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Capacitors in Series and Capacitors in Parallel

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A capacitor is a passive device which stores energy in an electric field and opposes the change in voltage.

An electric field can be created by placing two conducting plates in parallel and having one plate more positive than other as shown in fig.

Parallel Plate Capacitor

The material between two plates is non-conducting or insulating. This insulating material is called dielectric. An example of some dielectrics is air, ceramic, Formica and Teflon.

Capacitance

Capacitance is the property of a capacitor to store energy in an electric field. The energy is stored by the capacitance between parallel plates. The formula for capacitance is;

$C=\frac{Q}{V}$

The unit of capacitance is coulomb per volt, which is defined as farad (F).The farad is too large unit for practical circuits; Therefore capacitance values are expressed in microfarads () or pico-Farad ()

It is possible to obtain capacitance expression in terms of dielectric and other physical factors by considering the parallel plates of above figure.A charge is placed on the plates, each having a surface area of A. Then the flux density between the plates is

$D=\frac{Q}{A}\text{    }\cdots \text{    (1)}$                   

As a consequence of the charge, a voltage V occurs across the capacitor and the field intensity would be;

$\varepsilon =\frac{V}{d}\text{       }\cdots \text{    (2)}$                        

Whereas absolute permittivity of a dielectric medium would be;

\[\text{ }\!\!\varepsilon\!\!\text{ }~~=\frac{D}{\varepsilon }\text{       }\cdots \text{     (3)}\]             

D is the flux density and  is the electric field intensity. Substituting the preceding expression for D and  into equation (3) we find that

$\text{ }\!\!\varepsilon\!\!\text{ }=\frac{D}{\varepsilon }=\left( \frac{Q}{V} \right)\left( \frac{d}{A} \right)$

Q/V is capacitance; so;

\[C=~\text{ }\!\!\varepsilon\!\!\text{ }\frac{A}{d}\text{       }\cdots \text{     (4)}\]                  

Equation (4) indicates that the capacitance is measured by the geometric factors A and d and by the type of dielectric separating the plates. When the plate area is increased, the capacitance is increased. Similarly, when the separation d is decreased between the plates, the capacitance is increased.

The relative permittivity () is the ratio of absolute permittivity of a material to that for a vacuum and is defined as;              

${{\varepsilon }_{r}}=\frac{\varepsilon }{{{\varepsilon }_{o}}}\text{       }\cdots \text{     (5)}$

The relative permittivity is sometimes called dielectric constant. The dielectric constant for air is 1.0006. Now, substituting the expression of  from equation (5) into equation (4) , we obtain

$C={{\varepsilon }_{r}}{{\varepsilon }_{o}}~\frac{A}{d}$

Whereas C is in farads and A and d are in the units of square meter and meter, respectively.

Capacitors in Series

In a manner similar to that used for resistors and inductors, let’s apply Kirchhoff’s voltage law to the following figure as an initial step in determining the total capacitance.

Capacitors in Series

${{V}_{T}}={{V}_{1}}+{{V}_{2}}+{{V}_{3}}\text{      }\cdots \text{     (6)}$                                    

As we know that current through capacitor is proportional to rate of change of voltage across the capacitor. Therefore, it is necessary to express both sides of Equation (6) in terms of rate of change of voltage.

$\frac{d{{V}_{T}}}{dt}=\frac{d{{V}_{1}}}{dt}+\frac{d{{V}_{2}}}{dt}+\frac{d{{V}_{3}}}{dt}\text{        }\cdots \text{     (7)}$                                 

Since current flowing through capacitor is given by

${{i}_{C}}=C\frac{d{{V}_{c}}}{dt}$

We may rewrite equation (7) as      

\[\frac{{{i}_{T}}}{{{C}_{t}}}=\frac{{{i}_{T}}}{{{C}_{1}}}+\frac{{{i}_{T}}}{{{C}_{2}}}+\frac{{{i}_{T}}}{{{C}_{3}}}\text{        }\cdots \text{    (8)}\]                         

Dividing both sides of above equation by iT   yields;

$\frac{1}{{{C}_{T}}}=\frac{1}{{{C}_{1}}}+\frac{1}{{{C}_{2}}}+\frac{1}{{{C}_{3}}}$

Note that total capacitance for two or more capacitors in series is calculated quite differently from that series resistors and series inductors.

In case when only two capacitors are connected in series, the equivalent capacitance CT would be,

${{C}_{T}}=\frac{{{C}_{1}}{{C}_{2}}}{{{C}_{1}}+{{C}_{2}}}\text{        }\cdots \text{      (9)   }$                                         

Equation (9) is known as a product over sum rule. It may be used only when two capacitors are in series. In case, when two or more series capacitors are equal, the following expression may be used:

${{C}_{T}}=\frac{C}{N}$

Whereas C is the value of one of the equal capacitors and N indicates numbers of capacitors.

Capacitors in Parallel

The following figure shows three capacitors connected in parallel

Capacitors in Parallel

 

By applying Kirchhoff’s current law, we obtain

${{i}_{T}}={{i}_{1}}+{{i}_{2}}+{{i}_{3}}\text{    }\cdots \text{    (10)}$                        

As we know that;

${{i}_{C}}=C\frac{d{{V}_{c}}}{dt}$

Using the fact that VT is the voltage across the parallel capacitors, we may rewrite eq. (10) as;

${{C}_{T}}\frac{d{{V}_{t}}}{dt}={{C}_{1}}\frac{d{{V}_{T}}}{dt}+{{C}_{2}}\frac{d{{V}_{T}}}{dt}+{{C}_{3}}\frac{d{{V}_{T}}}{dt}\text{       }\cdots \text{      (11)}$        

Equation 11, can be written as

${{C}_{T}}={{C}_{1}}+{{C}_{2}}+{{C}_{3}}\text{       }\cdots \text{      (12)}$                

In other words, eq. (12) states that when capacitors are connected in series, the total capacitance is equal to the sum of individual capacitors. Unlike parallel resistors and parallel inductors, which are added only by their reciprocals, parallel capacitors are combined like series resistors or series inductors.

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About Ahmad Faizan

Mr. Ahmed Faizan Sheikh, M.Sc. (USA), Research Fellow (USA), a member of IEEE & CIGRE, is a Fulbright Alumnus and earned his Master’s Degree in Electrical and Power Engineering from Kansas State University, USA.