Passive Elements = R,L,C

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Last Updated: Monday, 21 August 2023 18:40

Published: Friday, 11 August 2023 13:21

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Passive Elements = Resistors, Capacitors, Inductors

Almost all the matter around us has the 3 passive elements built as intrinsic part of that matter. These 3 elements are Resistors, Capacitors, Inductors. They arise due to physical properties of resistance, capacitance and Inductance that gives rise to these 3 passive elements respectively.

Resistance:

Ohm's Law ad resistivity:

Conductivity of a material is defined by ohm's law as: J = σ*E where J=current density=I/A (I=current, A=Area), σ = conductivity of material and E=Electric field thru that material.

Resistivity (ρ) = 1/σ = inverse of conductivity.

For a material with uniform cross section A and length L, E=V/L (In a medium with no source/sink point, electric field is constant. It it weren't, charges would start accumulating at the point where electric fields are different)

=> I/A = σ*V/L

=> Resistance R = V/I = ρ*L/A => This is the classic Ohm's law that is stated in books, but it applies only to special case of homogeneous material with uniform cross section.

Units of ρ are Ω-m. The value of ρ comes by experimentally measuring it for different materials. Metals are good conductors and have lower ρ.

Cu=1.7*10^-8 ohm-m (Cu less resistive than Al and used in most conductors now, including wires on chip)

Al=2.6*10^-8 ohm-m

Resistivity is also written in units of ohm-mm^2/m, so Cu resistivity is 17 mΩ-mm^2/m => For a wire with area=1mm^2, Resistance is 17mΩ/m. Or for 1km of 1mm thick wire, Resistance is 17Ω. Gauge (AWG) is used in transmission wires to specify the thickness and current carrying capacity of wire (see in Power Transmission section).

Mobility of electron:

Other term used in connection with resistivity is mobility. Looking at atomic level, we see that any conductor has free flowing electrons which are moving randomly in all directions. When we apply a n Electric field, these electrons experience F = q*E and start accelerating as per Newton's law. They hit the atom lattice, and scatter off and lose some of their energy. Then it again starts accelerating in same direction, and this process continues. The final result is that the electron moves with a finite average velocity, called the drift velocity. This causes an electric current in the direction of the Electric Field. This motion is lot smaller than normally occurring random motion.

F= m*a=e*E (where m is mass of electron, e is it's charge)

Avg drift velocity (V_d) = a/τ = e*E/(m*τ) where τ=mean free time (i.e avg time between collisions).

So, V_d = -μ*E, where μ=e*/(m*τ) and is known as mobility of electron. SI unit of mobility is m^2/(V.sec) but is almost always expressed in cm^2/(V.sec)

I = -n*(A*L)*e/t = -n*(A*L)*e/(L/V_d) = -n*(A*L)*e*V_d (n=electron density per unit volume, Length L of distance covered in time t)

J=I/A = -n*e*V_d = -(-n*e*μ*E) =  n*e*μ*E

From Ohm's law J = σ*E  => σ = n*e*μ

So, conductivity of any material is related to the mobility of electron in that conductor as well as the concentration of free electrons. That is why different metals with same number of free electrons may have different resistivity as electron mobility might be different between them, depending on the lattice scattering. Thus electron mobility depends on following:

• Lattice structure: Depending on the lattice structure, scattering (or collision) may occur sooner, which will cause lower electron mobility.
• Velocity saturation for electrons may occur. Electrons don't keep going faster and faster with increasing Electric field. At some point, sufficiently high drift velocity is reached, and velocity maxes out. This is called Velocity Saturation (v_sat). v_sat is on the order of 1×10⁷ cm/s for both electrons and holes in Si. It is on the order of 6×10⁶ cm/s for Ge. So, basically μ starts decreasing as velocity sat approaches, and finally goes to very low values.
• Temperature affects electron mobility => With increasing temperature, phonon concentration increases and causes increased scattering, thus lowering the carrier mobility. In semi Conductors (Si, Ge), μ ∝ 1/T^2.  The power to T isn't exactly but varies based on material and whether it's electrons or holes. So, conductivity is supposed to decrease with increasing Temperature (as mobility decreases), but at the same time, carrier concentration may increase due to higher Temperature (true for semiconductors)

Typical electron mobility at room temperature (300 K) in metals like gold, Cu, Silver is 30–50 cm²/ (V⋅s). Carrier mobility in semiconductors is doping dependent. In Silicon (Si) the electron mobility is of the order of 1,000, while in germanium it's around 4,000.

Determine ρ of Cu using mobility:

1 mole of Cu has weight = 64g =>  1 atom of Cu has weight = 64g/(6.02*10^23 ) = 10^-22 g.

Density(Cu)=9g/cm^3 => # of atoms in 1cm^3 = 9g/(10^-22) = 10^23 atoms. Assume 1 free electron per Cu atoms

σ = n*e*μ = 10^23*1.6*10^-19*40 cm²/ (V⋅s) = 0.6*10^6 => ρ = 1/σ = 1.6 * 10^-6 ohm-cm = 1.6*10^-8 ohm-m, which matches closely to the ρ shown above (1.7*10^-8 ohm-m)

Capacitance:

Capacitance of a material is defined by gauss's law.

We saw insulators above which don't conduct electrons that well (i.e electrons drift very slowly in response to Electric field). Within insulators, we have a subset of materials which will get polarized when an Electric field is applied. This is called permittivity of the material. The electrons in such materials don't drift, but instead just shift slightly causing an internal Electric field. Such materials are called dielectric. A perfect dielectric is a material with zero electrical conductivity.

Dielectrics: https://en.wikipedia.org/wiki/Dielectric