When using an inductor in a circuit where quality factor of inductor is important its resistance becomes an important factor. Any resistance will reduce the overall quality factor of inductor. Even though inductors are often assumed to be pure inductors, they always have a finite amount of resistance, however low.

This DC resistance affects the quality factor of inductor, and is one of the major factors affecting this area of performance of the component. In view of this the inductor quality factor is widely specified for inductors to be used in RF applications.

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# Quality Factor of Inductor (Inductor Q)

Every inductor possesses a small resistance in addition to its inductance. The lower the value of this resistance R, the better the quality of the coil. The **quality factor** or the **Q factor of an inductor** at the operating frequency ω is defined as the ratio of reactance of the coil to its resistance.

## Inductor Q Factor Basics

When using an inductor in a circuit where the Q or quality factor is important its resistance becomes an important factor. Any resistance will reduce the overall inductor Q factor.

An inductor can be considered in terms of its equivalent circuit. This can be simply expressed as a perfect inductor with a series resistor.

**Where:**

- L is a perfect inductor
- R is the resistance of the inductor

The resistance within an inductor is caused by a number of effects:

**1. Standard DC Resistance**

The standard DC resistance will always be present (except in superconductors which are not normally encountered). This is one of the major components of resistance in any coil or inductor and one that can sometimes be reduced. Thicker wires, and sometimes silver or silver plated wires may be used.

**2. Skin Effect**

The skin effect affects the inductor Q because it has the effect of raising the resistance. The skin effect results from the tendency of an alternating current flow through the outer areas of a conductor rather than through the middle.

This has the effect of reducing the cross sectional area of the conductor through which the current can flow, thereby effectively increasing its resistance. It is found that the skin effect becomes more pronounced with increasing frequency.

To reduce the effects of the skin effect different types of wire can be used:

**Silver wire**: Silver or even silver plated wire can be used to reduce the effects of the skin effect. When compared to copper wire, silver wire has a lower resistance for a given surface area. To reduce the cost, silver plated wire can be used as the silver will be on the outside of the wire where most of the RF or alternating current is carried.**Litz wire**: Another form of wire that can be used is known as Litz wire. The name comes from the German word Litzendraht meaning braided, stranded or woven wire. It is a form of wire that consists of many thin strands of wire, each individually insulated and then woven together. In this way the surface area of the wire is considerably increased, thereby reducing the resistance to RF or alternating currents. Typically Litz wire is used for frequencies above about 500kHz, but below around 2 MHz.

**3. Radiated Energy**

When an alternating current passes through an inductor, some of the energy will be radiated. Although this may be small, it still adds to the losses of the coil and in exactly the same way as occurs in an antenna this is represented by a radiation resistance. Accordingly this is a component of the inductor resistance and will reduce the inductor Q factor.

**4. Core Losses**

Many inductors have ferrite or other forms of core these can introduce losses:

**Eddy currents**: It is a commonly known fact that eddy currents can flow in the core of an inductor. These are currents that are induced within the core of the inductor. The eddy currents dissipate energy and mean that there are losses within the inductor which can be seen as an additional level of resistance that will reduce the inductor Q factor.**Hysteresis losses**: Magnetic hysteresis is another effect that causes losses and can reduce inductor Q factor values. The hysteresis of any magnetic material use as a core needs to be overcome with every cycle of the alternating current and hence the magnetic field. This expends energy and again manifests itself as another element of resistance. As ferrite materials are known for hysteresis losses,, the effect on the inductor quality factor can be minimised by the careful choice of ferrite or other core material, and also ensuring that the magnetic field induced is within the limits of the core material specified.

Minimising the resistance effects reduces the losses and increases the inductor Q factor.

## Q Factor Formula

Thus for a inductor, q factor formula is expressed as,

Where:

- L is the effective inductance of the coil in Henrys and
- R is the effective resistance of the coil in Ohms.

As the unit of both resistance and reactance is Ohm, Q is a dimensionless ratio.

The Q factor may also be defined as:

Let us prove the above expression. For that let us consider a sinusoidal voltage V of frequency ω radians/seconds applied to an inductor L of effective internal resistance R as shown in Figure 1(a). Let the resulting peak current through the inductor be I_{m}. Then the maximum energy stored in the inductor

Figure 1. RL and RC circuits connected to a sinusoidal voltage sources The average power dissipated in the inductor per cycle

Hence, the energy dissipated in the inductor per cycle

Hence,

### Quality Factor of Inductor Conclusion

After going through the above inductor q factor basics we can now establish a quality factor of inductor formula. I hope you enjoy when reading this article, thank you.