ELECTRONIC DESIGN NOTES #4  CAPACITORS

The electronics world is ruled by the
MOS [Metal Oxide Semiconductor] technology. That is, tiny pieces of silicon (or selenium/germanium), thin layers
of metal (aluminum or copper) and even thinner layers of metal oxide (aluminum oxide or copper oxide).
Silicon, germanium, selenium materials are enhanced with metallic impurities, therefore forming semiconductor electrical paths using induced electrical fields. The
layers of metal create electrically conductive paths inside the tiny little chips,
while the oxide layers act as
insulatorsand they are also needed to form . . . capacitors!
Capacitors are used generously in electronics since they have many (AC
transient) functions. In this page
capacitors are presented
according to the following simplified structure:
1. Types of Capacitors 2. Useful Formulas 3. Equivalent Series and Parallel Capacitors 4. Growth and Decay in RC Circuits 5. Filtering Capacitors
NOTE The basic notions highlighted in this page are related to
a few electronic design topics presented in the first part, Hardware Design, of
LEARN HARDWARE
FIRMWARE AND SOFTWARE DESIGN.

1. TYPES OF
CAPACITORS
In electronics, capacitors are used in principal to filter spikes and
abrupt voltage variations;
however, capacitors are good for many other things. Let's summarize what we can do with capacitors on our PCBs:
1. to filter EMI (induced ElectroMagnetic Interference); 2. to control the amount of reactance or inductance in AC circuits; 3. to build RC/LC timing circuits; 4. to control the "slewrate" (the ramp) of electrical pulses; 5. to build resonant circuits; 6. to build frequency filtering circuits; 7. to build analogtodecimal circuits; 8. to achieve coupling inductance; 9. to build SMPS (SwitchedMode Power Supply) powerpumps; 10. to store energy; 11. many more.
The most general classification groups capacitors as being:
1. fixed
2. variable
Considering the way they are inserted in electrical/electronic circuits, capacitors are:
1. through hole
2. surface mount
3. big capacitors, or batteries of capacitors having specially designed mechanical fixtures and connectors
A capacitor is made of two metallic plates positioned at a certain distance, and having a dielectric material in between.
The nature of the dielectric influences capacitors' general construction.
Therefore, considering their dielectric capacitors can be:
1. ceramic 2. mica 3. paraffin (plain, or with paper) 4. polyethylene 5. tantalum (metal and oxide) 6. electrolytic (ammonia) 7. air (vacuum) 8. glass 9. porcelain 10. oil (transformer oil)
Variable capacitors come in a few types, as follows:
1. RF variable capacitors
2. trimmers
3. banks of capacitors
NOTE
Although they are used a lot in DC circuits, please note that
capacitors are ACtransient electrical components. As mentioned in
Design Notes 1, there are no instances of pure DC circuits; all of them have some AC characteristics.

2. USEFUL FORMULAS
In order to work with capacitors we need few mathematical formulas (as tools) close at hand. Here they are:

CAPACITOR FORMULAS 
Formula 
Description 
C [F] = Q [C] / U [V]

Capacitance 
K = C / Co (relative value; no unit) 
K = Dielectric constant
C = Capacitance of the capacitor specific to the actual dielectric
Co = Capacitance of the vacuum dielectric capacitor 
C [C] = (ε * A) / d 
C = Capacitance of the parallel plates type
capacitor
ε = dielectric permittivity
A = Area of one plate
d = distance between plates 
ε [C^{2}/N * m^{2}] = ε_{o}
* K 
ε = dielectric permittivity
ε_{o} = 8.85 * 10^{12} [C^{2}/N*m^{2}] = permittivity of vacuum 
Z [Ω] = U [V] / I [A] 
Ohm's Law in AC circuits (details
are presented in Design
Notes 1) 
Z [Ω] = √[R^{2} + (X_{L}  X_{c})^{2}]

Impedance (details
are presented in Design Notes 1) 
X_{c} [Ω] = 1 / (2 * PI * f * C)

Capacitive Reactance 
1 [pF] = 10^{12} [F]

Pico Farad 
1 [nF] = 10^{9} [F]

Nano Farad 
1 [uF] = 10^{6} [F]

Micro Farad 

NOTE
In case you are unfamiliar, the notations within square brackets represent the unit. For example,
X_{c} [Ω]
means: X_{c} is measured in ohms. That is an important aspect, because we could have instances when
the measurement unit is in miliohms, X_{c} [mΩ], or
kiloohms, X_{c} [kΩ], therefore the formulas we use
need to be scaled appropriately. For
that reason the measurement unit must be added explicitly within square brackets.

3. EQUIVALENT SERIES AND PARALLEL CAPACITORS
The equivalent of series resistors is calculated with:
1/C_{T} =
Σ 1/C_{i}
The equivalent of parallel resistors is calculated with: C_{T} = Σ
C_{i}
Calculation examples for series of three capacitors are presented next.


Fig 1: The equivalent capacitance of 3 capacitors in
series
1/C_{T} = 1/C_{1}+1/C_{2}+1/C_{3}
C_{T} = C1*C2*C3 / (C2*C3 + C1*C3 +C1*C2)
C_{T} = 24 / (12+8+6) = 24 / 26 = 0.923 uF
C_{T} is smaller than the smallest serial capacitor


Fig 2: The equivalent capacitance of 3 parallel
capacitors
C_{T} = C_{1}+C_{2}+C_{3}
C_{T} = 2uF+3uF+4uF = 9uF
C_{T} is greater than the greatest parallel capacitor


4. GROWTH AND DECAY IN RC CIRCUITS
A capacitor in series with a resistor forms a timing circuit. This function is used
a lot in
electronics.
In RC
circuits the time constant "T" is:
T [s] = R [Ω] * C [F]
In order to charge the capacitor to full capacity, however, it takes roughly 5 time constants (R*C) calculated
with the above formula. The "decay curve" behaves perfectly similar to the "growth" one
illustrated further down, having only an inverse second derivate (the curve holds water).
Fig 3: RC Charge Graph 

5. FILTERING CAPACITORS
This topic is described in Filter Design Notes #9.

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