When an interference source (output of module A, in the diagram) shares a ground connection with a victim (input of module B) then any current due to A’s output flowing through the common impedance section X-X develops a voltage in series with B’s input. The common impedance need be no more than a length of wire. Although it's easy to see that the wire resistance will create the impedance at DC, in fact the inductive part of its impedance will increase and become dominant with increasing frequency (see below), so high frequency or high di/dt components in the output will couple more efficiently.

The effect of this is that module A's output interferes (V_int) with the wanted (V_IN) input of module B; if the two signals are in the same frequency range they cannot be separated by filtering.

The solution is to separate the connections so that there is no common current path, and hence no common impedance, between the two circuits. This applies to any circuit which may include a common impedance, such as power rail connections. Grounds are the most usual source of common impedance because the ground connection is often assumed rather than explicitly shown on a schematic.

Wire impedance

The partial inductance L_p of a straight length of wire with a length approximately four times its separation from a ground plane return path is

L_p           =           0.002 · l · (2.3 log (4l/d) - 1) microHenries,           (1)

where l and d are length and diameter in cm.

Alternatively, for a track of width w and height h above a ground plane return path,

L_p          =          0.005 · ln (2π · h/w) microHenries per inch           (2)

These formulae, although different, provide approximately the same result. A useful rough rule of thumb for typical equipment wire of a few cm or more length, is either 10nH/cm or 25nH/inch. The wire resistance can be derived from its cross sectional area (CSA) and resistivity; for copper wire with a CSA in mm²,

R          =          0.017/CSA ohms/metre (3)

The impedance versus frequency of a 10cm length of wire of different diameters or PCB track of different widths (assumed to be 35µm thick) is graphed here: notice that the inductive impedance dominates whatever the width of track or diameter of wire, above a few kHz, and at lower frequencies for thicker wire. You can use the method below to calculate actual impedance of a given geometry from equation (1). Be aware that these formulae assume that the wire length is much greater than its diameter, and that it is "electrically short", i.e. significantly shorter than a quarter wavelength, so that transmission line effects can be neglected.

Circuit loop inductance

The above discussion refers to the partial inductance of a length of conductor, treated as separate from its return path. Strictly speaking, this is incorrect; the inductance should be calculated over the whole circuit loop, including both the go and return paths.

Engineers are well-acquainted with the issue of a small wire loop acting as a susceptible noise pick-up loop. As depicted in the image below, two oscilloscope probes have been positioned in close proximity to a noise source. Both probes feature ground leads connected to the measurement tip, but they differ in one crucial aspect - length of the ground lead. As a result, we observe a substantial disparity in the level of noise picked up by the larger loop (probe 1, channel 1). This set-up serves to illustrate two critical points: firstly, larger loops can readily couple electromagnetic interference (EMI) noise, and secondly, when conducting probe measurements, it is imperative to exercise extra caution to prevent measurement errors.

Of course, for the general case this requires an analysis of the geometry of the entire circuit, which may be complicated. But for the loop inductance of a long pair of wires (as in a two-wire or multicore cable) there is a simple calculation, as follows:

L          =          0.4 · μ_r · cosh^-1(s/d) μH per metre           (4)

          =          0.4 · μ_r · ln[(s/d)+ √((s/d)² -1)]

where s is the centre-to-centre separation distance of the wire pair and d is the diameter of each wire, s >> d, and μ_r is the relative permeability of any magnetic material in the path (= 1 if there is none).

Try your own calculation for wire impedance, based on equation (1) and (3):

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