ISSUE: October 2015 Leakage Inductance (Part 1): Friend Or Foe? by Ernie Wittenbreder, Technical Witts, Flagstaff, Ariz There are situations in which leakage inductance in a transformer or coupled inductor creates power losses and generates unwanted noise In these situations, the designer seeks to minimize leakage inductance as much as possible There are other situations in which leakage inductance provides a benefit Moreover, in certain situations leakage inductance plays a critically important role in the operation of the circuit to great benefit This three-part article series will attempt to foster a better understanding of leakage inductance, how to design around problems that leakage inductance creates, and how to use leakage inductance to advantage to reduce power losses, size, and cost Part 1 describes leakage inductance and the science and math behind it Part 2 will describe some of the problems created by leakage inductance and how these problems are dealt with Part 3 will describe where and how leakage inductance is beneficial This last part will also show some examples of relatively simple power supply circuits in which intelligent use of leakage inductance can lead to higher efficiency and higher power densities without creating leakage inductance-induced noise The Underlying Physics Let s start with the basics Fig 1 illustrates a wire that carries an electrical current The current in the wire produces a field of magnetic induction surrounding the wire as shown, which deflects a compass needle such that the compass needle aligns with the flux Fig 1 A wire carrying current flowing into the page produces clockwise magnetic flux surrounding the wire 2015 How2Power All rights reserved Page 1 of 9
For a long straight wire the magnitude of the field of magnetic induction associated with the current in the wire is proportional to the current in the wire and inversely proportional to the distance from the wire This relationship is described in quantitative terms by Ampere s law and the Biot-Savart law The fact that an electrical current produces a magnetic field is one of the key relationships between electricity and magnetism If instead of a long straight wire we have a square loop of wire connected to a voltage source to produce a current (as shown in Fig 2), then we will generate a magnetic field that is concentrated inside the loop of wire The magnetic flux in the loop is proportional to the current i in the wire If, instead of just one wire in the loop, we have N turns of wire in the loop then the total flux in the loop is which is proportional to the current i in the wire The constant of proportionality is called self inductance, L, so that Fig 2 A current in a loop of wire produces a magnetic field inside the loop In the loop of wire illustrated in Fig 3, an electromotive force (EMF), ε, is generated in the square loop of wire as the magnet is moved along the path indicated by the dotted line towards the loop According to Faraday s Law, the EMF is related to the rate of change in magnetic flux in the loop 2015 How2Power All rights reserved Page 2 of 9
where is the magnetic flux in the loop of wire Stated simply, a changing magnetic flux induces an EMF and an associated current in the loop of wire In more general terms, if instead of a single loop of wire we have a multi-turn coil of wire, then the EMF is increased by the number of turns, N, so (1) Fig 3 Current flows in the loop of wire as the magnet is moved towards the loop The direction of the current in the loop that results from the changing flux is significant Magnetic flux is associated with a magnet and it is also associated with current in a loop or coil of wire, as was shown above in Fig 2 According to Lenz s law the current induced in the loop will appear in such a direction that it opposes the change that produced it The motion of the magnet produces an increasing flux to the left and the current in the loop produces flux to the right, thereby cancelling the increased flux from the moving magnet The opposition of induced EMF is indicated by the minus sign in equation 1 If instead of a magnet we had a second loop of wire, wherein the second loop of wire was oriented so that the plane of the second loop was parallel to the first loop and placed in close proximity to the first loop, as shown in Fig 4, we would have another situation in which an EMF is induced in a first loop of wire by a changing magnetic field 2015 How2Power All rights reserved Page 3 of 9
Fig 4 A changing current (i 2 ) in a second loop of wire induces an EMF and associated current (i 1 ) in the first loop of wire nearby In the Fig 4 case, the field is created by current in a nearby second loop of wire A changing current in the second loop of wire produces changing magnetic flux that appears in the first loop of wire If initially there is no current in the second loop of wire, when the current initially appears it will create an increase in flux in the first loop much like the movement of the magnet and an EMF and current will be induced in the first loop of wire to oppose the flux created by the second loop The current induced in the first loop also creates magnetic flux that appears in the second loop, which generates an opposing current in the second loop The two loops of wire can be considered to be magnetically coupled An EMF can also be induced in an isolated loop of wire or multi-turn coil of wire in the absence of a moving magnet or nearby second loop of wire to generate a change in flux The application of a voltage to an isolated loop of wire, such as the loop illustrated in Fig 2, creates a current in the loop of wire which creates magnetic flux in the isolated loop The increasing magnetic flux in the loop that occurs at the instant that the voltage is applied to the isolated loop (when the switch is closed) induces an EMF and an associated induced current in the isolated loop to oppose the flux that created it The effect of the induced current is to initially cancel the current resulting from the voltage applied to the loop to create a net current that gradually increases from zero The flux in the loop is proportional to the net current in the wire and the constant of proportionality is called the self inductance, as was indicated above 2015 How2Power All rights reserved Page 4 of 9
In the general case of a multi-turn coil we have isolated coil is then where L is the self inductance Faraday s law for an The voltage that is applied to the coil divides between the induced EMF and voltage drops across other electrical elements in the coil such as wire resistance The current in the coil will ramp up until all of the applied voltage appears across the resistance As the current in the coil is ramping up more of the applied voltage appears across the resistance and less appears as induced EMF, so the induced EMF and the rate of current ramping decrease over time An inductor is composed of a loop of wire, usually with more than one turn, and usually wrapped around a material with relatively high magnetic permeability The magnetic material increases the magnetic flux in the loop and thereby increases the self inductance The operation of the inductor is described mathematically by In the case of Fig 4, we had two loops of wire wherein the EMF in the first loop of wire was induced by flux created in the second loop of wire In this case, flux in the first loop is created by the second loop of wire and is proportional to the current in the second loop of wire, i 2 And in this case, the constant of proportionality is called mutual inductance We induce an EMF in the first loop of wire due to a changing current in the second loop of wire Mathematically, We can also induce an EMF in the first loop due to changing current in the first loop, The total EMF induced in the first loop of wire is the sum of the EMF induced from the self inductance and EMF induced from the mutual inductance: (2) Similarly, the total EMF induced in the second loop of wire is (3) A transformer is a magnetically coupled pair of wire loops, usually in close proximity, and usually with the addition of a magnetic material that is wrapped by each of the wire loops Fig 4 is one example of a transformer In a transformer or inductor the loops of wire are referred to as windings Going forward we will refer to windings rather than wire loops 2015 How2Power All rights reserved Page 5 of 9
Defining Leakage Inductance Equations 2 and 3 above describe the relationships between currents and voltages in a transformer due to self and mutual inductance A transformer may have more than two windings If it has three windings then there are three winding pairs, a pair consisting of winding 1 and winding 2, a pair consisting of winding 1 and winding 3, and a pair consisting of winding 2 and winding 3 If there are four windings then there are six winding pairs and six mutual inductances In a transformer, flux from a first winding appears in a second winding A change in current in one of the windings produces a change in flux that induces an EMF in the other winding The EMF depends on the amount of flux produced by one of the windings that appears in the other winding One winding may produce some total amount of flux in which some fraction of the total flux produced appears in the other winding and the rest of the flux does not appear in the other winding, as shown in Fig 5 Some of the flux from one winding is coupled to the other winding and some of the flux is not coupled to the other winding The mutual inductance of the windings depends on the fraction of the flux from one winding that is coupled to the other winding in a winding pair This fraction is referred to as the coupling coefficient, k Fig 5 Some of the flux produced by winding 1 couples to winding 2 and some flux produced by winding 1 does not couple to winding 2 For a winding pair, the mutual inductances are equal and dependent on the self inductances of the windings, so There is an inductance associated with uncoupled flux or leakage flux, called leakage inductance, which is a property of the winding pair in the same way that mutual inductance and coupling coefficient are properties of the winding pair 2015 How2Power All rights reserved Page 6 of 9
Fig 6 illustrates an electrical circuit of a coupled pair of windings in a transformer There is a leakage inductance in the primary circuit indicated as L k1 wherein and a magnetizing inductance L M1 wherein There is also an ideal transformer An ideal transformer is one in which there is zero leakage, voltages and currents on opposite sides are related by the turns ratio and impedances on opposite sides are related by the square of the turns ratio, in the well-known fashion Fig 6 Leakage and mutual inductances in a winding pair If we were to measure the inductance of the first winding with the second winding open, we would measure the inductance to be L 1 With the second winding open there is no current in the second winding and no change in current in the second winding Similarly, if we measured the inductance of the second winding with the first winding open, the measurement would equal L 2 If we measured the inductance of the first winding with the second winding shorted, we would effectively be measuring the leakage inductance of the first winding in series with the magnetizing inductance of the first winding in parallel with the parallel combination of the leakage inductance of the second winding and the magnetizing inductance of the second winding referred to the first winding The two windings are now coupled and changing currents exist in both windings The leakage inductance of the second winding referred to the first winding is just the leakage inductance of the second winding multiplied by the turns ratio of the first winding to the second winding squared We can ignore terms containing the magnetizing inductances if the magnetizing inductances are much larger than the leakage inductances so that the measured inductance of the first winding with the second winding shorted is just the leakage inductance of the first winding plus the leakage inductance of the second winding multiplied by the turns ratio squared We call this the total leakage inductance referred to the first winding, L kt1 where 2015 How2Power All rights reserved Page 7 of 9
We can also measure the total leakage inductance referred to the second winding by measuring the inductance of the second winding with the first winding shorted Using the equations that we have above and the four measurements of the open and shorted inductances of the two windings we can calculate the magnetizing and leakage inductances of the winding pair Fig 7 illustrates electrical models of a transformer that are equivalents to the model illustrated in Fig 6 A key point here is that the leakage inductance appears as an inductance in series with the coupled magnetic portion of the transformer and the electrical behavior of the transformer is the same regardless of whether the model places the total leakage inductance in series with the primary winding, the secondary winding, or some combination of primary and secondary windings The leakage inductance always appears as a series element and it does not matter where the inductance is placed for modeling purposes The electrical behavior will be the same Fig 7 Two winding pair models equivalent to each other and to the winding pair model of Fig 6 For design purposes, there are a number of empirical formulas for calculating leakage inductance that have been developed that work well for typical manufactured transformers A formula offered by Bruce Carsten is, (4) where L kt1 is the total leakage inductance (henries) referred to the winding having N 1 turns, LMT is the length of mean turn (cm), is the total winding height (cm), b is the winding breadth, c is the space between windings (cm), and m is the number of winding sections (pairs of adjacent primary and secondary pairs), as illustrated in Fig 8 2015 How2Power All rights reserved Page 8 of 9
Fig 8 Winding dimensions and structure as they relate to equation 4 for calculating leakage inductance for a winding pair in a typical transformer By examining equation 4 and Fig 8 one can see how it s possible to design a transformer to control leakage inductance Increasing winding height and separation between windings increases leakage inductance Increasing winding breadth and the level of interleaving, m, of winding layers decreases leakage inductance Leakage inductance can be our foe or our friend In the second part of this article series, we will describe some of the problems created by leakage inductance and ways to deal with those problems or eliminate them In part 3, we will provide some examples of circuits in which leakage inductance is our friend and some other ways in which leakage inductance is an enabler of higher efficiency and higher density power conversion About The Author Ernie Wittenbreder is widely recognized in the industry from his seminars, publications, and many patents His areas of expertise include topology selection, design tradeoff decision making, high-efficiency design, soft switching, gate drivers, and design for EMC His applications experience spans a broad range including high reliability and extremely low-noise converters for military and aerospace, telecommunications, solid-state lighting, very high density power supply in package (PSIP) converters, and a wide variety of commercial and industrial applications Wittenbreder has developed a power converter computer simulation product and SiC gate driver IC He also has a great deal of litigation experience in patent infringement lawsuits Wittenbreder began his career as a physics professor and holds MS Physics and MSEE degrees and BS, math and physics For further reading on leakage inductance, see the How2Power Design Guide and enter leakage inductance in the keyword search And for more on magnetics design in general, see the Design Guide, locate the Design Area category, and click on the Magnetics link 2015 How2Power All rights reserved Page 9 of 9