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Capacitors and ESR

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1 Capacitor ESR Ratings

[edit] Capacitor ESR Ratings

[edit] ESR Equivalent Series Resistance

The ESR rating of a capacitor is a rating of quality. A theoretically perfect capacitor would be loss less and have an ESR of zero. It would have no in-phase AC resistance. We live in the real world and all capacitors have some amount of ESR. To understand why let us review what a capacitor is and what they are made of and how we rate them.

[edit] Measuring ESR

You can measure ESR with a meter called the Capacitor Wizard. Operations notes are at Capacitor Wizard

[edit] What is a Capacitor?

A capacitor consists of two conductive metal plates separated by an insulating dielectric. The dielectric can be made of glass, ceramic, Tantalum oxide, or plastics such as polyethylene or polycarbonate. Even air can be used as the dielectric. When the capacitor holds some energy in the form of extra electrons on one plate and electron holes on the other we say that the capacitor is charged.

[edit] Farads

Capacitance (C) is the amount of charge per volt of potential that a capacitor holds. (C =Q/V where Q = coulombs (the unit of charge) and V = Volts)

Capacitance is measured in farads, but most often a small fraction of a farad thus:

micro-farads uF millionths (10^-6) farads
pico-farads pF (10^-12) farads (sometimes called 'puffs' in engineering slang)

The energy stored in a capacitor is E = CV²/2 E is in joules.

Thus, the average power in watts is P_av= CV²/2t where t = time in seconds.

The maximum voltage rating and its capacitance determine the amount of energy a capacitor holds. The voltage rating increases with increasing dielectric strength and the thickness of the dielectric. The capacitance increases with the area of the plates and decreases with the thickness of the dielectric.

Thus, the capacitance of a capacitor (C) is related to the plate area (A), plate separation distance (d) and permittivity (ε) of the dielectric by the following equation:

C = εA/d Here A and d are based on meters as the unit and ε is in coulombs squared per Newton-meters squared notice the force unit involved - it explains why capacitor microphonics (remember the good old condenser microphone?) and a mechanical failure mode of capacitors).

[edit] Dielectric Constants

Dielectric constant (k) gets it's value by comparison of the charge holding ability of a vacuum where k = 1. Thus, k is the ratio of the capacitance with a volume of dielectric compared to that of a vacuum dielectric.

K = ε_d/ε₀Where ε_d is the permittivity of the dielectric and ε₀ is the permittivity of free space

Air has nearly the same dielectric value as a vacuum with k = 1.0001. Teflon, a very good insulator, has a value of k =2 while the plastics range in the low 2s to low 3s. Mica gets us a k =6. Aluminum oxide is 7, Tantalum's k is 11 and the Ceramics range from 35 to over 6,000.

Dielectric constants vary with temperature, voltage, and frequency making capacitors messy devices to characterize. Whole books have been written about choosing the correct dielectric for an application, balancing the desires of temperature range, Temperature stability, size, cost, reliability, dielectric absorption, voltage coefficients, current handling capacity (ESR). (Ivan Sinclair wrote a nice book on passives; unfortunately, it is out of print. This points to the fact that our universities are no longer teaching this material). See the [../kits/ccode.htm Secrets of capacitor codes page] for more information

[edit] Dielectric strength

Dielectric strength is a property of the dielectric that is usually expressed in volts per mil (V/.001") or volts per centimeter (V/cm). If we exceed the dielectric strength, an electric arc will 'flash over and often weld the plates of a capacitor together.

[edit] Q or Quality Factor

The Q of a capacitor is important in tuned circuits because they are more damped and have a broader tuning point as the Q goes down.

$Q = \frac{X_C}{R}$

where X_C is the capacitive reactance $X_C =\frac{1}{2\pi FC}$

and R is the soon to be defined term of ESR.

Q is proportional to the inverse of the amount of energy dissipated in the capacitor. Thus, ESR rating of a capacitor is inversely related to its quality.

[edit] Dissipation Factor

The inverse of Q is the dissipation factor (δ). Thus, δ = ESR/X_C and the higher the ESR the more losses in the capacitor and the more power we dissipate. If too much energy is dissipated in the capacitor, it heats up to the point that values change (causing drift in operation) or failure of the capacitor.

[edit] Ripple Current Rating

The ripple current is sometimes rated for a capacitor in RMS current. Remembering that P = I²R where R in this case is ESR it is plain to see that this is a power dispassion rating.

[edit] Dielectric Absorption

This is the phenomenon where after a capacitor has been charged for some time, and then discharged, some stored charge will migrate out of the dielectric over time, thus changing the voltage value of the capacitor. This is extremely important in sample and hold circuit applications. The typical method of observing Dielectric Absorption is to charge up a cap to some known DC voltage for a given time, then discharge the capacitor through a 2 ohm resistor for one second, then watch the voltage on a high-input-impedance voltmeter. The ratio of recovered voltage (expressed in percent) is the usual term for Dielectric absorption.

The charge absorption effect is caused by a trapped space charge in the dielectric and is dependent on the geometry and leakage of the dielectric material.

[edit] ESL

ESL (Equivalent Series Inductance) is pretty much caused by the inductance of the electrodes and leads. The ESL of a capacitor sets the limiting factor of how well (or fast) a capacitor can de-couple noise off a power buss.

The ESL of a capacitor also sets the resonate-point of a capacitor. Because the inductance appears in series with the capacitor, they form a tank circuit.

[edit] ESR Defined

ESR is the sum of in-phase AC resistance. It includes resistance of the dielectric, plate material, electrolytic solution, and terminal leads at a particular frequency. ESR acts like a resistor in series with a capacitor (thus the name Equivalent Series Resistance). This resistance often is the cause of failures in capacitor circuits. These circuits look just fine on paper, but the hidden resistance causes failure due to heat buildup.

To charge the dielectric material current needs to flow down the leads, through the lead plate junction, through the plates themselves - and even through the dielectric material. The dielectric losses can be thought of as friction of aligning dipoles and thus appear as an increase (or a reduction of the rate of decrease -- this increase is what makes the resistance vs freq line to go flat.) of measured ESR as frequency increases.

As the dielectric thickness increases so does the ESR. As the plate area increases, the ESR will go down if the plate thickness remains the same.

To test a Capacitors ESR requires something other than a standard capacitor meter. While a capacitor value meter is a handy device, it will not detect capacitor failure modes that raise the ESR. As the years go by, more and more designs rely on low ESR capacitors to function properly. ESR failed caps can present circuit symptoms that are difficult to diagnose.

[edit] Formulas at a glance

$\delta = \frac{ESR}{|X_C|}$

Capacitive reactance:

$X_C = \frac{1}{j \omega C} = \frac{1}{j2 \pi fC}$

Inductive reactance:

$X L = j ω L = j 2π f L$

Relationship between radian frequency $ω$ and frequency 'f':

$ω = 2π f$

$I = \frac{E}{\sqrt{R^2 + \left(\omega L -\frac {1}{\omega C}\right)^2}}$

$F_r = \frac{1}{2\pi \sqrt{LC}}$

Where k = dielectric constant, A = area, t = thickness of the dielectric, Q = coulombs the unit of charge, and V = Volts

$C = \frac{( 8.85 x 10^{-12}) kA}{t}$

$C = \frac{Q}{V}$

Where A (area) and d (thickness) use meters as the unit and $ξ$ is in coulombs (squared per Newton-meters squared), $ξ d$ is the permittivity of the dielectric, and $ξ 0$ is the permittivity of free space

$C = \frac{ \xi A}{d}$

$k = \frac{\xi_d}{\xi_0}$

Where energy E (in joules) stored in a capacitor is given by

$E = \frac{CV^2}{2}$

Thus, the average power in watts where t = time in seconds.

$P_{av} = \frac{CV^2}{2t}$

$Z_C = \sqrt{ESR^2 + X_c^2}$

[edit] Time Domain Refletometry (TDR) formulas

Characteristic Impedance of cable formulas Discontinuance of transmission characteristic impedance

Z_a = characteristic impedance through which the incident wave travels first and Z_b is the characteristic impedance through which the incident wave travels next. V_r is the reflected wave amplitude, V_i is the incident wave amplitude, and V_tis the transmitted wave amplitude.

$\frac{V_r}{V_i} = \frac{Z_b -Z_a}{Z_b + Z_a}$

$\frac{V_r}{V_t} = \frac{2 Z_b}{Z_b + Z_a}$

$Z_b = Z_a\left( \frac{V_i = V_r}{V_i - V_r}\right)$

Where Z₀ is the characteristic impedance: $\tau = RC = \frac{Z_0C}{2}$

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