Transmission line theory and behaviour holds a great deal of importance for EMC. It is relevant for the design of high frequency PCBs and it also explains a great deal of the effects seen when cables couple to their environment. Matching the transmission line formed by any cable will reduce its interaction with incoming and outgoing radiated fields.

Construction of a transmission line

Any pair (or more) of conductors can form a transmission line. Generally, intentional transmission lines are created from conductors that have a constant cross-section along their length, which makes for constant per-unit-length electrical parameters. These parameters are series inductance (L), parallel capacitance (C), series resistance (R) and parallel conductance (G). The line can be analysed as a series of elemental lengths each of length dL where dL is electrically small, i.e. of dimensions much less than a wavelength,  with lumped parameters LdL, CdL, RdL and GdL. Calculations are simplified if R and G are neglected, which is a reasonable assumption for short lines and low-to-mid frequencies, although for a full analysis they must be considered.

The per-unit-length inductance and capacitance depend on the conductor dimensions and the separation distance between them. Therefore the geometry of a transmission line has a direct bearing on its electrical parameters. The field surrounding the conductors has a transverse electromagnetic (TEM) structure, that is the field vectors are transverse or perpendicular to the axis of the line, along which the wave propagates. This is similar to the plane wave in free space, except that the fields are “anchored” to the conductors and are not independent of their position in the plane perpendicular to the direction of travel.

The per-unit length parameters give rise to a characteristic impedance Z₀ of the transmission line. This impedance is a function only of cross sectional geometry and not of length. A high capacitance - close proximity of large conductors - will give a low impedance, and vice versa a low capacitance will give a higher impedance. The calculation below will give you the impedance for a pair of parallel wires. An insulating dielectric increases ε_r and so reduces the impedance proportional to √ε_r.

Z₀ = 120·√(μ_r/ε_r)·ln(D/d + √((D/d)² - 1))          (1)

where D is the separation distance and d is the wire diameter

For a coaxial geometry the equation is

Z₀ = 60·√(μ_r/ε_r)·ln(D/d)                              (2)

where D is the outer screen inside diameter and d is the inner wire diameter

Losses from series resistance in the conductors increase with √F due to skin effect and this limits the usability of practical transmission lines as frequencies climb above 1GHz. μ_r is the relative permeability of any magnetic material in the line, which is not usually present, in which case μ_r = 1.

Because Z₀ is logarithmic with separation distance, it does not become very high when a wire is considered in near isolation, i.e. widely separated from its return conductor; the common mode Z₀ of a cable with respect to its environment can be shown to be around 150-200Ω, and this value is widely used in stabilising networks for various RF EMC tests.

Behaviour

The total voltage and current at any particular point z along the length of the line is composed of the sum of a forward-travelling wave and a backward-travelling wave. The ratio of this voltage and current (the input impedance of the line) can be defined in terms of a reflection coefficient which depends on the line load:

Z_in (z)  =  Z₀ · (1 + Γ(z))/(1 - Γ(z))           (3)                    

Γ(z) =  Γ_load · e^{j2β(z - L)}  (4)

where L is the position of the load (the far end of the transmission line), β = λ/2π (the phase constant), and the voltage reflection coefficient Γ(z) is the ratio of the voltages of the forward and reverse travelling waves.

The reflection coefficient at the load is given by

Γ_load  =  (Z_L - Z₀)/(Z_L + Z₀)          (5)

so that if Z_L = Z₀ the line is said to be matched, no reflection occurs, and the impedance, voltage and current along the length of the line are constant. If Z_L ≠ Z₀ then the line is mismatched and a standing wave of both voltage and current exists along its length. The impedance looking into the start of the line varies periodically with the ratio of L to λ. When the line electrical length^† is a quarter wavelength, the impedance at one end is transformed into its opposite at the other, through the line Z₀ - hence the description of this effect as a quarter-wave transformer. At a half wavelength, the impedance returns to the same at each end. Thus as the frequency varies for a given length of line, the impedance swings between high and low at one end if the other is mismatched. Only if the line is matched with Z_L = Z₀ will the input impedance remain constant with frequency.

This has consequences for the current injected into the line from a given source impedance, and hence for the radiating efficiency versus frequency of a line of a given length and termination. This is why radiated emissions and immunity of products are so strongly affected by cable length, termination and layout. Any cable will demonstrate this effect, though the cross-section against the external environment and hence the common-mode impedance of real cables acting as transmission lines is not constant; so the impedance-frequency characteristic is less pronounced, but still will cause significant variations in coupling across the spectrum.

^† Electrical length is the physical length of the line multiplied by the square root of the relative dielectric constant ε_r of the insulating material between the conductors (assuming no magnetic permeability within the line). Many dielectrics have an ε_r of around 2 – 2.5, so a multiplier of about 1.5 is typical.

The following videos demonstrate the concept of 1/4 and 1/2 wavelength antennas and cable resonance (one in the frequency domain, the other in the time domain). Georgia Tech has this excellent on-line animation page where you can watch switched DC waveforms and pulses propagate on the transmission lines. This animation provides insights from a time domain perspective, complementing the discussions we've had earlier, which primarily centred around frequency domain analysis. By examining the same phenomenon from these two different angles, readers can gain a deeper understanding of the topic.

Try your own calculation for the Z₀ of a pair of parallel wires, based on equation (1):

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