THE ROLE OF MAGNETIC COMPONENTS IN POWER ELECTRONICS Magnetic components are used to perform many functions in a power conversion system. These functions are broadly divided between power handling magnetics and signal handling magnetics. We focus on power handling magnetics since these are of unique concern to power electronic systems. These primarily include inductors and power transformers. Power Inductors In its minimal configuration, the power train of a switch-mode power converter circuit requires an inductor to perform basic energy storage / delivery from the source to load. By using PWM techniques charging and discharging voltages are applied during different intervals, which in turn cause the build-up of magnetic energy via the inductor s energy storage mechanism. During the discharge interval the magnetic energy is once again delivered back to the circuit. This continuous conversion from electrical to magnetic back to electrical energy is governed by the energy relationship E= ½ Li 2 (1) Here, the current i in an inductor L sets up a magnetic field H that induces a magnetic flux density B in the volume of the magnetic core. As changes in the inductor current are driven by the externally applied voltage, the energy level E is driven up or down, thereby storing or releasing energy back to the circuit. The change in the current level based on the applied inductor voltage v L is determined from v L = L di/dt ( 2 ) So it is evident from (1) and (2) that energy storage in the inductor is key to proper and efficient operation. This efficiency relies on storing energy in regions with high B.H products. As is well known, higher magnetic flux density is induced with higher permeability µ from B = µ H = µ 0 µ r H (3) The magnetic core is used to focus a high flux density into an adjacent nonmagnetic region ( e. g., an air gap ) with high field to obtain a high energy density. Small air gaps are very common way to introduce a high energy density region for energy storage and retrieval. These can be in the form of a single gap in a continuous high permeability flux path or several gaps distributed over the entire length of the flux path. The inductance of a powdered iron core is L= µ 0 N 2 Ace Lg + lm / µ r On the other hand, powdered iron and alloys coated with insulating epoxies and binder systems create a distributed gap effect. In this case L = µ 0 µ r N 2 Ace lm An important aspect of any inductor subject to dc current bias is saturation, and hence, is dc bias capability. This determines the maximum current level that can be carried without causing saturation of the magnetic flux density in the core. Using Ampere s circuital law around the core H = N I / lm
From above equations, Bmax = LImax / N Ace Transformer Action: The arrangement of two fixed coils wound on a core is known as a Transformer. A transformer is an electric device, without mechanically moving parts, for transferring electric energy from one or more circuits to one or more other circuits by electromagnetic induction. A very important point to remember is that, a transformer will not operate with a steady current in the primary. For electro-magnetic induction to occur, there must be a changing magnetic field which is produced when the current rises and falls to maximum then the least, then rises in the opposite polarity to maximum and then the least. Induction is caused by the changing magnetic field. For a transformer to operate continuously, the primary winding must be connected to an intermittent source of current or to some source of alternating current. An alternating current continuously changes in value and periodically reverses direction. Therefore, around a primary that is carrying an alternating current, the magnetic field is constantly building up and dying down, first in one direction then the other. These changes of the field mean that the magnetic lines continually cut through the secondary and induce voltage in it. Frequently, transformer cores are formed into a square or rectangular loop to provide a complete, closed path or circuit for the lines of magnetic flux. A solid block of steel with a square hole in the center has the primary coil wound around one leg and the secondary wound around the other leg. Produce by the current in the primary, the flux is carried almost entirely by the core and passes through the secondary winding to induce the desired emf (voltage). Design of HF transformers High frequency transformers transfer electric power. The physical size is dependent on the power to be transferred as well as the operating frequency. The higher the frequency the smaller the physical size. Frequencies are usually between 20 and 100kHz. Ferrite is mainly used as the core material. The first step to calculate a high frequency transformer is usually to choose an appropriate core. At first a core-weight or core-volume is determined depending on the transfer power and switching frequency. In the second step, the primary number of turns is calculated because this determines the magnetic flux density within the core. Then the wire-diameter is calculated, which is dependent on the current in the primary and secondary coils. Calculation of the minimum number of primary turns:
It is assumed that there is a square-wave voltage V 1 at the primary side of the transformer. This causes an input current I 1, which consists of the back transformed secondary current I 2 and the magnetizing current I M (see illustration 1). A core without an air-gap is used in order to keep the magnetizing current as small as possible. The square-wave voltage at the input of the transformer causes a triangular shaped magnetizing current I M which is almost independent of the secondary current (see also the equivalent circuit). The magnetizing current is approximately proportional to the magnetic flux φ i.e. to the magnetic flux density B. The input voltage V 1 determines the magnetic flux in the transformer core corresponding to Faraday's Law V = N d (φ)/dt (see illustration 2). For the transformer on the right in the diagram above, the following applies: The change B of flux density depends on the frequency f = 1/T and the number of turns N 1. The higher the frequency and the number of turns the lower the change of flux density. Now the minimum number of turns N 1 can be calculated to ensure that a certain change of flux density B is not exceeded. In general the following applies: the smaller the change in flux density B, the smaller the hysteresis losses. From this a suitable number of turns for N 1 results: Where A min is the minimum core cross-section. This determines the maximum flux density
TRANSFORMER LEAKAGE INDUCTANCE Leakage inductance is occurring due to the imperfect coupling of magnetic flux from one winding to another. In an ideal situation all magnetic flux generated by one winding should couple into the intended secondary winding. This could only happen if both windings were physically one and the same. So in all transformers only a portion of the flux generated is coupled, leaving a leakage part stray and uncoupled. This flux, like any flux generated by a set of windings, is represented in circuit models as an inductor in series with winding of concern. This so-called leakage inductance is really the manifestation of the redistribution of magnetic field energy in the region of transformer, whereby a portion of energy is recaptured into electrical current in the intended secondary. The uncaptured leakage field remains untapped and stored in certain regions of the transformer. The leakage inductance L leak is defined as ½ L leak i 2 = ½ B.H dv where L leak is referred to a either a primary or secondary winding and i is the current in that winding. If the primary and secondary windings of thickness h p and h s, respectively are separated by an insulation layer of thickness t ins all of a winding breath Ww ( fig.), the resulting leakage inductance due to stored field energy is L lk = µ 0 Np 2 M L T / Ww ( h p +h s /3 +tins ) Here, MLT is the mean length of turn. The use of interleaved primary and secondary windings will further reduce leakage inductance due to reduction of energy stored in the magnetic fields. This is due to a redistribution of the leakage field with lower peak intensities and, thus, lower energy content. The reduction of leakage effects is very important, especially in high frequency devices. Energy stored in the leakage field is usually dissipated when returned to the circuit, but also the finite amount of time required to do so limits the maximum transition speeds from fully on to fully off, thus limiting maximum permissible switching frequencies. SATURATION LIMITS As the hystersis loop reaches a maximum value of B, saturation occurs. The permeability drops quickly toward µ 0 as the saturation level, Bsat, is reached. Extra MMF will still increase B, but the rate of increase reduces drastically for a Ferro magnetic material above Bsat. Manufacturers routinely specify a value of Bsat at a particular value of H, since the saturation level is somewhat ambiguous. For good magnetic steels, the saturation value is about 2 T. for typical ferrites, Bsat = 0.3 t. powdered iron cores and many other magnetic alloys can operate to at least 1 T. The main reason for avoiding saturation in an inductor or transformer is that when the saturated permeability approaches µ 0, a core becomes magnetically transparent indistinguishable from outside air. Most of the flux then leaks through the air, and core offers no benefit. For a transformer, the relationship between input and output can break down under saturation, since the leakage flux does not couple the windings. A term usable flux density has been developed to bridge the disconnect between the saturation flux densities published by the magnetic material manufacturers at high magnetomotive force levels and the peak flux densities at which inductive components can maintain a desired inductive value. Usable flux density is defined as the calculated flux density at the point of minimum cross sectional area Amin in the magnetic core that corresponds to the dc bias condition at a desires operating temperature that results in a 10% - 15% decrease of the effective permeability from its unbiased value. A dc current I dc, equivalent to peak current, is typically used to measure the inductance for a specific peak current condition ; the maximum peak current density in the core, B pk max, is established as Where Lo is the zero-bias inductance. B pk max = L o Idc / Np Amin
THE HYSTERSIS LOOP AND LOSSES The magnetic properties are characterized by its hysterisis loop, which is a graph of flux density Versus magnetization force as shown below: Hysterisis Loop When a electric current flows through a conductor (copper wire), it generate a magnetic field. The magnetic field is strongest at the conductor surface and weakens as its distance from the conductor surface is increased. The magnetic field is perpendicular to the direction of current flow and its direction is given by the right hand rule shown below. When the conductor or wire is wound around a magnetic materials ( ferrite, iron, steel, MPP, sendust, high flux, etc.), and current flows through the conductor, a flux is induced on the magnetic materials. This flux is induced by the magnetic field generated by the current carrying conductor. The magnetic material's atomic parts got influenced by the magnetic field and causes them to align in a certain direction. The application of this magnetic field on the magnetic materials is called magnetization force. Magnetization force is called Oersted or A/m (amperes per meter). The units for Magnetization force is "H" The results of applying these magnetic field from the current carrying conductor causes the magnetic materials to have magnetic flux being formed inside the magnetic materials. The intensity of this flux is called flux density. Therefore flux density is defined as the flux per square area. Flux density is called gauss or Tesla. I Tesla is10,000 gauss, or 1mT is 10 gauss. The unit for Flux is "B"
Thus, the hysterisis loop is often called the BH curve. Understanding of the BH curve is extremely important in the designs of transformers, chokes, coils, and inductors. For a square wave application as in SMPS, the Flux density or B is given as: Note that B is a function of voltage (input voltage if calculated from primary windings, and output voltage if calculated from secondary side). Flux will reduce if you increase the number of turns, increase the switching frequency or increasing the size of the cores (increasing the area) The magnetization force or H is given as: Note that H is a function of input current. As the current swings from positive to negative the flux changes as well, tracing the curve. The permeability of a magnetic material is the ability of the material to increase the flux intensity or flux density within the material when an electric current flows through a conductor wrapped around the magnetic materials providing the magnetization force. The higher the permeability, the higher the flux density from a given magnetization force. If you look at the BH loop again, you will note that the permeability is actually the slope of the BH curve. The steeper the curve, the higher the permeability as shown below.
As the magnetization force increases (or the current over the conductor is increased), a point is reached where the magnetic material or core will saturate. See point "S" above on the curves. When that happens, any further increase in H, will not increase the flux. More importantly, the permeability goes to zero, as the slope now is flat. In this situation the magnetic material or core will fail to work as a transformer, chokes, or inductors. So, it is very important in a choke or inductor design, not to drive the core into saturation by increasing the current (AC or DC). Usually it is the DC current that saturate the cores since it is a constant current, and puts the cores to a certain flux level. In a transformer design, you must make sure that the maximum AC current swings from positive to negative is well below the saturation point. Another way to get saturation is by increasing the flux density which is normally achieved by increasing the voltage (see equation above). From the BH curve, you can see that when the permeability is high ( slope is steep), the cores will go into saturation faster. Conversely, when the permeability is low, the cores saturate at a much higher flux density. The nonlinear permeability and saturation effects in a ferromagnetic material can be shown in a B vs. H curve, called a magnetization characteristic. Let us start with a typical ferromagnetic core. First, heat it above Tc to eliminate all magnetization. The internal structure will consist of small, fully magnetized domains oriented at random, as shown in fig 1. The net flux measured outside the material is zero. Now, apply MMF to the material through a coil. As the current is slowly raised, domains begin to reorient. The flux rises quickly. At sufficient high current, most domains have flux aligned with the imposed H, and the rate of increase declines. Finally, virtually all domains are aligned, and further increase in flux are governed by the material s inherent paramagnetism. This behavior is shown in fig 2. The slope of the curve, by definition, is permeability. The slope close to the origin is termed initial permeability. When the MMF is reduced, many domains tend to remain aligned. Hence H must be reduced significantly before the domains begin to return to their initial unaligned arrangement. The magnetization curve is not fully reversible, a phenomenon known as hystersis. When the applied MMF is reduced to zero, some residual alignment remains, and an external remanent flux with density Br can be detected. The MMF must be reversed and driven somewhat negative before the net flux is again zero. The value of H needed to force the flux to return to zero is the coercive force, Hc. The hystersis loss in all core material increases with increases in ac flux density, Bac, and operating or switching frequency, f. the general form of the loss per unit volume (sometimes termed the specific loss), Pm, sp, is Pm, sp, = Kf a (Bac) d Eddy-Current Loss The core of a transformer is usually constructed of some type of ferromagnetic material because it is a good conductor of magnetic lines of flux. Whenever the primary of an iron-core transformer is energized by an alternating-current source, a fluctuating magnetic field is produced. This magnetic field cuts the conducting core material and induces a voltage into it. The
induced voltage causes random currents to flow through the core, which dissipates power in the form of heat. These undesirable currents are called eddy currents. The eddy current produced due to the resistive nature of the core and hence The eddy current loss is proportional to the square of the current in the winding. To minimize the loss resulting from eddy currents, transformer cores are LAMINATED. Since the thin, insulated laminations do not provide an easy path for current, eddy-current losses are greatly reduced. By using laminated core, (thin sheets of metal instead of a solid iron core) the path of the eddy current is broken up without increasing the reluctance of the magnetic circuit. The laminations lie in the same direction as the flux. Therefore, the insulating surfaces are directly across the path of the eddy currents. The resulting eddy current reduction improves the efficiency of the transformer. SKIN EFFECT LIMITATIONS When a magnetic core is made from conducting materials such as magnetic steels, time-varying magnetic fields applied to the core will generate circulating currents as is diagrammed in fig 1. Using right hand rule, it can be seen that these currents, generically termed eddy currents, flow in directions such that secondary magnetic fields are produced that oppose the applied ( primary ) magnetic field. These opposing fields tend to screen the interior of the core from the applied field, and the total magnetic field in the core decays exponentially with distance into the core as is shown in fig 2. The characteristic decay length in the exponential is termed the skin depth and is given by δ =Sqrt (2 /ωµσ) where f = ω / 2π is the frequency in hertz of the applied magnetic field, µ is the magnetic permeability of the core material, and σ is the conductivity of the magnetic material. If the cross-sectional dimensions of the core are large compared to the skin depth, then the interior of the core carries little or none of the applied magnetic flux as is diagrammed in fig 2. And the core is ineffective in its intended role of providing low reluctance return path for the applied magnetic field. Typical values of the skin depth are quit small even at low frequencies (typically 1 mm at 60 Hz) because of the large permeability of the materials and the skin depth becomes more of a problem as the applied frequency increases. Thus magnetic cores for inductors and transformers that utilize conducting magnetic materials are made from stacks of many thin laminations as shown in fig 3. Each lamination is electrically isolated from the other by means of a thin insulating coating on each lamination. The core stacking factor is defined as the ratio of the cross sectional area of the magnetic material to the total cross-sectional area of the core. The stacking factor will be less than 1. (typical values are 0.9 to 0.95 ) because part of the total area of the core is occupied by the insulating layers. Most magnetic steels have a small percentage of silicon added to the iron to increase the resistivity of the material and thus increase the skin depth. COPPER WINDINGS The conductor windings in an inductor in an inductor or transformer are made from copper because of its high conductivity. The high ductility of the copper makes it easy to bend the conductors into tight windings around a magnetic core and thus minimize the amount of copper and volume needed for the windings. High conductivity contributes to minimizing the amount of copper needed for the windings and thus to the volume and weight of the windings. COPPER FILL FACTOR
A typical cross-sectional view of a double-e core is shown in fig 4. The copper conductor from which the winding is made has a cross-sectional area Acu. The conductor may be composed of a single round wire or it may be a special multistandard conductor such as Litz wire in which each strand has a diameter on the order of a few hundred microns or less. Litz wire is used where otherwise the skin effect would present problems. The total number of turns N in the winding window of the core times the conductor area Acu gives the total copper area in the winding window. The total copper area will be less than the area Aw of the winding window for several reasons. First, the geometric shape of the conductors, usually circular, and the winding process prevents the N conductors from completely filling the window. Second, the conductor must be covered by an electrical insulator so that adjacent turns are not shorted to each other. This insulation takes up some of the winding window area. The ratio of the total copper area to the winding window area is termed the copper fill factor Kcu and is given by Kcu = N Acu / Aw Practical values of fill factor range from 0.3 for Litz wire to 0.5 0.6 for round conductors. WINDING LOSS DUE TO dc RESISTANCE OF WINDINGS The power Pcu,sp dissipated per unit of copper volume in a copper winding due to its dc resistance is given by Pcu, sp = ρ Cu ( J rms) 2 where Jrms = Irms / Acu is the current density in the conductor and Irms is the rms current in the winding. However, it is more convenient to express Pcu, sp as power dissipated per unit of winding volume, Pw,sp The total volume Vcu of the copper is given by Vcu = Kcu Vw, where Vw is the total winding volume. Using this result to express Pw, sp yields Pw, sp = Kcu ρ Cu (Jrms) 2