Designing an Inductor in Switched Mode Power Supply (S.M.P.S.) Systems for desired value of inductance (L) and magnetizing current (I) basically includes, selection of proper core from the energy storage capacity point of view. As in a transformer the core is selected from the power handling capacity point of view. After selecting the core for the inductor it further includes calculating nos. of turns, size of the conductor, calculating length of the air gap (Lg) to be provided in the core from keeping in mind.
This Documentation gives clear idea regarding necessity of Inductor in S.M.P.S systems, and basic concepts with regards to conduction Ohms law in bulk as well in point form, Magnetic ohms law, dielectric ohms law, defining inductance from their base, introduces concept of core saturation and the way one can get control over it, finally deriving the energy equation for an inductor from which one can select the appropriate core for their application by considering thermal and saturation constraints.
Necessity of an Inductor in SMPS Systems
The basic fundamental behind introduction of S.M.P.S. systems
before the general linear power supply systems is to, perform the energy transformation/conversion from the source to the load/sink in an optimal way with less losses and greater efficiency, by deploying the energy efficient devices/ non power dissipative devices like capacitors and semiconductor switches. The power/energy flow in the S.M.P.S. systems is not smoothly continuous because of the switching operation of semiconductor switches, so as a concern to make the energy/power flow continuously smooth we adopt an electro-magnetic reactive element called the inductor. The inductor is also deployed in the S.M.P.S system as in with a Snubber/Switching Aid Circuits to reduce the stress on the semiconductor Switches during the Turn-On time. In this document the designing of the inductor for smoothening the power flow will be discussed.
As stated earlier an inductor is an electro-magnetic element, which refers that on exciting an inductor with electrical energy it has the kinetic potential to store the energy in the form of Magnetic Energy. So the equivalent form of electrical and magnetic circuits for an inductor is as shown in figure (1) and figure (2) respectively.
From the electrical equivalent circuit shown in the Fig.1, the electrical circuit equation can be related as,
Where V, L, I are electrical circuit quantities.
Similarly from the magnetic equivalent circuit shown in the Fig.2, the magnetic circuit equation can be related as,
The above equation relates Faradays Law of Electromagnetism. Where V is the” Electrical circuit quantity and N, φ are the Magnetic circuit quantities.
From the above equation (i) and (ii) the Electrical and Magnetic equivalents of an inductor can be related as,
Where, LI = electrical flux linkages and Nφ = Magnetic flux linkages.
Derivation and Definition of Inductance (L)
Bulk Ohms Law
From most familiar basic electrical Ohms law in a bulk form, which states that the Driving Current (I) through a conducting material is directly proportional to the applied Driving Voltage (V) across the conducting material, provided the operating temperature is constant.
Therefore we can equate the bulk Ohms Law to V = IR ........(iv)
From the above, Ohm's Law at any particular point the space can be derived as below, which is so called as Ohm's Law in Point Form.
From the Bulk Ohms Law i.e. V = IR, can be rewritten as,
Magnetic Ohms Law at A Point
As an analogy to the Conduction Ohms law at any point in the space from the equation –(v), the Magnetic Ohms Law at any point in the space can be equated as,
Dielectric Ohms Law at a PointAs an analogy to the Conduction Ohms law at any point in the space from the equation – (v) , the Dielectric Ohms Law at any point in the space can be equated as,
To derive the equation for the Inductance we shall start deriving the “Bulk Magnetic Ohms Law” from the equation-6, so called “Magnetic Ohms Law at any point in the Space”.
From equation- (vi), we have
From the equation-(iii) we have,
From above Inductance (L) as be defined as, Flux linkages per ampere in a turn is called Inductance measured in henries (H).
Core Saturation and Air GapThe general purpose toroid core inductor with an Air Gap of length Lg is as shown in the Fig.7.
With the Air gap in circuit the equation-(ix) can be rewritten as,
(As the relative permeability of the core μr, is generally in terms of few thousands the overall core reluctance can be neglected in comparison with the Air Gap reluctance where μr = 1.)
From the Equation-10 it’s clear that “Providing an Air Gap in the Core makes the inductance to be independent of the Core Shape used, Core Reluctance and Core effective Length. It only depends on the Air Gap length and Area”. So, the advantage by providing Air Gap can be understood as following. The typical Flux linkages versus magnetizing current graph for ideal case is as shown below in fig.8.
Why dose Core Saturate?As shown in above graph the relation between flux linkages w.r.t. magnetizing current is linear provided it is Ideal condition. Practically in every magnetic material there undergoes a situation such that for a particular value of the magnetizing current all the magnetic dipoles in the magnet will get aligned up and there won’t be any further alignment of the dipoles to produce the net flux. Thus magnetic flux gets saturated. This condition in magnetic materials is called Saturation. This condition gives rise to the graphical relationship as shown in FIG.9.
What Happens if the Core Saturates?This Saturation Condition if allowed it leads to the damage of core, this is because “At the point of saturation as discussed the magnetic flux becomes leveled off and there is no net flux in the core, therefore the change flux linkages (Nφ) w.r.t. the magnetizing current decreases this causes the Inductance of the core to reduce, as the inductance is reduced the inductive reactance (XL) will reduce (XL = 2πfL), with this drop in inductive reactance the core magnetizing current further increases which further drops the inductance and the process become cumulative and core draws huge magnetizing currents leading to the thermal and dielectric breakdowns”.
How Air Gap Helps in Saturation?One can’t avoid the phenomenon of Saturation in the magnetic materials. As the Air Gap is provided in the core the core reluctance increases this makes the Flux to increase slowly w.r.t. the magnetizing current, hence giving a flexibility to operate the core at higher magnetizing currents which helps to “Store huge energy as of inductor” and as concerned to the transformer it helps to Transfer higher powers. Also from Equation-10 it’s clear that “Providing an Air Gap in the Core makes the inductance to be independent of the Core Shape used, Core Reluctance and Core effective Length. It only depends on the Air Gap length and Area”.
Energy Equation for an InductorThe familiar EE-geometry core is as show in the fig.10. We shall derive the energy equation for that core, howsoever if air gap is provided the inductance is not going to depend on the core geometry.
Where, Ac = effective core Area and Aw = Window Area
From the equation-(iii), considering the saturation constraints of the core we have
Also, volume of the conductor accommodated in the window can be equated as below,
From the equation (xi) and (xii)
This equation is called “ENEGRY EQUATION FOR AN INDUCTOR”. This forms the base to select appropriate core for an Inductor.
Example for Typical DesignLet, L= 20μH and Im = Irms = 5A a typical DC-DC converter operating at 20KHz.
Assumptions: B = 0.2T ; J = 3A/mm2 Kw = 0.35
From various range of cores available the product of AcAw is calculated and is tabulated below:
From the above table it’s clear that “E25.4/10/7” core suits our application.
No. of Turns:
The core area (Ac=Ae) is taken from the data sheet magnetic characteristics as shown in the below figure.
Size of the Conductor: