# Capacitors 1

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```					Passive Electronic Components
Lecture 7
Page 1 of 14
06-May-2011
Capacitors 1
Lecture Plan

1. Physical basics
2. Parameters specified
3. Typical constructions

1. Physical basics
1.1. Capacitor is the system of two conductive bodies electrically isolated from each
other. Every couple of isolated conductive bodies may be regarded as capacitor.
1.2. Suppose the external voltage source having potential difference V between its
terminals. When it is connected across the capacitor electrodes the portion of
electrons (charge q) is replaced from one electrode (positive) to another (negative)
to establish the same potential difference V between the electrodes as exists between
the terminals of the voltage source. This process is called “charging” of the
capacitor.
1.3. The work is done to separate the equal and opposite charges. The expended energy
is accumulated as electrical potential energy in the capacitor and may be stored in it
after disconnection of the voltage source. Capacitor is inherently an energy-storing
device.
1.4. There is direct proportionality between V and q:

q  CV .

1.5. The constant of proportionality C is called capacitance of the capacitor. C  q V .
1.6. The capacitance depends on:
 electrodes’ geometry (form, dimensions, positional relationship);
 properties of dielectric materials placed between the electrodes. (Electric field
displaces electric charges in the material's molecules, and thus additional energy may be
stored in the capacitor).

Let us apply Gauss Law in integral form to the positive electrode to calculate the charge:
                                                                    
q    0E  ds , where  is Gaussian surface around the positive electrode, ds is the surface element.

The potentials difference V between arbitrary points a  on positive electrode and a  on negative
a    
electrode may be calculated as V   E  dl . Then capacitance C may be expressed or in the terms of
a
                                                            
electric field E or in the terms of potential function  (remember that E   ):
                         
q
  0E  dS             0  dS
                      
C                                                   .
V         a                     
 E  dl
a
Passive Electronic Components
Lecture 7
Page 2 of 14
Compare the above equation with the similar expression for electrical conductance:
                
  E  dS      ds
1    I
  Sb             S            .
R U                    2  1
 E  dl
a

Example. Parallel-plate capacitor:

x                     +q
+ + + + + + + + + + + +
Gaussian surface 
d

- - - - - - - - - - - - - - -
-q

Suppose that:
       kx ;
    
     E     k  const inside the gap between the electrodes (axis x is normal to the
plane of each electrode);

     E  0 outside the gap. (It is simplification! Stray field is neglected).
                         
q
  0  ds            0k  ds
 0S
C         
        
           .
V                         kd  k  0        d

S – surface area of each electrode,
d – spacing between the electrodes,
 0  8.85 1012 F/m – permittivity of free space,
  1 – relative permittivity.

Pay attention that the above solution is precise only when electric field outside the gap between the
electrodes is zero. The more is the ratio of capacitor dielectric permittivity to ambiance permittivity
the weaker is stray field and the higher is precision of the solution. In the capacitors the mentioned
ratio commonly does not exceed four orders of magnitude (see table “Values of Relative Permittivity”
below). Compare with solution for electrical conductance of prism with base area S and length l (see
below) that is based as well on linear expression for electrical field:

1 S
   .
R   l

Practically the equation for resistor is more precise than the relevant equation for capacitor because the
ratio of resistive element conductivity to ambiance conductivity is several orders of magnitude higher
than the mentioned ratio of permittivities in capacitor.
Passive Electronic Components
Lecture 7
Page 3 of 14
1.7. Formulas for capacitance of two capacitors connected in parallel and in series may
be received by analogy with the respective formulas for conductance of two
resistors:

C  C1  C 2 – parallel connection.
1 C  1 C1  1 C 2 – series connection.
1.8. Capacitor electrodes may be metallic bodies of any shape. The classic construction
is a couple of metal foil (metal film) layers separated by insulating interlayer.
1.9. The insulator may be almost every conceivable dielectric material: wax, paper,
plastics, ceramics, glass, oils, minerals, metal oxide films, air, etc. The materials
may be used in combination: paper/wax, paper/epoxy, plastic/plastic, paper/oil,
plastic/oil, etc. Vacuum capacitors are made too, but the market is pretty well
limited to laboratory standards. Almost all commercial capacitors use some
dielectric material with >1.

Values of Relative Permittivity (Dielectric Constant)
Material                      @ 25C
Air or vacuum                               1
Paper                                   2.0…6.0
Glass                                   4.8…8.0
Mineral oil                             2.2…2.3
Mica                                    5.4…8.7
Quartz                                  3.8…4.4
Porcelain                               5.1…5.9
Plastics                                2.1…6.0
Aluminum oxide, Al2O3                      8.4
Tantalum oxide, Ta2O5                    24…27
Niobium oxide, Nb2O5                       42
Water                                      80
Ceramic                               15…20,000

Dielectric constants and minimum thickness of dielectrics used in various types of capacitors
Type of Capacitor               Dielectric      Dielectric constant       Dielectric thickness, m
Aluminum Electrolytic               Aluminum oxide              7…10                0.0013…0.0015 (per 1V)
Tantalum Electrolytic               Tantalum oxide             24…27                 0.001…0.0015 (per 1V)
Film (Metallized)                   Polyester                     3.2                        0.5…2
Ceramic (High  type)               Barium titanate         500…20,000                         1.5
Ceramic (Temp. compensation type)   Titanium oxide            15…250                           1.5

1.10.To get an idea of what a 1 Farad is, calculate the area that would be necessary in a
capacitor built to have 1 Farad with 1mm air gap between flat electrodes. The result
will be 113 square kilometers.

1.11.Equivalent circuit of a capacitor
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Lecture 7
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(Designations are explained below)
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Lecture 7
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2.   Parameters of capacitors.

2.1. Nominal capacitance value is the desired or claimed capacitance value of the
2.2. Capacitance tolerance is a maximum deviation of actual capacitance from the
specified nominal value expressed in percent. The manufacturer guarantees it at
standard (or stated) environmental conditions.
2.3. Voltage rating is a maximal voltage that can be continuously applied to the
capacitor at the temperatures within the applicable temperature category.
2.4. Equivalent series inductance (ESL or XL) is a parasitic inductance seen in series. It
is mostly of concern for power-supply filtering in high-speed digital systems, and in
RF applications.
2.5. Reactance is an opposition to the flow of an alternating or pulsating current caused
by the inductance and capacitance of the capacitor measured in Ohms:

1
X  X L  X C  L       .
C

2.6. DC leakage (DCL). Stray direct current of relatively small value that flows through
Isolation Resistance (IR) across the surface of insulator when a voltage is impressed
across it.
2.7. Equivalent series resistance (ESR, RS). RS is a parameter of capacitor model –
resistance of resistor connected in series with an ideal capacitor. ESR is expressed in
Ohms or milliohms. It may be represented as the square root of the difference
between the absolute value of impedance squared and the reactance squared:

2                2
       1          1 
 RS  LS 
2
  RS      
2                  2
Z
      C          C 
2                     2
        1            1 
Z   LS 
2                      2
RS                             Z      .
       C            C 

ESR is generally made up of the following components:
       metal losses (the resistance of the leads, end terminations, and electrodes);
       dielectric losses (losses caused by leakage current and high frequency losses)..

The listed components are frequency and temperature dependant. (Skin-effect and temperature
influence the resistance of electrodes and terminals. Dielectric loss is frequency and temperature
dependent too for the most materials). In aluminum electrolytic capacitors ESR increases
somewhat at low frequency due to electrochemical effects. But this change is small compared to
XC at low frequency. As long as ESR is frequency dependent the measurement frequency must
be always specified.

Maximal ripple voltage calculation:
A heat may be generated inside a capacitor by AC current and DC leakage. For practical purposes the
second factor is commonly insignificant except of high voltage application (for example 100 µA
leakage current at 10 kV applied voltage results in 1 W power dissipation).
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Lecture 7
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The power dissipated in the capacitor by AC current resulting from applied AC or ripple voltage is
calculated using the following formulas:

P  I 2 RS ;
U2
P       2
RS ;
Z

P  2  1  
2                          2
P          P                 1 
U          Z         RS   LS 
2
                      2fC   .
  RS        
RS         RS               C           RS                
                 

I – AC current, A (RMS);
RS – ESR, ;
U – AC or ripple voltage, V (RMS);
P – maximal dissipated power, W specified in data sheet (for example in the case of tantalum chip
capacitors P = 0.05…0.15W depending on the size);
Z – impedance,  (at frequency under consideration).

In film capacitor data sheets maximal rate of voltage change dV/dt is specified. It limits maximal
AC current I across capacitor:

dV d  q  I
   .
dt dt  C  C

2.8. Dissipation factor (DF, ) is a ratio of ESR to reactance. It is a matter of concern in
AC power applications. DF is usually expressed in % or as decimal fraction.

RS  R
        S  CR S .
X   XC

DF stems from the same loss factors as ESR:
         metal losses (the resistance of the leads, end terminations, and electrodes);
         dielectric losses (losses caused by leakage current and high frequency losses)..

Film-foil capacitors will have a lower dissipation factor than metallized film capacitors because the
thickness and therefore conductance of foil electrodes is greater. DF is frequency dependent
parameter. Indeed, both RS and XC are frequency dependent. The dissipation factor  is often listed as
tan  e where  e is the loss angle (note that for small  e tan  e   e ).

2.9. Quality factor (Q). It is reciprocal to dissipation factor: the ratio of reactance to
equivalent series resistance:
Passive Electronic Components
Lecture 7
Page 7 of 14
1       X   X    1
Q                C     .
        RS  RS CRS

2.10.Power factor (). It is the ratio of ESR to impedance, measured in percent:

RS
          100% .
Z

2.11.Impedance (Z) is total opposition offered to the flow of an alternating or pulsating
current, measured in Ohms. (Impedance is the vector sum of the resistance and the
reactance, i.e. the complex ratio of voltage to current). Minimum value of
impedance is RS. It is reached at resonant frequency  0  1 LS C (see right graph
below).
Z       
Consider the function Z   Z 0  in logarithmic coordinates ln       ; ln    where
Z0       0
Z 0  Z 0  :

2
           1 
2
         1 
Z               L S      ; Z  0   RS    0 LS         RS ;
2                                 2
RS                                     
        C                               0C 

Z  
2
 1                1 
 1            LS      
Z  0           RS              C 
2
 1             LS     LS C  0 LS C                    LS      0 
2
                   1   1
 1                             
  L C  L C 

     ;

 RS

C     0 S         S           RS
            C     0     
Z                     1     LS      
ln           ln       ln                ,    0 ;
Z0            0      R      C       
 S             
Z                       1       LS    
ln            ln       ln                ,    0 ;
Z0              0      R        C     
 S             
Z
ln           0,    0 .
Z0

It means that ln  Z Z 0  function has asymptotes - straight lines (see left graph below).
Passive Electronic Components
Lecture 7
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( f 0   0 2 )
2.12.Insulation resistance (IR). There is always direct current between capacitor
electrodes when the voltage is applied across them. The applied voltage and the
leakage current define insulation resistance. The leakage is relatively low in film
capacitors. Nevertheless, it may be a matter of concern even for film capacitors in
some analog applications (integrators, sample-and-hold circuits). Commonly the
lower is the dielectric constant, the higher is the insulation resistance. Data sheets
normally state it in [MF] units. To determine the IR for a given capacitor, divide
the [MF] value by the actual capacitance. Film capacitors and C0G ceramics
have the best IR. But low-quality ceramics can still work in some leakage-sensitive
applications if their capacitance value is low and their voltage rating is high (and if
their other disadvantages are acceptable). The parameter leakage current is used
instead of insulation resistance when the leakage is relatively high (in electrolytic
capacitors).

NOTE: Capacitors are commonly subjected to 2 insulation resistance tests: the first test determines the
insulation resistance measured across the terminals while the second one determines the insulation
resistance measured between the shorted terminals and exterior electrode attached to external
insulative surface of the capacitor.

2.13.Dielectric absorption (DA) that is sometimes called "soakage", "voltage retention".
In high-voltage power cables this phenomenon is called "return voltage". A
capacitor, once charged, stubbornly retains part of the charge, even after being
discharged (shorted for some seconds).

DA is modeled as an infinite series of RC networks in parallel with the primary capacitance, where the
RDA are mostly very large and the CDA are smaller than the nominal capacitance. However, while the
RC model is useful for predicting how capacitor will behave, it does not reflect the underlying
physics. DA is a source of an error in precision integrator and sample-and-hold circuit. Large high-
voltage oil-filled capacitors have exceptionally high DA. They can be shorted, yet retain enough
charge to be dangerous. They are normally shipped with a shorting jumper across the terminals that
should not be removed until the part has been installed. High-voltage capacitors in TV sets, the
parasitic capacitance in large oil-filled high-voltage transformers, some high-voltage cables are still
dangerous after being discharged.

There is a "standard" test for DA. The capacitor is charged by applying the rated voltage for one
minute. Then it is shorted for two seconds. After one minute retention with opened circuit, the
Passive Electronic Components
Lecture 7
Page 9 of 14
recovered voltage on the capacitor is measured using a high impedance meter. The DA is expressed
as the ratio in percent between the recovered voltage and the charging voltage. There are other
specifications too (for example, MIL-C-19978D calls for a 5 minute/5 second/1 minute time
sequence). These tests are useful for comparing the dielectrics. But the difference in DA of various
dielectrics as seen one minute after the shorting jumper is removed does not necessarily correlate with
the difference in DA as seen in the first few seconds or milliseconds. In general, teflon, polystyrene,
and polypropylene are the best (their DA is as low as 0.02%), while the electrolytic capacitors, high
dielectric constant ceramic capacitors, and oil-filled capacitors are the worst (1% and up).
Impregnated film caps have a DA that reflects the impregnant property rather than the film property.

3.   Typical constructions of capacitors.

Construction of capacitor electrodes

Capacitor electrodes

Smooth-surface                      Etched or porous

Flat                                       Cylindrical

Single-pair                                 Coaxial

Multiple                                  Wound

Capacitors with smooth-surface electrodes. The electrodes are foil, thick- or thin-film. The
dielectrics are:
 ceramics,
 polymer films,
 others (glass, mica, paper, etc.).

Cylindrical coaxial (tubular). Two constructions are available: (a) leaded (with wire terminals), (b)
feed-through.
Passive Electronic Components
Lecture 7
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Feed-through capacitors have almost zero self-inductance. As a result the insertion loss, or
attenuation curve, does not display a resonant peak as exhibited by axial leaded capacitors for
example. The construction allows the capacitor to be mounted through a bulkhead as shown
above. The shielding integrity of an enclosure is not compromised.

Wound capacitor

Flat electrodes. Discoidal capacitor and MLCC.

Etched or porous electrode capacitors. The electrode is:
 etched aluminum or tantalum foil,
 porous slug (sintered powder) of one of the following materials: tantalum, niobium,
niobium oxide (NbO),
 porous carbon.
The necessary part of etched or porous electrode capacitor is electrolyte that is used as second
electrode that matches precisely the form of the first electrode. It may be liquid (gel) or solid
substance (manganese dioxide, conductive polymer).
Passive Electronic Components
Lecture 7
Page 11 of 14

Capacitor dielectric is
a) solid isolative oxide formed on the surface of etched or porous Al, Ta, Nb, NbO electrode;
b) liquid “double layer” of hydrated ions (ions surrounded by polar molecules of water) in the
electrolyte of carbon-based capacitors.

Aluminum electrolytic capacitors.
An aluminum electrolytic capacitor consists of cathode aluminum foil, capacitor paper (electrolytic
paper), electrolyte, and an aluminum oxide layer, which acts as the dielectric, formed on the anode foil
surface. A very thin oxide layer formed by electrolytic oxidation (formation) has relatively high
dielectric constant and rectifying properties. When in contact with electrolyte, the oxide layer
possesses an excellent forward direction insulation property.

By etching the surface of aluminum foil, the effective area of the foil as compared to the initial area
can be enlarged 80…100 times for low voltage capacitors and 30…40 times for middle and high
voltage capacitors.
Since the oxide layer has rectifying properties, a capacitor has polarity. If both the anode and
cathode foils have an oxide layer, the capacitor would be bipolar (non-polar).
A 0.05…0.11 mm thick anode foil and a 0.02…0.05 mm thick cathode foil are facing each other
and are interleaved with a paper and wound into a cylindrical shape. This is called a "capacitor
element." At this stage, it has configuration of a capacitor when the paper and the aluminum oxide
layer are dielectrics, however, the unit has negligible capacitance. But when this capacitor element is
impregnated with liquid electrolyte the electrolyte being electrically connected with cathode foil gets
in touch with oxide dielectric layer. A capacitor attains a high capacitance value after that. Electrolyte
is functioning as a cathode. The basic characteristics required from an electrolyte are:
 high electrical conductivity,
 forming property (healing of any flaws in the dielectric oxide of the anode foil),
 chemical stability with the anode and cathode foils, sealing materials, etc.,
 superior impregnation characteristics,
 low vapor pressure.

Solid tantalum (Ta) capacitor. Solid niobium (Nb) capacitor.
Solid tantalum capacitors are manufactured from a tantalum metal powder. The typical particle sizes
are between 2 and 10 µm. The powder is compressed under high pressure around a tantalum wire to
form a “pellet”. The wire is the anode terminal of the capacitor. The pellet is sintered at high
temperatures (typically 1400 - 1800°C) in vacuum. This drives off any powder impurities. During
sintering the powder forms a sponge-like structure. This structure has high mechanical strength but is
also highly porous with a large internal surface area.

Then dielectric is formed over all surface area of the pellet. It is performed using electrochemical
process - anodization. The pellet is dipped into a very weak solution of phosphoric acid electrolyte and
Passive Electronic Components
Lecture 7
Page 12 of 14
DC voltage is applied between the pellet (positive electrode) and the electrolyte (negative electrode).
The dielectric (tantalum pentoxide Ta2O5) starts to form on the surface of the pellet. Its thickness is
controlled by the voltage applied during the formation process. The dielectric grows at a rate of
approximately 1.7 x 10-9m/V.

The next stage is providing the cathode electrode for the capacitor. This is achieved by pyrolysis of
solution of manganese nitrate Mn(NO3)2 into conductive solid - manganese dioxide MnO2. The pellet
is impregnated into an aqueous solution of manganese nitrate and then baked in an oven at
approximately 250°C to produce manganese dioxide coat over all dielectric surface of tantalum
“sponge”. This process may be repeated several times to build up a thick coat over all internal and
external surfaces of the pellet. The pellet is then dipped into graphite powder and silver polymer ink to
provide a good connection to manganese dioxide cathode plate.

Nowadays niobium (Nb) and niobium oxide (NbO) capacitors are being introduced. Niobium is much
more common element than tantalum. It should not have the supply and cost problems related to
tantalum. Nb and Ta capacitors are manufactured using almost the same manufacturing process.
Conductive oxide NbO is a new powder material for capacitors manufacturing. The advantages of
NbO capacitors that target low voltage (6…16V) range are lower price, lower weight, a much lower
tendency to ignite when compared with tantalum capacitors.

Electrochemical capacitors
Electrochemical capacitors are receiving increased attention in power sources for many applications.
They offer extraordinarily high power density compared with batteries as well as high cycle-life and
maintenance-free operation.
Electrical double-layer capacitor (EDLC). Suppose that two carbon electrodes are dipped in an
electrolyte and connected to a voltage source. It is known that such a galvanic cell does not conduct
DC current if the applied voltage is lower than particular threshold voltage. For example in the case of
0.9% (by weight) NaCl solution the threshold voltage is 2V. The reason of the phenomenon is
electrical double layer that forms on the electrode surface and consists of hydrated ions (ions
surrounded by polar molecules of water). Surrounding water molecules do not let an ion to touch the
electrode and discharge (see the picture below).
The model that gave rise to the term “electrical double layer” was first put forward in the 1850's by
Helmholtz. The attracted ions are assumed to approach the electrode surface and form a layer
balancing the electrode charge, the distance of approach is assumed to be limited to the radius of the
ion and a single sphere of water molecules surrounding each ion. The result is absolutely analogous to
an electrical capacitor that has two plates of charge separated by some distance with the potential drop
occurring in a linear manner between the two plates.
Passive Electronic Components
Lecture 7
Page 13 of 14
A double layer model including layers of solvent                   EDLC structure

Possibility of carbon activation is 2103 m2/g. “Dielectric” thickness (phase boundary) is 2…5 nm. Let
us evaluate possible capacitance of 1g carbon electrode. Suppose minimum possible value of relative
permittivity  = 1,  0  8.85 10 12 F/m. Then
a = 2103 m2/g = 2106 m2/kg             0 S             0am
C            ;   C            
m = 1g = 110-3 kg                       d                  d
=1                                S  am;             8.85  10 12  1  10 3  2  10 6
 0  8.85 1012 F/m                                                                       4( F )
 am                       5  10 9
d = 5nm = 510-9m                  C 0      .
d
         
 F m  m 2 / kg  kg


 F
        m               
C-?

The utilization of principle to store the electrical energy in an aqueous solution, have been first
proposed and claimed as an original development in the patent granted to H. I. Becker, General
Electric Co. in 1957 (U. S. patent 2,800,616 “Low voltage electrolytic capacitor”).
Commercial double layer capacitors originated from Standard Oil of Ohio Research
Center (Cleveland) in 1961 and 1962. But intensive manufacturing and use started in the beginning of
1990s.
Pseudocapacitor is based on chemical reduction/oxidation reactions on the surface of the metal oxide
(not carbon) electrodes. Commonly ruthenium dioxide is used as electrodes material. The electrode
behavior is equivalent to, and measurable as, a capacitance. This capacitance can be large but it is
faradaic and not electrostatic in origin. The technologies employing double-layer and
pseudocapacitance may be combined. Pseudocapacitance can increase the capacitance of an
electrochemical capacitor by as much as an order of magnitude over that of the double-layer
capacitance. However, its cost prevents its large-scale use. So it has been employed mainly in military
applications.
Passive Electronic Components
Lecture 7
Page 14 of 14
4. Marking
Large capacitor nominal value is commonly printed without coding like 10 F.
Smaller types often have 2-3 symbol marking that is read as pF. Last letter may designate the
tolerance. (Tolerance designation by letter is the same as in resistor marking). Example: 47 can be
assumed as 47 pF. A capacitor marked 104 is 10 with 4 more zeros or 100,000pF. 103J is a 10,000 pF
with 5% tolerance.

EIA marking code comprises one letter and one digit (see Table below). A1 means 10 pF.

For example 10 F/25 V capacitor marking might look like:

10F       10      10       25        1061   A7    EA7      E106   E10X2     106    106Y3
25V       25V      25       10        25J                                   25X2    25X2
1
Military
2
Letter X represents the date code
3
Letter Y represents the tolerance

```
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