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# Inductance in AC Circuits

## The Pure Inductance Circuit

Inductance was defined as the property of a circuit to oppose *change*
in current. This opposition results in induced EMF. The induced EMF is
proportional to the rate at which current is changing as well as the
magnitude of the inductance. This relationship can be given by the equation

If a sinusoidal current is flowing in the inductance shown in the figure below, the induced voltage across the inductance can be plotted versus time.

The plot of current in the figure below increases from time *t*_{0}
to *t*_{1}. The current is increasing at a decreasing rate, and
at time *t*_{1} the instantaneous rate of change of current is
zero. Therefore, *v*_{L} is zero at time *t*_{1}.
From time *t*_{1} to *t*_{2} the current is
decreasing; the rate of change of current is negative and
*v*_{L} is negative. At time *t*_{2}, the rate
of change of current is maximum, and thus *v*_{L} is maximum in
the negative direction. At time *t*_{3}, the rate of change of
current is zero, and thus *v*_{L} is again zero. From
*t*_{3} to *t*_{4}, the current is increasing,
the rate of change is positive, and *v*_{L} is positive, reaching
maximum at *t*_{4}, where the rate of change is maximum.

The maximum positive value of voltage occurs 90° ahead of the maximum positive value of current. The current is said to be lagging the voltage by 90 degrees. This phase relationship can be derived mathematically by applying the calculus.

From the previous equation, which is

where

by differentiation, d*i*/d*t* is found.

but

Therefore

By the general form of a periodic function (i.e. *V*_{m} sin (*ωt* + θ)),

and

Since the ratio of volts to amperes is defined as opposition to current in ohms,
the quantity *ωL* is measured in ohms. The quantity *ωL*
is called the **inductive reactance** and is symbolized *X*_{L}

If *V*_{m} = 1.414 *V* and
*I*_{m} = 1.414 *I* are substituted into the equation
*ωL* = *V*_{m} / *I*_{m}, it is seen
that the ratio of the effective values of voltage and current also equals
the inductive reactance

The reciprocal of inductive reactance is called **inductive susceptance**
and is given the symbol *B*_{L}. The unit of inductive
susceptance is the mho (or siemens S) when the frequency is in hertz and
the inductance is in henrys.

**Example 1:** An AC current with a frequency of 2 kHz and a maximum value
of 0.15 A flows in a coil having 175 mH inductance. (1) Find the maximum
voltage developed across the inductance. (2) Find the effective value of the
voltage across the inductance. (3) Write the periodic functions representing
the voltage and current.

Solution:

1.

2.

The effective value of the voltage can also be found by first calculating the effective value of the current

Then

3. If the current is taken as the reference,

The voltage is leading the current in an inductive circuit

**Example 2:** A voltage across an inductance is 40 V when the current is
120 mA. The frequency of current and voltage is 400 Hz. Find the inductance.

Solution: The magnitude of the inductive reactance can be found

The inductance can now be calculated.

The power relationship in an inductive circuit can be analyzed by writing the equation for power with instantaneous values

Applying this equation to the figure above, we see that the instantaneous power is
positive from *t*_{0} to *t*_{1}, negative from
*t*_{1} to *t*_{2}, positive from
*t*_{2} to *t*_{3}, and negative from
*t*_{3} to *t*_{4}. The instantaneous power is
plotted in the figure below. *Positive power* indicates that energy is
taken from the source, and *negative power* that energy is returned to
the source. Since over one complete cycle, from *t*_{0} to
*t*_{4}, as much energy is returned as is taken from
the source, the net energy taken from the source is zero. Power over a
complete cycle is therefore zero. This supports the definition that
inductance is the property of a circuit to store energy in the form of
a magnetic field. Thus, when current is increasing in magnitude, the
magnetic field is building up and storing energy from the source. When
current is decreasing in magnitude, the magnetic field is collapsing and
returning energy to the source.

The net power is also given by the equation *P* = *V**I* cos *θ*: