# Inductors in AC Circuits

Inductance of a coil In addition to the electric fields that occur, for instance, between the plates of a charged capacitor, electrical engineering also relies on a second kind of field, magnetic fields. While electric fields are generated by static charge, magnetic fields are generated by moving charge carriers, i.e. electric current. If an electric current passes through a coil comprising an array of conductive windings, magnetic field lines run through the coil as a result. The magnetic field's intensity is represented by the magnetic flux. If the flux through the coil changes (e.g. by varying the current), a self-induced voltage is generated in the coil. The intensity of this voltage depends on the rate of change of current as well as the size and nature of the coil. The coil's inductance L is a measure of its ability to generate self-induced voltages. The following relationship applies to extended coils:

µ0 is the magnetic field constant, µr the relative permeability of the material in the core of the coil, N the number of windings, l the length of the coil and A its cross-section (see the diagram below).

The unit of inductance is the Henry (H, 1 H = 1 Vs/A). A coil has an inductance of 1 H if a constant change of 1 A per second in the current flowing through a coil results in a self-induced voltage of 1 V. Energising and de-energising a coil If direct current is passed through a coil, the current characteristic after a brief transient phase is constant, so that no voltage is self-induced. In this case, the coil acts as an ohmic load, the (usually very low) resistance of which depends on the coil material's specific resistivity as well as the length and cross-section of the coil wire. When a coil is energised, its magnetic field builds up. The resulting change in flux gives rise to a self-induced voltage in the opposite direction to the applied voltage. Consequently, the current in the electric circuit does not rise abruptly (as in the case of a resistive load). Instead, it rises very gradually to a final value. The reverse process takes place on de-energisation of the coil. The decaying magnetic field now produces a self-induced voltage acting in the same direction as the previously applied voltage and capable of assuming very large values in coils with strong magnetic fields. The self-induced voltage tends to sustain the current through the coil, so that the current does not cut off suddenly, instead gradually dropping to zero. The diagrams below demonstrate the de-energising process. In the active state (left-hand illustration), a current I flows through the coil L. If the electric circuit is then broken (right-hand illustration), the energy stored within the coil's magnetic field tends to sustain the current. Because the current can no longer flow via the voltage source, it takes a detour via the resistor RL connected in parallel with the coil. In this process, the energy of the magnetic field is converted into thermal energy, causing the current to decay rapidly. The decay characteristic, as in the case of the capacitor, is exponential. The time constant in this case is the quotient of the inductance and the ohmic resistance.

The absence of a resistor RL would result in a very high voltage peak, easily capable of damaging sensitive components (e.g. in integrated circuits). For this reason, in practice inductors are usually connected in parallel with free-wheeling diodes which short-circuit voltage peaks across the coil to ensure that the coil's electromagnetic energy is converted into thermal energy.

__Series and parallel connection of coils__:

Inductors connected in series or parallel are governed by the same formulae as those applicable to resistive loads. The following diagram shows an array of inductors connected in series.

The total inductance here is:

In other words:

If several coils are connected in series, their total inductance is equal to the sum of the individual inductances. |

The following diagram shows an array of inductors connected in parallel.

The total inductance here is:

In other words:

If several coils are connected in parallel, the inverse of their total inductance is equal to the sum of the individual inverse inductances. |

__ Reactance of a coil__:

As in the case of a capacitor, the current and voltage of an ideal inductor coil (i.e. one with no active resistance) are displaced in phase by 90°, resulting in purely reactive power. The coil's higher AC resistance compared with its DC resistance is due to the *inductive reactance X* _{L}. This reactance arises from an opposing, self-induced voltage.

The higher the inductance *L *of a coil, the higher the opposing voltage and resulting reactance. Also, the faster the change in current - i.e. the higher the frequency - the higher the induced voltage. The formula for inductive reactance is:

In qualitative terms:

The higher the frequency and inductance, the larger the inductive reactance. |

**Example 1: **at a frequency of 50 Hz, a coil with an inductance of 2 H has the following reactance

**Example 2:** at a frequency of 40 Hz, a coil with a reactance of 12.5 W has the following inductance

__ Sinusoidal alternating voltage for a coil__:

As observed during energising and de-energising of a coil, current starts to flow through an inductor after a certain delay. Connecting a pure inductance (i.e. a coil with an ohmic resistance of zero) to a sinusoidal alternating voltage of the form

results in a sinusoidal coil current *i* which lags behind the voltage by an angle

The current is thus described by the following equation:

The corresponding characteristic is shown in the diagram below.

The coil current attains its maximum value when the voltage is zero, and vice versa.

The current flowing through a coil lags behind the applied voltage by an angle j = 90°. |

Though able to conduct alternating voltage in a similar way to a capacitor, a pure inductor only consumes reactive power due to the phase shift of 90°, causing the coil's magnetic field to build up and decay periodically. In reality, however, coils also have a very small ohmic resistance resulting in a phase shift slightly less than 90° and a corresponding consumption of active power.

**Energising and de-energising a coil:**

The processes of energising and de-energising a coil are already familiar from the treatment of *DC circuits*. If a coil is connected to a direct voltage source as shown in the diagram below (switch closed), a voltage *U*_{ind} is self-induced at the same time that a current starts to flow. According to Lenz's law, this voltage is in the opposite direction to the externally applied DC voltage *U*_{0}. As a result, the current through the coil rises with a delay described by an exponential function, reaching its maximum value after a period which depends on the inductance *L* and ohmic resistance *R* (the range labelled (1) in the timing diagram).

Turning off the external voltage (opening the switch) cuts off the power generating the coil's magnetic field and gives rise to a self-induced voltage in the coil which is opposite to the cause of the change in magnetic field (i.e. disconnection of the external voltage) and initially allows the current to keep flowing (the range labelled (2) in the timing diagram). The induced voltage thus causes the decay of the coil current to take longer.

The following animation illustrates these relationships.

**Phase shift in an RL element:**

An ideal coil comprises a pure reactance *X*_{L} 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 *U*_{R} (in phase with the current) and *U*_{L} (leading the current by 90°).

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

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

Replacing *X*_{L} with the relationship derived earlier results in the following:

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

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:

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