Any device relying on magnetism or magnetic fields to operate is a form of inductor. Motors, generators, transformers, and coils are inductors. The use of an inductor in a circuit can cause current and voltage to become out-of-phase and inefficient unless corrected.

EO 1.1DESCRIBE inductive reactance (XL).

EO 1.2Given the operation frequency (f) and the value of inductance (L), CALCULATE the inductive reactance (X_L) of a simple circuit.

EO 1.3DESCRIBE the effect of the phase relationship between current and voltage in an inductive circuit.

EO 1.4DRAW a simple phasor diagram representing AC current (I) and voltage (E) in an inductive circuit.

Inductive Reactance

In an inductive AC circuit, the current is continually changing and is continuously inducing an EMF. Because this EMF opposes the continuous change in the flowing current, its effect is measured in ohms. This opposition of the inductance to the flow of an alternating current is called inductive reactance (X_L). Equation (8-1) is the mathematical representation of the current flowing in a circuit that contains only inductive reactance.

where

The value of X_L in any circuit is dependent on the inductance of the circuit and on the rate at which the current is changing through the circuit. This rate of change depends on the frequency of the applied voltage. Equation (8-2) is the mathematical representation for X_L.

where

The magnitude of an induced EMF in a circuit depends on how fast the flux that links the circuit is changing. In the case of self-induced EMF (such as in a coil), a counter EMF is induced in the coil due to a change in current and flux in the coil. This CEMF opposes any change in current, and its value at any time will depend on the rate at which the current and flux are changing at that time. In a purely inductive circuit, the resistance is negligible in comparison to the inductive reactance. The voltage applied to the circuit must always be equal and opposite to the EMF of self-induction.

Voltage and Current Phase Relationships in an Inductive Circuit

As previously stated, any change in current in a coil (either a rise or a fall) causes a corresponding change of the magnetic flux around the coil. Because the current changes at its maximum rate when it is going through its zero value at 90 (point b on Figure 1) and 270 (point d), the flux change is also the greatest at those times. Consequently, the self-induced EMF in the coil is at its maximum (or minimum) value at these points, as shown in Figure 1. Because the current is not changing at the point when it is going through its peak value at 0 (point a), 180 (point c), and 360 (point e), the flux change is zero at those times. Therefore, the selfinduced EMF in the coil is at its zero value at these points.

Figure 1 Current, Self-Induced EMF, and Applied Voltage in an Inductive Circuit

According to Lenz's Law (refer to Module 1, Basic Electrical Theory), the induced voltage always opposes the change in current. Referring to Figure 1, with the current at its maximum negative value (point a), the induced EMF is at a zero value and falling. Thus, when the current rises in a positive direction (point a to point c), the induced EMF is of opposite polarity to the applied voltage and opposes the rise in current. Notice that as the current passes through its zero value (point b) the induced voltage reaches its maximum negative value. With the current now at its maximum positive value (point c), the induced EMF is at a zero value and rising. As the current is falling toward its zero value at 180 (point c to point d), the induced EMF is of the same polarity as the current and tends to keep the current from falling. When the current reaches a zero value, the induced EMF is at its maximum positive value. Later, when the current is increasing from zero to its maximum negative value at 360 (point d to point e), the induced voltage is of the opposite polarity as the current and tends to keep the current from increasing in the negative direction. Thus, the induced EMF can be seen to lag the current by 90.

The value of the self-induced EMF varies as a sine wave and lags the current by 90, as shown in Figure 1. The applied voltage must be equal and opposite to the self-induced EMF at all times; therefore, the current lags the applied voltage by 90 in a purely inductive circuit.

If the applied voltage (E) is represented by a vector rotating in a counterclockwise direction (Figure lb), then the current can be expressed as a vector that is lagging the applied voltage by 90. Diagrams of this type are referred to as phasor diagrams.

Example: A 0.4 H coil with negligible resistance is connected to a 115V, 60 Hz power source (see Figure 2). Find the inductive reactance of the coil and the current through the circuit. Draw a phasor diagram showing the phase relationship between current and applied voltage.

Figure 2 Coil Circuit and Phasor Diagram

Solution:

1. Inductive reactance of the coil

2. Current through the circuit

3. Draw a phasor diagram showing the phase relationship between current and applied voltage.

Phasor diagram showing the current lagging voltage by 90 is drawn in Figure 2b.

Summary

Inductive reactance is summarized below.

Inductive Reactance Summary

Opposition to the flow of alternating current caused by inductance is called Inductive Reactance (X_L).

The formula for calculating X_L is:

Current (I) lags applied voltage (E) in a purely inductive circuit by 90 phase angle.

The phasor diagram shows the applied voltage (E) vector leading (above) the current (I) vector by the amount of the phase angle differential due to the relationship between voltage and current in an inductive circuit.