In-Between Spaces

Inductance exists in the spaces between conductors.

Today I measured the inductance of four loops of wire. Each loop comprises the same length of insulated #10 AWG solid-copper wire (Figure 1). During testing, I probe the wires at their endpoints (bottom of figure), holding the wires vertically above the tester and well away from all other metal objects.

The leftmost loop, the round one, has a diameter of 10 in. It gives the largest inductance at 730 nH. Moving to the right, the inductance drops in each case until you reach the final loop, the twisted wire, at 190 nH.

I mention this simple experiment because I have all too often heard engineers say: “My via has an inductance of 1 nH,” or “My bypass capacitor has an inductance of 500 pH.” Those statements assume that you can ascribe discrete inductances to individual portions of a signal path.

That assumption is a good one when dealing with lumped-element components. According to Kirchhoff's laws for circuit analysis, the total inductance of two inductors in series should equal the sum of their independent inductances.

The correctness of Kirchhoff's analysis hinges upon a crucial precondition, namely that no significant electromagnetic fields inhabit the spaces between conductors (he is quite clear about this in the formulation of his laws). High-speed digital currents infuse the spaces between conductors with massive, fast-changing electromagnetic fields. These digital circuits do not meet Kirchhoff's precondition; therefore, Kirchhoff's laws are invalid in the high-speed domain.

In high-speed electronics, you must supplement Kirchhoff's laws with parasitic capacitance, due to electric fields, and parasitic inductance, due to magnetic fields.

Figure 2 illustrates the pattern of magnetic fields surrounding two wires. The wires carry equal and opposite currents, much like the hairpin structures in Figure 1. For a moment, imagine current I1 going out on one wire, changing direction at a hairpin turn, and returning as I2 on the other wire.

If you observe the system from a remote distance, the magnetic fields generated by I1 nearly cancel the equal-but-opposite magnetic fields generated by I2. The closer you bring the wires, the better the cancellation, and the smaller the overall magnetic-field energy.

The overall energy is important because inductance, L, represents nothing more and nothing less than the total magnetic-field energy, E, surrounding a current-carrying circuit. The precise relation between inductance and field energy is:

Philoshopical point

If the inductance of a circuit is related to its stored magnetic energy, and if the magnetic energy exists not inside the conductors but in the space between the conductors, then you might say that inductance itself is a property not of an individual conductor, but of the spaces between the conductors.

Relation to practical circuits

If the spacing between wires affects the pattern of magnetic-field cancellation, then that spacing affects the circuit inductance, as well. Ergo, you cannot ascribe inductance to just one part of a circuit without also specifying the shape and location of the complete signal-current path. Left unspecified, the remaining portions of the path could increase or decrease the inductance. All parts of a current path must be specified before you can determine the inductance of the whole.

For example, the inductance of a via depends on the location of nearby interplane connections. The inductance of a bypass capacitor depends on its proximity to the reference planes, and the layout of its mounting pads.