The March/April Morseman Problem


 Homer Cooper
 1 years ago
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1 The Problem as stated was: The March/April Morseman Problem A toroidal transformer is used in directional coupler versions of an SWR bridge. The primary winding is the single wire running through the toroid, the secondary is wound with equalspaced turns around the toroid. A diagram of the complete coupler, and an explanation of how it works, can be downloaded from my website 1 The figure shown in the column is figure 3, on the following page. The first diagram showing the coupler circuit from my website document is figure 1. Figure 1: The Bruene SWR detector, showing centretapped toroidal transformer. The mutual inductance between primary and secondary is found using Faraday s Law, with a straightforward integration. Questions: How can the primary wire shown be a winding? (A winding must be a closed loop, but this is just a single wire.) The wire is always shown threading through the exact centre of the toroid. Will it still work if it goes through offcentre? How will the mutual inductance change if this wire is offcentre? Before tackling these questions, we revise how mutual inductance is defined and calculated. Mutual Inductance Mutual inductance is a joint property of two closed conducting loops. Figure 2 shows two coils in proximity. Coil 1 carries a current, giving rise to the magnetic field shown by the field lines. Some of this field passes through coil 2. If the current in the first coil changes, the flux intercepted by the second coil will also change, and by Faraday s Law, a voltage will also be induced in its windings. This flux coupling is utilised in all transformers. The degree of coupling is a function of the geometry of the system, and is specified by their Mutual Inductance. Let the total flux passing through the windings of coil 1, having inductance L 1, be Φ 1. Let the total flux passing through coil 2 be Φ 2, and the voltage induced in its winding be V 2. Using Faraday s Law, Inductance is defined as L = dφ 1 = dφ 1. (1) 1 Download from the link on my website to The Bruene Directional Coupler and Transmission Lines. 1
2 Figure 2: Two coils in proximity coupled by mutual inductance. rearranging, dφ 1 = L 1 and V 2 = dφ 2 The negative signs imply that the induced voltage in each case opposes the flux increase. Let the proportion of flux intercepted by coil 2 be α. Then (2) (3) Φ 2 = αφ 1 (4) dφ 2 = α dφ 1 = +αl 1 = M (6) where M = αl 1 = the mutual inductance (7) (5) then M = dφ 2 (8) or, for a linear medium, M = Φ 2 I 1 (9) and the voltage induced across L 2 is V 2 = M Mutual Inductance of a Toroidal Coil Threaded by a Wire. In general mutual inductance is difficult to calculate. Only a few configurations, including that of the central wire toroidal transformer described here, are analytically tractable. Figure 3 shows such a toroid around which a coil (windings not shown) is wound. The toroid is threaded by a wire passing normally through its exact centre, part of a closed loop through which current flows. When an rf current flows in the wire, mutual inductance between the wire and the toroidal winding cause an induced voltage to form across the toroidal winding. The toroid has a rectangular crosssectional dimensions h and W, and inner radius R. For computational simplicity, the wire is assumed to be infinitely long, so that its magnetic field inside the toroid can be calculated using Ampere s Law. This law states that B.dl = µi (11) l In words, this says that the magnetic field component B.dl, integrated over any complete closed path l around a current I flowing anywhere through this path, is proportional to the current, with constant of proportionality µ, the permeability of the medium. (10) 2
3 Figure 3: An infinitely long straight wire passing through the centre of a toroid. Assume that a current I flows in the straight wire. The magnetic field lines due to the wire will be symmetric and circular, centred on the wire. Those between radii of R and R + W will pass through the toroid. We can calculate the flux through the toroid using this integration, evaluating the magnetic field using Ampere s Law. At radius r, the path length is r, and B is constant along it, so from Ampere s Law, rb(r) = µi (12) Rearranging, the magnetic field magnitude at radius r is B(r) = µi (13) r The flux passing through a small crosssection of the toroid from radius r to r + dr will be dφ 2 = µinh r integrating, Φ 2 = µinh Φ 2 = µinh but from equation 10, M = Φ 2 I so M = µnh dr (14) R+W dr (15) R r ( log e 1 + W ) (16) R log e ( 1 + W ) R Both plots of fieldlines shown next were produced using a numerical fieldcalculation program (see the Appendix) because analytic calculation for the righthand case is analytically intractible. Figure 4, left, shows the magnetic field lines surrounding a centrally placed wire through the toroid having a relative permeability of 20 (that is, µ = 20µ o. This value is rather low for a toroid used as a balun or SWR bridge detector, but this low value shows the lines better). The current in the wire is flowing into the page. In this case, all field lines are circular, and are either completely inside or outside the toroid. Those inside are closer together than those outside, indicating that the field strength is higher inside the toroid. The strength of the field is the same at all points in the toroid. Figure 4, right, shows the fieldlines resulting from an offcentre central wire. The fieldlines from the wire are circular at low radii, but as the radius increases some of them cut the surface of the toroid, where the permeability abruptly increases by a factor of 20. On entering the toroid, they are bent away from the normal, on leaving, towards the normal. Each fieldline is still continuous. The boundary conditions between the media, used to find the angle of diffraction, are derived from the two Maxwell equations (17) (18). B = 0 (19) B = J (20) 3
4 In words, the first one states that the divergence of the magnetic field is zero, the second states that the curl of the magnetic field equals the current density at the point where the curl is calculated. The macroscopic form of this law is Ampere s Law, equation 11 above. Figure 4: Magnetic fieldlines induced in the toroid by Left: A central wire, Right: An offcentre wire. The diffraction boundary conditions for fieldlines travelling between magnetic media are described in many references, for example The righthand plot of figure 4 also shows that the fieldlines inside the toroid are spaced more closely in the section nearest the wire, than those in the section directly opposite. This indicates that the field inside the toroid closest to the wire is higher than that directly opposite. In this figure, the difference in magnitude is about a factor of 2. Furthermore, the changing spacing of the field lines shows that magnetic intensity varies continuously around, and inside the toroid. The Effect of Wire Position on the Toroid s Inductance. If an evenlyspaced coil is wound around the toroid, its inductance turns out to be exactly the same for both situations. Physically, you can see this is reasonable because the larger inductance induced by the closelyspaced lines in the toroid near the wire might be exactly balanced by the smaller inductance induced by the more widelyspaced lines further away. And in fact, it is. Mathematically, this is a consequence of Ampere s law, stated again as B.dl = µi (21) l This, paraphrased, states that the total magnetic intensity integrated around any closed path is always proportional to the current flowing through the path. It does not matter where inside the loop the current passes! If the coil is wound uniformly around the whole toroid, this integrated intensity is therefore the same for both cases shown in figure 4, and since this total intensity is what determines the inductance, the inductance will also be the same! 4
5 Note, however, that the integration is not along a field line inside the toroid, since some lines enter and exit the toroid, and do not follow radial paths inside it. The integration is along a circular (radial) path, using the component of the field line resolved along this path. It s the constantlychanging direction of this component which makes an analytic solution for this case so intractable. But there s an interesting wrinkle to this, which affects the operation of the Bruene SWR bridge, shown in figure 1. The Bruene SWR Bridge A modified version of this toroidal transformer is the basis of the SWR measuring technique used in this Bridge, shown in figure 1. Here, the winding is also wound completely around the toroid, but is centretapped, and one half is used to induce a voltage proportional to the forward current, the other half the reverse current Assume that the centretap is positioned halfway between the maximum and minimum field intensity positions, that is, at an angle of 90 o to the line joining the wire in figure 4, right, to the closest point of the toroid (at about the 2 oclock position.) The lower coilhalf (nearest the wire) will then have a higher mutual inductance with the wire than the upper coilhalf (furthest from the wire), because the integrated field intensity through the lower portion will be higher than that through the upper. Hence, even if the detection components (diode rectifier and smoothing) are identical in the forward and backward circuits, the voltages induced in the two halves will always be unequal. In practice, this effect is small, and does not matter anyway, because adjustable resistances shown as R cal in figure 1 are always included to compensate for this effect  and for any difference in the diodes, capacitors and voltmeters used in the two sections. Note that figure 1 also shows that one of the capacitors used in the voltage divider section, at left, is variable. This is adjusted at the factory to make the reverse voltage zero when a pure resistance equal to the characteristic impedance of the line to be measured is used to terminate the bridge. You can also adjust this capacitor if you suspect the calibration, though most instructional pamphlets don t mention this  mine, for my MFJ bridges, don t, anyway. Appendix The plots of figure 1 were made using the program Vizimag, downloadable from You have to pay $39.75 (US) for a singleuser license, but you can use it free for 30 days. The program has an easilymastered graphical interface, and enables you to calculate, plot, and read magnetic field magnitudes and directions at any point for a variety of configurations involving wires, toroids, magnets, magnetic materials. You can also make animations! Gary ZL1AN 5
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