The external Q is the ratio between the energy stored in the resonator, and the energy lost in one radian (period over two pi) to the coupled circuits.  If you have two coupled circuits (an input and an output) then there will be two external Q s, which can be combined as 1/Qe1+1/Qe2=1/Qetot.  In an undercoupled circuit the plot over the frequency band of the resonance on a Smith chart will not go round the centre, whereas for an overcoupled circuit the plot will go round the centre.  This is to do with whether the resonator resistance or conductance is greater or less than the characteristic impedance of the transmission line coupled to it - the place where the plot crosses the real axis is the normalised resistance or conductance of the resonator.  Which of these you need to use depends on whether your resonator looks like a series or parallel resonance, i.e. whether it is open circuit (right hand side of Smith chart) or short circuit (left hand side of Smith chart) a long way from resonance.

The external Q is the ratio between the energy stored in the resonator, and the energy lost in one radian (period over two pi) to the coupled circuits.  If you have two coupled circuits (an input and an output) then there will be two external Q s, which can be combined as 1/Qe1+1/Qe2=1/Qetot.  In an undercoupled circuit the plot over the frequency band of the resonance on a Smith chart will not go round the centre, whereas for an overcoupled circuit the plot will go round the centre.  This is to do with whether the resonator resistance or conductance is greater or less than the characteristic impedance of the transmission line coupled to it - the place where the plot crosses the real axis is the normalised resistance or conductance of the resonator.  Which of these you need to use depends on whether your resonator looks like a series or parallel resonance, i.e. whether it is open circuit (right hand side of Smith chart) or short circuit (left hand side of Smith chart) a long way from resonance.

Thank you for your reply

 A measure of the level of coupling between a resonator and a feed is given by the coupling coefficient. To obtain maximum power transfer between a resonator and a feed line, the resonator should be matched to the line at the resonant frequency; the resonator is then said to be critically coupled to the feed. the coupling coefficient, g, as

g = Q0/Qe

g < 1: The resonator is said to be undercoupled to the feedline.

g = 1: The resonator is critically coupled to the feedline.

g > 1: The resonator is said to be overcoupled to the feedline.

David's definition is the same as mine:

The ratio Qo/Qe is Zo/R, where R is the equivalent series resistance of the resonator and Zo is the characteristic impedance of the feed-line.  In the parallel case it is G/Yo, so g or 1/g is the normalised impedance or admittance of the resonator at resonance.

If there are two feeds then one will need to be shorted or open to be able to 'see' or measure Qo as a second feed will just look like part of the resonator loss.

Thank you  #David for your reply

you say:"A measure of the level of coupling between a resonator and a feed is given by the coupling coefficient". but how this coupling coefficient can be calculated?. Also what is means Q0, is there relation between Q0 and Qu? and how i calculate Qo in the first step 

The internal Q, Qo, is the ratio between the energy stored in the resonator, and the energy lost in one radian (period over two pi) to the internal resonator losses.  For a simple LCR resonator it is sqrt(L/C)/R.

Q0 and Qu are the same thing, unloaded Q. From my first post: the coupling coefficient, g = Q0/Qe.

Q0 = ω0L / R   and Q0 = ω0L / Z0

.

one simple calculation for the resonator external quality factor is given by Daniel G. Swanson, Jr in his manuscript entitled with (Narrow-Band Microwave Filter Design). He mentioned that If you can simulate/ measure the reflected time delay at the tap point, you can compute: Qex = π f0 td/2

where f0 is in GHz and td is the reflected delay in ns. In the reference also you will find a detailed description how you can calculate it with the aid of EM simulations, I did it myself and it gives accurate results.

Hi ,

your answers was helpfull for me, but i have another question,

Is there any ways for calculation of external Q in HFSS ? i want to change  length of microstrip line into my PCB structure and measure external Q.

Hi,

I know how to calculate it in CST using eigenmode solver but I didn't calculate it in HFSS before. If you are interested I can contact you with more details.

Best wishes

If the resonance is a loop that touches the outside of the Smith chart, then the plot can be rotated until the real axis is lined up with the diameter of the resonance circle, then the three Qs are given by the bandwidths where the resonant loop crosses the following 3 sets of lines

1. for the loaded Q, 45 degree lines starting from the point where the resonant circle touches the edge  

2. for the external Q, the arcs of unit normalised reactance or susceptance from the same point.  

3. for the unloaded Q, arcs centered at the bottom and top of the smith chart joining the ends of the real axis.

Hi Maher Khaliel,

How to calculate Q factor in CST?

In Eigenmode solver, there is a box titled as Q-factor calculation in "Eigenmode solver parameter" window under Home ribbon. You check "Calculate external Q-factor". But before that you must define the port (by your choice) somewhere in your model through which the power leaks out (by selecting an appropriate face (using pick face) and clicking on the waveguide port button under simulation ribbon , for example). Also to check the value of the computed Q external, select Q-factor external and the corresponding mode in 3D Eigenmode result in Template-based post-processing.