Basic electricity and electronics

Phys2042 A.Y. 2022/23

Dr. Gashaw Beyene (Course Instructor)

Chapter 7
7. Inductance
7.1 Characteristics of Inductance
 Inductance is the ability of a conductor to produce induced voltage when the current varies.
A long wire has more inductance than a short wire, since more conductor length cut by
magnetic flux produces more induced voltage.
 Similarly, a coil has more inductance than the equivalent length of straight wire because the
coil concentrates magnetic flux.
 The symbol for inductance is L, and the unit is the henry (H).
 The wire for a coil can be wound around a hollow, insulating tube, or the coil can be
the wire itself.
 This type is an air-core coil because the magnetic field of the current in the coil is in air.
 With another basic type, the wire is wound on an iron core to concentrate the magnetic
flux for more inductance.
 Iron-core inductors are used in the audio-frequency range, especially in the AC power-line
frequency of 60 Hz (in our country also) and for lower frequencies in general.
Inductor Labeling
If the label reads 223K, the third number (3) is the power to be applied to the first two.
The K is not from kilo, representing a power of three, but is used to denote a tolerance of
±10% as described for capacitors. The resulting number of 22,000 is, therefore, in μH so
the 223K unit is a 22,000 μH or 22 mH inductor. The letters J and M indicate a tolerance
of ±5% and ±20%, respectively.
For molded inductors, a color-coding system very similar to that used for resistors is used.
The major difference is that the resulting value is always in μH, and a wide band at the
beginning of the labeling is an MIL (“meets military standards”) indicator. Always read the
colors in sequence, starting with the band closest to one end

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Chapter Seven Short note
 Basic electricity and electronics
Phys2042 A.Y. 2022/23
Dr. Gashaw Beyene (Course Instructor)

Self-Inductance
 The ability of a conductor to induce voltage in itself when the current changes is its self-
inductance or simply inductance.
 When the switch is closed, the current does not immediately reach its maximum value.
 Faraday’s law of electromagnetic induction can be used to describe the effect.
 As the current increases with time, the magnetic flux through the circuit loop due to this
current also increases with time.
 This increasing flux creates an induced emf in the circuit.
 As illustrated in Fig. below, henry is the amount of inductance that allows one volt to be
induced when the current changes at the rate of one ampere per second. The formula is

where is in volts and di/dt is the current change in amperes per second.
Again the symbol d is used to indicate an infinitesimally small change in current with time.
 The factor di/dt for the current variation with respect to time specifies how fast the
current’s magnetic flux is cutting the conductor to produce .

Inductance of Coils
 In terms of physical construction, the inductance depends on how a coil is wound.
Note the following factors.
1. A greater number of turns N increases L because more voltage can be induced. L
increases in proportion to N2. Double the number of turns in the same area and
length increases the inductance four times.
2. More area A enclosed by each turn increases L. This means that a coil with larger
turns has more inductance. The L increases in direct proportion to A and as the
square of the diameter of each turn.
3. The L increases with the permeability of the core. For an air core, r is 1. With a
magnetic core, L is increased by the r factor because the magnetic flux is
concentrated in the coil.
4. The L decreases with more length for the same number of turns because the
magnetic field is less concentrated.

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Chapter Seven Short note
 Basic electricity and electronics
Phys2042 A.Y. 2022/23
Dr. Gashaw Beyene (Course Instructor)

 These physical characteristics of a coil are illustrated in Fig. below. For a long coil, where
the length is at least 10 times the diameter, the inductance can be calculated from the
formula

How Opposes a Change in Current

 By Lenz’s law, the induced voltage must produce current with a magnetic field that
opposes the change of current that induces .
 The polarity of , therefore, depends on the direction of the current variation di. When di
increases, has polarity that opposes the increase in current; when di decreases, has
opposite polarity to oppose the decrease in current.
 In both cases, the change in current is opposed by the induced voltage. Otherwise,
could increase to an unlimited amount without the need to add any work.
 Inductance, therefore, is the characteristic that opposes any change in current. This is the
reason that an induced voltage is often called a counter emf or back emf.

 This current is increasing. By Lenz’s law, vL must have the polarity needed to oppose the
increase. The induced voltage shown with the top side negative opposes the increase in
current. The reason is that this polarity of can produce current in the opposite direction,
from minus to plus in the external circuit.
 Note that for this opposing current, is the generator. This action tends to keep the
current from increasing.
 In right Figure, the source is still producing electron flow into the top of the coil, but i is
decreasing because the source voltage is decreasing.
 By Lenz’s law, must have the polarity needed to oppose the decrease in current. The
induced voltage shown with the top side positive now opposes the decrease.
 The reason is that this polarity of can produce current in the same direction, tending to
keep the current from decreasing.

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Chapter Seven Short note
 Basic electricity and electronics
Phys2042 A.Y. 2022/23
Dr. Gashaw Beyene (Course Instructor)

Mutual Inductance
 Very often, the magnetic flux through the area enclosed by a circuit varies with time
because of time-varying currents in nearby circuits.
 This condition induces an emf through a process known as mutual induction, so named
because it depends on the interaction of two circuits.
 Consider the two closely wound coils of wire shown in cross-sectional view. The current i1
in coil 1, which has N1 turns, creates a magnetic field.
 Some of the magnetic field lines pass through coil 2, which has N2 turns. The magnetic flux
caused by the current in coil 1 and passing through coil 2 is represented by 𝜱12.
 In analogy to Equation of self-inductance, we can identify the mutual inductance M12 of coil
2 with respect to coil 1:

 Mutual inductance depends on the geometry of both circuits and on their

orientation with respect to each other.
 As the circuit separation distance increases, the mutual inductance decreases
because the flux linking the circuits decreases.
 If the current i1 varies with time, we see from Faraday’s law and the above Equation
that the emf induced by coil 1 in coil 2 is

( )
 If the current i2 varies with time, the emf induced by coil 2 in coil 1 is

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Chapter Seven Short note
 Basic electricity and electronics
Phys2042 A.Y. 2022/23
Dr. Gashaw Beyene (Course Instructor)

 In mutual induction, the emf induced in one coil is always proportional to the
rate at which the current in the other coil is changing.
 Although the proportionality constants M12 and M21 have been treated
separately, it can be shown that they are equal. Therefore, with M12 = M21 = M,

Transformers
 The transformer is an important application of mutual inductance. As shown in Figure
below, a transformer has a primary winding inductance LP connected to a voltage source
that produces alternating current, and the secondary winding inductance LS is connected
across the load resistance RL.
 The purpose of the transformer is to transfer power from the primary, where the generator
is connected, to the secondary, where the induced secondary voltage can produce current
in the load resistance that is connected across LS.

 Although the primary and secondary are not physically connected to each other, power in
the primary is coupled into the secondary by the magnetic field linking the two windings.
 The transformer is used to provide power for the load resistance RL, instead of connecting
RL directly across the generator, whenever the load requires an AC voltage higher or lower
than the generator voltage.
 By having more or fewer turns in LS, compared with LP, the transformer can step up or
step down the generator voltage to provide the required amount of secondary voltage.
 With unity coupling between primary and secondary, the voltage induced in each turn of
the secondary is the same as the self-induced voltage of each turn in the primary.
 Therefore, the voltage ratio is in the same proportion as the turns ratio:

 Efficiency is defined as the ratio of power out to power in. Stated as a formula,

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Chapter Seven Short note
 Basic electricity and electronics
Phys2042 A.Y. 2022/23
Dr. Gashaw Beyene (Course Instructor)

 Transformers can be used to change or transform a secondary load impedance to a new

value as seen by the primary.
 The secondary load impedance is said to be reflected back into the primary and is
therefore called a reflected impedance.
 The reflected impedance of the secondary may be stepped up or down in accordance with
the square of the transformer turns ratio.
 By manipulating the relationships between the currents, voltages, and turns ratio in a
transformer, an equation for the reflected impedance can be developed. This relationship
is

Example

7.2 Type of Inductance

Inductors, like capacitors and resistors, can be categorized under the general headings fixed or
variable.

Fixed inductor
Fixed-type inductors come in all shapes and sizes. However, in general, the size of an inductor is
determined primarily by the type of construction, the core used, or the current rating.

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Chapter Seven Short note
 Basic electricity and electronics
Phys2042 A.Y. 2022/23
Dr. Gashaw Beyene (Course Instructor)

Variable inductor
In each case, the inductance is changed by turning the slot at the end of the core to move it in and
out of the unit. The farther in the core is, the more the ferromagnetic material is part of the
magnetic circuit, and the higher the magnetic field strength and the inductance level.

Variable inductors with a typical range of values from 1 μH to 100 μH; commonly used in
oscillators and various RF circuits such as CB transceivers, televisions, and radios.

7.3 Inductors in DC Circuit

For inductors, a similar relationship exists between the induced volt age across a coil and the
current through the coil. For inductors, the average induced voltage is defined by

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Chapter Seven Short note
 Basic electricity and electronics
Phys2042 A.Y. 2022/23
Dr. Gashaw Beyene (Course Instructor)

Where indicates a finite (measurable) change in current or time. For the instantaneous voltage
across a coil can be derived by letting become vanishingly small. That is,

Inductors in series and in parallel

Inductors, like resistors and capacitors, can be placed in series or in parallel. Increasing levels of
inductance can be obtained by placing inductors in series, while decreasing levels can be obtained
by placing inductors in parallel.
For inductors in series, the total inductance is found in the same manner as the total resistance of
resistors in series.

For inductors in parallel, the total inductance is found in the same manner as the total resistance of
resistors in parallel.

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Chapter Seven Short note
 Basic electricity and electronics
Phys2042 A.Y. 2022/23
Dr. Gashaw Beyene (Course Instructor)

7.4. Inductors in AC circuits

Inductors in series
Since reactance is an opposition in ohms, the values of XL in series or in parallel are combined
the same way as ohms of resistance. With series reactances, the total is the sum of the individual
values.

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Chapter Seven Short note
 Basic electricity and electronics
Phys2042 A.Y. 2022/23
Dr. Gashaw Beyene (Course Instructor)

Inductors in parallel
The combined reactance of parallel reactances is calculated by the reciprocal formula.

The combined parallel reactance will be less than the lowest branch reactance. Any shortcuts for
calculating parallel resistances also apply to parallel reactances. For instance, the combined
reactance of two equal reactances in parallel is one-half either reactance.

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Chapter Seven Short note