Similarities and differences between Inductors and capacitors

Inductors and capacitors are both passive energy storage components – one stores energy in magnetic field while the other does so in electric field. These two components are quite interesting in their behaviour, and when these are compared with each other, the differences in behaviors are very peculiar. Their V-I characteristics (one with lagging power factor and the other leading) makes the use of capacitors a most favoured solution for improving the power factor in supply systems. The same differences facilitate and enable the operation of oscillators and resonance circuits. There are several such factors, which make a relative comparison of these two components an interesting study.

While current leads the voltage in capacitors, in inductors, it the voltage leading the current. Further, when these two are connected in parallel, their currents tend to cancel each other. On the other hand, if connected in series, voltages across them are of opposite polarities, with the result that each of these voltages is much higher than supply voltage (since their difference, or vector sum is the supply voltage itself).

It is known that capacitor acts as open circuit for steady DC voltages. Exactly opposite, an inductor is a short circuit for steady DC voltage.  On the other hand, at extremely high frequencies, capacitor becomes an almost like a short circuit, while inductor becomes an open circuit.

If connected across the output of rectifiers, capacitors are connected in parallel to output terminals of rectifier, while an inductor is put in series.  Capacitors and inductors are even connected in multiple stages in such application, for finer and finer filtering of ripples. 

Entire transmission, broadcast and communication industry depends on the complimentary behaviour of capacitor-inductor combination (oscillator circuits or resonant circuits).  Both capacitors and inductors are inseparable part of most electrical and electronic circuits.

The current dampening capability of inductors (reactors) is used with power capacitors to protect capacitors and switchgear from sudden current inrush while switching the capacitors in automatic power factor control (APFC) panels.

It will be interesting to go through the properties of capacitor and inductors in a comparative fashion.

Property	Capacitor	Inductor
Energy storage	Stores energy in electric field	Stores energy in magnetic field
Energy storage medium	Dielectric	Magnetic material
Behaviour in DC voltage	Acts as open circuit	Acts as short circuit
Response to change	Opposes change in voltage	Opposes change in current
Response to transients	Dampens surge voltage	Dampens surge current
Unit	Farad	Henry
Basic SI Dimensions of unit	Farad- M-1L-2T4I2	Henry – M2L2T-2I-2
Time Constant	CR
Reactance
Watt Loss component
V-I relationship
V-I lag/lead	Current leads voltage by 90°	Voltage leads current by 90°
Power factor	Leading	Lagging
Reactive Power	I2XC leading	I2XL  lagging
Energy Stored	½CV2	½LI2

It can be seen that all entities of capacitor and inductor are very closely similar, but like a mirror image of one other. It can be noted that in reactance equations, even the sign changes.  

Property	Capacitor	Inductor
Energy storage	Stores energy in electric field	Stores energy in magnetic field
Energy storage medium	Dielectric	Magnetic material
Behaviour in DC voltage	Acts as open circuit	Acts as short circuit
Response to change	Opposes change in voltage	Opposes change in current
Response to transients	Dampens surge voltage	Dampens surge current
Unit	Farad	Henry
Basic SI Dimensions of unit	Farad- M-1L-2T4I2	Henry – M2L2T-2I-2
Time Constant	CR
Reactance
Watt Loss component
V-I relationship
V-I lag/lead	Current leads voltage by 90°	Voltage leads current by 90°
Power factor	Leading	Lagging
Reactive Power	I2XC leading	I2XL  lagging
Energy Stored	½CV2	½LI2

SI Dimensional similarities

Let us see the dimensional equalities of Farad and Henry. For Henry, we get

Where C is Coulomb, F stands for Farad, Wb is Weber. Similarly, Farad has following equalities

These two dimensional equalities also have a stark similarity. If we multiply the units of capacitor and inductor, this leads to

Therefore,        

      This has the unit of frequency (cycles per second). .

This is what makes LC combinations useful in resonant circuits, oscillators etc. The resonance phenomenon is based on this relationship between capacitor and inductor.

Relationship between permittivity and permeability

There is a finite relation between magnetic permeability and permittivity of vacuum, the universal reference medium for most purposes. Magnetic permeability of vacuum, µ₀ is related with permittivity of vacuum ϵ₀ in free space by the relation

Where C ≈ 3 x 10⁸ m/sec is the speed of light. Magnetic permeability of free space µ₀, was derived in 1948 from Ampere’s Force Law, and definition of Ampere in terms of force between parallel wires of infinite length due to current flowing through them. The value of permittivity thus decided has the following value.

µ₀ = 4πx 10^-7 N/A²

Permittivity of free space, ϵ₀ calculated from the above relations, has the value

ϵ₀ = 8.854187817 x 10^-12 F/m

Similarity in Volt-time curve of capacitor and Current-time curve of inductor

When a capacitor is connected to a DC voltage source (battery) through a series resistor, as in the figure, it gets charged, following a typical charging voltage-time curve governed by time constant.

The voltage on capacitor at any time is given by

Charging current at any time is given by

Exactly in similar way, if an inductor is connected to a voltage source through a series resistance, as in the figure below, inductor current-time curve looks exactly similar, except that voltage in capacitor charging curve is replaced by current.

Inductor charging current at any time is

Voltage across inductor at any time is given by  

You can see the uncanny similarities in these equations for inductors and capacitors. A capacitor gets charged to 99.7 % of source voltage within five time constants, while inductor reaches current value of 99.7 % full steady state load current within five time constants.

Resonance Phenomenon

One of the most interesting cases is what happens at resonance frequency in an LC tank circuit. Consider the following circuit.

I_C waveform leads the voltage waveform of V by 90°, whereas I_L lags the voltage V by 90°. This means these two currents are always exactly in position to each other. Source current is therefore the difference in these two at every moment. At resonance, reactance of C is exactly same as L, but with opposite sign. It follows that at every moment, current I_L drawn by the ideal inductor L is exactly equal and opposite to I_C, meaning no current is drawn from source at any moment. Now, if the source is removed, both these currents will continue to flow. At any moment, energy being released by C is absorbed by L, and vice versa.The currents will continue even after the sources withdrawn. This LC circuit is an Oscillator, and energy stored keeps ‘oscillating’ between capacitor and inductor.