Modelling practical transmission lines

The RLGC transmission line model

Fig 1:

A transmission line can be represented as an infinite series of cascaded identical two port networks each representing an infinitely small section of the transmission line. The small networks represent:

the distributed resistance of the conductors is represented by a series resistance per unit length R;
the distributed inductance is represented by a series inductance per unit length L;
the capacitance between the conductors is represented by a shunt capacitance per unit length C; and
the conductance of the dielectric material separating the two conductors is represented by a conductance per unit length G.

This model is described in most transmission line text books, and online at Telegrapher's Equation.

Solution of the model gives two key equations that describe transmission line behaviour.

where γ is the complex propagation coefficient, and Z₀ is the complex characteristic impedance.

R,L,G, and C may be frequency dependent. In practical transmission lines at HF and above the following assumptions are often appropriately used:

inductance per unit length as constant (due partially to skin effect and a fully effective outer conductor));
capacitance per unit length as constant;
resistance per unit length is subject to skin effect and is proportional to the square root of frequency; and
conductance per unit length is due to dielectric loss and is proportional to frequency.

Because of Skin Effect, the effective resistance of a round conductor of homogenous material and more than about three Skin Depths in radius, is proportional to the square root of frequency. Conductor losses in that case are proportional to square root of frequency.

The reversal of electric field in a lossy dielectric expends a certain amount of energy, and when excited by an alternating voltage, the dielectric loss is usually proportional to the number of times that the field reverses in a given time period. Dielectric Loss in that case is proportional to frequency.

Skin Effect is well developed where the conductor radius more than about three skin depths.

Solving the equation for γ reveals that conductor loss is approximately proportional to R where R<<ωL, and dielectric loss is approximately proportional to G where G<<ωC. For most purposes, error is usually acceptable where R<ωL/3 and G<ωC/3.

Fig 2: Loss characteristics of example coaxial lines

Fig 2 shows the conductor and dielectric component losses for two popular coaxial cables of similar overall diameter. Both cables have a nominal Z₀ of 50Ω, RG213 uses a solid polyethylene dielectric and LMR400 uses a foamed polyethylene dielectric which has lower dielectric loss but more importantly a larger diameter centre conductor which reduces conductor loss.

As a result of the HF characteristics of the conductor and dielectric materials described above, Matched Line Loss at HF and above is very well approximated by the model MatchedLineLoss(dB)=k₁*f^0.5+k₂*f.

Taking RG213 as an example, MatchedLineLoss(dB)=5.929e-6*f^0.5+8.234e-11*f. The model is limited in this case to frequencies above:

30kHz for well developed Skin Effect; and
17kHz so that loss is proportional to R (R<ωL/3).

The lower frequency limit for application of the loss model depends on the line characterisitics.

Materials exceptions

Note that not all materials are ideal in the sense discussed above, for example:

conductors that are not homogenous, or very small diameter conductors might not have well developed skin effect, and may not exhibit R∝f^0.5;
dielectric materials may exhibit a different loss factor at different frequencies; and
energy coupled through ineffecitive shield structures.

The model is very suited to common practical coaxial cables with copper conductors and solid or foamed Polythene dielectric used at HF and above. The vast majority of commonly used RF grade coaxial cables fall into this category. Note though that some types of cables may have shield structures that are not fully effective at higher or lower frequencies (open weave braid, and thin metalised plastic shields respectively).

The model may not predict behaviour of cables with copper clad steel, tinned, or silver plated steel centre conductors well at HF because the Skin Effect criteria are violated (cladding or plating thickness not greater than three skin depths). Cables with small diameter conductors, even though homogenous, may not satisfy the skin depth requirement at lower frequencies.

Developing a loss model from measured Matched Line Loss

Line specifications usually include Matched Line Loss measured at a range of frequencies. Measured transmission line loss for a line terminated in its nominal Z₀ is usually a good fit to the model MatchedLineLoss(dB)=k₁*f^0.5+k₂*f. Measured loss vs f can be used to find k1,k2 values for the loss model (MatchedLineLoss(dB)=k₁*f^0.5+k₂*f) such that the residual sum of squares of error (SSE) is minimised.

In the case of a simple tranmission loss test, strictly speaking, the line is not usually perfectly matched for, amongst other reasons, Z₀ is a function of frequency for most practical lines and is not usually exactly equal to |Z₀|+j0. This overestimates loss, though the error introduced is usually relatively small and its contribution is usually insignificant compared to other error sources (eg manufacturing tolerances, temperature variation etc), and the model remains a good predictor of line behaviour for most practical lines at HF and above.

It is important to identify measurements at lower frequencies that significantly depart from the loss model. These will have significant difference from the value predicted by the model, and exclusion of them from the model will reduce the residual SSE.

Note that many manufacters publish data obtained from the loss model rather than the raw measurement data, extremely low residual SSE is a good indicator.

Developing a RLGC model

The valid loss model along with nominal Z₀ (R_n) and velocity factor vf (both obtainable from manufacturer's data, or by measurement) can be used to determine R(f), L, G(f) and C.

Returning to the loss model (MatchedLineLoss(dB)=k₁*f^0.5+k₂*f) for a moment, if k₁>0 and k₂>0 (as they are for all practical materials) there is a finite frequency f'=(k₁/k₂)^2 at which conductor loss equals dielectric loss. At f',Z₀=(L/C)^0.5 (it is purely real, purely resistive, R_n), and R and G can be determined.

At f', R=2*R_n/20*ln(10)*k1*k1/k2, and R(f)=2*R_n/20*ln(10)*k1*f^0.5. This expression will give R∝f^0.5 even below the lower accuracy limit of the loss model.
Likewise, at f', G=2/R_n/20*ln(10)*k2*(k1/k2)^2, and G(f)=2/R_n/20*ln(10)*k2*f. This expression will give G∝f.
C=1/(C₀*vf*R_n)
L=1/(C₀*vf)*R_n

where C₀ is 299,792,458.

The RLGC values can be used to find γ and so predict loss at all frequencies where R∝f^0.5 and G∝f, even if outside the valid range of the loss model.

Solving the RLGC model

RF Transmission Line Loss Calculator uses a table of k1,k2 obtained from published or measured data at HF and above, and is a flexible tool for solution of the transmission line equations.

In some cases, data points at lower frequencies are excluded because they are a bad fit to the loss model. This has only occured with coaxial cables that use copper clad steel (CCS) or plated steel inner conductors. The effect is not limited to coaxial cables, the effect should be exhibited by two wire lines using CCS conductors at lower frequencies.

The calculated loss model reveals the underlying model for R∝(f^0.5); and G∝(f). These R and G values are used, along with nominal Z₀ and velocity factor to solve the Telegrapher's Equation to calculate Z₀, γ and then Matched Line Loss, and loss under mismatched conditions, etc.