Contents
 Overview
 Schematic Symbol
 Uses
 Types
 Dielectric Constants Of Common Materials
 Capacitors In Series And In Parallel
 Formulas
 Charge
 Energy
 Force
 Single Disc Capacitance
 Sphere Capacitance
 Parallel Plate Capacitance
 Concentric Cylinder Capacitance
 Equivalent Series Resistance (ESR)
 Leakage Currents
 How Leakage Current Is Specified
 Why Leakage Currents Are Important
 Measuring The Leakage Current Of A Capacitor
 Voltage Dependence
 Decoupling
 Mains Line Filters
 Dielectric Soakage
 Charge Pumps (Bootstrapping)
 Capacitor Packages
 Feedthrough Capacitors
 Repairing Electrolytics
 Energy Loss While Charging A Capacitor
Overview
Capacitors are a passive electronic component that stores charge between two conductive surfaces. The conductive surfaces (plates) are usually very close together, with a dielectric inbetween, to maximize the capacitance for a given size. They are a popular component, that along with resistors, make up the two most commonly used components on a circuit board. They come in a large range of sizes, from water tank sized caps to small SMD capacitors and capacitors formed from tracks on PCB’s.
The mechanical equivalent of a capacitor is a spring. The larger the capacitance, the stronger the spring. This is using the forcevoltage mechanical equivalent.
Schematic Symbol
Below is a table of most of the capacitor schematic symbols in use.
Uses
Some of the wellknown uses for capacitors are:
 Analog filtering
 Energy storage (supercapacitors)
 Charge pumps (voltage boosting, or a bootstrap capacitor)
 High power energy supplies (e.g. for powering coil and rail guns)
 Oscillators (along with resistors and/or inductors)
 Touch/proximity sensors and screens
 Distance measurement (e.g. the technique used in digital calipers)
A feedforward capacitor is the name given to capacitor between the VOUT and ADJ pins of a linear regulator to improve stability.
Types
Ceramic
Range  1pF – 100uF 
Polarized  No 
Typical Marking 

Uses 

Ceramic capacitors are named after the tiny disc of ceramic material they use for their dielectric. Values above 1nF are usually made from stacked ceramic plates and are called ‘multilayer monolithics’.
Ceramic capacitors can generate audible noise when operated at certain frequencies. This is due to the electrostrictive (not piezoelectric, as many people suggest) effect of the internal dielectric materials in a monolithic ceramic capacitor. This causes the capacitor to mechanically vibrate (the movement is in the order of 1pm1nm), which creates acoustic noise. This noise can commonly be heard in switchmode power supplies and other highfrequency switching devices. Tantalum and electrolytic capacitors do not exhibit this effect, and can be used as a replacement when this noise is undesirable.
Ceramic Dielectrics
The following table lists the common ceramic dielectric codes.
Class 1 (Temperature Compensating)
Standard  Symbol  Temperature Range  Capacitance Tolerance 
JIS  CH  25°C to +85°C  +/ 60ppm/°C 
EIA  C0G  55°C to +125°C  +/ 30ppm/°C 
EIA  NP0  55°C to +150°C  +/ 30ppm/°C 
Class 2 (Temperature Stable)
The following are based on the EIA RS198 standard.
First Character (lower temperature letter)  Second Character (upper temperature letter)  Third Character (change in capacitance over temperature) 

X = 55°C (67°F)  4 = +65°C (+149°F)  P = ±10% 
Y = 30°C (22°F)  5 = +85°C (185°F)  R = ±15% 
Z = +10°C (+50°F)  6 = +105°C (221°F)  S = ±22% 
7 = +125°C (257°F)  T = +22/33%  
8 = +150°C (302°F)  U = +22/56%  
9 = +200°C (392°F)  V = +22/82% 
The most common codes from the above table are X5R, X7R, and Y5V.
The following table lists the class 2 codes defined by the JIS standard.
Standard  Symbol  Temperature Range  Capacitance Tolerance 
JIS  JB  25°C to +85°C  +/ 10% 
There is also the two codes JB (which is similar to X5R) and CH (which is similar to C0G) produced by TDK. They are similar to the codes mentioned except optimised for a smaller temperature range.
Insulation Resistance
The insulation resistance limits for military MLCC’s are:
IR > \( 10^{11}\Omega \) or \( 10^3 M\Omega \cdot uF \), whichever is less, at \( +25^{\circ}C \)
IR > \( 10^{10}\Omega \) or \( 10^2 M\Omega \cdot uF \), whichever is less, at \( +125^{\circ}C \)
IR requirements for commercial MLCC’s are about two times less.
Electrolytic
Range  100nF – 5000uF 
Polarized  Yes (but some special ones aren’t) 
Typical Marking  Because of their large size, the capacitance is usually printed in it’s absolute form on the cylinder. 
Uses 

Electrolytic capacitors uses a very thin electrically deposited metal oxide film (\(Al_2 O_3\))as their dielectric. They have a high capacitance density (well, that was before supercaps came along). They are usually cylindrical in shape, and come in throughhole (axial and radial) and surfacemount types.
In overvoltage conditions, holes can be punched through the dielectric layer and the capacitor will begin to conduct. The good news is that if the overvoltage dissappears quickly enough (e.g. just a surge or spike), the capacitor can selfheal. The bad news is that a if the capacitor heats up enough, the dielectric can boil, create vapours, and the cap explodes. Most electrolytics have a specific “weak spot” on the case which is designed to break in an overpressure situation. This can make quite a bang, and can be dangerous if you happen to be peering closely at the circuit while this happened.
Tantalum
Range  100nF2mF (from 47nF to 10mF on DigiKey as of Jan 2014) 
Polarized  Yes (mark indicates POSITIVE side) 
Typical Marking  Capacitance is usually printed directly onto capacitor 
Uses 

Tantalum capacitors are actually special type of electrolytic capacitor. But they deserve their own category because of their special properties and widespread use. The have lower ESR, lower leakage and higher temperature ranges (up to 125°C) than their electrolytic counterparts.
Most tantalum capacitors are made with a solid electrolyte, and therefore are not prone to the electrolyte evaporation/drying up problems normal electrolytics have. This makes them able to retain their rated capacitance for years, if not decades.
Construction
At the heart of a tantalum capacitor is a pellet of tantalum (\(Ta_2 O_5\)).
Packaging
Tantalum capacitors come in both throughhole and SMD packages.
Price
Tantalum capacitors tend to be more expensive than any other commonly used capacitor (electrolytic, ceramic), and so are usually reserved for applications when a large amount of capacitance with low ESR is needed in a tight space.
Issues
The MOST RETARDED THING about tantalum capacitors is that the polarity indicator is a stripe, next to the POSITIVE end. Whoever decided on this when they were first made needs to be shot. It goes against pretty much all other stripypolaritymark thingies, which all indicate which end is the negative end (think electrolytics, diodes, e.t.c). So, be very careful and vigilant when using these, for it is so easy for forget this rule!
Tantalum capacitors are more susceptible to reverse and overvoltage than their electrolytic counterparts. At a high enough voltage, the dielectric breaks down and the capacitor begins to conduct. The current can generate planty of heat, and here’s the best part, it can start of a minithermite reaction between tantalum and manganese dioxide. Some slightly better news to offset this is that at lowenergy breakdowns, tantalum capacitors can actually selfheal and stop the leakage current.
Because of their large operating temperature range, stability, and high price, they are often found in medical and space equipment.
Film Capacitors
Synoyms  MKT, MFC’s (metallized film capacitors), power (film) capacitor 
Range  1nF – 10uF 
Polarized  No 
Dielectric  Polyester, Polycarbonate 
Typical Marking  Because of their large size, the capacitance is usually printed in it’s absolute form on the block somewhere. 
Uses 

MKT capacitors are a variant of polyester capacitors usually in a bright yellow, rectangular block.
Metallized film capacitors have a selfhealing effect when an overvoltage even occurs, while others such as ceramic capacitors do not. This makes them safer to use in highpower applications.
Polyester (Green Cap)
Range  1nF – 10uF 
Polarized  No 
Dielectric  Polyester, Polycarbonate 
Typical Marking  Value in pF with multiplier (similar to resistors) 
Uses 

Polyester capacitors use polyester plastic film for their dielectric. They have similar properties to disc ceramic capacitors. They are sometimes called green caps because they have a green outer plastic coating to protect them. The problem with that is that not all polyesters are green! Quite a few are brown, among other colours.
Supercapacitors
Range  10mF1000F 
Polarized  Yes (mark indicates negative side) 
Typical Marking  Capacitance is usually printed directly onto capacitor 
Uses 

Supercapacitors are actually a special type of electrolytic capacitor.
They typically range from 10mF up to 1000F (in a single capacitor). Stacks of these capacitors can produce capacitances as high as your imagination.
You have to be careful, the leakage current of large supercapacitors (10F and greater) can be quite high (100’s uA or mA’s!). Even worse, some datasheets don’t even mention the leakage current! The ESR of a supercapacitor usually decreases with increasing capacitance.
Throughhole and SMD super capacitor packages exist.
Door Knob Capacitors
Door knob (or barrel) capacitors are a form of ceramic capacitor named after their lookalike appearance to a door knob. They are usually rated for high voltages (kV’s), and used in RF applications. They hav a low dielectric loss and linear temperature coefficient of capacitance. They are typically used in the frequency range from 50kHz100MHz.
Dielectric Constants Of Common Materials
Sorted by alphabetic order.
Material  Dielectic Constant (value or range, no unit)  Notes: 

Air  1  See below for data on how temperature, humidity, and pressure influences the dielectric of air. 
Bakelite  4.45.4  
Ethanol  24  
Formica  4.64.9  
Glass  7.68.0  This is common window glass 
Mica  5.4  
Mylar  3.2  
Paper  3.0  
Paraffin  2.1  
Plexiglass  2.8  
Polyethylene  2.3  
Polystyrene  2.6  
Porcelain  5.15.9  
Quartz  3.8  
Rubber  2.8  Hard rubber 
Teflon  2.1  
Vacuum  1.0  
Vinyl  2.84.5  
Water  76.580  Distilled water 
The Dielectric Of Air
The dielectric of air changes with humidity, pressure and temperature.
Temperature  5ppm/C 
Relative Humidity  1.4ppm/%RH 
Pressure  100ppm/atm 
Capacitors In Series And In Parallel
The behaviour of capacitors when connected together in series and in parallel is exactly the opposite behvaiour of what resistors and inductors exhibit.
Capacitors In Parallel
Capacitors in parallel can be treated as one capacitor with the equivalent capacitance of:
$$ C_{total} = C1 + C2 $$
That is, in parallel, the total equivalent capacitance is the sum of the individual capacitances. This is shown in the below diagram.
Connecting capacitors in parallel increases the capacitance. Parallelconnected capacitors occurs everywhere in circuit design. A classic example is bulk decoupling for a switchmode power supply, which will typically have more than one large capacitor connected in parallel on the input.
One of the benefits of connecting many capacitors in parallel rather than using one large capacitor is that you will usually get a lower ESR (equivalent series resistance).
Capacitors In Series
Capacitors in series with each other can be treated as one capacitor with a capacitance:
$$ C_{total} = \frac{1}{\frac{1}{C1} + \frac{1}{C2}} $$
It is usually easier to remember this equation as:
$$ \frac{1}{C_{total}} = \frac{1}{C1} + \frac{1}{C2} $$
This is shown in the following diagram.
Notice how the total equivalent capacitance is less than any one capacitor in the series string. Connecting capacitors in series reduces the capacitance.
One of the benefits of connecting capacitors in series is that each capacitor only sees a portion of the total applied voltage, hence you can apply a higher voltage than the max rated voltage for any single capacitor. However, care must be taken to make sure the capacitors don’t build up a charge imbalance, which could cause a single capacitor to take more than it’s fair share of voltage, and blow up! A balancing circuit can be made by connecting a highvalue resistor(e.g. \(1M\Omega\)) across each capacitor. This causes any unbalanced buildup of charge to dissipate through the resistors, at the expense of increasing the leakage current of the circuit (remember, capacitors have an internal leakage current also). This is similar to how a battery cell charge balancing circuit works.
Formulas
Charge
The charge stored on the plates of a capacitor is related to the voltage and capacitance by:
$$Q = CV$$
where:
\(Q\) = charge stored in plates (Colombs)
\(C\) = capacitance (Farads)
\(V\) = voltage (Volts)
If using this formula, see the Capacitor Charge Calculator.
Energy
The energy stored in a capacitor is:
$$E = \frac{1}{2}CV^2$$
where:
\(E\) = energy stored in the capacitor (Joules)
\(C\) = capacitance (Farads)
\(V\) = voltage across the capacitor (Volts)
As shown by the equation, the energy stored in a capacitor is related to both the capacitance and voltage of the capacitor. A typical 100nF, 6.5V capacitor can store 2.11uJ. Not much huh! If you are really considering capacitors for their energy storage capabilities, you must look at supercapacitors, which have typical values of 100F and 2.5V (as of 2011). This gives 313J of energy, which is useful amount for powering something.
If using this formula, see the Capacitor Energy Calculator.
Force
The force exerted on the two parallel plates of a capacitor is:
$$F = \frac{\epsilon_0 AV^2}{2d^2}$$
where:
\(F\) = outwards force extered on each parallel plate of the capacitor, in Newtons
\(\epsilon_0\) = the permittivity of free space
\(A\) = overlapping area of the two plates, in meters squared
\(V\) = voltage across the capacitor, in Volts
\(d\) = seperation distance between the two plates, in meters
Single Disc Capacitance
The capacitance of a single thin plate, with a ground at ‘infinity’ (or more practically, just very far away) is:
$$C = 35.4 \times 10^{12} \epsilon_r d$$
where:
\(C\) = capacitance (Farads)
\(\epsilon_r\) = relative dielectric constant (1 for a vacuum)
\(d\) = diameter of the thin plate (meters)
Sphere Capacitance
The capacitance of a single sphere, again, with a ground at infinity is:
$$C = 55.6 \times 10^{12} \epsilon_r d$$
where:
\(C\) = capacitance (Farads)
\(\epsilon_r\) = relative dielectric constant (1 for a vacuum)
\(r\) = radius of sphere (meters)
Parallel Plate Capacitance
The capacitance of two parallel plates is approximately
$$C = \epsilon_r \epsilon_o\frac{A}{d}$$
where:
\(\epsilon_o\) = electric constant (\(8.854 \times 10^{12}Fm^{1}\))
\(\epsilon_r\) = dielectric constant of the material between the plates (no unit)
\(A\) = overlapping surface area of the parallel plates (meters squared)
\(d\) = distance between the plates (meters)
Concentric Cylinder Capacitance
The capacitance of two concentric cylinders as shown in the diagram above is:
$$\frac{2 \pi \epsilon_o \epsilon_r}{\ln (\frac{b}{a})} L$$
where:
\(a\) = radius of inner cylinder (meters)
\(b\) = radius of outer cylinder (meters)
\(L\) = length of both cylinders (meters)
and all other variables as previously mentioned
Equivalent Series Resistance (ESR)
Ceramic SMD capacitors have very low ESRs. In fact, in certain applications, this can be a bad thing (such as the input/output stabilization capacitors for linear regulators and DC/DC converters), and either tantalums are used or resistance has to be added in series with the capacitor. Since usually only milliOhms is required, this can be done with an appropriately sized PCB track which is usually snaked to the capacitor terminal.
Electrolytic capacitors typically have a large ESR (there are special lowESR types, but they still don’t compare to ceramic caps).
Capacitor Type  Typical ESR (at 1kHz) 

Supercap (1100F)  40.1Ω 
Since the ESR is proportional to the capacitor’s plate area, for a similar capacitor designs, the ESR decreases with increasing capacitance.
Leakage Currents
Leakage currents are present in all types of capacitor. Leakage current is the sum of electrical losses from energy required to build up the oxide layers, weaknesses in the dielectric, tunnel effects, and cross currents. They are typically increase proportionally to the capacitance of the capacitor. We can reduce the leakage current down to two main factors, the absorption current \( I_{abs} \), and the intrinisic leakage current \( I_{il} \).
$$ I_{leakage} = I_{abs} + I_{il} $$
Absorption currents are due to quantum tunneling of electrons at the metal/ceramic barrier! Absorption currents, \( I_{abs} \) reduce with time and have weak temperature dependence, while intrinsic leakage currents \( I_{il} \) remain constant with time but exponentially increase with temperature.
Desorption currents (depolarization) flow when the voltage on a capacitor is decreased (e.g. when it is shorted). These currents can actually recharge a previously discharged capacitor, sometimes up to dangerous voltages (people experimenting with coil/rail guns can have this problem)!
Capacitors that have had a relatively constant voltage across them for a decent amount of time typically exhibit far less absorption current than one which has not been charged in the shortterm past. This is due to a phenomenon called ‘selfhealing’, in where a charged capacitor will heal defects in the electrolyte. Uncharged electrolytic capacitors may have weakened electrolyte due to ‘dissolution’, the destruction of the dielectric when no charge is present.
The leakage current through a capacitor can be modeled with a resistor in parallel with the actual capacitance, as shown in the image below:
How Leakage Current Is Specified
For electrolytics, the maximum leakage current is usually specified in terms of the capacitance.
$$ I_{leakage} = xC $$
where:
\( I_{leakage} \) = the leakage current, usually specified in units of mA (this is up to the manufacturer and their choice of constant)
\( x \) = a fixed constant (e.g. 0.5)
\( C \) = the capacitance of the capacitor, and again, choice of units is up to the manufacturer
Note how when specified this way, the current is completely independent on voltage. The leakage current for electrolytic supercaps in the range of 1 to 100F is typically 0.5C (mA), where C is the rated capacitance in Farads.
The leakage current for MLCC capacitors is specified by an insulation resistance. To work out the leakage current, you just use Ohm’s law as follows:
$$ I_{leakage} = \frac{V}{R_{insulation}} $$
where:
\( V \) = the voltage across the capacitor
\( R_{insulation} \) = the insulation resistance as specified on the capacitors datasheet
Note how when leakage current is specified this way, it is dependent on the voltage.
Ceramic capacitors are rated with an initial minimum insulation resistance (e.g. 500MR) and then a lower minimum resistance rated over its entire life time (e.g. 50MR).
Why Leakage Currents Are Important
Leakage current becomes an important parameter to consider when designing longlife battery powered circuits. This is especially true for circuits powered of primary batteries with high internal resistance, such as lithium thionyl chloride baterries (LiSOCl2), because large (>100uF) capacitors can be required to help provide energy during high pulse current situations. These capacitors can have significant leakage current.
Measuring The Leakage Current Of A Capacitor
Because of the small current’s/total energy involved, you can’t really measure the leakage current of a capacitor with standard multimeter. One way is to use a dedicated highresistance meter, commonly called a megaohm meter or insulation resistance tester.
Voltage Dependence
Some types of capacitors have a capacitance which changes depending on the applied voltage (well, technically, all do, but I’m talking about a significant/useful change).
The good news is this can be manipulated to make things such as voltagecontrolled oscillators (VCO’s), in where the capacitance is part of a resonant circuit, and the resonant frequency is changed by modifiying the voltage on the capacitor, hence changing the capacitance. Diodes also offer this feature and can be used to make FM radio signals by modulating a highfrequency waveform.
The bad news is that this also adversely affects the capacitance in situations where you want it to stay constant. This can actually be a very significant problems, especially with small package size ceramic capacitors (such as 0603 and 0805 SMD chip capacitors). An excellent explanation on this effects if Maxim Integrated’s “Temperature and Voltage Variation of Ceramic Capacitors, or Why Your 4.7uF Capacitor Becomes a 0.33uF Capacitor“. The following graph is from Maxim’s page, and just serves as an example to show by how much the capacitance can vary in normal operation conditions!
This can upset opamp gains, frequency cutoff points of filters, and the time constant of RC oscillators.
Decoupling
Capacitors are commonly used for decoupling, as this following schematic shows (taken from the RaspberryPi PCB design).
Mains Line Filters
Capacitors used on mains lines for filtering are usually rated with the “XY” scheme.
Capacitors rated with an X are deemed suitable for connecting between two main voltage AC lines (linetoline). They pose no risk if they either fail open or closed circuit. Capacitors with a Y are deemed suitable for connecting between line and neutral. These capacitors do pose a risk if they fail closed circuit, as this would make the ground (and hence chassis) “hot”.
They are also given a number to represent there impulse test rating, as shown in the table below.
Classification:  Impulse Voltage (V): 

X1  4,000 
X2  2,500 
Y1  8,000 
Y2  5,000 
Dielectric Soakage
A weird and little known about property of capacitors is their ability to seemingly ‘create’ energy and charge themselves up when left in certain conditions. This can be particularly dangerous with high voltage capacitors such as the old oilfilled paper capacitors, which would charge themselves up and then give anyone a shock who was unfortunate enough to get too close.
It’s called dielectric soakage because it’s essentially a property of the dielectric which retains some of the charge if a capacitor is discharged quickly and then left open circuit. The voltage climb can be up to 10% of the original voltage on the capacitor.
Charge Pumps (Bootstrapping)
A charge pump (also commonly called bootstrapping), is a way of using capacitors to generate a voltage higher than the supply. A typical chargepump circuit has two capacitors and two diodes, and requires an oscillating input.
It is commonly used as a simple way of driving the gate of a ndoped silicon switch (such as a NChannel MOSFET or IGBT) when being used as a high side driver. Although using a PChannel as the highside driver would not require this voltage stepup, NChannel MOSFET’s are preferred in most cases because of their lower onresistance and price. When using a capacitor to charge the gate of a NChannel MOSFET, you must make sure the capacitor stores enough charge to transfer to the MOSFET gate while raising the voltage enough drive the onresistance as low as you need. A general rule of thumb is that the capacitor should store 100x more charge than the gate charge as stated on the MOSFET’s datasheet. The charge stored in a capacitor can be calculated using the following equation:
$$Q = CV$$
where:
\(Q\) is the charge (Coulombs)
\(C\) the capacitance (Farads)
\(V\) the voltage (Volts)
However, if you really want to optimise the charge pump capacitor, this rule does not suffice. The Fairchild Semiconductor Application Note, “Design And Application Guide Of Bootstrap Circuit For HighVoltage GateDrive IC” has an indepth analysis of the bootstrap capacitor and surrounding circuit.
The following circuit shows a charge pump circuit used to generate 5V from a +5V PWM signal.
Capacitor Packages
Throughhole capacitors can usually be used in a surface mount fashion by lying the caps down on the board, bending the legs 90 degrees, and soldering them onto pads on the PCB.
Ceramic chip capacitors usually have a three letter code which describes the max cap temp, min cap temp, and change over temperature. The following table shows what the three letter code means for “Class II” and “Class III” ceramics. This basically covers all ceramic caps except the NP0/COG capacitors which belong to Class I.
1st Character: Low Temp 
2nd Character: High Temp 
3rd Character: Change Over Temp (max) 


Char  Temp (°C)  Num  Temp (°C)  Char  Change (%) 
Z  +10  2  +45  A  ±1.0 
Y  30  4  +65  B  ±1.5 
X  55  5  +85  C  ±2.2 
–  –  6  +105  D  ±3.3 
–  –  7  +125  E  ±4.7 
–  –  8  +150  F  ±7.5 
–  –  9  +200  P  ±10 
–  –  –  –  R  ±15 
–  –  –  –  S  ±22 
–  –  –  –  T  +22, 33 
–  –  –  –  U  +22, 56 
–  –  –  –  V  +22, 82 
Feedthrough Capacitors
Feedthrough (or feedthru) capacitors are special threeterminal capacitors (sometimes with four connections) used for suppression of RF noise. They are also known under the more general name of an EMI suppression filter or threeterminal capacitor.
Their advantage over a standard decoupling capacitor to ground is lower parasitic series inductance, which offers a lower impedance path for RF noise to ground.
Schematic Symbol
Component Package
Many smaller, PCB suitable feedthrough capacitors come in chip packages, such as the 0603 or 0402 size. They can be distinguished from normal capacitors by the fact that the package will have three or four terminals rather than the standard two.
Uses
Feedthrough capacitors are commonly used in RC, LC, π and ttype filters when good RF performance is required.
Repairing Electrolytics
When electrolytic capacitors get old, they can dry out and stop working properly (for example, their capacitance can reduce and/or maximum dielectric voltage drop). There are many sources quoting that electrolytic capacitors can be repaired by ramping the voltage slowly up to its full rated voltage. It is meant to repair the aluminium oxide dielectric layer.
This can be done with power supply set the rated voltage and a highpower (e.g. 5W) 30kΩ resistor in series with the capacitor, as shown in the image above. Connect the circuit and measure the voltage across the capacitor. It should start at 0V and increase as an inverse exponential as the current through the capacitor decreases. Wait until the voltage across the capacitor gets above 90% of the rated voltage before disconnecting the circuit, as shown in the image below. If the voltage stabilises below 90% of the fullrated voltage, the capacitor is stuffed and can be thrown out.
Energy Loss While Charging A Capacitor
An interesting phenomenon occurs when charging a capacitor from a fixed voltage source (e.g. battery or power supply). Assuming a realworld world situation, there is always going to be some resistance between the voltage source and capacitor. Exactly the same amount of energy is dissipated through this resistance as is stored in the capacitor when charging it up to the voltage source voltage \(V_{cc}\). It doesn’t matter how large or small this resistance is! The resistance could just be the ESR of the capacitor, or it could be a dedicated resistor placed in series to limit the inrush current.
Here is an example schematic:
This quite significant and fixed energy loss has implications when it comes to charging caps in lowpower circuits (e.g. running of a battery), and sizing resistors (including NTC thermistors) for limiting the inrush current to capacitors, normally as part of the front end to a power supply or motor driver.
The Proof
Let’s start from the basics.
We know total energy in the circuit is the integral of power over time:
$$ E_{in} = \int{P_{in} \cdot dt} $$
Using the basic equation for electrical power (\( P = VI \) we can write:
$$ P_{in} = V_{in} \cdot I_{in} $$
Substituting the power equation into integral equation, we get an equation for the energy in terms of the voltage and current:
$$ E_{in} = \int{ V_{in} I_{in} \cdot dt } $$
If we assume a fixed DC voltage source, \( V_{in} \), that does not vary over time, we can bring it outside the integral:
$$ E_{in} = V_{in} \int{ I_{in} \cdot dt } $$
Now using the equation for charge in it’s integral form \( Q = \int{I \cdot dt} \) we can write:
$$ Q_{in} = \int{I_{in} \cdot dt} $$
We can substitute the charge equation into for energy equation:
$$ E_{in} = V_{in} Q_{in} $$
Assuming the cap was fully charged to \( V_{in} \) (o.k., this would take an infinite amount of time, but lets just be realistic and say 99.9% is fully charged), and using the basic formula for the energy in a capacitor (\( E = \frac{1}{2}CV^2 \)), we can write:
$$ E_{cap} = \frac{1}{2}CV_{in}^2 $$
Now, all of the input charge \( Q_{in} \) referred to in charge equation got to the capacitor since the resistor and capacitor share the same current. The equation for the energy in the capacitor can be rewritten in terms of charge (using \( Q=CV \)):
$$ E_{cap} = \frac{1}{2} V_{in} Q_{in} $$
We now have an equation for the energy given out by the voltage source, and an equation for the amount of energy given to the capacitor. Subtracting the two will give the amount of energy lost to the resistance during charging:
$$ E_{res} = E_{in} – E_{cap} $$
$$ = V_{in} Q_{in} – \frac{1}{2} V_{in} Q_{in} $$
$$ = \frac{1}{2} V_{in} Q_{in} $$
Woah, hang on a moment, this is the same as the energy in the capacitor!
This implies that when charging a capacitor from a fixed DC source, you dissipate just as much energy as heat as you store in the capacitor. It does not matter what the resistance is (it could just be the resistance of the wires and the ESR (equivalent series resistance) in the capacitor).
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