## Q-factor

Q-factor is Not Q-value. in Q-factor, the Q is for Quality.

Q-factor is a dimensionless factor for showing the degree of reasonace of a coil or an Oscillator.

Higer value means a better coil. The definition is the resonance frequency over the Full Width Half Maximum (FWHM).

$Q = f/\Delta f$

High value means:

1. high sensitivity
2. noise reduction due to narrow band of absorption.
3. over damped with long decay time.
4. higher energy stored
5. lower energy loss

For complex electric circuit, the Q factor is:

Q = | im Z/ re Z|

It seem that there is a conflict between impedance matching.

## Impedance matching and Smith Chart

the circles with center on x-axis of the Smith chart represent the real part of impedance.

the circles with center on the x = 1 are the imaginary part of the impedance.

and a circle with radius 1 and centered at (0,0) represent a coaxial line with 50Ω, the (-1,0) is zero length and (1,0) is ¼ wavelength from the load.

thus, we can construct the input impedance by Smith Chart very easily.

AND i always think, the normalized input impedance should be 1 in order to give minimum reflection. However, when the source impedance is not 50Ω, it is not the case. the impedance matching for normalized source impedance is

$z_{in} = z_s^*$

this is come form the Maximum Power Theory.

anyway, to construct the input impedance, we just need to  remember when circuit element is in series, we use normal Smith chart, if the circuit element is in parallel, we use a 180 degree rotated Smith chart, since adding by reactance is more easy.

there is an iphone app for this: rfCalc

it is very easy to use!

the question is how do i know the source impedance!???

## [ Be-12(p,n)B-12 ] the detector circuit

i joined a scattering experiment for $^{12}Be(p,n)^12B$. today the Beam-Line-Detector team was finished the BLD, and SHARAQ group is trimming the beam to the SHARAQ detector. i was worked on the neutron detector array, but i was absented for all preparation job due to a polarized proton target.

here is the detector circuit:

when the neutron hits on the plastics scintillator, it create s proton and further release photons. these photons is result of photoelectric effect and Compton scattering. the photons have different time on reaching the 2 ends of the scintillator due to the position. Thus, by analysis the timing, we can identify the position.

The Photo-Multipler Tube convert photon into electric signal. This signal go to a Splitter, the splitter simply copy the signal. one output is go to a Discriminator, which create a digitized signal. the PMT at the 2 ends will give different timing, and the Discriminator signals will be combined after the AND gate. If the neutron hit the scintrillator near either ends, the AND gate make sure that the later signal will be the trigger signal.

the trigger signal will enable the Charge-Digital-Converter (QDC) start integrate the delayed signal for 300ns. Since from the raw signal to the AND gate takes time, thus a delay will ensure a correct detection of the total energy for each signal. However, the signal will have exponential decay when traveling alone the PMT. the signal decay is:

$S = S_0 e^{- k x}$

For signal created at position r from the mid-point of the scintrillator , the 2 signals are:

$S_1 = S_0 e^{- k (L-r)}$

$S_2 =S_0 e^{-k ( L+r)}$

where L is the half length from the mid-point of the scintrillator. thus, if we use a geometric mean, we can get the original signal strength.

$S_0 = \sqrt{S_1 S_2} e^{k L}$

the trigger signal also combined with the discriminator signal for determination of the time. the trigger signal will define the t=0. by this, we can calculate back the location of the neutron. If we define the mid-point of the scintrillator be x=0. a neutron hits on position r will take different path and the time different for the scintrillator is:

$\Delta t_s = 2 r / c$

where c is the speed of light. we also have to add the time different due to the cable, thus, the total time different is:

$\Delta t = t_1 - t_2 = \frac{2r}{c} + t_0$

by fitting the statistic, the average of the position should be zero, thus the mean of Δt is the offset by the cable.

However, as we mention before, the signal will decay. since we are using Threshold Discriminator, different signal strength will give different time even for same start time. This is called Walk effect. For lower signal strength, the effect is stronger. and give larger Walk time. since the Walk time is always additional, the statistics of the QDC-TDC graph will has a tail at one side only.

In fact , there is a technique the tackle the Walk effect, which is by a Constant Fraction Discriminator. the principle is that, if we copy the signal and apply a negative fraction on 1 of them and summing up. the point of zero is always the same no matter the amplitude or the signal strength.

the Walk effect can be calculated by assuming the falling of the signal is just like a line with slope -m. with threshold -θ, and a decay ratio k. the time are:

$t_1 = \frac{\theta}{m}$

$t_2 = \frac{\theta}{k m} = \frac{1}{k} t_1$

$\Delta t = t_1-t_2 = \frac{k - 1}{k} \frac{\theta} {m}$

if we plot the signal strength versus time different, we have a curve in the form

$y = \frac{k-1}{k} \theta \frac{1}{x}$

From the equation, a smaller decay ratio, which corresponding to the location closer to the mid-point of the  scintrillator, the bigger the $1- \frac{1}{k}$ and give a border curve.

## Smith Chart ( quick guide )

for the coaxial cable is ¼ wavelength, everything in transmission line theory will be simplified.

the input impedance will be

$Z_{in} = Z_0 \frac{1-\Gamma}{1+\Gamma}$

where Γ is the characteristic impedance:

$\Gamma = \frac{ Z_L - Z_0 }{Z_L + Z_0}$

and the reflection wave and input wave ratio is :

$\rho = \left| \frac{V_-}{V_+} \right| = |\Gamma|$

these are 3 important equations on smith chart.

we can use normalized impedance, which is defined as

$z_{in} = Z_{in}/Z_0$

$z_L = Z_L / Z_0$

than  2 of the  3 equations will be normalized to :

$z_{in}= \frac{1-\Gamma}{1+\Gamma}$

$latex \Gamma = \frac{ z_L – 1 }{z_L + 1}$

by setting $z_{in} = r + i x$, we have :

$\Gamma = \frac{ 1- r^2 - x^2 }{(1+r)^2+x^2} - i \frac{2x}{(1+r)^2+x^2}$

by some algebra, we have :

$z_{in} = 1/ z_L$

which is the result for ¼ wavelength cable. by this,we have:

$z_L = \frac{r}{r^2+x^2}- i \frac{x}{r^2+x^2}$

after many equations, for impedance matching, we have following equivalent statements.

• impedance matching
• $z_L = 1$
• $\Gamma = 0$
• $z_{in} = 1$
• $\rho = 0$ , no reflected wave

OK. the math is over. The Smith Chart is the Cartesian coordinate for Γ, real axis on horizontal, imagine axis on vertical. and the circles, are the transformed coordinate of the input impedance. the transformation is called Möbiüs  transform.

a good impedance matching can be found at the origin of the chart, where $\Gamma = 0 + 0 i$, and if we read the input impedance coordinate, it is $z_{in} = 1 + 0 i$, which mean, impedance matched.

However, there is never so ideal in real world. the input impedance always has some imaginary part, or real part not equal to 1. so, what if, both of them are different by 0.2 or 10 Ohm?

## [Pol. p target] fix the NMR coil position

my colleague was improve the system by reducing the reflected wave. and also found the maximum condition for the timing. the microwave trigger should be delayed by 167 us and have a phase 180.

then, i get the polarization signal in FID area be 340 unit or up! compare to BG, which is just 25 unit. But in order to measure the polarization dependence or any dependence, i hope to have a small fluctuation on measurements.

when i try to fix the NMR coil by using optics mounting, i found that the input impedance shown on the network analyzer is very sensitive to position, only a very tiny different can change the impedance alot. That is why the NMR signal is not so reproducible, or with a large fluctuation.

when the experiment going well, suddenly, the signal gone, it only give signal like background. i checked the coil, put it out and found it is quite distorted. fix it in a good shape and do again. nothing.

then i thought, may be the network analyzer. then i restart. but don’t know how to operate…..

finally, it is the tuner circuit broken.

anyway, today found a way to stabilized the NMR coil. as long as moving it slowly, it can reproduce similar input-impedance with 5 ohm different. that result is done with a coil-before-fixed. Now the coil is in nice shape. it goes and leaves the sample more smoothly.

Todo:

1. setup the network analyzer and use it to get back the NMR signal
2. testing the stability of the signal
3. fin the crystal orientation.

## An answer on pervious post

In pervious lablog, the reflected wave of the NMR pulse still is about half of the input pulse, that means, the power is reflected. And we had tuned the tuner to be “impedance matched”.

The problem is, when the tuner is not impedance matched but just frequency matched.

We tune the tuner by the help of network analyzer and tuning for a resonance depth at 12.8MHz.

The tuner contains a parallel capacitor and a series capacitor with the NMR coil. The resonance depth at 12.8MHz show that we matched the parallel capacitor with the insurance of the coil. The total impedance of the tuner and coil is :

$i \left(1 - \frac{\omega^2}{\Omega^2} \left( \frac{C_s}{C_p}+1\right) \right) / \left( C_s \omega \left( 1 - \frac{\omega^2}{\Omega^2} \right) \right)$

Thus the amplitude of the total impedance is determined by the series capacitor.

In conclusion, the total impedance is not matching with the coaxial cable. This gives the reflected wave.

## [Pol. p target] try to polarize sample

the microwave system was tuned again. the minimum resonance deep of the reflected microwave is 60mV at cw mode. when it is pulse mode, during the triggered microwave, the reflected wave is still large (8.4V), and we can see a clear charge and discharge signal. we also correct the microwave trigger.

the NMR coil is 60 turns now, compare to the old one is 10 turns. thus it is more sensitive. we cannot find a polarized signal. the NMR signal either has large noise or large reflected wave. may be the impedance is not matching or the resonance deep is not enough.

ToDo:

1. minimize the NMR noise
2. reduce the reflected NMR pulse.
3. polarize target
4. optimization
5. laser polarization measurement
6. spin echo
7. polarization transfer.