
ELECTRONIC DESIGN NOTES #5  INDUCTORS, TRANSFORMERS

In this page inductors
and transformers are presented together, because they are closely related. The
simplified structure employed to describe inductors/transformers 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
NOTE The basic
notions highlighted in this page are related to a few electronic
design topics presented in the first part, Hardware Design, of
LEARN HARDWARE FIRMWARE AND SOFTWARE
DESIGN.

1. ELECTROMAGNETIC FIELD
In addition to creating an Electrostatic Field, electric current flowing through conductors
also creates an Electromagnetic
Fieldperfectly similar to the (mechanical) magnetic one, plus a few
additional electrical characteristics.
Specific to both the magnetic and electromagnetic fields is, the "magnetic
flux lines" are distinct (discrete): they
go out from the conventional North Pole, then they enter the conventional South Pole.
Therefore, in one particular point we
can measure the number of magnetic field/flux lines (Φ) per unit of area (A), and that results in Magnetic Field
Density (also known as the Electromagnetic Induction):
Magnetic Field Density:
B [tesla] = Φ [weber]
/ A [m^{2}]
1 [T] = 1 [Wb/m^{2}] = 10^{4}
[G (Gaussthis is an older notation)]
Each conductor develops an electromagnetic field when electrical current passes through it. However, in order to amplify
the electromagnetic field we need to coil our conductor a few times: the result is a solenoid having a number
of turns (N), a length (L), and an inside coil area (A).
Note that inside our solenoid (the core) there is air, but we could easily introduce various metallic alloys
instead. Now, different core materials have the quality of increasing, or
of decreasing, the 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*PI*10^{7} [Wb/A*m]
A few formulas are of particular interest, therefore they are listed in the following table.

ELECTROMAGNETIC FIELD FORMULAS 
Formulas 
Explanations 
B = μ_{o}*I / 2*PI*r 
Induction around a single straight
conductor (wire)
B = electromagnetic induction [T]
μ_{o} = magnetic permeability in air
l = the current through conductor in [A]
r = distance of the reference point to the center of the conductor in [m] 
B = μ_{o}*I / 2*r 
Induction inside a single loop
The sense of the B vector may be found using the righthand rule
r is radius of the loop in [m] in this case 
B = μ_{o}*N*I / 2*r

Induction inside multiple (N) loops 
B = μ_{o}*N*I / L 
Induction at the center of a long
solenoid
L = the length of the solenoid 
B_{m} = χ*N*I / L 
Induction inside metallic cores
χ is magnetic susceptibility 
B [Tesla] = Φ [Wb] / A [m^{2}]
B = B_{o} + B_{m}
B = (μ_{o}*N*I / L) + (χ*N*I / L)
B = μ * H
B = (μ_{o }+ χ) * (N*I / L) 
Magnetic Flux Density in metallic
cores
B_{o }= magnetic flux density in air_{
}B_{m} = magnetic flux density in metal 
H = N*I / L 
Magnetic Field Intensity
H is in [A*turns/m] 
μ = μ_{o} + χ
μ = B / H 
Permeability of magnetic materials
μ_{o} = magnetic permeability in vacuum/air
χ = magnetic susceptibility in metal 
B = F / I * L
F [N] = I*L*B 
F =
electromagnetic force in [N]
I = current [A]
L = length of conductor [m]
B = magnetic induction [Tesla] 
F [N] = μ*I_{1}*I_{2}*L / 2*PI*d 
Electromagnetic force between two
parallel conductors
A. if the currents flow in opposite directions in the two conductors, F is repulsion
B. if currents have the same direction, F is attraction
I_{1} is current in conductor 1 [A]
I_{2} is current in conductor 2 [A]
L = common length of the 2 conductors [m]
d = distance between the 2 conductors [m] 

2. ELECTROMAGNETIC INDUCTION
Suppose we have a coil and it is wired in series with a ampmeter. 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 ampmeter that electrical current is flowing in one sense or the other in our basic, rudimentary
circuit!
The voltage that appears inside the coil when it is subjected to crossing/moving magnetic field lines is
named the Electromotive Force (U_{emf}), and it is produced by
the (Electro) Magnetic Induction.
Faraday's Law of Induction The electromotive force induced in a conductor is equal to the rate of change of the
magnetic flux
through that conductor.
The Electromotive Force is: U_{emf} = dΦ/dt
(with Φ = B * A) dΦ/dt = magnetic flux lines variation in time
For one closed coil, or a ring, the above equation may be also written
as:
The Electromotive Force of one coil is: U_{emf} = B * L
* v
B = Magnetic Flux Density L = the length of the conductor v = the perpendicular velocity across magnetic lines.
If v is not perpendicular, if it comes at an angle (x) to
the perpendicular then:
The Electromotive Force of one coil is: Uemf = B*L*v*sin x
If we have a coil having N turns, then the relation becomes:
The Electromotive Force of multiple turns is: Uemf = B * N * L * v
Now, we used a moving permanent magnet to induce power in our (one turn) coil, but we
could get exactly the same effect if
we use another coil, fixed this time, which is generating an alternative electromagnetic field.
Suppose we have two separate circuits: the first one is described aboveit is just a ring or a oneturn 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 as follows:
The Electromotive Force of Mutual Inductance: U_{emf} = M * (di/dt)
M is Mutual Inductance, a constant specific to the system of coils used. It is the only unknown in the above
relation; therefore, we calculate it with:
Mutual Inductance: M = Uemf / (di/dt)
M is expressed in [H] Henry Uemf is in [V] di/dt is in [A/s]
The tricky part is, if we have only the second coil [described above in the second
instance] it is capable to
selfinduce Uemf! In this case the SelfInduction is:
Self Induction:
L = Uemf / (di/dt)
L is expressed in [H] Henry Uemf is in [V] di/dt is in [A/s]
The above relation defines the inductance (L) of a coil as being an AC circuit element, due to
the current variation in
time (di/dt). In continuous DC circuits, capacitors and inductors do exactly nothing. Sorry, we need to
clarify this: in continuous DC circuits a capacitor creates an electrostatic field and an inductor creates an
electromagnetic field, but they do not interfere with the DC circuit. However,
if the DC current varies, or if
we bring close another moving magnetic or electrostatic field, our initial DC circuit becomes an active/transitive AC one.
NOTE
Please discover the Impedance and Inductive Reactance formulas in previous Design Notes.

3. 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 follow next.


Fig 1: The
equivalent inductance of 3 series inductors
L_{T} = L_{1} + L_{2} + L_{3}
L_{T} = 2mH +3mH + 4mH = 9mH 

Fig 2: The equivalent inductance of 3 inductors in
parallel
1/L_{T} = 1/L_{1} +1/L_{2 }+ 1/L_{3}
L_{T} = L1*L2*L3 / (L2*L3 + L1*L3 +L1*L2)
L_{T} = 24 / (12 + 8 + 6) = 24 / 26 = 0.923mH 

4. CURRENT GROWTH/DECAY IN INDUCTIVE CIRCUITS
A inductor in series with a resistor forms a timing circuit, just as the capacitor does.
The time constant T of RL circuits is: T [s] = L [H] / R [Ω]
In order to reach 100% of the 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: The rising current curve graph in LR circuits. 

5. TRANSFORMERS
The beauty with AC current flow is, we can easily step up or down the voltage, and in order to do that we use transformers. A
transformer is an electric device having two coils placed on a closed loop metallic core.
Suppose we have a transformer with a primary coil having N_{p} number of turns, and a secondary coil with
N_{s} number of turns. Our transformer uses U_{p} as the primary voltage, and it outputs U_{s}
in the secondary coil. We work with the following set of equation:
TRANSFORMER FORMULAS
Us / Up = Ns / Np
which is exactly the same as
Us * Np = Up * Ns
In the same time we have
Us / Up = Ip / Is
which is, again, exactly the same as
Us * Is = Up * Ip
Now, because
Up = Np * dΦ/dt
then
Us = Ns * dΦ/dt
Using the set of formulas presented above we can calculate the dimensions of the metallic core. That should be all you
need to work successfully with transformers. However, the manufacturers of metallic cores have
datasheets
specific to their products, therefore you do need to consult them.

6. POWER FACTOR IN AC CIRCUITS
The power in AC circuits is
P =
I^{2} * Z and it has the following subcomponents:
Apparent Power: P_{A} [VA] = √[P_{R}^{2} + (P_{XL}  P_{XC})^{2}]
True Power: P_{R} [W] = P_{A} * cos(x)
Reactive Power: P_{X} [VAR] = P_{A} * sin(x) = P_{XL } P_{XC}
Fig 4: The Power Factor in AC circuits. 

The above picture presents the following:
P_{A} is
Apparent Power in [VA] and you can see in Fig 4 where it comes from: it is the red vector P_{R} is
True Power in [W] given by the (pure) resistance of the AC circuit P_{XC} is
Reactive Capacitive Power in [VAR], and it is lagging PR by (PI/2) P_{XL} is
Reactive Inductive Power in [VAR]; it is ahead of PR by (+PI/2) P_{X} is the difference P_{XL}P_{XC}. It is mandatory that Reactive Inductive Power is
greater than Reactive Capacitive Power due to the issues presented in Design Notes 2 cos(x) is the
Power Factor (Pf) VAR is read as "Volt Ampere Reactive" and it is used to mark Inductive and Capacitive reactive
currents [only the current is considered as being reactive]
Power factor:
P_{f} = cos(x) = P_{R }/ P_{A} = True Power / Apparent Power
NOTE Because they are opposed by PI (180 degrees),
the Inductive and Capacitive reactances compensate each other.
However, it is better to have slightly inductive AC circuits rather than capacitive ones, because capacitive
reactance is very dangerous.
Of course, too much inductive reactance is also not very good, therefore we employ
adjustable banks of capacitors
to control it.

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