When two circuits are in close proximity to each other, coupling between them is described as "near field" or "induction field" coupling. It can be separated into its magnetic and electric components.

Magnetic field coupling

A current flowing through a long wire creates a magnetic field around that wire. The field amplitude is described by the Biot-Savart law:

δH          =          I · (δs · sin Θ)/r²

where δH is the field strength in A/m from a current element of length δs at a distance r from the element carrying a current I and making an angle Θ with it.

This is the field in free space from a single wire. Since circuits in general require that current flows in a loop, the proximity of the return conductor to this wire will be an important factor in reality, which is neglected in the equation above. The return current (equal in amplitude but opposite in direction) will reduce the actual measured field unless the return conductor is infinitely far away.

If a current-carrying source wire is routed near to a victim wire, the mutual inductance M between them creates a voltage in the victim circuit just as it does from the common impedance. V₁₂ is induced in the second circuit by the action of I₁ and M, in the same way that a voltage is induced across the partial self-inductance of a length of wire. As with that process, if I₁ is sinusoidal with frequency F (Hz) then

V₁₂          =          -j · 2F · M · I₁          (1)

Alternatively,

 V           =           -M · dI₁/dt          (2)

M depends on the areas of the source and victim current loops, their orientation and separation, and the presence of any magnetic screening (note that this is not the same as electric field screening). M is difficult to calculate in general, but some common situations can be described, such as the case of two long wires above an infinite ground plane which carries both return currents, as shown:

M          =          10⁶· (µ₀/4π) · ln(1 + 4h₁·h₂/s²)  µH/m          (3)

or a flat cable configuration of three wires with the centre wire as the return,

M          =           10⁶· (µ₀/2π) · ln(s/d)  µH/m          (4)

For s, h, d << length; units unimportant. The factor µ₀/2π is equivalent to 2·10^-7. Short lengths of cable loomed together will have values of mutual inductance in the range of from 0.1 to 3μH; you can calculate the value of M using the table at the end of this page.

To reduce the coupling you can create magnetic screening by including a magnetically permeable material such as mu-metal between the conductors, but a simple conductive screen such as copper or aluminium, which has no magnetic properties, will have little effect on the magnetic flux. In fact, a small attenuation of at most a few dB can be created, which is due to induced eddy currents in the screen partially cancelling the source magnetic field.

Mutual inductance between two separate circuits leads to (usually unintentional) coupling between them and so is undesirable. But mutual inductance between two conductors which are part of the same circuit is normally highly desirable, and is the basis of many EMC mitigation techniques, as we shall see.

Electric field coupling

Changing voltage on one conductor creates an electric field which may couple with a nearby conductor and induce a voltage on it. The voltage induced on the victim conductor in this manner is

V           =           C_C · dV_L/dt · Z_IN,           (5)

or

V          =          V_S · Z_IN/(1/(2·π·F·C_C) + Z_IN) (6)

where C_C is the coupling capacitance and Z_IN is the impedance (to ground) of the victim circuit.

The value of C_C is a function of the distance between the conductors, their effective areas and the presence of any dielectric material. The capacitance per unit length between two conductors of equal diameter d spaced D apart in free space is given by

          C    =    0.0885 · π/{cosh^-1(D/d)}   pF/cm           (7)

and the capacitance between two plates of overlapping area A cm² spaced apart by d cm in free space is given by

          C    =    0.0885 · A/d   pF          (8)

The value 0.0885 in the above equations accounts for the permittivity of free space; in the presence of a dielectric, multiply the result by the dielectric constant ε_r (relative permittivity) of the material.

Typically, two parallel insulated wires 0.1” apart show a coupling capacitance of about 50pF per metre; the primary-to-secondary capacitance of an unscreened medium-power mains transformer is 100-1000pF. Note that stray capacitance between ground points will complete the coupling path even if the two circuits are not directly ground-referenced to each other.

Screening using a conductive material between the two circuits will potentially have a very beneficial effect, by rearranging the coupling capacitances, but requires a careful analysis of the screen's connection to the reference point. Too much inductance in the screen connection creates a resonance with the capacitances which actually enhances coupling.

You can see the effect of a screen as follows:

if the screen (green) is halfway between the two circuit nodes (red), then we can say approximately that C_C1 = C_C2 = 2 · C_C. If it is not connected to any reference point but is floating, then the net coupling capacitance is (2 · C_C/2), i.e. there is no change to the coupling at all;

if the screen is now connected to the common reference, and the connection inductance is neglected, then whatever the coupling from C_C1, there is no interference voltage on the screen and the coupling via C_C2 is unimportant; screening is perfectly effective;

but if we include the reference connection inductance L, then the C_C1-L-C_C2 circuit forms a high-pass resonant filter which will increase coupling at its resonant frequency, even though it will still act as a screen at lower frequencies. Above resonance, coupling will tend towards the level shown by no screen at all.

This electric field screening effect, while important, is by no means the only aspect of properly shielded enclosures, which are considered more fully elsewhere.

Notice that the above electric coupling analysis assumes a common (ground) reference. This isn't necessary for magnetic coupling. If the source and victims don't share a common reference, there will still be electric coupling; but the analysis then has to take into account the mutual coupling between the two reference nodes. In a distributed system this may be quite a complex network of impedances.

Summary

Electric-field coupling between conductors is due to a voltage source and can be modelled as a mutual capacitance.

Magnetic-field coupling between conductors is due to a current source and can be modelled as a mutual inductance.

Magnetic coupling is directly proportional to the area enclosed by the circuit. This can be understood as the return current has the effect of partially cancelling its signal current, and hence reducing the magnetic coupling effect. It is unaffected by the impedance of the victim circuit

Electric coupling is proportional to the area of overlap of the relevant circuit nodes, and the impedance of the victim circuit.

In practice, both electric and magnetic coupling will coexist in any real circuit, although one or the other may dominate. Mixed coupling at low frequencies can be evaluated by superposition, that is, summing the effects of mutual capacitance and mutual inductance, taking into account the phase of each path. At higher frequencies, the line is no longer electrically short and you have to use a more complete model that takes into account line length. In this case, the voltages induced in the victim circuit may differ from one end to the other, leading to the terms Near End Crosstalk (NEXT) and Far End Crosstalk (FEXT).

Try your own calculations for inductive coupling of a pair of wires over a common ground plane, based on equations (1) and (3):

or for capacitive coupling of a pair of wires, based on equations (6) and (7):

Click here for Questions