# Inductors in AC Circuits: Inductive Reactive and Phasor Diagrams

## In an AC circuit with an inductor, the voltage across an inductor "leads" the current because of the Lenz' law.

#### Key Points

• With an inductor in an AC circuit, the voltage leads the current by one-fourth of a cycle, or by a 90º phase angle.

• The rms current Irms through an inductor L is given by a version of Ohm’s law: $I_{rms} = \frac{V_{rms}}{X_L}$. XL is called the inductive reactance, given as $X_L = 2\pi \nu L$.

• Phasors are vectors rotating in counter-clockwise direction. A phasor for an inductor  shows that the voltage lead the current by a 90º phase.

#### Terms

• A law of electromagnetic induction that states that an electromotive force, induced in a conductor, is always in such a direction that the current produced would oppose the change that caused it; this law is a form of the law of conservation of energy.

• A representation of a complex number in terms of a complex exponential.

• Root mean square: a statistical measure of the magnitude of a varying quantity.

#### Figures

1. ##### AC Voltage Source in Series with an Inductor

(a) An AC voltage source in series with an inductor having negligible resistance. (b) Graph of current and voltage across the inductor as functions of time.

2. ##### Phasor Diagram

Phasor diagram for an AC circuit with an inductor.

3. ##### Phasors for Inductors in AC Circuits

Suppose an inductor is connected directly to an AC voltage source, as shown in Figure 1. It is reasonable to assume negligible resistance because in practice we can make the resistance of an inductor so small that it has a negligible effect on the circuit. The graph shows voltage and current as functions of time. Figure 1(b) starts with voltage at a maximum. Note that the current starts at zero, then rises to its peak after the voltage driving it (as seen in the preceding section when DC voltage was switched on).

When the voltage becomes negative at point a, the current begins to decrease; it becomes zero at point b, where voltage is its most negative. The current then becomes negative, again following the voltage. The voltage becomes positive at point c where it begins to make the current less negative. At point d, the current goes through zero just as the voltage reaches its positive peak to start another cycle. Hence, when a sinusoidal voltage is applied to an inductor, the voltage leads the current by one-fourth of a cycle, or by a 90º phase angle.

Current lags behind voltage, since inductors oppose change in current. Changing current induces an emf . This is considered an effective resistance of the inductor to AC. The rms current Irms through an inductor L is given by a version of Ohm’s law: $I_{rms} = \frac{V_{rms}}{X_L}$ where Vrms is the rms voltage across the inductor and $X_L = 2\pi \nu L$ with $\nu$ the frequency of the AC voltage source in hertz. XL is called the inductive reactance. Because the inductor reacts to impede the current, XL has units of ohms (1 H=1 Ω⋅s, so that frequency times inductance has units of (cycles/s)(Ω⋅s)=Ω), consistent with its role as an effective resistance.

### Phasor Representation

The voltage across an inductor "leads" the current because of the Lenz's law. Therefore, the phasor representing the current and voltage would be given as in Figure 2. Again, the phasors are vectors rotating in counter-clockwise direction at a frequency $\nu$ (you can see that the voltage leads the current). Subsequent Atoms will discuss how these phasors can be used to analyze RC, RL, LC, and RLC circuits.

Figure 3

#### Key Term Glossary

AC
Alternating current.
##### Appears in these related concepts:
atom
The smallest possible amount of matter which still retains its identity as a chemical element, now known to consist of a nucleus surrounded by electrons.
##### Appears in these related concepts:
circuit
A pathway of electric current composed of individual electronic components, such as resistors, transistors, capacitors, inductors and diodes, connected by conductive wires or traces through which electric current can flow. T
##### Appears in these related concepts:
current
The time rate of flow of electric charge.
##### Appears in these related concepts:
DC
Direct current; the unidirectional flow of electric charge.
##### Appears in these related concepts:
frequency
The quotient of the number of times n a periodic phenomenon occurs over the time t in which it occurs: f = n / t.
##### Appears in these related concepts:
hertz
In the International System of Units, the derived unit of frequency; one (period or cycle of any periodic event) per second. Symbol: Hz
##### Appears in these related concepts:
Hertz
Measurement of sound frequency.
##### Appears in these related concepts:
inductor
A passive device that introduces inductance into an electrical circuit.
##### Appears in these related concepts:
Inductor
A device or circuit component that exhibits significant self-inductance; a device which stores energy in a magnetic field.
##### Appears in these related concepts:
Law
A concise description, usually in the form of a mathematical equation, used to describe a pattern in nature
##### Appears in these related concepts:
ohm
in the International System of Units, the derived unit of electrical resistance; the electrical resistance of a device across which a potential difference of one volt causes a current of one ampere; symbol: Ω
##### Appears in these related concepts:
phase
Any one point or portion in a recurring series of changes, as in the changes of motion of one of the particles constituting a wave or vibration; one portion of a series of such changes (in distinction from a contrasted portion) as the portion on one side of a position of equilibrium (in contrast with that on the opposite side).
##### Appears in these related concepts:
phasor
A representation of a complex number in terms of a complex exponential.
##### Appears in these related concepts:
reactance
The opposition to the change in flow of current in an alternating current circuit, due to inductance and capacitance; the imaginary part of the impedance.
##### Appears in these related concepts:
resistance
The opposition to the passage of an electric current through that element.
##### Appears in these related concepts:
rms
Root mean square: a statistical measure of the magnitude of a varying quantity.
##### Appears in these related concepts:
rms current
the root mean square of the current, Irms=I0/√2 , where I0 is the peak current, in an AC system
##### Appears in these related concepts:
rms voltage
the root mean square of the voltage, Vrms=V0/√2 , where V0 is the peak voltage, in an AC system
##### Appears in these related concepts:
sinusoidal
In the form of a wave, especially one whose amplitude varies in proportion to the sine of some variable (such as time).
##### Appears in these related concepts:
vector
A directed quantity, one with both magnitude and direction; the between two points.
##### Appears in these related concepts:
voltage
The amount of electrostatic potential between two points in space.