EEE251 BASIC ELECTRICAL ENGINEERING

Lecture 9:
Inductors

Instructor: N. Thwala

June, 2023
Lecture Objectives
• Become familiar with the basic construction of an inductor, the factors that affect the
strength of the magnetic field established by the element
• Be able to determine the transient (time-varying) response of an inductive network and
plot the resulting voltages and currents.
• Understand the impact of combining inductors in series or parallel.

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Inductance
• Sending a current through a coil of wire establishes a magnetic field through and
surrounding the unit. This coil is called an inductor.
• The inductance level determines the strength of the magnetic field around the coil due
to an applied current.
• The higher the inductance level, the greater the strength of the magnetic field.
• Inductors are designed to set up a strong magnetic field linking the unit, whereas
capacitors are designed to set up a strong electric field between the plates.
• Inductance is measured in henries (H).
• 1 henry is the inductance level that will establish a voltage of 1 volt across the coil due
to a change in current of 1 A/s through the coil.

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Inductance
Inductance is dependent on the area within the coil, the length of the unit, and
the permeability of the core material. It is also sensitive to the number of turns
of wire in the coil as dictated by:

Substituting for the permeability results in:

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Inductance
• If we break out the relative permeability as follows:

we obtain the following useful equation:

• Which states that: The inductance of an inductor with a ferromagnetic core is μr times
the inductance obtained with an air core.
• The size of an inductor is determined primarily by the type of construction, the core
used, or the current rating.
• The standard values for inductors employ the same numerical values and multipliers
used with resistors and capacitors. In general, therefore, expect to find inductors with
the following multipliers: 1 mH, 1.5 mH, 2.2 mH, 3.3 mH, 4.7 mH, 6.8 mH, 10 mH,
and so on.

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Induced Voltage vL
• Faraday’s law of electromagnetic induction:
If we move a conductor through a magnetic field so that it cuts
magnetic lines of flux as shown in Fig., a voltage is induced
across the conductor that can be measured with a sensitive
voltmeter. The faster you move the conductor through the
magnetic flux, the greater the induced voltage.
The same effect can be produced if you hold the conductor still and move the magnetic field across the
conductor. Note that the direction in which you move the conductor through the field determines the
polarity of the induced voltage.

Also, if you move the conductor through the field at right angles to the magnetic flux, you generate the
maximum induced voltage. Moving the conductor parallel with the magnetic flux lines results in an
induced voltage of zero volts since magnetic lines of flux are not crossed.

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Induced Voltage vL
If we now move a coil of N turns through the magnetic field as
shown in Fig., a voltage will be induced across the coil as
determined by Faraday’s law:

The greater the number of turns or the faster the coil is moved through the magnetic flux pattern, the greater
the induced voltage.
Lenz’s law, which states that an induced effect is always such as to oppose the cause that produced it. The
inductance of a coil is also a measure of the change in flux linking the coil due to a change in current
through the coil. That is,

The following notation is used for the induced voltage across an inductor:

The equation states that: the larger the inductance and/or the more rapid the change in current through a
coil, the larger will be the induced voltage across the coil.
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R-L Transients: The Storage Phase
A great number of similarities exist between the analyses of inductive and
capacitive networks. That is, what is true for the voltage of a capacitor is
also true for the current of an inductor, and what is true for the current of a
capacitor can be matched in many ways by the voltage of an inductor.
The circuit in Fig. is used to describe the storage phase. Note that it is the same circuit used to describe the
charging phase of capacitors, with a simple replacement of the capacitor by an ideal inductor.

Throughout the analysis, it is important to remember that energy is stored in the form of an electric field
between the plates of a capacitor. For inductors, on the other hand, energy is stored in the form of a magnetic
field linking the coil.

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R-L Transients: The Storage Phase
At the instant the switch is closed, the choking action of the coil prevents an instantaneous
change in current through the coil, resulting in iL = 0 A as shown in Fig. (a). The absence
of a current through the coil and circuit at the instant the switch is closed results in zero
volts across the resistor as determined by vR =iRR=iLR=(0 A)R=0 V, as shown in Fig. (c).
Applying Kirchhoff’s voltage law around the closed loop results in E volts across the coil
at the instant the switch is closed, as shown in Fig.(b).

Initially, the current increases very rapidly as shown in Fig. (a) and then at a much slower
rate as it approaches its steady-state value determined by the parameters of the network
(E/R). The voltage across the resistor rises at the same rate because vR =iRR=iLR. Since the
voltage across the coil is sensitive to the rate of change of current through the coil, the
voltage will be at or near its maximum value early in the storage phase. Finally, when the
current reaches its steady-state value of E/R amperes, the current through the coil ceases to
change, and the voltage across the coil drops to zero volts. At any instant of time, the
voltage across the coil can be determined using Kirchhoff’s voltage law in the
following manner: vL =E – vR

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R-L Transients: The Storage Phase
• The equation for the transient response of the current through an inductor is the following:

with the time constant now defined by:

• The equation for the voltage across the coil is the following:
and for the voltage across the resistor:

• We assume that: the storage phase has passed and steady-state conditions have been established
once a period of time equal to five time constants has occurred.
• Current cannot change instantaneously in an inductive network.
• The inductor takes on the characteristics of an open circuit at the instant the switch is closed.
• The inductor takes on the characteristics of a short circuit when steady-state conditions have been
established.
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Example 1
Find the mathematical expressions for the
transient behaviour of iL and yL for the circuit
in Fig. below if the switch is closed at t = 0 s.
Sketch the resulting curves.

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Example 1: Solution
The resulting waveforms:

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Initial Conditions
Since current through a coil cannot change instantaneously, the current through a coil begins the
transient phase at the initial value established by the network (see Fig.) before the switch was closed. It
then passes through the transient phase until it reaches the steady-state (or final) level after about five
time constants. The steady-state level of the inductor current can be found by substituting its short-circuit
equivalent (or Rl for the practical equivalent) and finding the resulting current through the element.

Current iL transient equation:

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R-L Transients: The Release Phase
After the storage phase has passed and steady-state conditions are established, the switch can be opened
without the sparking effect or rapid discharge due to resistor R2, which provides a complete path for the
current iL (see Fig.)

The voltage vL across the inductor reverses polarity and has a magnitude determined by:

Fig. P
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R-L Transients: The Release Phase
• As an inductor releases its stored energy, the voltage across the coil decays to zero in the following
manner:
• The current decays from a maximum of Im = E/R1 to zero.
• The voltage vR and vR are expressed as follows:
1 2

• If the switch is opened before iL reaches its maximum value, then the equation for the decaying
current changes to:
where Ii is the starting or initial current. The voltage across the coil is defined by the following:

with

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Example 2
For the network in Fig.:
a. Find the mathematical expressions for for five time constants of the storage phase.
b. Find the mathematical expressions for if the switch is opened after five time constants of the
storage phase.
c. Sketch the waveforms for each voltage and current for both phases covered by this example. Use the defined
polarities in Fig. P

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Example 2: Solution

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Example 2: Solution
c. The waveforms:

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Thévenin Equivalent: 𝝉 = L/RTh
Example 3:
Switch S1 in Fig. has been closed for a long time. At t = 0 s, S1 is opened at the same instant that S2 is
closed to avoid an interruption in current through the coil.
a. Find the initial current through the coil. Pay particular attention to its direction.
b. Find the mathematical expression for the current iL following the closing of switch S2.
c. Sketch the waveform for iL .

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Solution

Fig. A

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Instantaneous Values
• The instantaneous values of any voltage or current can be determined by simply
inserting t into the equation. From these equations:

we can derive:

• The average induced voltage for inductors is defined by:

If the current increases with time, the average current is the
change in current divided by the change in time with a positive
sign. If the current decreases with time, a negative sign is
applied.

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Example 4
Find the waveform for the average voltage across
the coil if the current through a 4 mH coil is as
shown in Fig.

The waveform for the average voltage across the coil is shown in Figure
below:

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Inductors in Series and 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:

• For two inductors:

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Example 5
• Reduce the network in Fig. A to its simplest form.

Fig. A

Fig. B
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Steady-State Conditions
• Recall that the term steady state implies that the voltage and current levels have reached their final
resting value and will no longer change unless a change is made in the applied voltage or circuit
configuration.
• Our assumption is that steady-state conditions have been established after five time constants of the
storage or release phase have passed.
Example 6
Find the current IL and the voltage VC for the network in Fig.

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Energy Stored By An Inductor
An ideal inductor stores energy in the form of a magnetic field. And the energy stored is:

Example 7: Find the energy stored by the inductor in the circuit in Fig. when the current through it has
reached its final value.

Solution:

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 Next Lecture:
Sinusoidal Alternating Waveforms
(AC)

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