# Solenoid's field intensity calculation dependency on width/diameter of coil?

Textbook formula for field intensity in solenoid coil is

H = (N * I) / l


H magnetic field intensity in ampere-turns

NI ampere-turns

l is length between the coil poles (along the axis of the field flux)

This formula does not take into consideration the width (or diameter) of the coil. Apparently it is based on assumption that diameter is smaller than the length and hence does not significantly impact this calculation.

I am considering a single electromagnet as a model for a BLDC motor' stator coil. Those are often more wider coil diameter and of short coil length. Wider coil surely weakens the intensity of the field inside the coil. Imagine the flux lines getting thinner as the coil loops get wider. How to reflect this fact in the calculation of the field? How to adjust that formula to include the width/diameter of the coil?

Also, a bit aside from the main question, please give me a hint How to calculate the attracting force developed in such a wide solenoid in Newtons, knowing field intensity and, say, attracting an iron cylinder of known mass and permeability ? Note: for the sake of simplicity assume air core in solenoid.

Edit: if my second question seems to spill into larger area of expertise please disregard it and simply suggest the answer to the main question as it is important for finding if the range of the field stays below the saturation point.

• There are formulas that are better for this. Do some research. – Andy aka Apr 8 '18 at 23:23
• @Andy I did research before posting. Keywords such as "solenoid coil magnetic field intensity formula" gave me 2 pages of the same referrals to the above formula. My question is of a practical importance - I need a practically usable formula to calculate field density for a solenoid which has more width than length, hence both parameters must be included in the formula. Any non-practical approximations are beyond this topic. If you have the answer please share - this might become the only place on Internet to find this answer. – VladBlanshey Apr 9 '18 at 16:25

You are embarking on a complex problem for which fea was developed. The coil alone will not give an accurate result. The entire magnetic circuit path (loop) must be described. In short; you must find the total iron length, iron area, air gap area and gap-length, an equation describing H vs. B of the iron, and the coil amp-turns. Then compute the total amp-turns drops in the magnetic circuit by starting with an arbitrary value for flux. If NI drop totals are higher than the coil NI, then lower the flux (many iterations may be required). When the true flux value is thus found, compute the air gap energy (the air gap NI x flux/2). Now you must rotate the armature slightly which changes the air gap overlap area and compute it all again. The difference in air gap energy for the two conditions is the energy of rotation (torque x radian angle). Solve for torque.

• What do you mean by "air gap" in air core-based solenoid? – VladBlanshey Apr 17 '18 at 19:27
• where do you apply torque and angular angle in a simple air-core coil ?How does a rotation of coil around the air effect field calculation? – VladBlanshey Apr 17 '18 at 19:32

The Apr 8 question infers that you are calculating the coil for a brushless d.c. motor. The coil field cannot be calculated independently of the rest of the motor. An air gap is where flux transfers between the coil pole and the armature. It is usually only about 0.2mm wide. In the calculatons the armature is rotated about 1-2 degrees to find two related values of air gap energy in order to calculate torque. In doing all the math you will also find B and H.

• 1. the question is about solenoid, not motor 2.yes the coil field of the solenoid can be calculated - all textbooks explain this, my question only changes single parameter in otherwise typical task 3.there is no air gap in the stated question 4. there is no rotation in solenoid neither torque (please check sources on internet about simple coil and plunger making it a solenoid). I regret mentioning motor in this question as it does not constitute the essence of the question but only gave you a chance to step aside to another topic. – VladBlanshey Apr 19 '18 at 2:03

You have a hole in the basic knowledge. That hole drives you to ask nonexistent.

The magnetic field strength or flux density of a winding isn't one number, it's a complex spatial vector field, every point in the space has it's own vector direction and strength which aren't writable with any ready to use formulas, it's only possible to calculate them numerically.

If there exists iron somewhere, it affects to the field. How much? It's solvable only numerically. Only in some special cases the effect of the Iron can be expressed with formulas.

Calculate numerically = to solve the field differential equations in given geometry by dividing the space, wires and possible iron to small enough pieces and approximating the derivatives. Finite element analysis (FEA) is the general name of the practical numeric calculation methods.

There are some well known usable formulas for the inductances of some simple coils. The following link points to one for tight short solenoids, so tight that the winding is more a multi turn wire loop with no axial length:

https://technick.net/tools/inductance-calculator/circular-loop/

Inductance is the total magnetic flux of a coil divided by the current. Inductance is a single number => it has no data of the magnetic vector field. And the formula is ok only in two cases:

• the space is totally free of iron or
• the space is full of iron.

If there's a piece of iron lurking somewhere, the inductance is more than if the space was ironless, but less than if the space was full of iron. No more info available.

Inductance is still useful. You can calculate two limits for it and you can measure the inductance if there's a substantial piece of iron somewhere.

Why useful? The magnetic field energy is 0,5 * L * I^2 where L=the inductance and I is the coil current. The magnetic force to a piece of iron has the direction to where the piece should move to increase the inductance as much as possible per moved millimeter. The force can be calculated as vector gradient of total field energy versus the placement of the iron piece.

In practice measure the inductance before and after moving the iron piece a short distance. Calculate the field energies after you have the inductances (assume some practical current). Then divide the energy difference by the movement distance. That's your force. Rotational torque is the energy difference divided by a small rotation angle (radians).

In motors and other systems where the iron moves, you cannot calculate the current as the applied supply voltage/resistance. Induced voltage decreases the total voltage. Refer electrical motor theory for exact torque formulas.

• I'd like to know my "hole" in basic knowledge -that's my purpose to be here. Can you help me by answering questions? What is "non-existent" in my posted question? Solenoid? Example of a simple formula of field intensity calculation taken from a textbook? Or may be making a solenoid wider than it is long is not possible? Please be specific. Thanks – VladBlanshey Apr 19 '18 at 16:11
• Special vectored field is indeed a reality in every electromagnetic interaction including the basic solenoid described in every textbook. Yet this formula I quoted is surely known to you since school. Do you reject using such formulas today because you've seen a field simulation on the FEA screen? Do you know that practical engineers are still using those formulas today to find the field limits of the field as they did over 100 years ago? My posted problem is only different in the shape of the solenoid - not in a phenomenon. – VladBlanshey Apr 19 '18 at 16:21
• @VladBlanshey Nonexistent = a writable explicit "no need to find integrals" formula for magnetic field (=vector B or H as a function of x, y and z) of a solenoid when there's a piece of iron somewhere near. Even with no iron the field formula can be written only for the midpoint. General formula without iron has integral which can be calculated only numerically and the "piece of iron somewhere" case needs iterative numerical differential equation solution. But we have never seen any drawings of your case, which can be a solvable special case. that someone here knows. Make one drawing. – user287001 Apr 19 '18 at 17:08
• It's existent, well known, and applied not only in coil's midpoint but anywhere within coil except endpoints as flux is quite uniformed inside. Read a classic explanation here hyperphysics.phy-astr.gsu.edu/hbase/magnetic/solenoid.html#c2 and here en.wikipedia.org/wiki/Solenoid#Quantitative_description Quote "As shown above the magnetic flux density B within the coil is practically constant and given by B = u * NI / L ". There you can find explanation why it is true for a core of any permeability. Similar with wide coil loops but I am not allowed provide here a partial answer. – VladBlanshey Apr 19 '18 at 18:42
• about "special case" to need a drawing: solenoids pictures are shown in all above reference links (or many more if you look for google images). The only "special" about my case as was stated originally - the width is larger than length - enough data for a basic imagination. – VladBlanshey Apr 19 '18 at 18:52