Fundamentals of Inductor

Introduction

An inductor is an insulated or uncovered wire in the form of a coil, and it is a passive and two-terminal device. It stores energy in the form of a magnetic field. The wire wounded on a magnetic material forms an inductor. A constant current flowing through the inductor will create a magnetic field, and magnetic fields do not like to change. The inductor is a device that tries to prevent the change in the current flow. When the current flow is changing to time, the Inductor will generate a force and try to maintain the current flowing through it.

Hydraulic or Water analogy

The Paddlewheel or turbine shown in figure 1 has mass. The turbine will not rotate immediately after applying the inlet water flow from a pump. If the water is not flowing through the turbine, the pressure will develop nearby the inlet portion, and it gives force to rotate the turbine. The turbine starts moving and spins in proportion to the water flow rate. Now the pump is switched OFF, the water circulates in the pipe, and the turbine rotates continuously. If the water supply is from a sump and a closed circulating path does not exist, switching OFF the pump will reduce the flow rate slowly and makes it zero.

Figure 1. Water analogy for Inductor

Relating to the analogy, the inductor is a passive device, it has an impedance, and it requires a power supply to operate. Initially, the current flow through the inductor is zero. Now a Direct Current (DC) supply of 10V is connected to an inductor. The current will start flowing from source to inductor. The inductor gets a sudden change in current from 0 amps to some other value, it will oppose this change in current flow, and 10V develops across inductance.

The voltage developed across inductance will give a force. The force induces current flowing through the inductor, now voltage across the inductance will gradually reduce to 0V, and the current flow will become constant. So, the inductor will act as a wire for DC supply under steady-state conditions. Various types of inductors are shown in figure 2.

Figure 2.  Type of Inductors

The inductance of Inductor

The current (i) flow in an inductor introduces the magnetic field or flux density (Φ) around it. The increase or decrease in current through the inductor changes the magnetic field and produces an opposing electromotive force (emf). The emf will restrict the current flow. The Φ developed due to current I entirely depend on the shape of the circuit or inductance L

L = Φ/i

Φ = Li  ——- (1)

Differentiating equation (1)

dΦ/dt  = – L(di/dt)  = voltage across the inductance (e)  ——–(2)

Emf  = e = – NdΦ/dt  (Using Faraday’s and Lenz’s Laws) —– (3)

Where N= number of turns in the coil or inductor,

dΦ/dt = Rate of change of flux density,

The negative sign indicates that the Emf opposes the circulating current.

From equation (2) and (3),

The Inductance (L) and the number of turns in the inductor (N) has a direct relationship.

The magnetomotive force (MMF) is directly proportional to the magnetic flux density (Φ). The proportionality constant is magnetic resistance or reluctance (Ɽ).

Ғ = mmf = ⱤΦ

The Ɽ and Φ  are proportional numbers turn in the coil and the current (I) respectively.

Ғ = ⱤΦ = NI

Φ = Ni/Ɽ ——- (4)

Differentiating the equation (4)

 dΦ/dt  = (N/ Ɽ) x (di/dt) – (N/ Ɽ)  x (dⱤ/dt)

Where reluctance Ɽ is approximately constant and the result of differentiating constant  is zero.

dΦ/dt = (N/ Ɽ)x (di/dt)  ——- (5)

Substituting equation (5) in (3), and neglecting the negative sign,

Voltage across inductance = e = N2/ Ɽ x (di/dt).

Inductance = L = e/(di/dt) =  N2/ Ɽ  —— (6)

The inductive transducer measures a change in physical quantities in terms of change in inductance using the equation (6)

The energy in an Inductor

The inductor stores energy in the form of a magnetic field. The instantaneous power is calculated as P = v x i = exi = iL(di/dt)   from equation (6).

Pdt = iL(di)

The integration of power (P) with limits 0 to t seconds and 0 to i yields the stored energy.

W = Li2/2,  The current flowing through the inductor generates this energy.

The response of Inductor to AC and DC inputs:

DC input to Inductor

The voltage across inductor   e =L(di/dt), (From the equation (6)). Based on basic mathematics, differentiating a constant is zero. Constant current or Direct Current (DC) will not create a voltage across an inductor, the voltage across inductance is zero. The inductor acts as a short circuit for DC, and it offers zero impedance.

Alternating Current input to Inductor

Taking the Laplace transform of the voltage equation,

E = Ls xI.

The impedance = E/I = Ls —— (7)

Where the letter s is a complex number and  s = α + jω

Where α = Constant

ω = Frequency in radian.

The commonly known AC signal is sinusoidal, it does not have a stable state, and its real part or α is zero.

From equation (7)

The inductor impedance (Z) = Ljω —— (8)

In practice, the frequency in Hertz is used. The conversion factor from radian (ω) to frequency is ω =2πf, and substituting this in equation (8)

Z= jL2πf

The positive half cycle of the AC supply introduces current flows in one particular direction of the inductor. In the negative half cycle, the current will flow in the opposite direction. The inductor resists the change in current flow and develops a voltage across it.

 The voltage across the inductor is directly proportional to the rate of change of current or frequency. If the supply frequency is maximum, (a) the voltage across the inductor and impedance of the inductor will increase to a maximum value (from equation (8)), (b) current through the inductor is minimized. The inductor behaves like an open circuit for very high-frequency AC signals.

V= E =Ljω

Phase angle of signal   = tan-1[(Imaginary Part)/ (Real part)].

                                    = tan-1[Lω/0] = + 90o

In the inductor, when AC supply is applied,  voltage leads the current by 90o.

Mutual Inductance

Let there are two coils in proximity to each other. The application of the AC supply induces a current flow in the primary coil. The current flow will induce a magnetic field. The magnetic field lines slice the secondary coil and induce an emf proportional to the interacting magnetic field.  The primary coil current induces an emf in the secondary coil,  it is called mutual inductance (Figure 3).

Figure 3. Mutual Inductance

Conclusion

The inductor is one of the significant parts of human life. Most of the electrical appliances have an inductor like an electric fan, refrigerator, washing machine, and motor. The majority of electrical loads (motors and generators) used in industries are inductive loads. The transformers and power generators are using the coils. The basic knowledge about inductor is essential for every individual.