Simple calculation of the self inductance of two coaxial solenoids in series?

[individual61]

Hi all,

The situation is this: I need to calculate the self inductance of two solenoids wired in series, of equal radius and a given separation d. They are coaxial. The length h of each is known.

Most double-solenoid references I found looked at the mutual inductance, and not the self-inductance of this setup. This is not a transformer, but rather a magnetic quadrupole trap for cold atoms. The coils must be wired in series (and not from separate sources) so as to have identical currents. The ultimate goal is to be able to calculate the turn-on time of the trap for different winding numbers.

Just as background, the coils will probably be made from hollow copper tube (1/8 in OD ACR tubing), will probably not contain more than one layer of winding, the radius will be about 1.5 cm, and the separation, about 3 cm.

Attempt at a solution: Nothing too concrete yet. This isn't as simple as treating the system as an RL circuit with series inductors, because the magnetic field of one coil is very much a part of the flux of the other.

I was considering some very crude boundary values for the total self inductance using the simple solenoid formula:

$$L = \mu_0 n^2 l A$$

where n is the number of turns per unit length, l the length of the solenoid, A its cross-sectional area.

I can calculate the value for a full solenoid using l = h + d + h and the physical winding density, then perhaps using the average winding density (N + N turns along h + d + h length).

Anyway, ideas and suggestions are welcome. As I mentioned, I am just aiming for a back-of-the-envelope calculation to see of the coils will have the right response time.

Cheers,

Paul

 

[bjacoby]

Hi all,

The situation is this: I need to calculate the self inductance of two solenoids wired in series, of equal radius and a given separation d. They are coaxial. The length h of each is known.

Most double-solenoid references I found looked at the mutual inductance, and not the self-inductance of this setup. This is not a transformer, but rather a magnetic quadrupole trap for cold atoms. The coils must be wired in series (and not from separate sources) so as to have identical currents. The ultimate goal is to be able to calculate the turn-on time of the trap for different winding numbers.

Just as background, the coils will probably be made from hollow copper tube (1/8 in OD ACR tubing), will probably not contain more than one layer of winding, the radius will be about 1.5 cm, and the separation, about 3 cm.

Attempt at a solution: Nothing too concrete yet. This isn't as simple as treating the system as an RL circuit with series inductors, because the magnetic field of one coil is very much a part of the flux of the other.

I was considering some very crude boundary values for the total self inductance using the simple solenoid formula:

$$L = \mu_0 n^2 l A$$

where n is the number of turns per unit length, l the length of the solenoid, A its cross-sectional area.

I can calculate the value for a full solenoid using l = h + d + h and the physical winding density, then perhaps using the average winding density (N + N turns along h + d + h length).

Anyway, ideas and suggestions are welcome. As I mentioned, I am just aiming for a back-of-the-envelope calculation to see of the coils will have the right response time.

Cheers,

Paul

The turn-on time will be related to the inductance and the resistance of the coils and the charging source. L/R is the time constant of the device. Note that the LOWER the resistance of the coils the LONGER it will take to turn on!

L comes from the total self inductance of the coils. but since you have two coils mutually coupled the total inductance is L₁ + L₂ $$\pm$$ 2M.

M can be plus or minus depending upon how the windings are wired.

Thus you only need the self-inductance of a single layer solenoid and the value of the mutual coupling between two single layer coils spaced apart.

To get these values I urge you to get a copy of the Dover book Inductance Calculations by Frederick W. Grover. In there you'll find Nagaoka's formula for single layer coils which is more than accurate enough for what you want. You'll find as well, a chapter on mutual inductance of coaxial single layer coils. Note the book is "old school" requiring table look-ups etc.

Good luck!

PS. A close-spaced coil on a 1 inch form 6 inches long with 1/8" wire (roughly 8 gauge wire) (45 turns) gives about 10 micro Henries inductance and a resistance of about .00877 Ohms. Of course that is just the value for a single coil with no mutual coupling to a second coil. So if the coils are identical you'll have 20 micro Henries plus or minus twice the mutual term. And resistance will be about 0.0175 Ohm. Just how fast do you intend to pump this thing? By the way, note that M never gets larger than L so the maximum possible inductance of two of these coils is 40 micro Henries.

 

[individual61]

Hi bjacoby,

Thanks for the reply. After posting this I accidentally came across a copy of that book in the library, and I worked out the inductances as follows:

First, the self inductance of a single coil, square cross section, can be calculated from a table or formula given.

Then, the mutual inductance of the two coils is just N^2 times the mutual inductance of two rings set the same distance apart. For that calculation, for two rings, one must use a slightly modified radius, given that the real coils have a certain cross section.

Finally, as you said, everything is assembled into

Ltotal = 2 L - 2 M.

In summary, if you want to calculate inductances of an enormous range of coils and so on, get hold of that book. Pity he doesn't give more formulas instead of tables, but considering it was published in 1946, it is understandable. :P

p.