Why Do I Sometimes See Negative Mutual Inductance in 2D Extractor? J. Eric Bracken, Ph.D. Ansoft LLC Last revised: 3 June 2009 Introduction Sometimes 2D Extractor returns an inductance (L) matrix whose entries cause concern. Some users are surprised to see a negative off‐diagonal term in the inductance matrix (mutual inductance) because these entries are often all positive. Please be assured that these results are correct and not a cause for worry. This note attempts to explain why this is so. Discussion Negative mutual inductance is quite common in field solvers (like the 3D inductance solver in Q3D) that compute partial inductance. All that is required to generate a negative partial mutual inductance is to take a parallel set of conductors and reverse the current/voltage reference direction in one of the conductors, perhaps by swapping the source and sink terminals. Negative mutual inductance can also occur in field solvers like 2D Extractor that compute loop inductance, but the reasons for this may be less clear. To better understand how this can happen, consider the common coupled microstrip configuration shown below in Figure 1. A three‐conductor coupled microstrip geometry.. The thickness of the FR4_epoxy substrate is 50 microns. The copper signal conductors are 15 microns thick, 50 microns wide, and spaced 100 microns apart. The copper ground plane below is 15 microns thick and 1000 microns wide. Figure 1. A three‐conductor coupled microstrip geometry. When the loop inductance matrix is computed in 2D Extractor at 1 GHz, all of the entries of the inductance matrix are positive: Figure 2. Inductance matrix at 1 GHz. However, if we set up a frequency sweep analysis and solve down to a low frequency like 1 Hz, we see a significant negative mutual inductance value between conductor 1 and conductor 3: Figure 3. Inductance matrix at 1 Hz. If we plot the mutual inductance versus frequency, we see that it transitions from negative to positive at about 2.5 MHz: Mutual Inductance L13
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Figure 4. Plot of mutual inductance L13 versus frequency. The question is: why does this happen? To better understand the phenomenon, let’s reconsider the basic definitions of self and mutual inductance in terms of electromagnetic fields. When a current flows, it produces a magnetic field. This magnetic field has an associated energy density at each point in space. By integrating the energy density over all space, we can compute the total energy stored in the magnetic field. The total energy stored in an inductor can also be computed from circuit theory. By equating the circuit theory energy to the magnetic field energy, we obtain a formula for the self inductance of a current path in terms of the magnetic field ·
: where the integral is taken over all space, and we have assumed that the total current is 1 Amp. It is helpful to recast this expression using the vector potential and the current distribution associated with the magnetic field. It is possible to show that an equivalent formula for self inductance is ·
The advantage of this expression is that the integrand is only non‐zero in conducting regions, because the current density goes to zero elsewhere. In fact, we only need to look at the conductors in the current path of interest (signal line and return) to evaluate it. So the integral becomes ·
·
Here denotes the cross‐section of conductor , and denotes the cross‐section of the ground return conductor. It is worth noting that the current density within the signal conductor will be positive, while the current density in the ground conductor will be negative because it is flowing in the opposite direction. The expression for mutual inductance modification: is similar to the above formula for self inductance, with one ·
·
The integrand is now “mixed up”: it uses the vector potential from current path multiplied by the current density from path . The current density for signal path is only non‐zero over the cross‐
section of signal conductor (where it is positive) and the cross‐section of the ground return conductor (where it is negative.) · must be positive This formula shows that for a mutual inductance to be positive, the integrand over most of the cross‐sectional area of the signal and return conductors. If the sign of the current and the vector potential differ over a large part of these conducting regions, then the mutual inductance will be negative. Consider the plot of the vector potential at 1 GHz shown below: Figure 5. Plot of the vector potential at 1 GHz. The field is concentrated in the region surrounding the active line (conductor 1), and drops off rapidly as we move away from it. The physical reason for this is that the current on the ground plane is bunching up beneath conductor 1 in order to minimize the impedance of the loop, which is dominated by the loop’s self inductance at high frequency. Note also that the vector potential is positive everywhere, tending toward zero at the ground plane. Therefore, regardless of the position of the conductors (and their associated current) the first term in the mutual inductance equation will be positive, and the second term will be zero, resulting in a mutual inductance that is positive overall. Now let’s plot the vector potential at 1 Hz: Figure 6. Plot of the vector potential at 1 Hz. The plot looks quite different: magnetic fields now fully penetrate the conductors, and they are nonzero over a much larger region. The reason for the more dispersed magnetic field is the ground plane. It is carrying the return current for conductor 1. At low frequency, this current is free to spread out uniformly over the entire ground plane to minimize the self resistance of the current loop. We see from the plot that the vector potential becomes negative in the region around conductor 3. As usual, the current density in the signal conductor is positive, so this means that the first term of the mutual inductance equation will be negative for conductor 3. The reason for the negative vector potential is the ground plane: it’s carrying a negative current density, and it’s closer to conductor 3 than conductor 1 is, so it has a greater influence over it than the active signal line. In the ground plane the vector potential is generally non‐zero, being positive (with a large magnitude) near the signal line and negative (with a small magnitude) further away. Because the current density in the ground plane is uniform and negative, this means that the second integral in the mutual inductance equation will be negative too, so we get a negative mutual inductance here. A careful examination of the field lines in Figure 6 will reveal that the vector potential does not tend toward zero as we go further away from the excited signal line, but stays negative and actually increases slightly in magnitude. That leads to the question: what would happen to the mutual inductance if we had a larger ground plane? Would this be sufficient to change the sign of the ground plane integral and make the mutual inductance positive once again? It turns out that this is indeed true. We increased the width of the ground plane from 1000 microns to 3000 microns and re‐simulated it. The resulting vector potential is plotted below: Figure 7. Plot of the vector potential at 1 Hz using a much wider (3000 um) ground plane. The region of negative vector potential is pushed much further out from the signal lines. The resulting mutual inductance is now positive for all frequencies, as shown in the graph below: Mutual Inductance L13
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Figure 8. Mutual inductance L13 with a 3000 um ground plane. 1.00E+008
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Clearly the low‐frequency mutual inductance between the lines is strongly affected by the size of the ground plane used. To get some idea how important this affect is, we performed a parametric sweep of the ground plane width in 2D Extractor from widths of 1000 to 3000 microns in steps of 500 microns. Figure 9 below shows the results for the mutual inductance : Mutual Inductance L13 vs. Ground Plane Width
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Figure 9. Mutual inductance L13 vs. frequency for different ground plane widths. From the plot we see that the mutual inductance here is positive as long as the ground plane width is about 1500 microns or greater. Therefore it appears from this example that a reasonable guideline to follow to avoid negative mutual inductances is to make the ground plane at least 5 times wider than the maximum horizontal extent of the signal traces (350 microns in this example.) You might be tempted to try making it huge (perhaps 50 times wider than the signal line extent), but this would only waste computation time. The low‐frequency mutual inductance will always be a strong function of the ground plane size, because the ground plane controls the extent of the low‐frequency current loop. We should also point out that for frequencies of 10 MHz or higher, the ground plane size has little effect on the result. This is expected, because at high frequencies the fields concentrate strongly in the region surrounding the signal line to minimize the inductive impedance. The low‐frequency mutual inductance to the nearest neighbor line is also affected by the width of the ground plane, as shown in Figure 10 below. The variation is less than that for but is still significant. Even the self inductance has a significant dependence on the ground plane width (again only at low frequency). This is shown in Figure 11 below. Mutual Inductance L12 vs. Ground Plane Width
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Figure 10. Mutual inductance L12 vs. frequency for different ground plane widths. Self Inductance L11 vs. Ground Plane Width
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Figure 11. Self inductance L11 vs. frequency for different ground plane widths. 1E+008
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Summary Negative mutual (loop) inductances can and do occur in 2D Extractor. An analysis of the fundamental electromagnetic definition for mutual inductance shows that this is not a bug in the field solver, but a physically reasonable result. The negative mutual inductance is only observed at low frequencies, and only when a relatively narrow ground return conductor is present.