Since I still don’t have the quadrature differential amplifier assembled, I’ve tested a single coil pair (2 opposite side coils connected in series) using the function generator and the oscilloscope. Here the steps:
- The coil pair is connected to the CH1 of the oscilloscope. The oscilloscope probe was set to 10x to reduce the stray capacitance, which now decreased to 13pF
- A single turn copper wire loop with a diameter of 50mm was placed in the center of the RF coil structure. It was axially rotated until finding the maximum signal. In this position, if turned 90º, we would receive no signal (it would be received by the other coil pair)
- Then, I’ve connected the output of the function generator directly to the previous copper wire loop. The output impedance of the function generator has a standard 50Ω impedance, and the copper wire loop, a short circuit. So, configuring the output with 1V square wave at 2kHz, the current to the single turn coil is 1V/50Ω=20mA. In this way, I’m generating a pulsed magnetic field, and thus I can see the output response of the coil pair
The single copper wire loop:
The setup:
And finally, the output resonating signal:
I’ve added an external 100pF capacitor to tune to the desired 210kHz, which added to the probe capacitance of 13pF means a total of 113pF.
I calculated the generated magnetic field by using the single wire loop equation:
Here in Catalunya, we have an earth magnetic field of around 45μT, which means I’m generating a pulse about 90 times smaller. Despite this, we can see an output response of about 0.5VPP. Not bad!
Q Factor
The theoretical values for the Q factor given the 2 measured coil parameters are:
- Coil pair 1: 4.44mH and 16.8ohm. It means a Q Factor of 349 @ 210kHz
- Coil pair 2: 4.48mH and 16.8ohm. It means a Q Factor of 351 @ 210kHz
One way to measure the real Q is by using the resonating decaying signal. It consists of counting how many cycles it takes to reach half the initial voltage, and then, multiply that value by 4.53. Taken the previous acquisition, we see a Q=13·4.53=58, about 6 times lower than theoretically expected. In other words, the effective AC resistance became about 100Ω. The reason is the high frequency losses.
For such a “low” frequency and the wire diameter used, the skin effect is not a problem. But the proximity effect really is, where the magnetic field generated by the current flowing by the wire itself induces currents to the other adjacent wires.
For an unloaded RF coil (meaning there is no water sample), the Q factor usually goes from 50 to 600, while a loaded RF ranges from 10 to 100. So, we are in the worst-case range. The number of turns increases the proximity effect losses, although it also increases the sensitivity.
One might think that the highest the Q factor the better, because it is the ratio between the stored energy and the energy lost per cycle. But that’s not actually true, some NMR labs even intentionally decrease the Q by adding a series resistor. The thing is that the highest the Q factor, the sharper the frequency response, and the lowest the bandwidth. And we need some bandwidth to acquire the MRI signal when we apply the gradient magnetic fields.
The bandwidth can directly be calculated from the Q factor, but I wanted to measure it to make sure. So I swept a constant current sinusoidal signal through the single wire loop, and measured the RMS voltage at the output of the LC resonator. Here the result:
In this case, there is no surprise and the theoretical value corresponds with the real one. The Q factor is defined as:
I don’t really know the NMR implications about not having a so high Q factor, but what I do know is that 4kHz of bandwidth is a perfect span for the gradient magnetic fields, as already seen in the next gradient section.
Finally, from the above plot we can directly get the sensitivity:
