Inductors and capacitors

An Inductor is an energy storage lement.
It is a two-terminal circuit element that is a model of a real device that consists of
a coil of resistance-less wire wound around a material.

                                                            
Inductance
L is the inductance of the device.
L = (moN2A) /(l + 0.45d)     (if l >d/2 & core is non-ferromagnetic)               
where

• N is the number of turns of the wire
• A is the cross sectional area of the loops
• l is the length of the coil
• d is the diameter of the wire
• mo = the permeability of free space = 4p x 10-7 H/m.
  In this case the material inside the coil is a vacuum.

Inductance is a property of the device that measures the ability of the device to store energy in the form of a magnetic field.
The units of inductance are Volt-Seconds per Ampere or Henrys. 

The voltage-current characteristic for an inductor is:      
v = L  di/dt
-or-
i = 1/ L  òtot  v dt + i(to)

Two observations from the v-i characteristic:

• There is no voltage across an inductor if the current through it is not changing with time.
  That is why we say that “inductors act as a short circuit to a dc current”.
  
• Since there can not be infinite voltages, the current through an inductor cannot change instantaneously. (However, the voltage can.) This means that iL(t+) = iL(t-)

The energy stored in an inductor is:
(wL)0-->t   =   wL(t)  = ½ L i2  –   ½ L i02
From this we see that the energy stored in an inductor is only a function of the current through it.
So even if the voltage is 0, there can be some energy in an inductor.

An inductor is a passive element.
From the expression for energy stored in an inductor we see that (wL)-inf-> t is always > 0. 

Equivalent Inductors, Leq
Combinations of inductors can be reduced to an equivalent inductance.
They combine like resistors.

Practical Model of an Inductor
A more practical model of an inductor is an ideal inductor in series with a small resistance.

A Capacitor is an energy storage element.
It is a two-terminal circuit element that is a model of a real device that consists of
two parallel conducting plates separated by a non-conducting material.

           
Capacitance
C is the capacitance of the device.
C = e A / d
where

• A is area of the plates
• d is the distance between the plates
• e is the dielectric constant, a property of the material between the plates
  The dielectric constant is a property of a material that is a measure of the materials ability to store energy per unit volume for unit voltage difference.

Capacitance is a property of a device that measures the ability of the device to store energy in the form of an electric field. 
The units are Coulombs per Volt or Farads.

The voltage-current characteristic for a capacitor is:
i = C  dv/dt
-or-
v = 1/ C òtot  i dt + v(to)

Two observations from the v-i characteristic:

• There is no current through a capacitor if the voltage across it is not changing with time.
  That is why we say that “capacitors act as an open circuit to a dc voltage”.
  
• Since there can not be infinite currents, the voltage across a capacitor cannot change instantaneously. (However, the current can.) This means that vC(t+) = vC(t-)

The energy stored in a capacitor is:
(wC)0-->t  =   wC(t)  = ½ C v2  –   ½ C v02
From this we see that the energy stored in a capacitor is only a function of the voltage across it.
So even if the current is 0, there can be some energy in a capacitor.

A capacitor is a passive element.
From the expression for energy stored in a capacitor we see that (wC)-inf-> t is always > 0. 

Equivalent Capacitors, Ceq
Combinations of capacitors can be reduced to an equivalent capacitance.
They combine like conductors.

Practical Model of a Capacitor
A more practical model of a capacitor is an ideal capacitor in parallel with a large resistance.

A note about Linearity:
The voltage-current relationships for capacitors and inductors are linear. 
Hence these will hold for circuits with inductors and capacitors:

• KVL
  
• KCL
  
• Thevenin’s Theorem
  
• Norton’s Theorem
  
• Principal of superposition