Coax Traps are LC traps using the inherent capacitive properties of coaxial cable, wrapped around a form to create an inductive coil. The resonant frequency of an LC trap is defined by the following formula:
where is the frequency in Hertz, is the inductance in Henries and is the capacitance in Farads. We like to work in , and , so lets take care of that right now.
Inductance is given by the following:
Where is the number of turns, is the diameter of the coil in inches, and is the length of the coil in inches.
Capacitance, , can be found by using the following:
Where is the capacitance of the coax per unit of length and is the length of the coax. The unit type of is based upon it’s input. For example, if is in , should also be in and will be returned in .
Ok… That’s nice. So what do we do with that information? Well, let’s start with what we know and what we want to find out. Our inputs should be the frequency of the trap , the diameter of the form we’re wrapping around , the diameter of the coax we’ve chosen and the capacitance of the coax . The rest must be found. Unfortunately, none of our formulae are directly solvable with just these pieces of data, so we’re going to need to do some manipulation. Let’s start with the simple ones.
In other words, the diameter of the coil is equal to the diameter of the form plus the diameter of the coax. We measure our diameter from the center of the coax on one side of the form to the center of the coax on the other side. On to the length of the coil.
The length of the coil is equal to the number of revolutions time the diameter of the coax, since each revolution of the coil is one coax next to another.
The length of the coax is equal to (formula for the circumference of a circle) times the number of revolutions in the coil. Although, in practice, the coax length winds up being about an inch longer because of the two tails protruding through the form. So…
Now lets clear the unknown variables from the formula for Inductance:
Things are starting to shape up here. What we really need to know now is the number of turns in our coil. I’m not going to lie… this part sucks. We might need to use all of our cunning. Our objective is to generate a formula that expresses in terms of , , and . It would also be nice if we could do this using an algorithm that could be programmed into a calculator.
Substitute and in our resonance equation.
This looks ok. We have only one unknown variable here: . I can already tell that we’re not going to be able to isolate it. This is going to be a polynomial… and I don’t think it’s quadratic. Let’s shuffle things around a bit.
And that, my friends, is a cubic equation. We can solve this, but we’re going to need some help. Let’s start with the basic form of a cubic equation.
Just like quadratic equations, there is a formula to solve it. Only this one is exponentially more complicated (pun intended.)
Pressing on… let’s assign values for , , and according to our formula and replacing with . ( shows up again later in this algorithm, but it’s not the same !)
Now, we could just plug those numbers in to our cubic formula and find value, but as it turns out, there’s a better way. It’s described here. I’ll plug our values for a,b,c,d in where appropriate.
According to our algorithm, there are three ways this can go, depending upon the value of , and possibly and . How we proceed will depend upon whether , or the special case when , and . Since we aren’t working with real values, we don’t know which of these cases are in scope. It’s possible that they all are. I know from experience that and are possible. My end goal is to build a calculator, so I think the safest bet is to include all of these cases in the program.
Let’s start with . In this case, only one root is real. The other two roots are imaginary and useless to us.
Now let’s look at , (but both and .) In this case, there are three real roots, so we are going to wind up with three formulae. Some may and probably will be negative. We can discard those, although we need to solve them first to be sure. If I were a real math wiz, or just less lazy, I might be able to eliminate one or more of these solutions based upon the range and domain of each equation, but I’m CompSci, so I’ll just potentially waste some processor cycles. If this were an algorithm that were being run in a loop, I make reconsider this for efficiency’s sake.
And finally, the special case where , and .
And that’s it! Given values for resonant frequency, coax capacitance, coax diameter, and coil form diameter, we can calculate the number of turns necessary… not that you’d ever want to solve it by hand.