Capacitors and Inductors

Capacitors

Just like the Resistor, the Capacitor, sometimes referred to as a Condenser, is a simple passive device that is used to “store electricity”. The capacitor is a component which has the ability or “capacity” to store energy in the form of an electrical charge producing a potential difference (Static Voltage) across its plates, much like a small rechargeable battery.

There are many different kinds of capacitors available from very small capacitor beads used in resonance circuits to large power factor correction capacitors, but they all do the same thing, they store charge.

 

In its basic form, a Capacitor consists of two or more parallel conductive (metal) plates which are not connected or touching each other, but are electrically separated either by air or by some form of a good insulating material such as waxed paper, mica, ceramic, plastic or some form of a liquid gel as used in electrolytic capacitors. The insulating layer between a capacitors plates is commonly called the Dielectric.

Capacitance

The capacitance (C) of the capacitor is equal to the electric charge (Q) divided by the voltage (V):

C is the capacitance in farad (F)
Q is the electric charge  in coulombs (C), that is stored on the capacitor
V is the voltage between the capacitor's plates in volts (V)

Capacitance of plates capacitor

 

The capacitance (C) of the plates capacitor is equal to the permittivity (ε) times the plate area (A) divided by the gap or distance between the plates (d):

C is the capacitance of the capacitor, in farad (F).
ε is the permittivity of the capacitor's dialectic material, in farad per meter (F/m).
A is the area of the capacitor's plate in square meters (m²].
d is the distance between the capacitor's plates, in meters (m).

Capacitors in series

 

The total capacitance of capacitors in series, C1,C2,C3,.. :

 

 

Capacitors in parallel

The total capacitance of capacitors in parallel, C1,C2,C3,.. :

C_Total = C₁+C₂+C₃+...

 

Capacitor's current

The capacitor's momentary current i_c(t) is equal to the capacitance of the capacitor,
times the derivative of the momentary capacitor's voltage v_c(t):

 

Capacitor's voltage

The capacitor's momentary voltage v_c(t) is equal to the initial voltage of the capacitor,
plus 1/C times the integral of the momentary capacitor's current i_c(t) over time t:

 

Energy of capacitor

The capacitor's stored energy E_C in joules (J) is equal to the capacitance C in farad (F)
times the square capacitor's voltage V_C in volts (V) divided by 2:

E_C = C × V_C ² / 2

Inductors

An inductor is a passive electronic component that stores energy in the form of a magnetic field. In its simplest form, an inductor consists of a wire loop or coil. The inductance is directly proportional to the number of turns in the coil. Inductance also depends on the radius of the coil and on the type of material around which the coil is wound.

Inductors in a parallel configuration each have the same potential difference (voltage). To find their total equivalent inductance (L_eq):

 


 

The current through inductors in series stays the same, but the voltage across each inductor can be different. The sum of the potential differences (voltage) is equal to the total voltage. To find their total inductance:


 

These simple relationships hold true only when there is no mutual coupling of magnetic fields between individual inductors.

Stored energy

Neglecting losses, the energy (measured in joules, in SI) stored by an inductor is equal to the amount of work required to establish the current through the inductor, and therefore the magnetic field. This is given by:

where L is inductance and I is the current through the inductor.


This relationship is only valid for linear (non-saturated) regions of the magnetic flux linkage and current relationship. In general if one decides to find the energy stored in a LTI inductor that has initial current in a specific time between and can use this: