Quality Factor of Resonant Circuit | Loaded vs Unloaded Q

Introduction : The quality factor (Q factor) of a resonant circuit is a key parameter that measures energy storage relative to energy loss. In real world circuits, both unloaded and loaded Q factors are used to evaluate performance, bandwidth and efficiency. Understanding the difference between these two helps engineers design stable and high performance resonant systems.

Quality Factor (Q-Factor)

The quality factor (Q) of a resonant circuit is a dimensionless parameter that measures how efficiently the circuit stores energy compared to how much energy it loses per cycle. It indicates the sharpness of resonance, selectivity, and bandwidth of the circuit.

The Q-factor is a measure of frequency selectivity of resonant or non-resonant circuit. The general definition of Q-factor can be expressed by following formula.

For resonant circuit,

Where,

• Fr = resonant frequency
• BW = f2-f2 i.e. half power or 3 dB bandwidth

At resonant frequencies, electric and magnetic energy are equal and in time quadrature. This means, when magnetic field is maximum, electric field will be minimum and vice versa. The energy is always stored in the volume while dissipated at the surface. Hence Q-factor can also be expressed by following equation.

In practical resonant circuits, two types of Q factors are defined:

• Unloaded Quality Factor
• Loaded Quality Factor

Unloaded Quality Factor (Qu)

It represents the quality factor of a resonant circuit without any external load connected. It accounts only for internal losses such as resistance of inductors, dielectric losses, and conductor losses. The formula for unloaded Q factor (Qu) can expressed as follows.

• Indicates intrinsic performance of the resonator
• Higher Qu means lower internal losses.

External Quality Factor (Qe)

When cavity resonator is connected with external load or circuit, its Q factor is called “external Q factor” and denoted as “Qe” or “Qext”. It is associated with only external losses occurred due to presence of external load. It is expressed by following formula.

Where,

• k = coupling factor
• k = 1 for criticial coupling ;
• k < 1 and k = 1/p ; for under coupling ; ( p = standing wave ratio)
• k > 1 and k = p ; for over coupling.

Loaded Quality Factor (QL)

It is the quality factor of the resonant circuit when an external load is connected. It includes both internal losses and power extracted by the load. The formula for loaded Q factor (QL) can expressed as follows.

• Always lower than unloaded Q
• Determines actual bandwidth of the circuit
• BW = Half power or 3 dB bandwidth with load
• Fr = Resonant frequency

Qu can also be expressed similarly, Qu = Fr/BW

Relationship Between Unloaded Q and Loaded Q

Where, Qe is external Q (due to load coupling)

Loaded Q is associated with both internal and external losses occurred when cavity is connected with external load. It is expressed as follows.

Summary: The quality factor of a resonant circuit quantifies energy efficiency and resonance sharpness. The unloaded Q reflects intrinsic losses of the resonator, while the loaded Q includes external loading effects and determines usable bandwidth. In practical RF and microwave systems, loaded Q governs real world performance, whereas unloaded Q indicates the theoretical efficiency of the resonant structure.