Todoids: Its Magnetic Field and its Inductance
Daniel Hercules @ 2010 June 20
Part I: Magnetic Field
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1. Declaring some necesary variables.
In the figure 1, we can see the normal sketch of a toroid for which we are going to calculate its magnetic field inside and its inductance. For such calculus we need to declare some variables that are vital for the solution of the problem. The variable R[1] is the internal radius from the central axis of the toroid giving in meters while the variable named R[2] is the external radius of the toroid. Also, H[1] is the height given also in meters for the lateral side of the toroid. Is also important to declare the relative permeability of the core; this variable is givin by K[i] and is just an integer that takes the value of 1 in the case that is not modified. The variable mu[0] is the permeability of vacumm which is substracted from the library of Scientifical Constants. The Current is given in amperes and is expresed on the variable II[i].
The equation that describes the behaviour of the magnetic field inside the toroid is given on the equation 1.
(Eq. 1)
Remembering the units in which are going to work in this example then we continue with the declaration of the variables and then the calculus of the parameters of interest.
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2. Calculating the total number of turns of the toroid
First we need to decide which of the methods to use. Remember just to execute one of them because the result of each of the procedures is going to be recorded in the variable N[1] for all the procedures and executing all of them provokes wrong results. You can calculate the number of turns by giving the external diameter of the conductor and asuming that the number of turns is given by the amount of turns that you can have due to the internal circumference of the toroid or, you can simply state the number of turns of the toroid in the respective section. The variable DD[e] is the external diameter of the conductor given in meters and C[i] is the circumference that is going to be calculated on the procedure by means of the previous data given above.
2.1. Calculating the number of turns using the external diameter of the conductor
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(2) 
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(3) 
2.2. Recording the number of turns in the case that such number were given for the problem
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(4) 
3. Defining the radius in which the calculation is desired.
This radius is the on in which we desire to make the calculation. Is given in meters as well and it should be a measure that is contained in the interval R[1] < R[C] < R[2]. There is included a method to double check that this number is in between that range of radius.
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(5) 
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(6) 
If the result from the later expresion is false, then there is something wrong with the radius in which the calculation is going to be performed or the logical order of the R[1] and R[2].
4. Calculation of the Magnetic Field inside a Toroid.
Taking in count the equation 1 that previously stated, we proceed to execute the function with the data given and to obtain a result that is going to be expressed in Teslas.
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(7) 
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(9) 
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Part II: Its Inductance
1. Transversal Area of the Toroid: The inductance of a toroid is defined by the equation 2. First we need to calculate the area of the transversal section of the toroid. In the figure given above, the toroid has a rectangular transversal area that can also be substituided by a circular area by executing the proper command and also by taking in count that either the magnetic field and the inductance is calculated in reference to the diameter that crosses the toroid on the center of it following an straight line from the axis to the exterior of the toroid (perpendicular to H[1] in the case that we consider a rectangular one, or the same direction but on the middle of it, in the case that we consider a circular one). The area is going to be saved in the variable named A[1].
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(10) 
1.1. Circular Area: In this case, the diameter of the circle is going to be the difference between R[2] and R[1].
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(11) 
2. Integrating the Magnetic Flow inside the toroid. The lower and upper limit of the integral are in fact R[1] and R[2] respectively.
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(12) 
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(14) 
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(15) 
That is how we obtain the inductance and the Magnetic field of a Toroid. This document can be modified to obtain other results that the student might want to implement and also can be edited to obtain some other information from its calculations.
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