- Quality Factor Definition: The quality factor (Q factor) is defined as the ratio of reactance to resistance, indicating efficiency at a given frequency.
- Inductor Quality Factor: The Q factor of an inductor is the ratio of its reactance to its resistance, calculated as Q = ωL / R.
- Energy in Inductor: The energy stored in an inductor is based on the peak current through it, with power dissipation determined by its resistance.
- Quality Factor of Capacitor: The quality factor of a capacitor is the ratio of its reactance to its series resistance, given by Q = 1 / (ωRC).
- Lossy Capacitor: A lossy capacitor can be modeled with a capacitance and high parallel resistance, influencing its efficiency.
Quality Factor of Inductor
An inductor stores magnetic energy, but its winding, core and nearby conductors also cause loss. These losses can be represented by an effective series resistance. At a stated angular frequency ω, the quality factor (Q factor) of an inductor compares the magnitude of its inductive reactance with that effective resistance.
For the series equivalent model, quality factor is expressed as
Here L is the effective inductance in henrys and R is the effective series resistance in ohms. Both values can change with frequency, signal level, DC bias, temperature and the measurement fixture. Because reactance and resistance are both measured in ohms, Q is dimensionless.
Q can also be defined from energy:
This is 2π times the maximum stored energy divided by the energy dissipated in one cycle. To derive the inductor expression, consider a sinusoidal voltage V at angular frequency ω applied to an inductor L with effective series resistance R, as shown in Figure 1(a). Let Im be the peak current.
The maximum magnetic energy stored in the inductor is
Figure 1. RL and RC circuits connected to a sinusoidal voltage sources
The average power dissipated by the effective resistance is 
The energy dissipated in the inductor during one cycle is therefore
Substitution into the energy definition gives
Quality Factor of a Capacitor
Figure 1(b) shows a capacitor C represented by a small equivalent series resistance R. The Q-factor, or quality factor of a capacitor, at angular frequency ω is the magnitude of its capacitive reactance divided by its series resistance.
Thus,
For this simple series model, Q = 1/(ωCR) and the dissipation factor D = 1/Q. Q is dimensionless because reactance and resistance are both measured in ohms. The energy definition of Q also applies. With a sinusoidal applied voltage, the maximum electric energy stored in the capacitor is
Here Vm is the peak voltage across the capacitance C.
The circuit current satisfies
so
Here Im is the peak current through C and R.
The maximum energy stored in C is therefore 
The energy dissipated per cycle is
Substitution gives the capacitor quality factor:
A lossy capacitor can instead be represented by a capacitance C and a high parallel resistance Rp, as shown in Figure 2. This is an alternative equivalent model, not a single physical model for every capacitor.
For the parallel model, the maximum stored energy is 
Here Vm is the peak applied voltage. The average power dissipated in Rp is
Figure 2. Alternative method of representing a lossy capacitor
The energy dissipated per cycle is 
Therefore, Q = ωCRp, as also shown by 
