# The March/April Morseman Problem

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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 equal-spaced 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 centre-tapped 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 off-centre? How will the mutual inductance change if this wire is off-centre? 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 cross-sectional 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) Re-arranging, the magnetic field magnitude at radius r is B(r) = µi (13) r The flux passing through a small cross-section 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 field-lines shown next were produced using a numerical field-calculation program (see the Appendix) because analytic calculation for the right-hand 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 field-lines resulting from an off-centre central wire. The field-lines 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 field-line 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 field-lines induced in the toroid by Left: A central wire, Right: An off-centre wire. The diffraction boundary conditions for field-lines travelling between magnetic media are described in many references, for example The right-hand plot of figure 4 also shows that the field-lines 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 evenly-spaced 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 closely-spaced lines in the toroid near the wire might be exactly balanced by the smaller inductance induced by the more widely-spaced 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

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