I = P/V
a = I/J
R = ρL/a
where ρ is the resistivity of the transmission line material, L is the length, and a is the cross sectional area.
P_{L}= I^{2}R
Generating Station feeding Transformer
So, what is a magnetic circuit? By magnetic circuit we mean this kind of a construct. You have a core of some regular shape on which some winding is put. Now if there is no current flowing into this coil we do not expect any flux lines to be generated in this core; however, when you switch this source on and a current does flow then magnetic lines of force will circulate inside this core. The flux density or the magnet field intensity inside the core will depend on the magnitude of the current i into the number of turns in the coil N. This quantity multiplication of N and i is called the magneto motive force or MMF, and this magneto motive force causes a flux to circulate.
We all know that if a conductor carries some current i then at a point near the conductor there will be a magnetic field. A magnetic field can be characterized by a magnetic field intensity vector H or a magnetic flux density vector B. The relationship between these two is very simple. B is equal to mu naught mu naught H, where mu naught is the permeability of free space and is given by 4 pi into 10 to the power minus 7; mu R is a dimensionless quantity. It is called the relative permeability of the material with which the core is made; for free space mu R is one; for ferromagnetic material it can be in thousands.
Now the magnetic flux density, the incremental magnetic flux density at any point near a conducting wire due to a small length of the wire d L is given by the BiotSavart’s law, which says you can find out the incremental flux density d B; from the formula d B equal to mu naught mu r by 4 pi into i dl cross r by r cube, where r is the position vector of the point from the incremental length dl. Please note that the d B is a vector, and its direction will be determined by the cross product of i dl and r. In this case r is in this direction, and i dl is in this direction; therefore, the direction of B will be out of this paper on the top.
The total flux density B at that point due to length of the wire can be obtained by integral equal to integral over length mu naught mu r by 4 pi i dl cross r by r cube. While this gives a general formula for determining the flux density near a conducting wire in practice this integration is difficult to perform except for very simple structure of the current carrying conductor.
Therefore, for solving magnetic circuits we use a different formula which is the ampere’s circuital law. This says that the line integral of magnetic flux intensity H over a closed path is equal to the current total current enclosed. There may be several of them the total current enclosed; that is integral over a closed path H dot dl equal to the total current enclosed. This law is very convenient for finding out the flux density and the magnetic field intensity in magnetic circuits; let us see why.
The structure of a magnetic circuit as we have said consists of a regular core and a winding. Now when this winding is excited there will be flux line inside the core. Since the relative permeability of the core is normally much higher than air or free space almost all the flux lines will be confined inside the core. Also since the total amount of flux crossing the core at any position of the core is fixed the flux density along this flux line is constant, and hence the H over this path is constant. Now if we choose the closed path of the circuital law to be something inside this core we know that H over this length is constant; therefore, we can find out H from the simple formula that if there are total N number of conductors carrying current I then the total current enclosed is N I, and if the length of the mean length of the path to the core is L then H is given by NI by L.
So, in a magnetic circuit finding out the value of H and hence the value of B which is nothing but mu naught mu r H is relatively simple. This of course makes the assumption that the leakage flux outside the core is almost negligible. Now this gives you a method of relating the flux with the MMF in I. That is the total flux a phi is equal to the flux density B multiplied by the crosssectional area of the core A which is then is given by mu naught mu r; H is given by N I by L into A, or this is given by N I divided by mu naught. Phi is equal to B into A which is mu naught mu r; H is N I by L mu naught mu r A.
So, if N I is the MMF then flux phi can be written as phi equal to MMF N I by reluctance R where the reluctance R of the core is given by L divided by mu naught mu r A. So, if we extend the analogy that the MMF N I is equivalent to a source of potential, and flux is equivalent to a current in a magnetic circuit then the quantity reluctance is equivalent to resistance in an electrical circuit; therefore, a magnetic circuit can be represented by a MMF source N I which sends flux through a reluctance R.
The corresponding equivalent electrical circuit is a voltage source E sending a current through a resistance R; that is why this kind of a structure with a core with a winding on it is referred to as a magnetic circuit because of its similarity with an electrical circuit. In practice though this magnetic circuit is somewhat more complicated than a electrical circuit. In a electrical circuit the resistance is normally constant; however, in almost all practical magnetic circuits the core material is made up of ferromagnetic material.
One property of this ferromagnetic material is that the relationship between the flux density and H magnetic field intensity is not linear; that is if we plot the flux density B versus the magnetic field density H. If it was linear then it would have been a straight line; however, for most material most practical ferromagnetic material it is not so. Initially it follows a straight line, but as v increases after some point this starts deviating from a straight line. In fact, B does not increase as fast as H, and beyond some point even if you keep on increasing H you practically do not increase.
This is due to the special constructional feature of this magnetic material which are made up of very tiny molecular level magnetic dipoles; however, almost all ferromagnetic material exhibits these characteristics which is called the B H characteristics, and that B H characteristics has a prominent saturation phenomenon where the relationship between B and H is nonlinear, and after a critical value of B further increasing H does not really increase B to a very great extent. Therefore, for ferromagnetic material the practical operating point that realizes the two potential of the material is somewhere in this junction in this region where the B H curves starts bending. This is called the knee point of operation.
Because given this point even if you put a large MMF that is H there will not be much increase in the magnetic flux density or the total magnetic flux circulated; obviously, in this region the relative permeability of the material which is mu R equal to B divided by mu naught H does not remain constant; therefore, reluctance of the magnetic core which is given by R equal to L divided by mu naught mu r A does not remain constant for all values of B. This is what makes analysis of magnetic circuit somewhat more difficult compared to analysis of an electrical circuit, let us see why.
A magnetic core need not always be made of a single material. In many cases a magnetic core in addition to the ferromagnetic core material we will also possibly include an air gap. Now the relative permeability of the air gap and the magnetic core are very different. So, how we are going to find given a number of turns and the current flowing through it; how are we going to find out what is the total flux circulated? In this case where the magnetic circuit has two different types of material with two different relative permeability. In an electrical case it would have been simple. This is basically a series circuit; we could have drawn it for electrical case; we could have drawn it as a MMF source N I, reluctance of the magnetic path magnetic core R core in series with the reluctance of the air gap or air.
The problem with ferromagnetic material is that the value of R c is not known at priority, because it depends on the value of the flux density as we have already seen. The value of R a more or less remains constant, it is not a function of the flux density it is linear; the B H characteristics of air gap is linear. Therefore solving this involves is a little more involved, and let us see how we will solve this kind of a circuit. Although the field intensity may be different the total flux crossing the core material and the air gap of course same. If we neglect the fringing effect at the air gap then the flux density in the core and flux density in the air should also be same. The total MMF N I can be written as flux magnetic field intensity H c in the core multiplied by the length of the core plus field intensity in the air gap multiplied by the length of the air gap.
But the relationship between intensity in the air gap and the flux density is constant; therefore, N I equal to H c into L c plus H is equal to B air by mu naught since mu R for air gap is one into L a, but B a is equal to B c. Therefore, we can write N I equal to H c L c plus B c by mu naught L, a or B c equal to mu naught into N I minus H c L c by L a. This is one relationship relating B c to H c which is linear. The other relationship between B c and H c is given by the B H curve of the material.
Now we plot them together H c on this axis and B c on y axis. This is the intrinsic B H characteristics of the material, and the other one given by the circuit which can be considered as a magnetic load line. The intersection of these two curves will give you the operating point H c 0 and B c 0. So, you see although solution of a magnetic circuit like this is somewhat more involved than solving an equivalent electrical circuit, but it can still be handled in a graphical manner, and this approach is used. At this point we think that we have discussed sufficiently about magnetic circuit. So, we will move to a different topic which is the induced voltage in a coil.
Let us say we have a magnetic core, and we are using a coil to set up magnetic flux in this direction. If the current flowing is into the coil is direct current then this flux will be a direct flux; however, if the core is excited with alternating current then the flux will also be alternating. We have seen the relationship between the flux and the current which is linear if the reluctance of the material can be assumed to be constant. The question now comes if I put another coil around this leg, what will happen? This has been studied by Michael Faraday, and the famous law says that there will be a voltage induced in this coil one, two. The magnitude of the induced voltage e will be proportional to the rate of change of flux density d phi d t multiplied by the number of turns in the coil N, and the polarity of the induced voltage will be such that it will try to oppose the very cause which it is due. Let us look at it a little more carefully what that means.
Let us say this is our coil, and there is a flux phi in the upward direction. This is a coil with let us say N turns, and there is a flux phi in this direction, and let the phi be varying in a sinusoidal fashion phi equal to phi max sin omega t. So, by Faraday’s law the induced voltage e will be such that magnitude e will be proportional to N d phi d t equal to N omega phi max cos omega t, and the direction of the polarity of the induced voltage will be such that it will tend to oppose the very cause which it is due; what is the cause that the changing of the flux. Now let us see at t equal to 0 the flux was 0, and as time progresses t becomes positive; this flux tries to increase.
So, in order to oppose that cause, what the induced voltage will try to do? If you close this switch this induced voltage e will send a current through this loop which will tend to cancel this current which will tend to cancel this flux; therefore, the circulating current will be generating a flux which will be in the opposite direction. So, what will be the direction of that current which we can cancel this flux; obviously, the current will be in this direction. Therefore, this coil with this kind of a flux can be replaced by an e. m. f source e where the terminal two at time 0 will be positive, and the terminal one at time 0 will be negative, and the potential e 21 in that case will be given by N omega phi max cos omega t. So, if we want to find out the r m s value then E 21 equal to omega is given by 2 pi f into N phi max divided by root 2.
Or the induced voltage E equal to 2 pi by root 2 is 4.44 f phi max N. So, this is the formula of induced r m s voltage in a coil with number of turns N which links an alternating flux I. Generally alternating quantities in ac circuit analysis are represented by phasors; therefore, if we draw the phasors and take the flux phasor to be the reference phasor then the induced voltage E will be leading it by an angle 90 degree. Here I would like to bring to your attention some of the conventions that are used by the authors of different books. The quantity that we have been calling induced voltage in fact by many authors is called the counter induced voltage, because here we have got this e by simply saying e equal to N d phi d t.
And we have incorporated the last part of the Faraday’s law that is it opposes the very cause which it is due by taking its proper polarity. This many authors call the counter induced voltage. The actual Faraday’s law says e equal to minus N d phi d t. So, in our case this e will be opposing the supply voltage in the case where e is taken with a negative sign. This is called the induced voltage; therefore, in our case the k v L of the coil will be written as v equal to e; in this case the k v L will be written as v plus e equal to 0 that is the only difference. In fact, if there is no scope of confusion then we will keep calling this as the induced voltage although many authors in some text books will be calling it the counter induced voltage.
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