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Theorem on typical L-network for impedance matching in NMR tanks

Source tree on Github

Raw copy pasta for mathematica

I think I have a result that is useful in the practice of NMR and applies to a particular probe design. This circuit is common in practice

Please note however unlike this drawing, for this result we should model the resistor in series with the inductance. Excuse the confusion.
 The inductive load is tuned and matched to the characteristic impedance of a transmission line Z0 (usually 50 ohms) by the two variable capacitors.

If the series losses in the coil are set to

r = Z0 / 4,


C_tune =~ C_match

for any reasonable L and f where f is frequency of operation and f > 500 KHz. For f < 500 KHz the approximation begins to break down.

Typical Values for L are

200 nH < L < 30 uH

Typical values for f are

1 MHz < f < 200 MHz

I do not know why this works, but I discovered it while investigating the space in mathematica.

 See source code here

Is this correct?

Discussion on Reddit:
POSTED BY: Allen Majewski
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Suppose we can utilize transmission line transformers to reduce the effective Z0 from 50 ohms to something less allowing higher Q.
Say Z0 = 20 and r = 5. Then I get really nice agreement for the caps with L = 30 uH down to around 1 MHz. This would be excellent for sweeping and snooping for unknown quadrupolar resonances in this band.

14N NQR often appears around there. Locating pure NQR spectra precisely would in many cases clarify NMR studies.  The pursuit of pure NQR is difficult however because the frequency is unknown.  Reducing the parameter space to a single value would make sweeping much more efficient.

Any shortcuts  and tricks to allow easy sweeping could greatly acceleratre understanding of NQR in yet unstudied samples.  NQR is indicative of internal field geometry in solids and is thus important in the identification of phase transitions.
POSTED BY: Allen Majewski
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