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Inductors

# Series LR Circuit

The action of current and voltage in a purely resistive circuit is for all practical purposes instantaneous. In other words, when a switch is closed in a circuit containing only a DC source and a resistor, current and voltage reach a maximum value almost instantaneously.

It has been mentioned previously that inductance affects circuit action only
when there is a *change* in circuit current. It can then be assumed that
the technician is primarily concerned with inductance in AC and not steady
state DC circuits. This assumption is correct. But since DC current and
voltage can increase and decrease in magnitude it is informative to study
the effects of the inductor when voltage or current is increasing or decreasing.

### Growth Current in Series LR Circuit

All inductors and batteries have some internal resistance. For ease of explanation this resistance can be lumped together or treated as an individual resistor. This process of lumping circuit elements is called idealization. The circuit in the figure below is idealized and consists of a perfect voltage source, perfect switch, perfect inductor, and a resistance which represents the combined inherent resistances of all the elements.

It has been stated that inductance is the property of a conductor that opposes an increase or decrease in current. Using the circuit and waveforms of the figure above, a voltage and current analysis of the series RL circuit will now be made.

At time zero (*t*_{0}) the switch (SW) would be in the number one
(1) position. As can be seen the circuit is open. The waveforms in the figure above,
view (B) show that at (*t*_{0}) there is no voltage across the coil
(*v*_{L} is zero), there is no voltage across the resistor
(*v*_{R} is zero), and there is no current flowing (*i* is zero).

The waveforms show the above conditions remain unchanged until time one
(*t*_{1}). At *t*_{1} the switch is thrown to the
number two (2) position. At this instant the current will attempt to increase
to its maximum Ohm's law value, in this case:

The current attempts to attain a value of one amp (1 A) in zero elapsed time.
Thus, the current is undergoing its maximum rate of change (roc) at this
instant (*t*_{1}). At the exact instant (*t*_{1}),
there is a simultaneous displacement of electrons in all parts of the circuit,
and while this displacement is immeasurable as a current flow, it does cause
the coil to produce a back EMF which is almost equal to the EMF of the source
(*V*_{b}). An inspection of the waveforms (figure above, view B)
will show the above statements in a graphical form. At
*t*_{1}, *v*_{L} (back EMF) has increased to nearly
the source value, *v*_{R} shows no measurable voltage drop, and
*i* shows no measurable amount of current flow. A brief instant of
time after *t*_{1}, current flow starts and a measurable drop
occurs across *R*. As part of the source voltage is now dropped
across the resistor, less voltage is applied to the coil. As a result, a
decrease of back EMF occurs which permits an increase of current.

Going back to t1, it was mentioned that the rate of change of current (roc) was maximum. This can be proven by computing the rate of change of current at various instants of time as follows:

Note: This roc only applies at the exact instant of time (*t*_{1}).

As the current increases, the *v*_{R} drop increases, and
the induced voltage of the coil decreases.
In fact, examination of the *v*_{R} waveform indicates that at
time two (*t*_{2}) *v*_{R} has increased to about
6.3 volts. According to Kirchhoff's law for voltage the sum of the voltage drops
(*v*_{L} + *v*_{R}) must equal *V*_{b}.
Therefore, *v*_{L} must equal
*V*_{b} - *v*_{R}. At *t*_{2} then:

substituting approximate values from waveforms:

The approximate roc at *t*_{2} may now be found:

From the above it can be seen that as time increases the roc decreases. As time increases more voltage is dropped across the resistor and less across the inductor. Since:

it follows that Δ*i* / Δ*t* decreases with time.

The action of the circuit is continued up to time three (*t*_{3}).
It can now be seen from the waveforms that *v*_{L} has
decreased to nearly zero, *v*_{R} is now nearly equal to the
source voltage, and the circuit current has just about reached maximum.
Close inspection of the waveform shows that at *t*_{3}:

Actually the current will never reach the maximum value of 1 A. After a
predictable length of time however, the magnitude of the current is so close
to its theoretical maximum value it can be considered to have reached this
value. This final point is reached in the figure above (view B) very shortly
after *t*_{3}. After *t*_{3} a steady state value is
established by the current, and since for all practical purposes there is no
more change, there is effectively no more back EMF being produced. The energy
taken to overcome the back EMF of the coil is now stored
in the magnetic field which exists around the coil.

As long as the switch is maintained in position (2) the conditions of the
circuit will remain as follows: *v*_{L} equals to zero,
*v*_{R} equals the voltage source and
*i* is equal to its maximum Ohm's law value.

### Decay Current in Series LR Circuit

At time four (*t*_{4}) the switch (SW) is moved instantaneously
to position (3). Since the source *V*_{b} is now removed from the
circuit the current will *attempt* to stop instantly. Again the
term attempt has been used because of the action of the inductor in opposing
any change. At *t*_{4} the waveform depicts the coil as developing
a large back EMF, only this time the polarity is in
the opposite direction because the back EMF is developed by the collapsing
of the magnetic field. At *t*_{4} the circuit of the figure above
(view A) consists of a basic circuit with the resistance connected across
the coil, which is now acting as the source of voltage. The equivalent circuit
during decay time is shown in the figure above, view C. Current through the
resistance is maintained in the original direction due to the action of the
collapsing flux field.

The waveforms in the figure above (view B) indicate that at the instant of
time *t*_{4} the coil (acting as the source),
is supplying maximum current (*i* at *t*_{4} is 1 A) and the
circuit satisfied Kirchhoff's voltage law. (The voltage drop,
*v*_{R} = 10 V, is equal to the source, *v*_{L} = 10V.)
The energy originally supplied to the coil is stored in the field surrounding
the coil, and as the field collapses more and more of this energy is returned
to the circuit. The less energy the field contains, the less rapidly it will
collapse, and the less rapidly it collapses the less back EMF will be produced.

The action of the circuit during decay is shown more clearly by the
waveforms. It can be seen that at *t*_{5} the field has expended
a large part of its stored energy. The coil voltage *v*_{L}
(absolute value) has decreased to a value of approximately 3.7 V. To satisfy
Kirchhoff's voltage law the resistive voltage drop *v*_{R} must
also have decreased to 3.7 V which in turn indicates a decrease in circuit
current. Since current will only be maintained during the time the field is
collapsing, the waveforms indicate the above circuit action will continue until
*t*_{6}, when for all practical purposes the field has completely
collapsed. This time *v*_{L} has decreased to zero,
*v*_{R} has decreased to zero, and *i* has decreased to zero.
It will be noted that the waveforms of resistor's voltage and circuit
current are identical and therefore, in some graphs, may be
illustrated by the same curve.

### LR Time Constant

It was stated, during the explanation of roc, that the current would reach
its maximum value in a certain predictable length of time. Specifically,
the time it takes the current in a circuit,
containing only resistance and inductance to increase to 63.2 percent
(or decrease to 36.8 percent) of its maximum value is known as the
**time constant**. The time constant is determined by the
ratio of circuit inductance to circuit resistance. The mathematical equation is:

where: *τ* (tau) - time constant in seconds, *L* - circuit
inductance in henrys, *R* - circuit resistance in ohms

**Example**. Find the time constant of a circuit in which the inductance
is 2 H and the resistance is 10 Ω.

The current in an LR circuit does not rise in a linear manner. The instantaneous current
magnitude with respect to time follows what is called an **exponential curve**.

The exponential curve is a result of the fact that a current, in approaching
a maximum value in a series of time constant, will only increase 63.2 percent
of the remaining value in *each* time constant. In other words, if the
final maximum value of current in a circuit is to be 1 A,
then the first time constant the current will increase to 63.2 percent of 1 A or
0.632 A. During the second time constant the *change* in current will
be 63.2 % of the difference between the final value and the value at the end
of the first time constant:

According to the explanation of the exponential curve and the definition
of a time constant, the current will increase 63.2 percent of the
*remaining* value during the second time constant.
This will be 63.2 percent of 0.368 A or approximately 0.233 A.

Therefore, at the end of the two time constants the current, in its rise toward maximum, will have a value equal to the current increase in the second time constant or:

Following the above procedure again the remaining value of current will be:

63.2 percent of this value, the amount or rise in current during the
3^{rd} time constant, is:

At the end of the third time constant the current will equal:

In the above calculations the increase of current in each succeeding time constant is seen
to be less and less. For normal applications, **after five time constants
have passed the current differs from its final value by such a negligible amount
that it is considered to have reached its final value**.

### Universal Time Constant Chart

Since the growth and decay of current in any LR circuit follows the exponential curve, a curve developed for a specific circuit can be made to apply to any LR circuit by merely changing the values of time and current.

The chart above presents the magnitude of voltage and current in a series LR circuit of any integral time constant during growth and decay. The curve A represents inductor growth current or resistor voltage during growth of inductor current. The curve B represents inductor current during decay, inductor voltage during inductor current growth, and resistor voltage during decay of inductor current.

The curve representing the growth current in a series LR circuit is a graph of the general exponential formula given below:

where:

*i* - instantaneous current in amps

*I*_{max} - maximum value of current in amps

e - epsilon, the base of natural logarithms, (rounded off to 2.718)

*τ* - time constant (*L*/*R*) in seconds

*t* - elapsed time in seconds

This equation is just one variation of the exponential formula used in LR circuits. Other variations are listed below for convenience:

Equations for LR circuits during current growth:

where:

*V*_{b} - source voltage in volts

*v*_{R} - instantaneous resistor voltage in volts

*v*_{L} - instantaneous coil voltage in volts

All other factors as defined previously.

Equations for LR circuits during current decay:

where:

*I*_{s} - circuit current at start of decay, in amps

*R* - circuit resistance, in ohms

All other factors as defined previously.