Capacitance and Capacitors

I.  Elementary Characteristics

     In its most elementary state a capacitor

    consists of two metal plates separated by

    a certain distance d, in between the plates

    lies a dielectric  material with dielectric

    constant έ = εoε , where εo is the dielectric of air.

 

The dielectric material allows for charge to accumulate between the capacitor plates.  Air (actually vacuum)

 

has the lowest dielectric value of εo = 8.854 * 10-12 Farads/meter where the Farad is the unit for capacitance.

 

All other materials have higher dielectric values, since they are higher in density and can therefore accumulate

 

more charge.

 

Capacitance is defined to be the amount of charge Q stored in between the two plates for a potential difference

 

or voltage V existing across the plates.  In other words:

 

The capacitance C is given by     C = Q/V      (electrical definition)

 

The Physical meaning of capacitance can be seen by relating it to the physical characteristics of the two plates,

 

so that, the capacitance is related to the dielectric of the material in between the plates, the square area of a

 

plate and the distance between the plates by the formula:      

 

C = εoε A/d   (physical definition)

 

Clearly, the larger the area of the plate the more charge can be accumulated and hence the larger the

 

capacitance.  Also, note that as the distance d increases the Capacitance decreases since the charge cannot be

 

contained as 'densely' as before.

 

Both definitions of Capacitance are compatible, although for our purposes we will be referring mostly to the

 

electrical definition.

 

 

Capacitance and Energy

                                                                                                                           

Perhaps  the two most important differences between capacitors and inductors  and resistors are that both capacitors and inductors values depend on frequency whereas resistors don't, and they can both store energy whereas resistors dissipate the energy.   This is what the picture on the right means to illustrate since touching such a

capacitor will certainly transfer a great deal of charge into our bodies!!

As the capacitor is fully charged the energy is given by:

W = 1/2 * C*V    , where W stands as the symbol for energy and V is the voltage across the plates.

 

Types of Capacitors

The simple two-plate capacitor model falls short in representing all capacitors since we have different types

such as: ceramic disc capacitors, electrolytic capacitors, polyester capacitors, tantalum capacitors and surface mount capacitors.  Each type is selected according to several criteria, essentially: the maximum voltage the capacitor can hold, the value of the dielectric, dimensions and tolerance ratings.

                                                                                                                          

           

                Polyester type            Electrolytic type            surface mount       ceramic disc caps.           

 

The surface mount capacitors are about  the size of a dime or smaller  and therefore certainly used

 

with great advantage on the layout of any PC Board requiring many components.

 

Note that although the Physical definition for different types of capacitors will change depending on their

 

structure and dimensions, however the Electrical definition  C = Q/V  holds for any capacitor type.

 

 

II. Capacitor Circuits

 

1. Basic Capacitor Circuit

      Here, the basic electrical parameters are given.

 

      The voltage V which can be seen as the battery  or voltage across

 

      the capacitor, the capacitance C,  and the current I.

 

 

Before the battery is connected across the capacitor  the voltage across the capacitor is 0 volts.  Once the battery is connected the capacitor will start to charge up and since  I = Q/T (the current is the charge going through a wire over a period of time, by definition) ,  a small change in charge over a small period in time can be expressed as dQ/dT or dq/dt and hence the current must be expressed more properly as I = dq/dt at a specific point of time.  But, since C = Q/V or Q = C*V a very small change in charge will become dq = C*dV, that is

the capacitance times a small change in voltage.  Therefore the current I can be expressed as:

I = C*dV/dt which is a basic equation we will need to come back to later on.  Once the capacitor is fully charged we can state that V = Q/C where Q is the total final charge and obviously this will have to equal the

voltage across the battery.

Of course we have described here only the ideal situation  since any capacitor has losses and more realistically

a resistor should be included in series with the capacitor.  Also, we have omitted to include a switch in the circuit to now when the capacitor starts charging up.

 

 

2.  Capacitors in Parallel

    For two capacitors in parallel the voltage across either one is

 

    the same, namely V.  The charge in C1 is Q1 and the charge in

 

    capacitor C2 is Q2.  By the electrical definition  of capacitance

 

    we can also state that:  C1 = Q1/V  and  C2 = Q2/V.

 

 

The total capacitance of the circuit Ctotal will be given by the total charge Qtotal over the voltage or Qtotal/V.

 

But the total charge for this circuit is the sum of Q1 and Q2 since both capacitors are in parallel and the charge

 

becomes additive.  Hence, the total capacitance will be given by:  Ctotal = (Q1+Q2)/V    or equivalently

 

Ctotal = Q1/V  +  Q2/V  =  C1 +  C2.

 

In the case of more than two capacitors in parallel the total charge Ctotal = (Q1 + Q2 +Q3 + ... Qn)/V

 

where n denotes the total number of capacitors in the circuit.  This equation becomes:

 

Ctotal = C1 + C2 + C3 + ... + Cn .

 

Note that capacitors in parallel add in the same way that resistor add in series!!

 

 

3. Capacitors in Series

    For the case of capacitors in series the total voltage V splits

   

    into the voltage V1(across C1) and the voltage V2(across C2).

 

    The total charge Q will be the charge on the total capacitance.

 

    As in any series circuit the current I is the same throughout.

 

 

    By definition again the total Capacitance Ctotal = Q/V or Q/(V1 + V2).  By taking the reciprocal, i.e:

 

    inverting both sides, we get that:  1/Ctotal  = (V1 + V2)/Q = V1/Q  +  V2/Q  or:  1/(Q/V1)  +  1/(Q/V2);

 

    (May want to grab a piece of paper and pencil to work through the Math!)

 

    Hence  1/Ctotal  = 1/C1  +  1/C2    and for the case of more than two capacitors, we have that:

 

    1/Ctotal  = 1/C1  +  1/C2  +  1/C3 + ... + 1/Cn ,  where n is the total number of capacitors.

   

    Thus, we NOTE that capacitors in series add as resistors in parallel !!!

 

    Again, these are only some of the basics for capacitor circuits and certainly we will have to take a look at

 

    the capacitor and resistor in series so as to complete our exploration of Capacitor Circuits in D.C.

 

    electronics.

 

   

DC Electronics

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