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Phase Shift in an RL Element

An ideal coil comprises a pure reactance XL where the voltage leads the current by a phase angle j = 90°, the two vectors thus forming a right angle (illustrations on the left below). Connecting an ohmic resistance and inductance in series to form an RL element produces, as in the case of the RC element considered earlier, an overall resistance comprising an active and a reactive component. In this case, the voltage leads the current by a phase angle j that lies somewhere between 0° and 90° depending on the frequency, resistance and inductance (illustrations on the right below). The voltage vector U here is determined by geometric addition of the partial voltages UR (in phase with the current) and UL (leading the current by 90°). 

INDUCTORS IN AC CIRCUITS

The following diagram shows the triangle of resistances for an RL element.  

INDUCTORS IN AC CIRCUITS

The impedance Z can be calculated easily from the triangle of resistances, which being right-angled, results in the following relationship:

INDUCTORS IN AC CIRCUITS

Replacing XL with the relationship derived earlier results in the following:

INDUCTORS IN AC CIRCUITS

The phase angle j can also be determined from the triangle of resistances:

INDUCTORS IN AC CIRCUITS

If the values of w or f, R and j are known, this equation can be used to determine inductance by resolving in terms of L. The following relationship is then obtained:

INDUCTORS IN AC CIRCUITS

Alternatively, the phase shift between the coil voltage UL and supply voltage U can be measured. The inductance is then derived from the following formula:

INDUCTORS IN AC CIRCUITS

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