**Introduction to Inductive Reactance Formula**

Electrical *resistance* —the extent of the opposition that a medium offers to DC—is a scalar quantity, because it can be expressed on a one-dimensional scale. Resistance is measured in units called *ohms.* Given a certain DC voltage, the electrical current through a device goes down as its resistance goes up. The same law holds for AC through a resistance. A component with resistance has “electrical friction.” But in a coil of wire, the situation is more complicated. A coil stores energy as a magnetic field. This makes a coil behave sluggishly when AC is driven through it, as if it has “electrical inertia.”

**Coils And Current**

If you wind a length of wire into a coil and connect it to a source of DC, the coil becomes warm as energy is dissipated in the resistance of the wire. If the voltage is increased, the current increases also, and the wire gets hot.

Suppose you change the voltage source, connected across the coil, from DC to AC. You vary the frequency from a few hertz (Hz) to many megahertz (MHz). The coil has a certain *inductive reactance* (denoted X _{L} ), so it takes some time for current to establish itself in the coil. As the AC frequency increases, a point is reached at which the current cannot get well established in the coil before the polarity of the voltage reverses. As the frequency is raised, this effect becomes more pronounced. Eventually, if you keep increasing the frequency, the current will hardly get established at all before the polarity of the voltage reverses. Under such conditions, very little current will flow through the coil. Inductive reactance, like resistance, is expressed in ohms. But the “inductive ohm” is a different sort of ohm.

The inductive reactance of a coil (or *inductor* ) can vary from zero (a short circuit) to a few ohms (for small coils) to kilohms or megohms (for large coils). Like pure resistance, inductive reactance affects the current in an AC circuit. But unlike pure resistance, inductive reactance changes with frequency. This affects the way the current flows with respect to the voltage.

*X* _{L} Vs Frequency

If the frequency of an AC source is given (in hertz) as *f* , and the inductance of a coil is specified (in units called *henrys* ) as *L,* then the inductive reactance (in ohms), X _{L} , is given by:

X _{L} = 2 *πfL*

Inductive reactance increases linearly with increasing AC frequency. Inductive reactance also increases linearly with increasing inductance. The value of X _{L} is directly proportional to f; X _{L} is also directly proportional to L. These relationships are graphed, in relative form, in Fig. 9-8.

**Fig. 9–8.** Inductive reactance increases in a linear manner as the AC frequency goes up, and also as the inductance goes up.

Inductance stores electrical energy as a magnetic field. When a voltage appears across a coil, it takes a while for the current to build up to full value. Thus, when AC is placed across a coil, the current lags the voltage in phase. The current can’t keep up with the changing voltage because of the “electrical inertia” in the inductor. Inductive reactance and ordinary resistance combine in interesting ways. Trigonometry can be used to figure out the extent to which the current lags behind the voltage in an inductance-resistance, or *RL,* electrical circuit.

*Rl* Phase Angle

When the resistance in an electronic circuit is significant compared with the inductive reactance, the alternating current resulting from an alternating voltage lags that voltage by less than 90° (Fig. 9-9). If the resistance *R* is small compared with the inductive reactance X _{L} , the current lag is almost 90°; as *R* gets relatively larger, the lag decreases. When *R* is many times greater than X _{L} , the phase angle, *ø* _{RL} , is nearly zero. If the inductive reactance vanishes altogether, leaving a pure resistance, then the current and voltage are in phase with each other.

The value of the phase angle *ø* _{RL} , which represents the extent to which the current lags the voltage, can be found using a calculator that has inverse trigonometric functions. The angle is the arctangent of the ratio of inductive reactance to resistance:

**Fig. 9–9.** An example of current that lags voltage by less than 90°, as in a circuit containing resistance and inductive reactance.

*ø* _{RL} = arctan( *X *_{L} /R )

**Inductive Reactance Practice Problems**

**Practice 1**

Find the phase angle between the AC voltage and current in an electrical circuit that has 50 ohms of resistance and 70 ohms of inductive reactance. Express your answer to the nearest whole degree.

**Solution 1**

Use the above formula to find *ø* _{RL} , setting X _{L} = 70 and *R* = 50:

Practice problems for these concepts can be found at: Waves and Phase Practice Test

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From Trigonometry Demystified: A Self-Teaching Guide. Copyright © 2003 by The McGraw-Hill Companies, Inc. All Rights Reserved.