**CAPACITOR IN SERIES AND IN PARALLEL AND RESISTOR IN SERIES AND IN PARALLEL**

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**INTRODUCTION**

Capacitor and resistor are the two basic components used in electrical and electronic circuits that are further classified into active and passive components. Active components control the flow of energy and are capable of introducing net energy into the circuit, whereas passive components cannot rely on a source of power and are incapable of controlling current by means of another electrical signal.

Resistors and capacitors come under the category of passive components, except resistors limit the flow of current in a circuit, whereas capacitors provide reactance to the flow of current and are used to store electrical charge. They are the most essential components employed in various electrical or electronic circuits.

**THE CONCEPT OF CAPACITOR **

The capacitor is a component which has the ability or “capacity” to store energy in the form of an electrical charge producing a potential difference (*Static Voltage*) across its plates, much like a small rechargeable battery.

There are many different kinds of capacitors available from very small capacitor beads used in resonance circuits to large power factor correction capacitors, but they all do the same thing, they store charge.

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In its basic form, a capacitor consists of two or more parallel conductive (metal) plates which are not connected or touching each other, but are electrically separated either by air or by some form of a good insulating material such as waxed paper, mica, ceramic, plastic or some form of a liquid gel as used in electrolytic capacitors. The insulating layer between a capacitors plates is commonly called the **Dielectric**.

Due to this insulating layer, DC current cannot flow through the capacitor as it blocks it allowing instead a voltage to be present across the plates in the form of an electrical charge.

The conductive metal plates of a capacitor can be either square, circular or rectangular, or they can be of a cylindrical or spherical shape with the general shape, size and construction of a parallel plate capacitor depending on its application and voltage rating.

**CONCEPT OF RESISTOR**

**Resistor**, the most basic yet essential passive component of an electronic circuit, is nothing but a device that restricts the flow of current. So why is it called passive? It is called so because it dissipates power rather than generating it in a circuit.

Resistor is an electrical component that reduces the electric current. The resistor’s ability to reduce the current is called resistance and is measured in units of ohms (symbol: Ω). If we make an analogy to water flow through pipes, the resistor is a thin pipe that reduces the water flow.

Resistors can be connected in series; that is, the current flows through them one after another. The circuit shows three resistors connected in series, and the direction of current is indicated by the arrow.

** **CAPACITORS IN SERIES

Much like resistors are a pain to add in parallel, capacitors get funky when placed in *series*. The total capacitance of *N* capacitors in series is the inverse of the sum of all inverse capacitances.

If you only have **two** capacitors in series, you can use the “product-over-sum” method to calculate the total capacitance:

Taking that equation even further, if you have **two equal-valued capacitors in series****,** the total capacitance is half of their value. For example two 10F supercapacitors in series will produce a total capacitance of 5F (it’ll also have the benefit of doubling the voltage rating of the total capacitor, from 2.5V to 5V).

**Example for Series Capacitor Circuit:**

Now, in the below example we will show you how to calculate total capacitance and individual rms voltage drop across each capacitor.

As, per the above circuit diagram there are two capacitors connected in series with different values. So, the voltage drop across the capacitors is also unequal. If we connect two capacitors with same value the voltage drop is also same.

Now, for the total value of capacitance we will use the formula from equation (2)

**So, C _{T} = (C_{1} * C_{2}) / (C_{1} + C_{2})**

**Here, C _{1} = 4.7uf and C_{2} = 1uf**

**C _{T} = (4.7uf * 1uf) / (4.7uf + 1uf)**

**C _{T }= 4.7uf / 5.7uf**

**C _{T }= 0.824uf**

Now, voltage drop across the capacitor C_{1} is:

**VC _{1 }= (C_{T} / C_{1}) * V_{T}**

**VC _{1 }= (0.824uf / 4.7uf) * 12**

**VC _{1 }= 2.103V**

Now, voltage drop across the capacitor C_{2} is:

**VC _{2 }= (C_{T} / C_{2}) * V_{T}**

**VC _{2 }= (0.824uf / 1uf) * 12**

**VC _{2 }= 9.88V**

# CAPACITORS IN PARALLEL

The voltage ( Vc ) connected across all the capacitors that are connected in parallel is **THE SAME**. Then, **Capacitors in Parallel** have a “common voltage” supply across them giving:

V_{C1} = V_{C2} = V_{C3} = V_{AB} = 12V

In the following circuit the capacitors, C_{1}, C_{2} and C_{3} are all connected together in a parallel branch between points A and B as shown.

When capacitors are connected together in parallel the total or equivalent capacitance, C_{T} in the circuit is equal to the sum of all the individual capacitors added together. This is because the top plate of capacitor, C_{1} is connected to the top plate of C_{2} which is connected to the top plate of C_{3} and so on.

The same is also true of the capacitors bottom plates. Then it is the same as if the three sets of plates were touching each other and equal to one large single plate thereby increasing the effective plate area in m^{2}.

Since capacitance, C is related to plate area ( C = ε(A/d) ) the capacitance value of the combination will also increase. Then the total capacitance value of the capacitors connected together in parallel is actually calculated by adding the plate area together. In other words, the total capacitance is equal to the sum of all the individual capacitance’s in parallel. You may have noticed that the total capacitance of parallel capacitors is found in the same way as the total resistance of series resistors.

**RESISTORS IN SERIES**

Individual resistors can be connected together in either a series connection, a parallel connection or combinations of both series and parallel, to produce more complex resistor networks whose equivalent resistance is the mathematical combination of the individual resistors connected together.

A resistor is not only a fundamental electronic component that can be used to convert a voltage to a current or a current to a voltage, but by correctly adjusting its value a different weighting can be placed onto the converted current and/or the voltage allowing it to be used in voltage reference circuits and applications.

Resistors in series or complicated resistor networks can be replaced by one single equivalent resistor, R_{EQ} or impedance, Z_{EQ} and no matter what the combination or complexity of the resistor network is, all resistors obey the same basic rules as defined by *Ohm’s Law** and Kirchhoff’s Circuit Laws.*

Resistors are said to be connected in “Series”, when they are daisy chained together in a single line. Since all the current flowing through the first resistor has no other way to go it must also pass through the second resistor and the third and so on. Then, resistors in series have a **Common Current** flowing through them as the current that flows through one resistor must also flow through the others as it can only take one path.

Then the amount of current that flows through a set of resistors in series will be the same at all points in a series resistor network. For example:

In the following example the resistors R_{1}, R_{2} and R_{3} are all connected together in series between points A and B with a common current, I flowing through them.

### Series Resistor Circuit

As the resistors are connected together in series the same current passes through each resistor in the chain and the total resistance, R_{T} of the circuit must be **equal** to the sum of all the individual resistors added together. That is

and by taking the individual values of the resistors in our simple example above, the total equivalent resistance, R_{EQ} is therefore given as:

R_{EQ} = R_{1} + R_{2} + R_{3} = 1kΩ + 2kΩ + 6kΩ = 9kΩ

So we see that we can replace all three individual resistors above with just one single “equivalent” resistor which will have a value of 9kΩ.

Where four, five or even more resistors are all connected together in a series circuit, the total or equivalent resistance of the circuit, R_{T} would still be the sum of all the individual resistors connected together and the more resistors added to the series, the greater the equivalent resistance (no matter what their value).

This total resistance is generally known as the **Equivalent Resistance** and can be defined as; “*a single value of resistance that can replace any number of resistors in series without altering the values of the current or the voltage in the circuit*“. Then the equation given for calculating total resistance of the circuit when connecting together resistors in series is given as:

## Resistors in Series Example

Using Ohms Law, calculate the equivalent series resistance, the series current, voltage drop and power for each resistor in the following resistors in series circuit.

All the data can be found by using Ohm’s Law, and to make life a little easier we can present this data in tabular form.

Resistance | Current | Voltage | Power |

R_{1} = 10Ω |
I_{1} = 200mA |
V_{1} = 2V |
P_{1} = 0.4W |

R_{2} = 20Ω |
I_{2} = 200mA |
V_{2} = 4V |
P_{2} = 0.8W |

R_{3} = 30Ω |
I_{3} = 200mA |
V_{3} = 6V |
P_{3} = 1.2W |

R_{T} = 60Ω |
I_{T} = 200mA |
V_{S} = 12V |
P_{T} = 2.4W |

Then for the circuit above, R_{T} = 60Ω, I_{T} = 200mA, V_{S} = 12V and P_{T} = 2.4W

## The Voltage Divider Circuit

We can see from the above example, that although the supply voltage is given as 12 volts, different voltages, or voltage drops, appear across each resistor within the series network. Connecting resistors in series like this across a single DC supply has one major advantage, different voltages appear across each resistor producing a very handy circuit called a **Voltage Divider Network****.**

This simple circuit splits the supply voltage proportionally across each resistor in the series chain with the amount of voltage drop being determined by the resistors value and as we now know, the current through a series resistor circuit is common to all resistors. So a larger resistance will have a larger voltage drop across it, while a smaller resistance will have a smaller voltage drop across it.

The series resistive circuit shown above forms a simple voltage divider network were three voltages 2V, 4V and 6V are produced from a single 12V supply. Kirchhoff’s Voltage Law states that “*the supply voltage in a closed circuit is equal to the sum of all the voltage drops (I*R) around the circuit*” and this can be used to good effect.

# RESISTORS IN PARALLEL

Unlike the previous series resistor circuit, in a parallel resistor network the circuit current can take more than one path as there are multiple paths for the current. Then parallel circuits are classed as current dividers.

Since there are multiple paths for the supply current to flow through, the current may not be the same through all the branches in the parallel network. However, the voltage drop across all of the resistors in a parallel resistive network IS the same. Then, **Resistors in Parallel** have a Common Voltage across them and this is true for all parallel connected elements.

So we can define a parallel resistive circuit as one where the resistors are connected to the same two points (or nodes) and is identified by the fact that it has more than one current path connected to a common voltage source. Then in our parallel resistor example below the voltage across resistor R_{1} equals the voltage across resistor R_{2} which equals the voltage across R_{3} and which equals the supply voltage. Therefore, for a parallel resistor network this is given as:

In the following resistors in parallel circuit the resistors R_{1}, R_{2} and R_{3} are all connected together in parallel between the two points A and B as shown.

### Parallel Resistor Circuit

In the previous series resistor network we saw that the total resistance, R_{T} of the circuit was equal to the sum of all the individual resistors added together. For resistors in parallel the equivalent circuit resistance R_{T} is calculated differently.

**FUNCTIONS OF CAPACITOR AND RESISTOR**

A resistor is a little resistance package which controls the flow of current to other components in an electrical circuit. It’s not only used to amplify signals but to limit the flow of current, adjust signal levels, terminate transmission lines, etc. It limits the current flow to a safe value.

A capacitor consists of two or more parallel conductor plates with an insulator between them. The function of a capacitor is to keep the positive and negative charges separated from each other. The effect of a capacitor is known as capacitance.

**CONCLUSION **

Resistors are in parallel when each resistor is connected directly to the voltage source by connecting wires having negligible resistance. Each resistor thus has the full voltage of the source applied to it. Each resistor draws the same current it would if it alone were connected to the voltage source (provided the voltage source is not overloaded).

Capacitors are stubborn components; they’ll always try to resist sudden changes in voltage. The filter capacitor will charge up as the rectified voltage increases. When the rectified voltage coming into the cap starts its rapid decline, the capacitor will access its bank of stored energy, and it’ll discharge very slowly, supplying energy to the load. The capacitor shouldn’t fully discharge before the input rectified signal starts to increase again, recharging the cap. This dance plays out many times a second, over-and-over as long as the power supply is in use.

Thus, summarize, there are many different types of resistor available from low cost, large tolerance, general purpose carbon type resistors through to low tolerance, high cost, precision film resistors as well as high power, wire wound ceramic resistors. A resistor regulates, impedes or sets the flow of current through a particular path or it can impose a voltage reduction in an electrical circuit.

In many electrical circuits resistors are connected in series or parallel. A designer might for example combine several resistors with standard values (E-series) to reach a specific resistance value. For series connection, the current through each resistor is equal. There is only one path for the current to follow. The voltage drop however, is proportional to the resistance of each individual resistor.

**REFERENCES **

Dorf, Richard C.; Svoboda, James A. (2001). Introduction to Electric Circuits (5th ed.). New York: John Wiley & Sons.

Harter, James H. and Lin, Paul Y. (1982) Essentials of electric circuits. Reston Publishing Company. pp. 96–97.

Ulaby, Fawwaz Tayssir (1999). Fundamentals of Applied Electromagnetics. Upper Saddle River, New Jersey: Prentice Hall.

Wu, F. Y. (2004). “Theory of resistor networks: The two-point resistance”. Journal of Physics A: Mathematical and General. 37 (26): 6653.

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Peter Hezekiah Lawson (Sir Pee). The CEO of Sir Pee Integrated Services and www.libraryguru.com and www.projectvilla.com.ng. A reputable researcher, ICT Instructor and a publisher of many research works in Education.