## A precise analysis of an L-network for impedance matching below 3 MHz. The desire for “an ideal” characterization of the circuit parameter space for utility in fabrication by hand. The reduction to a pure mathematics problem.

### Introduction

This is a full description of a situation often encountered by scientists in the process of fabrication of the NMR probe. The analysis requires some tedious complex algebra, a bit of circuit theory, and enforces a matching condition. I tried to write this so that one may infer the cirucit theory from context. If there is a problem, just ask.

We will examine the impedance of this reactive L network

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### Goals of this challenege:

Characterize the parameter space of the variables Cm, Ct, ω, L, r and produce some useful set of tables for lab, in which the relationship between Cm and Ct is known for a given ω L and r. Furthermore to ponder the level of greed allowed. Which parameters limit others? Compare this with what the laboratory reality is.

——

One must always strive for impedance matching conditions to be satisfied, which for us means 50 ohms real. So we must

### enforce

Im_Z = 0

Re_Z = R := 50 ohms

—–

These requirements are nasty if you allow the impedance of the coil to have a small (but very physical and influential) real part r

Z_coil = j ω L + r

So the total impedance is

Z_tot = -j / (Cm ω) + Z_coil || Z_Ct = Re_Z + j Im_Z = R + j 0 = 50

Note:

* Z_m is the impedance of the matching cap only
* Z_t is the impedance of the tuning cap only
* the notation A || B means “A parallel B” and A || B = ( 1/A + 1/B)^(-1)

Since Z_coil has a real and an imaginary part, the expression for total impedance is a headache.

So I did it by hand, and with mathematica, and iteratively found what I consider decently short code with reasonably concise expressions. Here we go.

—–

Clone the mathematica stuff here

``` git clone https://github.com/Altoidnerd/NMR-Tank-Circuits```

In which there is a file where I do in fact show the real and imaginary parts of Z_tot are:

real part (which we denote Re_Z…please note the sloppyness. Here w is ω)
``` Re_Z = r/((r^2 + L^2 w^2) (r^2/(r^2 + L^2 w^2)^2 + (T w - (L w)/(r^2 + L^2 w^2))^2))```

and imaginary part (Im_Z)

``` ImZ = (-(1/(M w)) - (T w)/(r^2/(r^2 + L^2 w^2)^2 + (T w - (L w)/(r^2 + L^2 w^2))^2) + (L w)/((r^2 + L^2 w^2) (r^2/(r^2 + L^2 w^2)^2 + (T w - (L w)/(r^2 + L^2 w^2))^2))) ```

where we eliminated the need for subscripts but denoting Cm := M and Ct := T.

How do we make useful data from these equations? To answer this, we must first assess what the experimenter can really control.

* coils are hard to wind and have prescibed results. In general, the parameter r is less than 1 ohm, but its actual value is not constant through frequency sadly. It must be treated as such.

* A typical coil inductance L satisfied 1.0 uH < L 30 uH. Intermediate values such as 8 uH tend to be the most difficult to fabricate. A coil inductance of 8uH I find would be useful for lower frequencies, below 3MHz, which are currently causing me problems. It is here the equations become extremely sensitive.

* The capacitance T and M can within reason, be expected to continuously vary between 0 < T,M < 1 nF and even more reasonably if the upper boundary is around 300 pF.

* the frequency is going to satisfy 1 MHz < f < 30 MHz; so ω = 6.28 f so we can say about, that
1 e7 < ω < 3e8

I have made many charts. Got any brilliant ideas?

——–

A typical annoying situation in lab would be:

Drat. To reach the target frequency, we must either replace the capacitors with larger ones,
or exchange the coil with one of larger inductance. Which will take me less time?

I usually do not know in fact. I either make a intuitive guess, prepare some primitive tests, or try a bit of each.

The code in the github repo above will give you some parameter sliders. You can try plotting M, and T vs ω as L and that little tiny r are varied…I still must get to the bottom of these matters, such as, the qualitative effect of increasing r at fixed ω and L etc. How to encapsulate all such desirable relations in a single concise set of diagrams is what I truly seek, from the kind theorists of who may read this.

——-

### Final thoughts.

I have studied this problem up down left right…I wrote some interesting special cases down here, but I believe there is more to be known about these equations that could be of service to the designer.

## Theorem for impedance matching in NMR tanks using L-networks facilitating broadband frequency sweeps for unknown quadrupolar resonances in solids.

Locating pure NQR spectra precisely would in many cases clarify NMR studies.  Furthermore NQR is indicative of internal field geometry in solids and is thus useful in the identification of quantum phase transitions.

The pursuit of pure NQR is difficult however because the resonant frequency is sample specific and is often unknown. Unlike in the case of NMR, the frequency cannot be controlled in the laboratory, but is rather a property of a material that is a fingerprint of the local environment of the nucleus in question.

In general, operating a pulsed spectrometer at various frequencies requires the corresponding adjustment of the two capacitors shown below. Reducing the parameter space to a single value would make sweeping much more efficient. Any shortcuts  and tricks to allow easy sweeping could greatly accelerate understanding of NQR in yet unstudied samples.

This general probe topology is common in the practice of nuclear resonance.

The inductive load is tuned and matched to the characteristic impedance of a transmission line Z0 (usually 50 ohms) by the two variable capacitors C1 and C2.

Postulate: If the series losses in the coil are set to

R = Z0 / 4,

then

C_1 =~ C_2

regardless of the value of Z0, and for any reasonable and f where f is frequency of operation and > 500 KHz. For < 500 KHz the approximation begins to break down for feasible values of L.

Suppose we can utilize transmission line transformers to reduce the effective Zo from 50 ohms to something lower, allowing a higher Q.

If for example the effective characteristic impedance of the tank Z0 = 20 ohms, then one could set r = 5 ohms externally. This results in 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, as 14N NQR often appears below 5 MHz.

Source tree on Github

https://github.com/Altoidnerd/NMR-Tank-Circuits

Raw copy pasta for mathematica

https://raw.githubusercontent.com/Altoidnerd/nmr-tank-circuits/master/mathematica/A-network.m