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Capacitors and ESR

Capacitor ESR Ratings

ESR Equivalent Series Resistance

The ESR rating of a capacitor is a rating of quality. A theoretically perfect capacitor would be lossless 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, what they are made of, and how we rate them.

Measuring ESR

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.

Farads

Capacitance (C) is the amount of charge per volt of potential that a capacitor holds.

C = Q / V

where Q = charge (measured in coulombs) and
V = potential difference (measured in 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 is the 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 -capacitance of a capacitor (C) is related to

the plate area (A)
Plate separation distance (d)
permittivity (ε) of the dielectric by the following equation:

$C = \varepsilon A/d$

Here A and d are based on meters as the unit and ε is in Farads per meter (or 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).

Dielectric Constants

Dielectric constant (k) gets its 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.

$k = \varepsilon_{\rm d}/\varepsilon_0$

where ε_d is the permittivity of the dielectric
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 27
- 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 Secrets of capacitor codes page for more information

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). It is the maximum potential difference across a unit thickness of the dielectric before it breaks down and allows a spark. If we exceed the dielectric strength, an electric arc will "flash over" and often weld the plates of a capacitor together.

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.

$Q = \frac{X_{\rm C}}{R}$

where X_C is the capacitive reactance

$X_{\rm C} =\frac{1}{\omega C}=\frac{1}{2\pi f C}$

and R is the soon-to-be-defined term of ESR. Q is inversely proportional to the amount of energy dissipated in the capacitor.

Thus, ESR rating of a capacitor is inversely related to its quality.

Dissipation Factor

The inverse of Q is the dissipation factor (tan(δ)). Thus, tan(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.

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 dissipation rating.

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.

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 bus. 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.

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.

Dielectric losses can be thought of as friction of aligning dipoles and appears as an increase (reduction of the rate of decrease) of the measured ESR as frequency increases. This increase is what makes the resistance vs freq line go flat above a certain frequency.

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 capacitor's 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.

Formulas at a glance

Dissipation Factor

Where:

δ is the angle between the capacitor's impedance vector and the negative reactive axis.
DF is the Dissipation Factor (sometimes expressed as a percentage).
ESR stands for Equivalent Series Resistance
X_C is the capacitive reactance

$DF = \tan \delta = \frac{ESR}{|X_{\rm C}|}$

Capacitive Reactance

Reactance is used to compute amplitude and phase changes of sinusoidal current. It is denoted by the symbol , $\scriptstyle{X}$ and can be used in place of resistance in many calculations - It can be thought of as the effective AC resistance at some frequency. , $X_{\rm C} = \frac{-1}{ \omega C} = \frac{-1}{2 \pi fC}$

The -1 above is because the reactance is negative from the following vector math: Both reactance , $\scriptstyle{X}$ and resistance , $\scriptstyle{R}$ are required to calculate impedance , $\scriptstyle{Z}$ .

In some circuits one of these may dominate, but an approximate knowledge of the minor component is useful to determine if it may be neglected.

$Z = R + jX\,$ where :j² = − 1

The 'magnitude' , $\scriptstyle{|Z|}$ phase , $\scriptstyle{\theta}$ of the impedance depend on the combined action of the resistance and the reactance. , $\scriptstyle{|Z|}$ is the ratio of the voltage and current amplitudes, while the , $\scriptstyle{\theta}$ is the voltage–current phase difference.

$|Z| = \sqrt{ZZ^*} = \sqrt{R^2 + X^2}$ where Z^* is the complex conjugate of Z

$\theta = \arctan{X \over R}$

If , $\scriptstyle{X > 0}$ the reactance is inductive
If , $\scriptstyle{X = 0}$ then the impedance is purely resistive
If , $\scriptstyle{X < 0}$ the reactance is capacitive The reciprocal of reactance ( 1 / X) is susceptance - not a term you will likely meet up with.

Inductive reactance:

X_L = ωL = 2πfL Relationship between angular frequency ω and frequency f

ω = 2πf

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

Resonance Frequency

F_r = Frequency of Resonance

$F_{\rm r} = \frac{1}{2\pi \sqrt{LC}}$

Dielectric Constant to Capacitance

Where

k = dielectric constant (, $\varepsilon_d/\varepsilon_0$ ; dimensionless)
A = area (square meters)
t = thickness of the dielectric (meters) * Q = charge (coulombs)
V = potential difference (volts)

$C = \frac{\varepsilon_0 kA}{t}$ , $C = \frac{Q}{V}$

, $\varepsilon_d$ is the permittivity of the dielectric
, $\varepsilon_0$ is the permittivity of free space =(, $8.85 \cdot 10^{-12} {\rm F}\;{\rm m}^{-1}$ )

$C = \frac{ \varepsilon A}{t}$ , $k = \frac{\varepsilon_d}{\varepsilon_0}$

Stored Energy

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

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

Average power

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

$P_{\rm av} = \frac{CV^2}{2t}$ , $Z_{\rm C} = \sqrt{ESR^2 + X_{\rm C}^2}$

Time Domain Reflectometry (TDR)

Characteristic Impedance of cable formulas

Discontinuance of transmission characteristic impedance

Where

Z_a = characteristic impedance through which the incident wave travels first
Z_b = the characteristic impedance through which the incident wave travels next.
V_r = the reflected wave amplitude,
V_i = the incident wave amplitude
V_t is the transmitted wave amplitude. , $\frac{V_{\rm r}}{V_{\rm i}} = \frac{Z_{\rm b} -Z_{\rm a}}{Z_{\rm b} + Z_{\rm a}}$ , $\frac{V_{\rm r}}{V_{\rm t}} = \frac{2 Z_{\rm b}}{Z_{\rm b} + Z_{\rm a}}$ , $Z_{\rm b} = Z_{\rm a}\left( \frac{V_{\rm i} + V_{\rm r}}{V_{\rm i} - V_{\rm r}}\right)$

Where:

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

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