RF Probe Building

(Written by Jianlong (Ken) Zhu)

RF Probe

We designed our own RF probe with the aid of a manual, Experimental Pulse NMR: A Nuts and Bolts Approach (Eiichi Fukushima and Stephen B. W. Roeder). Below are some takeaways from the section “V.C.4. Probes” (p.p. 373-385):

  • A good coil for RF probe has a large inductance, which is given by the equation: $L = \frac{n^2 a^2}{23a+25b}$ μH, where n is the number of loops, and a and b are the radius of the loop and the length of the coil measured in centimeters.

  • The spacing between the turns of the coil should approximately equal to the diameter of the wire.

  • The length of the coil should approximately equal to the diameter of the loop.

  • “#14 wire is about the smallest that can be recommended for a free standing coil.”

Hand-Made Coil

Our desired resonance frequency is 21 MHz and the capacitance is in the range of 50-1000 pF, the maximum possible inductance for our case is 1.149 μH.

We wanted to fit the coil inside a test tube whose inner radius is 1.3 cm, and also we wanted to keep the length of the coil, b, below 2.5 cm.

Using the maximum L and the desired values for a and b, our Python calculator returned a wire diameter of 0.75mm - way thinner than the recommended #14 wire (1.628mm).

a = 0.56
b = 2.5
L = 1.14877
n = ((23*a+25*b)*L/a**2)**0.5

print('n: '+str(n))
wireWidth = b/(2*n)
print('wire width: ' + str(wireWidth) + ' cm')

Therefore we decided to compromise the inductance and coil length for a thicker wire. We chose a #18 wire (1.024 mm), made L = 1.0 μH, and our Python optimization program returned the values n = 18.52 and b = 3.79 cm.

wireWidth = 0.1024
a = 0.56
n = 0
L = 1

while abs(n-((23*a+25*b)*L/a**2)**0.5)>0.01:

So the specs of our hand-made coil became:

Wire: AWG #18 Loop radius: a = 0.56 cm Coil length: b = 3.8 cm
Number of loops: n = 18.5 Inductance: L = 1.0 μH


We used a Mathematica calculator to obtain the desired capacitance for our tuning and matching capacitors, 62 pF and 10 pF.

This notebook finds the ideal tuning (C1) and matching (C2) capacitances for a desired resonant frequency given the inductance and resistance of the coil and assuming a 'low resonance' circuit (i.e. matching capacitor in series and tuning capacitor in parallel with the inductor).

Clear[C2, C1, R, L, w, w0, f0, Z, Q];
(* Desired resonance frequency (Hz) *)
f0 := 18.735 10^6;
w0 := 2*Pi*f0; 
(* Inductance of coil (H) *)
L := 1 10^(-6);
(* Resistance of coil (Ohms) *)
R := 1(* Theoretically from 90 cm of AWG-18 copper wire: 0.0210*0.9 *);

Q := w0 L/R;
StringJoin["Q = ", TextString[Q]]

(* Low frequency circuit impedance *)
Z := (I w L + R)/(1 - w^2 C1 L + I w C1 R) + 1/(I w C2);
(* Current thru coil *)
IL := 1/(Z + 50)/(1 - w^2 C1 L + I w C1 R);
(* Find values of w and C2 that maximize current through coil (by
finding the minimum of the inverse curve) *)
FindMinimum[-Abs[IL] /. w -> w0, {C1, 5 10^(-9), 500 10^(-9)}, {C2, 
  5 10^(-9), 500 10^(-9)}]

(* Checks that the above values for C1 and C2 give the expected
impedance *)
StringJoin["{Re[Z], Im[Z]} = ", 
 TextString[{Re[Z], Im[Z]} /. {w -> w0, C1 -> 6.19738 *10^(-11), 
    C2 -> 1.02239*10^(-11)}]]

(* Plots resonance peak with matching impedance *)
Plot[Abs[IL] /. {C1 -> 6.19738 *10^(-11), C2 -> 1.02239*10^(-11)}, {w,
   0.90 w0, 1.10 w0}, AxesLabel -> {"w", "Current thru L"}, 
 PlotRange -> All]  

Desired resonance curve Theoretical Resonance Curve


The next step was to solder the parts together according to the circuit diagram below:

Low Frequency Circuit

Product Handmade Coil

Similarly, we assembled another RF probe with a factory-made coil whose inductance is 0.36 μH.

Factory-Made Coil

The RF probe with our hand-made coil yielded a quality factor (Q) of 19.75, and below is the resonance curve: Measured Resonance Curve