When an ac voltage is placed across an inductor and starts to increase (either positively or negatively) from zero, it takes a fraction of a cycle for the current to follow. Once the voltage starts decreasing from its maximum peak (either positive or negative) in the cycle, it again takes a fraction of a cycle for the current to follow. The instantaneous current can’t quite keep up with the instantaneous voltage, as it does in a pure resistance. Thus, in a circuit containing inductive reactance, the current is said to lag the voltage in phase.

#### Pure Inductance

**In a pure inductive reactance, the current lags the voltage by 90°**

Suppose that you place an ac voltage across a coil, with a frequency high enough so that the inductive reactance, XL, is much larger than the resistance, R. In this situation, the current is 1⁄ 4 of a cycle behind the voltage. That is, the current lags the voltage by 90°, as shown in Fig. above figure. At very low frequencies, large inductances are normally needed in order for the current lag to be a full 90°. This is because any coil has some resistance; no wire is a perfect conductor. If some wire were found that had a mathematically zero resistance, and if a coil of any size were wound from this wire, then the current would lag the voltage by 90° in this inductor, no matter what the ac frequency. When the value of XL is very large compared with the value of R in a circuit—that is, when there is an essentially pure inductive reactance—the vector in the RL plane points straight up along the XL axis. Its angle is 90° from the R axis, which is considered the zero line in the RL plane.

#### Inductance with Resistance

**In a circuit with inductive reactance and resistance, the current lags the voltage by less than 90°**

When the resistance in a resistance-inductance (RL) circuit is significant compared with the inductive reactance, the current lags the voltage by something less than 90° (above figure). If R is small compared with XL, the current lag is almost 90°, but as R gets larger relative to XL, the lag decreases. The value of R in an RL circuit can increase relative to XL because resistance is deliberately placed in series with the inductance. It can also happen because the ac frequency gets so low that XL decreases until it is comparable to the loss resistance R in the coil winding. In either case, the situation can be schematically represented by an inductance in series with a resistance (following figure).

If you know the values of XL and R, you can find the angle of lag, also called the RL phase angle, by plotting the point R + jXL on the RL plane, drawing the vector from the origin out to that point, and then measuring the angle of the vector, counterclockwise from the resistance axis. You can use a protractor to measure this angle, or you can compute its value using trigonometry. Actually, you don’t have to know the actual values of XL and R in order to find the angle of lag. All you need to know is their ratio. For example, if XL = 5 Ω and R = 3 Ω, you get the same RL phase angle that you get if XL = 50 Ω and R = 30 Ω, or if XL = 20 Ω and R = 12 Ω. The angle of lag is the same for any values of XL and R in the ratio 5:3.

**Schematic representation of a circuit containing resistance and inductive reactance**

#### Pure Resistance

As the resistance in an RL circuit becomes large with respect to the inductive reactance, the angle of lag gets small. The same thing happens if the inductive reactance gets small compared with the resistance. When R is many times greater than XL, the vector in the RL plane lies almost on the R axis, going east (to the right). The RL phase angle in this case is close to 0°. The current is nearly in phase with the voltage.

In a pure resistance, with no inductance at all, the current is precisely in phase with the voltage (following figure). A pure resistance doesn’t store and release energy as an inductive circuit does, so there is no sluggishness in it.