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
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
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
* 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
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.
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.