Inductors

Take the simplest circuit possible having one switch, one limiting resistor, and one led; all are in series and connected to +5V and 0 V poles of a chemical battery. Our circuit is a DC logic implementation: when the switch is closed the led will light, and the state of the DC circuit is True; when the switch is opened, the led is Off, and the state of the DC circuit is False.

Now, when we open or close the switch two AC "things" happen:
1. The DC circuit is energized from 0 V to +5 V, or backwards, and both are AC transitive states;
2. the switch will introduce signal bouncing (spikes) and they are again AC variations.

The above circuit needs as a minimum a capacitor to filter signal bouncing produced by the switch. Again, I want to point out that, practically, there are no pure DC circuits. Here come our capacitors and inductors: both are AC circuit reactive elements, and they help us deal with transient AC states.

In this page we will study inductors and transformers, because they are closely related. The structure needed to present them is:
1. Electromagnetic Field
2. Electromagnetic Induction
3. Equivalent series and parallel inductors
4. Current growth/decay in inductive circuits
5. Transformers
6. Power Factor in AC circuits

ELECTROMAGNETIC FIELD

In addition to creating an Electrostatic Field, electric current flowing through conductors creates an Electromagnetic Field, perfectly similar to the magnetic one, plus few (electrical) characteristics.

Specific to both magnetic and electromagnetic fields is, magnetic flux lines are distinct (discrete) and they go out the North Pole, then they enter the South Pole.

In one particular point, we can measure the number of magnetic field lines (Φ) per unit area (A), and that results in Magnetic Field Density (also known as Electromagnetic Induction):

Magnetic Field Density: B [tesla] = Φ [weber] / A [m²]

1 T = 1 Wb/m² = 10⁴ G [gauss--this is an older notation]

Each conductor develops electromagnetic field when current passes through it. However, in order to amplify electromagnetic field we need to coil few times our conductor: the result is a solenoid, having a number of turns (N), a length (L) and a coil area (A).

Note that the inside our solenoid (the core) is air, but we could easily introduce various metallic alloys instead. Now, different core materials have the quality of increasing or decreasing magnetic field density, based on their characteristic Magnetic Permeability (μ):

Magnetic Permeability in vacuum ( and also in air, because it is almost the same) is:
μ_o = 4*π*10^-7 [Wb/A*m] or
μ_o = 4*π*10^-7 [T*m/A]

Few cases are of particular interest, and are listed here:

ELECTROMAGNETIC FIELD FORMULAS

ELECTROMAGNETIC INDUCTION

Suppose we have a coil and it is wired in series with a amp-meter. That is all: there is no power source and nothing else. Next, we bring a magnet close to our coil, and we start moving it in and out the coil: we will immediately notice on our amp-meter that current is flowing in one sense and the other in our circuit!

The voltage that appears inside the coil when it is subjected to crossing (moving) magnetic lines is named Electromotive Force (U_emf), and it is produced by (Electro) Magnetic Induction.

Faraday's Law of Induction
The electromotive force induced in a conductor is equal to the rate of change of magnetic flux through that conductor.

Electromotive Force: U_emf = dΦ/dt (with Φ = B * A)

dΦ/dt = magnetic flux lines variation in time

For a closed loop circuit having only wires (a closed coil, or a ring) the above equation may be also written as:

Electromotive Force for one coil: U_emf = -B * L * v

B = Magnetic Flux Density
L = length of the conductor
v = perpendicular velocity across magnetic lines. If v is not perpendicular, and it comes at an (x) angle to perpendicular then:

Electromotive Force for one coil: Uemf = -B*L*v*sin x

If we have a coil with N turns, then the relation becomes:

Electromotive Force for multiple turns: Uemf = -B * N * L * v

Now, we have used a permanent magnet to induce power in our (one turn) coil, but we get exactly the same effect if we use another coil, generating an alternative electromagnetic field.

Suppose we have two separate circuits: the first one is described above and it is a ring or one-turn closed coil; the second one is a coil powered by alternative current. The new Induced Electromotive Force is named Mutual Inductance in this case, and it is calculated with:

Electromotive Force for Mutual Inductance: U_emf = -M * (di/dt)

M is Mutual Inductance, and it is a constant specific to the system of coils we have used. It is the only unknown in the above relation, and we calculate it with:

Mutual Inductance: M = -Uemf / (di/dt)

M is expressed in [H] Henry;
Uemf is [V];
di/dt is [A/s].

The tricky part is, if we have only the second coil (described above in the second case) it is capable to self-induce Uemf! In this case Self-Induction is:

Self Induction: L = -Uemf / (di/dt)

L is expressed in [H] Henry;
Uemf is [V];
di/dt is [A/s].

The above relation clearly defines the Inductance (L) of a coil as being an AC circuit element due to the current derivate related to time. Practically, in continuous DC both the capacitor and the inductor do nothing. Sorry I need to clarify this: in continuous DC a capacitor creates an electrostatic field and an inductor creates an electromagnetic field, but they do not interfere with the DC circuit. However, when the DC current varies, or if we bring close another moving magnetic or electrostatic field, our DC circuit becomes an active AC one.

EQUIVALENT SERIES AND PARALLEL INDUCTORS

The equivalent of series inductors is calculated with: L_T = Σ L_i
The equivalent of parallel inductors is calculated with: 1/L_T = Σ 1/L_i

Calculation examples for three inductors:

	>Fig 1: The equivalent inductance of 3 series inductors LT = L1 + L2 + L3 LT = 2mH +3mH + 4mH = 9mH
	Fig 2: The equivalent inductance of 3 inductors in parallel 1/LT = 1/L1 +1/L2 + 1/L3 LT = L1*L2*L3 / (L2*L3 + L1*L3 +L1*L2) LT = 24 / (12 + 8 + 6) = 24 / 26 = 0.923mH

CURRENT GROWTH/DECAY IN INDUCTIVE CIRCUITS

A inductor in series with a resistor form together a timing circuit, just like the capacitor does.

Time constant T: T [s] = L [H] / R [Ω]

In order to reach 100% current, it takes 5 time constants (L/R) calculated with the above formula. The Decay curve behaves perfectly similar to the Growth one, having only an inverse second derivate (the curve holds water).

Fig 3: Rising current curve