a simple way to represent a Parallel Circuit. The schematic version is very useful when we want to compute the values, and understand the concept. <math>R </math> represents a resistor and the <math>Battery</math> represents the total <math>emf</math> in the circuit.
+===Middling===
+'''Question'''
+'''Calculate the total current of the following circuit.'''
-[[File:paralleldiagram.jpeg]]
+[[Image:pc4.png]]
-This diagram is the actual Parallel Circuit we could see in real life. The actual battery could represent the total emf in the previous schematic sketch, and the light bulb would act as a resistor from the previous parallel circuit.
-==Examples==
+'''Solution'''
-These are practical examples we could solve through the formula, and the concept we've learned through this Wiki Page.
+Start by finding the total resistance of the circuit.
-===Simple===
+:::<math>\frac{1}{R}_{total}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}+\frac{1}{R_4}</math>
-'''Calculate the total resistance of this parallel circuit.'''
+:::<math>\frac{1}{R}_{total}=\frac{1}{75&Omega;}+\frac{1}{150&Omega;}+\frac{1}{25&Omega;}+\frac{1}{100&Omega;}</math>
-[[File:simplep.png]]
-===Solution===
+:::<math>{R}_{total}=14.286&Omega;</math>
-:::<math>\frac{1}{R}_{Total}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3} </math>
+From here, we can use Ohm's Law to find the total current <math>I_{total}</math>
-Using this formula we could derive the total resistance of a parallel circuit.
+:::<math>I=\frac{V}{R}</math>
-Plugging in 90 for <math>R_1 </math>, 45 for <math>R_2 </math>, and 180 for <math>R_3 </math>, we get
-:::<math>{R}_{Total}=25.7143 ohm</math>
+:::<math>I_{total}=\frac{V}{R_{total}}</math>
-===Middling===
-'''Calculate the current running through <math>R_1 </math>.'''
+:::<math>I_{total}=\frac{9V}{14.286&Omega;}</math>
-'''Total current running through the circuit is 5 ampere. <math>R_1 </math>= 4 ohm, and <math>R_2 </math>= 6 ohm'''
-[[File:middlep.gif]]
+:::<math>I_{total}=0.63A</math>
-===Solution===
+This problem can be solved in one step by combining Ohm's Law with what we know about the resistance of a parallel circuit (see [[#A Mathematical Model|A Mathematical Model]] above)
+:::<math>I_{total}=V(\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}+\frac{1}{R_4})</math>
-In order to find out the current running through <math>R_1 </math>, we can use the current divider law.
-First calculate the total resistance.
-:::<math>\frac{1}{R}_{Total}=\frac{1}{R_1}+\frac{1}{R_2}</math>
+:::<math>I_{total}=(9V)(\frac{1}{75&Omega;}+\frac{1}{150&Omega;}+\frac{1}{25&Omega;}+\frac{1}{100&Omega;})</math>
-Rearranging this formula we get
-:::<math>{R}_{Total}=\frac{R_1R_2}{R_1+R_2}=2.4 ohm </math>
+:::<math>I_{total}=0.63A</math>
-Now plug in all the value to the following equation.
-:::<math>I_{Total}=5A </math>
-:::<math>I_1=I_{Total}\frac{R_{Total}}{R_1}=3A</math>
 ===Difficult===
-'''Calculate the current running through  <math>R_2 </math>.'''
+'''Question'''
+'''Calculate the total current of the following circuit.'''
+[[Image:pc5.png]]
+'''Solution'''
+This problem relies on understanding both series and parallel circuits. See [[Series Circuits]] if any of the following assumptions are unclear.
+This easiest way to solve this problem is to calculate the total resistance of the circuit, and then use Ohm's Law to calculate the total current. Since resistors in series can be simplified into a single resistor with resistance equal to the sum of all the resistors in series, we can simplify the circuit by combining R1 and R5, as well as R3 and R4.
+[[Image:pc6.png]]
+Since R2 and R3+R4 are connected in parallel, we can use what we know about parallel circuits to further simplify the circuit.
+:::<math>\frac{1}{R}_{total}=\frac{1}{R2}+\frac{1}{R3+R4}</math>
-[[File:difficultpp.png]]
+:::<math>R_{total}=35.714&Omega;</math>
-===Solution===
-First we have to find out the total resistance of this circuit. Since we have both series and parallel component in the circuit, we have to add all the components to find out the total resistance.
+[[Image:pc7.png]]
-:::<math>R_{Parallel}=\frac{R_2R_3}{R_2+R_3}=18.75 ohm </math>
-:::<math>R_{Total}=R_1+R_{Parallel}+R_4=58.75 ohm </math>
-Now using the Ohm's law we could determine the total current running through the circuit.
+Finally, these two simplified resistors are connected in series, so we can find the total resistance.
-::<math>V_{Total}=I_{Total}R_{Total}</math>
+:::<math>R_{total}=(R1+R5)+(R2+R3+R4)</math>
-::<math>V_{Total}=10V</math>
-::<math>I_{Total}=\frac{V_{Total}}{R_{Total}}</math>
-::<math>I_{Total}=0.17A</math>
-We can now use the current divider law to calculate the current running through <math>R_2 </math>.
-:::<math>I_2=I_{Total}\frac{R_{Parallel}}{R_2}=0.10625A</math>
+:::<math>R_{total}=160.714&Omega;</math>
-To check our answer, calculate the current running through <math>R_3 </math>
-:::<math>I_3=I_{Total}\frac{R_{Parallel}}{R_3}=0.06375A</math>
+Now we can use Ohm's Law to calculate the total current.
-:::<math>I_{Total}=I_2+I_3=0.17A</math>
+:::<math>I=\frac{V}{R}</math>
-Since the total current in a parallel circuit equals to the sum of the current running through each branch, we could verify our answers.
-==Connectedness==
+:::<math>I_{total}=\frac{V}{R_{total}}</math>
+:::<math>I_{total}=\frac{9V}{160.714&Omega;}</math>
+:::<math>I_{total}=0.056A</math>
-Circuit, including parallel, series, RC, RL... , could be applied to many areas. Circuits are used for many the power sources in the world. In every day's life we encounter using  electrical events from turning on lights, using a mp3 player, and so on. Majoring in computer engineering, I have to deal with many circuits, and apply it to the real world. I choose this topic to cover more details about the circuit, and since a lot of students had hard time solving the circuit problem in Test 3. I'm interested in this area, because I can really understand what I can't easily visualize it in real life. By studying circuit, I can understand how power or electricity works, and even computer! In detail, I'm interested in the application of a simple circuit to design a computer. How integrated circuits are used to build a hardware and software.
+==Connectedness==
-http://scienceofeverydaylife.discoveryeducation.com/views/other.cfm?guidAssetId=D1507F6E-09C3-4E7B-B1E9-16708E402009
-This website provides some very useful video that shows how simple circuits works in the real world.
+Circuits can be applied to many areas. However, it is rare that we can find a true parallel circuit in the real world. Many times, real world examples combine both series and parallel circuits. But with some generalizations, we can find examples of parallel circuits in our everyday lives. For example, your phone's battery supplies voltage to the various components and devices within your phone, such as the processor, memory, display, etc., and these are all connected in parallel. Electricity to houses is typically set up in parallel so that one appliance or light can be used without having to use all of them(which would be necessary in series to complete the circuit). On a larger scale, electric or power grids distribute electricity in parallel to the consumers that need it.
 ==History==
-Integrated Circuit was invented by two inventors name Jack Kilby, and Robery Noyce in the mid 1950s. The work was not done in a corporation. They just came up with the same idea in the same time. Jack Kilby was working for Texas Instrument, whereas Robery Noyce was an research engineer who founded for Semiconductor Corporation. To design a complex computer machine, they needed something more small with more data, and information compared to original transistors, resistors, and capacitors. So they designed an integrated circuit that has many functions within a single chip. It was used to deal with complex matters with minimum cost. From the start of 1960s, many company started to use a chip, replacing the original transistors, and resistors. This integrated circuit is known as one of the most important invention in human kind.
+Electric circuits would not be possible without a constant source of current. Therefore, the battery was a necessary invention for the circuit to be developed. The invention of the battery is generally attributed to Alessandro Volta, who stacked pairs of zinc and copper soaked in brine, and realized that it produced current. The SI unit for electric potential, the ''volt'', is named after him in his honor.
-I wanted to introduce this specific circuit, because this is what I'm really interested in in my area computer engineering. (Designing the core of the computer)
-[[File:jackpp.jpeg]]
+These, and all later batteries, produce ''direct current'', or DC for short, which supplies a constant current in one direction. Circuits created using these batteries are very similar to the circuits discussed on this page. In fact, these circuits formed the basis for Thomas Edison's invention of the incandescent light bulb, and later his attempt at distributing electricity across a power grid.
-[[File:robertpp.jpeg]]
-== See also ==
+However, DC has an inherent disadvantage in that there is a large power loss over long distances due the resistance of the wires used to transport the electricity. As such, Edison's rival and contemporary Nikola Tesla used ''alternating current'', or AC, to efficiently distribute electricity over long distances. AC periodically changes direction, and this property allows it to travel long distances with little power loss, and it also generates relatively little heat during the process (another problem of DC). Using electric devices known as transformers, it is possible to change voltage levels in an AC circuit, and this forms the foundation for the electricity that is sent to your home. You will learn later about how a transformer works when you study magnetic induction.
-You could refer to other topics where you can build more idea about the circuits.
+After the invention of lighting, and power grids that could effectively distribute electricity, the further development of home appliances and other electric devices meant that the influence of electric circuits would continue to spread. Nowadays, it is hard to imagine how different our lives would be if we did not have electricity.
-:[[Series Circuit]]
+== See also ==
-First, you should know clearly about the series circuit in order to understand the basic circuit design. In real life we could see a parallel circuit combined with the series circuit, as the example shown in before.
-:[[RC]]
+See [[Series Circuit]] for more information about simple circuits.
-When we thoroughly understand the general idea of a circuit, it's very useful to refer to more complicated circuit as RC cirucit.
+See [[RC]] to learn about how a capacitor interacts with a circuit.
+See these topics to deepen your understanding of circuits and electricity.
+:[[Ohm's Law]]
+:[[Current]]
 :[[Node Rule]]
 :[[Loop Rule]]
 :[[Power in a circuit]]
-:[[Current]]
-:[[Ohm's Law]]
-These pages would provide some basic information that we could refer to understand.
+===Further reading===
+Introduction to Electric Circuits 9th edition James A. Svoboda Richard C. Dorf
-===Further reading===
+Basic Electric Circuit Theory Issaak D. Mayegoyz W.Lawson
-Books, Articles or other print media on this topic
+Matter & Interactions Vol. II: Electric and Magnetic Interactions, 4th Edition R. Chabay B Sherwood
 ===External links===
-Internet resources on this topic
+The following are some sites that further expand upon both series and parallel circuits.
+*[http://www.physicsclassroom.com/class/circuits/Lesson-4/Parallel-Circuits Parallel Circuits]
+*[http://www.regentsprep.org/regents/physics/phys03/bparcir/ Basics]
+*[http://science.howstuffworks.com/environmental/energy/circuit3.htm History of Circuit]
+*[http://www.allaboutcircuits.com/textbook/direct-current/chpt-5/what-are-series-and-parallel-circuits/ All about parallel and series circuit]
+*[https://learn.sparkfun.com/tutorials/series-and-parallel-circuits Series and Parallel Circuits]
+*[https://www.swtc.edu/Ag_Power/electrical/lecture/parallel_circuits.htm Parallel Circuits]
+*[http://science.howstuffworks.com/environmental/energy/circuit.htm How Circuits Work]
 ==References==
-This section contains the the references you used while writing this page
+*[https://learn.sparkfun.com/tutorials/series-and-parallel-circuits Series and Parallel Circuits]
+*[https://www.swtc.edu/Ag_Power/electrical/lecture/parallel_circuits.htm Parallel Circuits]
+*[http://science.howstuffworks.com/environmental/energy/circuit.htm How Circuits Work]
+*[https://www.physicsclassroom.com/Physics-Interactives/Electric-Circuits/Circuit-Builder/Circuit-Builder-Interactive Physics Classroom]
+*[https://www.circuitlab.com/circuit/g2qzxs/resistors-in-series-and-parallel/ Circuit Lab]
-[[Category:Which Category did you place this in?]]
+[[Category:Simple Circuits]]