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Magnetic Coupling in Transmission Lines and Transformers Explore the similarities of these closely-related structures and discard some widespread misunderstandings. Gerrit Barrere, KJ7KV During my investigation of transformers • If two wires carrying differential current pass gram like Figure 1 follow the direction of and transmission lines and the marriage of through a core, the flux they produce can- the vector field, and the density of spacing the two, a number of questions arose that I cels, resulting in zero flux in the core. Yet between lines indicates the field magnitude. could not answer. For instance: the core has a significant effect on low- In general, the magnitude of the magnetic • If the characteristic impedance of a frequency behavior. How can this happen field along a given line is not constant; but in transmission line is the familiar with no flux? a symmetrical case like Figure 1, it is. I slowly pieced together explanations for Flux is a smooth and continuous phenom- ZLC0 = (square root of inductance per length over capacitance per length), those and other aspects of lines and transform- enon, like swirling water. The lines in vector field diagrams merely indicate the trends of why doesn’t this change when the line is ers, which I haven’t seen in the literature; surrounded by ferrite, which increases in- perhaps you will find them helpful too. ductance? Through most of this discussion I will be re- • One source stated that the electrical length ferring to transmission lines as two wires, but the same ideas apply to any form of line ex- of a line is increased by adding ferrite, cept coaxial cable, which is treated separately. thus extending the low-frequency re- sponse. But electrical length also depends The Basics on per-length inductance (electrical First, I present a review of terminology and length is physical length times frequency the basics. An isolated wire carrying a current, times ), and my measurements LC I, produces a magnetic field H external to the showed that neither impedance nor elec- wire, as shown in Figure 1. The strength of this trical length vary with ferrite loading. field varies as 1/r, where r is the distance from • What exactly are “conventional transformer the center of the wire. It also produces a mag- currents” and what is the difference be- netic field internal to the wire, which goes from tween this and “transmission line mode”? 1/r at the surface to zero at the center of the • How can “choking reactance” of a trans- wire. For frequencies low enough that skin ef- mission line transformer affect differen- fect is negligible, the profile of magnetic field tial transmission line signals? This should strength, H, versus radius, r, inside the wire is Figure 1 — Magnetic field around a be strictly a common-mode effect. linear, as shown by the low-frequency line in current-carrying wire. • Typically, transmission line transformers Figure 2. Note that the field reaches a peak right are wound with something like a half at the surface of the wire, at radius r. Also note meter of line. At HF and even VHF this that a smaller diameter wire will create a larger is a small fraction of a wavelength, so peak H field at its surface, since the peak is how can they be considered transmission proportional to 1/r. lines at all? How can they develop sig- The magnetic and electric fields are vec- nificant longitudinal voltage if they are tor fields [Indicated here in italic boldface electrically short lines? type — Ed.] since at every point there is a magnitude and a direction to them. The mag- netic field always forms closed paths or loops 16816 12th Pl NE of magnetic flux, and the electric field always Shoreline, WA 98155 terminates perpendicular to a (perfect) con- Figure 2 — Magnetic field strength [email protected] ductor. The lines of flux shown in a field dia- variance with distance from center of wire. 28 Sep/Oct 2006 Reprinted with permission; © ARRL the flux field. Flux does not actually occur ing the two, so mutual inductance increases. since the tightly-curved H lines near the wires as discrete lines. In a single inductor, current through it causes rapidly begin pointing in opposite directions. The magnetic field H produces a magnetic voltage to be produced across it. When two The combined H field is also nearly zero ev- flux density field, B, by the relationship B = inductors are linked by mutual inductance, erywhere else, since the fields nearly cancel μH, where μ is the permeability of the me- current in one produces voltage in the other (they would perfectly cancel if the wires some- dium in which the flux occurs (μ is the prod- due to the flux from the one, which links the how occupied the same region in space). uct of the relative permeability μr and the per- other. The combined magnetic field of two close meability of free space μ0). Thus if a given H Figure 3 shows two wires in cross sec- wires carrying differential current is mark- field passes through high-permeability mate- tion, with current entering the page in wire edly different from a single wire. An intense, rial, a greater flux density will result than if it A and no current in wire B. The current in short line of magnetic field is created between passes through air or copper, which have a μr wire A causes the magnetic flux shown. None the wires, and the field is nearly zero every- of 1. This flux density may vary along the path of the flux inside loop fa links with wire B, where else. of an H field loop, too, depending on the per- so it contributes to the self-inductance of wire This altered field pattern, however, does meability in portions of the path. Keep in mind A only, and not to the mutual inductance be- not affect the individual inductance of the that flux density is not the same as total flux tween A and B. All the flux outside loop fb wires or their mutual inductance, because — when flux density is integrated (summed) does link with wire B, however, and contrib- inductance is a property that exists because over an area, it results in the total flux passing utes to both the self-inductance of A and the of conductor geometry and the permeability through that area. mutual inductance. Note that the mutual in- of the surrounding medium only, regardless Refer to Figure 2 again. Since the wire ductance gets much larger as the wires be- of any current or what the overall magnetic and air have permeability of 1, this is also a come very close, because of the 1/r increase field pattern looks like. That explains how picture of the flux density field, B, because in flux magnitude. The amount of mutual ferrite-loading a coaxial cable can affect its of the current in the wire. The total flux pro- inductance approaches the self-inductance as inductance, despite the fact that its external duced is proportional to the integral of this B the wires approach perfect coupling. magnetic field is nearly zero (see below). field, which is the area underneath the line Flux from more than one source present in A final note on inductance: For a particu- in Figure 2 (between the line and the hori- the same region of space adds in a vector fash- lar wire carrying a particular current, a spe- zontal axis). The total flux inside the wire is ion at every point. This applies to magnetic or cific total H field is produced. If the perme- constant for low frequencies, since the area electric fields. The resulting field is still a vec- ability of the field path is increased, the in- of the triangle from 0 to any R is 0.5 . R . tor field, and at every point the contributions ductance of the wire is increased. This is be- 1/R = 0.5. The external total flux increases from the individual fields are added vectorially. cause the B field, and therefore the total flux dramatically with shrinking R, however, since This may change the direction and magnitude produced by the current, increases. This is each reduction of ΔR results in increased of the resulting field considerably. what happens when the wire is passed external flux of approximately ΔR/R (the area Imagine now that the same magnitude of through a ferrite block, for example. This also under the added slice when R changes to R – current is present in both wires of Figure 3, distorts the H field, because the ferrite forms ΔR). So, the total external flux produced by with current coming out of the page in one and a low-reluctance path for the flux, so the field a wire at a given current increases greatly as into the page in the other. This is simply a dif- concentrates inside the ferrite since it is an the wire size is reduced. ferential pair. The H field for one wire will be easier path to follow. Inductance is a measure of the magnetic clockwise and counterclockwise for the other. flux linking a conductor relative to the amount Now when these wires are brought close to- Transformer Operation of current producing the flux. Flux “links” a gether, the H fields add such that an intense A transformer is an assembly of inductors wire by circling it, like a link of a chain. Re- line of H field is produced perpendicular to (windings) that share flux — often nearly all member that flux always forms closed loops, the line joining the centers of the wires, right of their flux.

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