COROLLARY THEOREMS - ELECTRONINC DESIGN NOTES: Capacitors

ELECTRONIC DESIGN NOTES #4

Capacitors

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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). The silicon is enhanced with impurities therefore forming semiconductor electrical paths using induced electrical fields. The layers of metal create electrically conductive paths inside the tiny little chips. The oxide layers act as insulators, and they are also needed to form ... capacitors!

Capacitors are used generously in electronics since they have many (AC) functions. In this page they are presented according to the following 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 electronic design topics presented in the first part Hardware Design of Learn Hardware Firmware and Software Design.

Types of capacitors

In electronics, capacitors are used in principal for filtering spikes and sudden voltage variations; however they are good for many other things. Let's summarize what we can do with capacitors on our PCBs:
1. to filter EMI (induced Elector-Magnetic Interference);
2. to control the amount of reactance or inductance in AC circuits;
3. to build RC timing circuits;
4. to control the "slew-rate" (the ramp) of electrical pulses;
5. to build resonant circuits;
6. to build frequency filtering circuits;
7. to build analog-to-decimal circuits;
8. to achieve coupling inductance;
9. to build SMPS (Switched-Mode Power Supply) power-pumps;
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 two metallic plates positioned at a certain distance, and having a dielectric material in between. The nature of the dielectric influences capacitors' general construction; for example:
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 few types:
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 AC electrical parts. As mentioned in Design Notes 1, there are no instances of pure DC circuits; all of them have some AC characteristics.

USEFUL FORMULAS

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

USEFUL CAPACITORS 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 with 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 [C²/Nm²] = ε_o * K**	ε = dielectric permittivity ε_o = 8.85 * 10^-12 [C²/N*m²] = permittivity of vacuum
Z [Ω] = U [V] / I [A]	Ohm's Law in AC circuits (details in Design Notes 1)
Z [Ω] = √[R² + (X_L - X_c)²]	Impedance (details in Design Notes 1)
*X_c [Ω] = 1 / (2PIfC)**	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, capacitive reactance (X_c), and the symbol ([Ω]) is "the measurement unit"; it tells us that X_c is measured in ohms. That is an important aspect, because we could have instances when the measurement unit is mili-ohms or kilo-ohms, therefore the formulas we use have to be scaled appropriately. For that reason the measurement unit must be added explicitly within square brackets.

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 three capacitors:

	Fig 1: The equivalent capacitance of 3 capacitors in series 1/C_T = 1/C₁+1/C₂+1/C₃ C_T = C1C2C3 / (C2C3 + C1C3 +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 = C₁+C₂+C₃ C_T = 2uF+3uF+4uF = 9uF C_T is greater than the greatest parallel capacitor

GROWTH AND DECAY IN RC CIRCUITS

A capacitor in series with a resistor forms a timing circuit. This function is used intensively in electronics.

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 Growth

FILTERING CAPACITORS

This topic is presented in Filter Design Notes. Important details are also in Learn Hardware Firmware and Software Design book.

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