DIELECTRIC PROPERTIES
2.1 Dielectric constant
2.1.1 Definition
The Dielectric constant (or relative permittivity) is an intrinsic property of dielectric material. Under given conditions is a measure of the extent to which it concentrates electrostatic lines of flux. It is the ratio of the amount of stored electrical energy when a potential is applied, relative to the permittivity of a vacuum. The relative static permittivity is the same as the relative permittivity evaluated for a frequency of zero. The relative permittivity describes the ease by which a dielectric medium may be polarized. The capacitance of a capacitor is proportional to εr.
The dielectric constant (relative static permittivity) is represented as εr or sometimes κ or K. It is defined as
εr= ε/ε0
Where ε is the static permittivity of the material, and ε0 is the electric permittivity of the free space(which is equal to 8.854*10-12 F/m)
The relative permittivity is complex and frequency dependent, which gives the static relative permittivity for ω = 0.
When two electric charges are placed inside a dielectric, the electrostatic force between the charges is changed by a factor 1/εr. Empirically it is observed that the force between two charges never increases (the force stays the same or decreases) by the presence of any dielectric medium, hence the relative permittivity εr ≥ 1. Only the vacuum has εr = 1 (exact). All dielectrics have values larger than one (but some materials, such as non-dense inert gases, have relative permittivity very close to the vacuum value of unity).
Other terms for dielectric constant are relative static permittivity, or relative dielectric constant, or static dielectric constant.
If σ is the charge density on the plate (charge per surface, in SI units C/m2), the strength E of the field E is given by
E = σ/ (2 ε0εr),
Where ε0 is the electric permittivity of free space. When the charge density σ is positive the electric field (a vector) E points away from the plate. Of course, plates of infinite size do not exist, but this formula is applicable when the height is much smaller than the dimensions of the plate, so that border effects can be neglected.
In parallel-plate capacitors border effects can usually be ignored and because both plates have the same charge density (but of opposite sign), the electric field inside a capacitor, filled with a dielectric with εSr, is double that of one plate
E = σ/ (ε0εr).
If the plates have surface area A, they carry a total charge Q = σ A (positive on one plate, negative on the other),
Q = ε0εr A E.
The distance between the plates is d,
then the voltage difference V between the plates is (E / d).
The capacitance C of a capacitor is by definition (Q / V), so that we find that the capacitance of a parallel-plate capacitor is linear in the relative permittivity εr:
C = εr C0 with C0 ≡ (ε0 A) / d.
Clearly, C0 is the capacity with vacuum between the plates, and one may define
εr is equal to the ratio of the capacitance of a capacitor filled with the dielectric to the capacitance of an identical capacitor in a vacuum without the dielectric material.
Because εr > 1, the insertion of a dielectric between the plates of a parallel-plate capacitor always increases its capacitance, or ability to store opposite charges on each plate.
The relative permittivity is defined as a macroscopic property of dielectrics.
The static relative permittivity of a medium is related to its static electric susceptibility, χe by
εr = 1+ χe
The relative static permittivity, εr, can be measured for static electric fields as follows: first the capacitance of a test capacitor, C0, is measured with vacuum between its plates. Then, using the same capacitor and distance between its plates the capacitance Cx with a dielectric between the plates is measured. The relative dielectric constant can be then calculated as
εr = Cx / C0
For time-variant electromagnetic fields, this quantity becomes frequency dependent and in general is called relative permittivity.
2.1.2 Generalizations
The relative permittivity of a dielectric is a function of temperature and, when the dielectric is a gas. For low-symmetry dielectrics (solids and liquid crystals) it may happen that the constant is a second rank tensor, which is represented with respect to a Cartesian coordinate system by a 3 × 3 matrix.
For frequency-dependent (time-dependent) electric fields the relative permittivity is in general a function of the angular frequency ω.
Electric field above infinite plate
An infinitesimal surface element in cylinder coordinates times the surface charge density σ gives an infinitesimal charge in the plate,
dQ= σrdrdΨ
where σ is assumed constant over the plate and for convenience sake we take it positive (hence the field points in the positive z direction).
2.2 CAPACITANCE
2.2.1 Definition
In electromagnetism and electronics, capacitance is the ability of a body to hold an electrical charge. Capacitance is directly proportional to the surface area of the conductor plates and inversely proportional to the separation distance between the plates.
C= (Aε/d)
Where ε= ε0 εr
ε0 is the permittivity of free space where ε0 = 8.854x d-12 F/m
d is the separation between the plates, measured in metres.
2.2.2 Charge Stored
The amount of charge (symbol Q) stored by a capacitor is given by:
Charge, Q = C × V
Where Q = charge in coulombs (C)
C = capacitance in farads (F)
V = voltage in volts (V)
2.2.3 Energy stored
The energy (measured in joules) stored in a capacitor is equal to the work done in moving dq from one plate to the other against the potential difference V = q/C requires the work dW.
When they store charge, capacitors are also storing energy:
Energy, E = ½QV = ½CV²
where E = energy in joules (J).
Note that capacitors return their stored energy to the circuit. They do not 'use up' electrical energy by converting it to heat as a resistor does.
2.2.4 Capacitive Reactance Xc
Capacitive reactance (symbol Xc) is a measure of a capacitor's opposition to AC (alternating current).
Capacitive reactance, Xc = 1 / (2fC)
Where Xc = reactance in ohms ()
f = frequency in hertz (Hz)
C = capacitance in farads (F)
The reactance Xc is large at low frequencies and small at high frequencies. For steady DC which is zero frequency, Xc is infinite (total opposition), hence the rule that capacitors pass AC but block DC.
2.2.5 Time constant
Time constant = R × C
where Time constant is in seconds (s)
R = resistance in ohms (Ώ)
C = capacitance in farads (F)
The time constant is the time taken for the charging (or discharging) current (I) to fall to 1/e of its initial value (Io). 'e' is the base of natural logarithms, an important number in mathematics e = 2.71828
2.2.6 Uses of Capacitors
Capacitors are used for several purposes:
Timing -for example with a 555 timer IC controlling the charging and discharging.
Smoothing -for example in a power supply.
2.3 Loss tangent (tanδ) and Q-factor
2.3.1 Definition
The loss tangent is a parameter of dielectric material that quantifies its inherent dissipation of electromagnetic energy. The term refers to the angle in a complex plane between the resistive (lossy) component of an electromagnetic field and its reactive (lossless) component.
2.3.2 Electromagnetic Field Perspective
For time varying electromagnetic fields, the electromagnetic energy is typically viewed as waves propagating either through free space, in a transmission line, or through a waveguide. Dielectrics are often used to mechanically support electrical conductors and keep them at a fixed separation. Maxwell’s equations are solved for the electric and magnetic field components of the propagating waves that satisfy the boundary conditions of the specific environment's geometry. In such electromagnetic analyses, the parameters permittivity ε, permeability μ, and conductivity σ represent the properties of the media through which the waves propagate. The permittivity can have real and imaginary components such that
ε = ε' − jε'' .
The component ε′ represents the familiar lossless permittivity, is s measure of how much energy from an external electric field is stored in material, which is given by the product of the free space permittivity and the relative permittivity,
ε′ = ε0 εr.
ε′ is mostly greater than 1 for solids and liquids.
where ε″ is the imaginary amplitude of permittivity attributed to bound charge and dipole relaxation phenomena, which is a measure of how dissipative or lossy a material is to an external field. ε″ is usually greater than 1, bt less than ε′.
2.3.3 Loss tangent component
The loss tangent is then defined as the ratio (or angle in a complex plane) of the lossy reaction to the electric field E in the curl equation to the lossless reaction:
tanδ= (ω ε''+σ)/ ω ε'
For dielectrics with small loss, this angle is << 1 and tan(δ)~δ. For electromagnetic wave, the power decays with propagation distance z as
P = Poe − δkz ,
where Po is the initial power,
k= ω (µ ε') ½ = (2π/λ)
ω is the angular frequency of the wave, and
λ is the wavelength in the dielectric.
Also, a similar analysis could be applied to the permeability where
μ = μ' − jμ'' ,
with the subsequent definition of a magnetic loss tangent
tanδ = (μ''/ μ')
where C is the lossless capacitance.
2.3.4 Q-factor
A capacitor's loss tangent is sometimes stated as its dissipation factor, or the reciprocal of its quality factor Q, as follows
tanδ = DF= 1/Q
DF= Dissipation factor
Graph representing losses and Q-factor
Dielectrics should be selected in such a way that the DF of dielectrics be low or Q be high in order to reduce the losses.
2.4 Polarization of Dielectrics
2.4.1 Definition
If a material contains polar molecules, they will generally be in random orientations when no electric field is applied. An applied electric field will polarize the material by orienting the dipole moments of polar molecules. This decreases the effective electric field between the plates and will increase the capacitance of the parallel plate structure. The dielectric must be a good electric insulator so as to minimize any DC leakage current through a capacitor.
The amount of charge stored in a capacitor is the product of the voltage and the capacity. The voltage can be increased, but electric breakdown will occur if the electric field inside the capacitor becomes too large. The capacity can be increased by expanding the electrode areas and by reducing the gap between the electrodes One method for increasing capacity is to insert dielectric material (an insulating material with no free charges) between the plates that reduces the voltage because of its effect on the electric field.
When the molecules of a dielectric are placed in the electric field, their negatively charged electrons separate slightly from their positively charged cores because of forces in opposite direction. With this separation, referred to as polarization, the molecules acquire an electric dipole moment. A cluster of charges with an electric dipole moment is often called an electric dipole.
Polarization
The electric dipole moment p of two charges +q and −q separated by a distance d is a vector of magnitude p = qd with a direction from the negative to the positive charge. An electric dipole in an external electric field is subjected to a torque τ = pE sin θ, where θ is the angle between p and E. The torque tends to align the dipole moment p in the direction of E. The potential energy of the dipole is given by Ue = −pE cos θ, or in vector notation Ue = −p · E.
The electric susceptibility χe relates the polarization to the electric field as P = χeE. In general, χe varies slightly depending on the strength of the electric field. The dielectric constant κ of a substance is related to its susceptibility as κ = 1 + χe /ε0; it is a dimensionless quantity.
2.4.2 Effects of dielectrics
The presence of a dielectric affects many electric quantities. A dielectric reduces by a factor K the value of the electric field and consequently also the value of the electric potential from a charge within the medium. The insertion of a dielectric between the electrodes of a capacitor with a given charge reduces the potential difference between the electrodes and thus increases the capacitance of the capacitor by the factor K. For a parallel-plate capacitor filled with a dielectric, the capacity becomes C = Κε0A/d. A third and important effect of a dielectric is to reduce the speed of electromagnetic waves in a medium by the factor √K .
Effect of polarization
2.5 DIELECTRIC A.C CONDUCTIVITY
2.5.1 Definition
Dielectric conductivity sums all the dissipative effects and represents actual conductivity caused by migrating the charge carriers. For good dielectric materials, the conductivity should be very low, because basically dielectrics act as insulators and insulators should not allow the passage of electricity or current through them. The dielectric conductivity (σ) can be calculated using the following relation:
σ = ε0ωtanδ = ε''ε0ω
where, ε0 (8.854*10-12 F/m) is the permittivity of free space and ω (=2πf) is the angular frequency. This is the frequency at which the prepared dielectric materials vibrate and allow only such frequencies to pass through them, blocking the others.
2.6 TEMPERATURE COEFFICIENT OF RESONANCE FREQUENCY
2.6.1 Definition
The temperature coefficient of resonance frequency (τf) represents the variation of resonance frequency with change in temperature (in ppm/C). Generally all the dielectric materials used in communication field work at particular frequency i.e. they block all the frequencies except the selected frequency and showing an infinite attenuation at other frequencies. But with slightly change in temperature, the desired resonance frequency of dielectrics changes producing losses at the desired frequency and interfering with other frequencies. So while selecting the dielectric materials, it is taken into consideration that the temperature coefficient of resonant frequency should be very-very small for the selected material.
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