Series Circuits

claimed by Cameron Whiteman spring 2017

Main Idea

A series circuit is the most simple type of electrical circuit in which components are placed in succession of one another. A circuit in series only has one loop that can be drawn throughout.

The electrical connection is not branched in any way and any disruption of the circuit causes the entire circuit to lose current. One can visualize this circuit as simply a closed loop. simple

circuits often contain multiple components of differing resistances and functions which can be solved for using the loop rule. These components include: resistors, switches, capacitors, inductors, and of course, batteries.

The loop Rule

When attempting to solve a series circuit, the simple rule to follow is the Loop rule. This is derived from Kirchoff's laws as well as Ohm's law and allows for electric potential, current, or

resistance(s) to be solved for. To begin, draw a loop beginning from the positive terminal of the battery and traveling in a mathematical loop on a path to the negative terminal. More

difficult circuits may have multiple loops and require more calculation, however, in the end this rule should always be true. The sum of the current at each resistor multiplied

by its resistance must be equal to the electric potential of the battery. In other words, sum of the current going into the circuit must be equal to sum of the current leaving.

A Mathematical Model

Kirchhoff's Current and Voltage Laws apply in a series circuit.

Through Kirchhoff's Current Law, we know that the sum of all current going in must equal the sum of all current going out.

[math]\displaystyle{ \sum{I}_{in} - \sum{I}_{out} = 0 }[/math]

Since there are no nodes for the current to split up, the current throughout a series circuit will always be the same through each component.

Through Kirchhoff's Voltage Law, the sum of all voltage in a closed system must be zero.

[math]\displaystyle{ \sum{V}_{Battery} - \sum{V}_{Components} = 0 }[/math]

Ohm's Law is extremely useful in finding the voltages, resistances, and current throughout the series circuit.

Ohm's Law gives us the following formula:

[math]\displaystyle{ V=IR }[/math]; it can be rearranged to yield [math]\displaystyle{ I=\frac{V}{R} }[/math] and [math]\displaystyle{ R = \frac{V}{I} }[/math]

Total Resistance in a series circuit is the sum of all resistances. It can be used to find the overall current in the circuit, which can then be used to find individual resistances.

Total resistance is described by:

[math]\displaystyle{ R_T=\sum_{n=1}^N {R}_{Series}=R_1+R_2+R_3+...R_N }[/math]

When solving circuits, the voltage across each component can be used in relation with Kirchhoff's Voltage Law in order to find unknown values.

The voltage across an inductor is related by the following equation:

[math]\displaystyle{ \Delta V_{inductor} = L\frac{dI}{dt} }[/math]

The voltage across a capacitor can be determined using the following formula:

[math]\displaystyle{ \Delta V_{capacitor} = \frac{Q}{C} }[/math]

The voltage across a battery is found through the following equation:

[math]\displaystyle{ \Delta V_{battery} = Ɛ_{mf} }[/math]

A Computational Model

The best way to visualize a series circuit is to draw a schematic, which is a simplified representation of the circuit in real life.

Resistors are usually represented in a schematic with

In series, individual resistances can be added up throughought and the sum is equal to the entire resistance across the whole circuit.

[math]\displaystyle{ R_\text{total} = R_1 + R_2 + R_n }[/math]

Batteries are represented in a schematic by

Batteries are what contain the electrochemical energy to power whatever you have connected to your circuit. In series, the total voltage

of the system is equal to the sum of the voltages of the batteries connected, assuming that they are attached in the same direction

(positive terminal to negative terminal). If not, then they cancel each other out and a "net voltage" can be determined.

Switches can be open or closed. An open switch is represented by

Inductors are represented in a schematic by

The total inductance of a system is similar to that of resistors and is equal to the sum of the individual inductances across the circuit.

Capacitors are represented by

The following equation can be utilized to calculate the total capacitance of a system.

[math]\displaystyle{ \frac{1}{C_\mathrm{total}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_n} }[/math].

Examples

For these examples, find the values specified.

Simple

Find the current in the circuit.

Middling

Find the voltage across the battery if the current in the circuit is 0.5 A

Difficult

The voltage across [math]\displaystyle{ R_1 }[/math] is 5 volts. The voltage across [math]\displaystyle{ R_2 }[/math] is 6 volts. What are the resistances of [math]\displaystyle{ R_1, R_2, }[/math] and [math]\displaystyle{ R_3 }[/math] if the current measured across [math]\displaystyle{ R_3 }[/math] is .65 A and the voltage of the battery is 16V?

Solutions to Examples

Simple

[math]\displaystyle{ I=0 }[/math] Since the circuit is open, there is no way for current to flow through the circuit.

Middling

1. Find Total Resistance

[math]\displaystyle{ R_T=R_1+R_2+R_3=10+35+15=60 }[/math] Ohms

2. Use Ohm's Law (current is given already)

[math]\displaystyle{ V=IR=0.5\bullet60=30 }[/math] Volts

Difficult

1. Note that current across a series circuit is the same.

Current = 0.65 A

2. Sum of battery voltage is equal to the sum of all voltage across resistors.

[math]\displaystyle{ \sum{V}_{Battery} = \sum{V}_{Components} }[/math]

[math]\displaystyle{ 16 = 5 + 6 + V_3 }[/math]

[math]\displaystyle{ V_3 = 5 }[/math]

3. Use the rearranged Ohm's Law to find the resistances.

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

[math]\displaystyle{ R_1=\frac{5}{.65}=7.69 }[/math] Ohms

[math]\displaystyle{ R_2=\frac{6}{.65}=9.23 }[/math] Ohms

[math]\displaystyle{ R_3=\frac{5}{.65}=7.69 }[/math] Ohms

Connectedness

Series circuits are the most basic type of circuits.

They are used in all electronics; even parallel circuits can be simplified into a series circuit!

Some realistic applications include making lights, motors, and other electrical appliances work.

The most commonly used example for a series circuit is in lighting. However, Series is generally not the

ideal setup as the loss of one bulb causes the current to cease and loss of power to all bulbs. On a larger

scale, the idea of series is used in everyday household appliances. The device, whether it be a lamp or toaster,

is connected into the wall and a switch either flips on to connect the circuit, allowing current to flow or there is

a disconnect in the system, causing the entire unit to fail.

The most common usage of series circuits in physics appears when breaking down much more complex and detailed circuits

that might even be in parallel. When doing this, Kirchoff's laws are utilized to break up the circuit into multiple individual loops

that, at which point, can be treated as series circuits. This trick can be extremely useful for solving difficult circuits and allows even

the most confusing problems to be broken down into the simplest form.

History

Series circuits date as far back as when the first battery was invented.

In the 1800's Alessandro Volta invented the first battery; it was originally used to produce hydrogen and oxygen from water.

Around the 1880's, however, light bulbs were commercialized and used to illuminate cities- none of this could be done without the basic circuit.

Electric circuits involve both direct current, DC, and alternating current, AC. Direct current, when all of the current only flows in one direction, was first used by Thomas Edison for his electric power transmission. Alternating current, ;when the current alternates directions, was first discovered by Nikola Tesla while he was looking for a way to allow power transmissions to go longer distances than with DC.

Gustav Robert Kirchoff was the main contributor to understanding how circuits worked. He was born March 12, 1824 and died on October 17, 1887. As a renowned German physicist,

he proposed what are now universally accepted laws surrounding the way that circuits work as student in 1845.

See also

Ohm's Law

Resistors and Conductivity

Current

Ammeters,Voltmeters,Ohmmeters

Power in a circuit

RL Circuit

LC Circuit

Further reading

Matter & Interactions, Vol. II: Electric and Magnetic Interactions, 4nd Edition by R. Chabay & B.Sherwood (John Wiley & Sons 2015)

External links

All About Circuits

BuildElectronicCircuits

History of Electrical Circuits

References

"All About Circuits - Electrical Engineering & Electronics Community." All About Circuits - Electrical Engineering & Electronics Community. Web. 30 Nov. 2015.

"Build Electronic Circuits - Electronics Explained in a Simple Way." Build Electronic Circuits. Web. 30 Nov. 2015.

Matter & Interactions, Vol. II: Electric and Magnetic Interactions, 4nd Edition by R. Chabay & B.Sherwood (John Wiley & Sons 2015)

Soclof, Sidney. HowStuffWorks. HowStuffWorks.com. Web. 30 Nov. 2015.