nmr-utils/Probe.ipynb

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from numpy import *

In [15]:

%matplotlib inline
import matplotlib.pyplot as plt
e0 = 8.8541878188e-12

In [103]:

def capacitor(ri, ra, l, er=9):
    C = 2*pi*er*e0*l/log(ra/ri)*1e12
    #print(f"{C:.2f}pF")
    return C*1e-12

In [104]:

def capacitor_query(ri, ra, l, er=9):
    C_min = capacitor(ri,ra,l,er=1)
    C_mid = capacitor(ri,ra,l/2,er=1) + capacitor(ri,ra,l/2,er=er)
    C_max = capacitor(ri,ra,l,er=er)
    print("----Params----")
    print(f"ri = {ri*1e3:.2f}mm\nra = {ra*1e3:.2f}mm\nl = {l*1e3:.2f}mm\ner = {er:.1f}")
    print("----Result----")
    print(f"C_min = {C_min*1e12:.2f}pF")
    print(f"C_max = {C_max*1e12:.2f}pF")
    print(f"C_mid = {C_mid*1e12:.2f}pF")

Capacitor

First we need to know the capacitor and the range we have available, we need the inner and outer diameter r_i, r_a, length l, and \epsilon_r

In [148]:

capacitor_query(4e-3,8e-3,10e-2,2)

----Params----
ri = 4.00mm
ra = 8.00mm
l = 100.00mm
er = 2.0
----Result----
C_min = 8.03pF
C_max = 16.05pF
C_mid = 12.04pF

Resonance frequency of an LC circuit: f_0 = \frac{1}{2\pi\sqrt{LC}}

Solve for L:

L_{RF}= \frac{1}{(2\pi f_0)^2C}

In [143]:

def probe_design(f0, C_mid=12e-12):
    L_RF = 1e6/((2*pi*f0)**2*C_mid)
    print(f"L_RF={L_RF:.2f}µH") 
    L_duplex = 50/(2*pi*f0)*1e6
    print(f"L_duplex={L_duplex:.2f}µH") 
    C_duplex = 1/(2*pi*f0*50)*1e12
    print(f"C_duplex={C_duplex:.2f}pF") 
    
    return L_RF*1e6

def probe_range(L_RF, C_min, C_max):
    f_min = 1/(2*pi * sqrt(C_max*L_RF))
    f_max = 1/(2*pi * sqrt(C_min*L_RF))
    print(f"f_min = {f_min:.1f}MHz\nf_max = {f_max:.1f}MHz")

Now we can calculate the needed coil inductance for the resonance circuit. Here, we also calculate the $\Pi$-circuit C and L paramters for the duplexer (the "$\lambda/4$" lumped circuit)

In [150]:

L_RF = probe_design(25e6, C_mid=12.04e-12)

L_RF=3.37µH
L_duplex=0.32µH
C_duplex=127.32pF

In [147]:

probe_range(L_RF, 8e-12, 16e-12)

f_min = 21.7MHz
f_max = 30.6MHz

Coil64

Calculate the coil windings:

Give the wire diameter, incl. isolation thickness, former diameter (5mm)

Coil64 v2.2.32 - One layer close-winding coil

Input:

Inductance L: 3.38 microH
Frequency f: 25 MHz
Former diameter D: 5 mm
Wire diameter d: 0.3 mm
Wire diameter with insulation k: 0.436 mm
Wire material Mt: Copper

Result:

Number of turns of the coil N = 56.719 
Length of wire without leads lw = 96.894 cm
Length of winding l = 25.165 mm
Weight of wire m = 0.614 g
DC resistance of the coil Rdc = 0.236 Ohm
Reactance of the coil X = 530.929 Ohm

Self capacitance Cs = 0.356 pF
Coil self-resonance frequency Fsr = 218.021 MHz
Coil constructive Q-factor Q = 171 
Loss resistance ESR = 2.67 Ohm

Additional results for parallel LC circuit at the working frequency:
=> Circuit capacitance: Ck = 11.635 pF
=> Characteristic impedance: ρ = 531 Ohm
=> Equivalent resistance: Re = 77.531 kOhm
=> Bandwidth: 3dBΔf = 171.199 kHz

Input data for LTSpice:
Inductance: 3.380μ
Series resistance: 236.334m
Parallel resistance: 78.065k
Parallel capacitance: 0.356p

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