This section shows how PUL parameter functions are obtained using measured data across a given frequency range.^{9, 21} It shows that lumped circuit (electrically short) characteristics are used, and transmission line effects (electrically long) are ignored. This yields: *L = L(f)*, *R = R(f)*, *C = C(f)*, and *G = G(f)*.

**CFR MODELING**

This section shows how CFR is calculated as a voltage transfer function using transmission line equations.^{9} This calculated CFR can then be compared to a measured voltage transfer function of the cable. If the comparison is a match, it implies that: 1) the PUL parameters of the cable were correctly measured and calculated and 2) the analytical expression for the CFR is correct and accurately calculated. Once it is confirmed that the analytical CFR expression is correct, *G(f)* can be set to zero to determine its influence on the cable CFR.

The CFR in terms of transmission line theory is equal to the output voltage divided by the input voltage, where *x =* ℒ at the end of the line (output) and where *x* = 0 at the beginning of the line (input).^{9, 21, 22}

Equation (7) computes the voltage at any point on the line.^{9, 23}

Where x is the distance along the line of length ℒ. The source voltage is V_{S}, Z_{S} is the source impedance and Z_{0} is the characteristic impedance of the cable. Z_{0} is a function of the PUL parameters, which in turn depends on the frequency where:^{24}

In Equation (7) there are two reflection coefficients. One for the source (Γ_{S}) and one is for the load side (Γ_{L}). With Z_{L} is the load at the end of the cable. The reflection coefficients are:

Equation (7) also contains the complex and frequency dependent propagation constant γ, where:

Equation (11) expresses the input voltage at the sending end of the transmission line by substituting *x* = 0 in Equation (7), which yields:

Equation (12) expresses the output voltage at the receiving end of the transmission line by substituting *x* = ℒ in Equation (7).

Therefore, the division of Equation (12) by Equation (11) describes the CFR as:^{9}

Note that in Equation (13), the source impedance and source voltage are cancelled. The CFR equation relies only on the load reflection coefficient *(Γ _{L}*), the length of the cable (ℒ) and the propagation constant (

*γ*).

**CFR RESULTS**

In this section, CFR’s are given for the two cables measured. These are:

• The CFR measured with a spectrum analyzer and tracking generator. This is used as a reference for the analytical equations.

• The CFR calculated by using the measured *L = L(f), R = R(f), C = C(f)* and *G = G(f)*. This shows a good correlation with the measured CFR and shows that the analytical CFR is probably correct.

• An analytically determined CFR with *G = G(f) = 0*. This shows the CFR that results if the PUL conductance is omitted from the model.

**Figure 11** shows the CFR for 10 m of the 0.8 mm^{2} twisted pair cable. Up to 1 MHz, the cable is electrically short with no transmission line effects. After 1 MHz, transmission line effects are seen as the CFR “oscillates” in relative amplitude response. It is not entirely clear why the analytical CFR’s are not a closer match with the measured CFR. There is a maximum of 4 dB discrepancy between 1 MHz and 10 MHz. It should, however, be remembered that *R(f), L(f), G(f) and C(f)* are at best estimates of the real physical functions and that the CFR is sensitive to these functions. The results are assumed to be close enough to deem the analytical CFR calculation correct.

Assuming the analytical CFR that includes *R(f), L(f), G(f) and C(f)*, is correct, and a comparison can be made between the CFR with *G(f)* and with *G(f) = 0*. It can be clearly seen that the bandwidth prediction with* G(f) = 0* is larger than the real bandwidth. This causes an error in the channel capacity estimation as given in Equation (2).

**Figure 12** shows the CFR for a 10 m, 1.5 mm^{2} flat twin and earth cable. The results are like the 0.8^{ }mm^{2} twisted pair. There does, however, seem to be a closer correlation between the measured CFR and the calculated CFR, which includes *R(f), L(f), G(f) and C(f)*. The maximum discrepancy between the two CFR’s is 2 dB between 1 MHz and 10 MHz.

As with the 0.8 mm^{2} twisted pair, the bandwidth when *G(f) = 0* is larger than the true CFR, especially at frequencies up to 100 MHz. These results clearly show that conductance is a critical parameter in determining the CFR of a power cable, and it cannot be neglected.

**CONCLUSION**

The CFR is an important characteristic in the determination of data transfer speed on a PLC power cable. Cable characteristics up to 100 MHz are investigated. It is shown that CFR can be computed using classical transmission line theory. This computed CFR shows good agreement with the measured CFR if the correct PUL parameters (*R(f), L(f), G(f) and C(ff*)) are used.

Literature using classic transmission line theory, however, usually makes two assumptions: 1) PUL conductance *GG* is negligible and 2) PUL parameters are constant with frequency (i.e., they are frequency independent). Both assumptions are shown to be problematic and can lead to errors when determining the CFR of an indoor power cable.

This article establishes that the omission of the PUL conductance parameter leads to an erroneous CFR and leads to a larger computed CFR bandwidth as opposed to what is measured.

It is also established that to correctly calculate the CFR, the PUL parameters and their frequency dependence must be known. Short circuit and open circuit tests are performed on two indoor power cable types and their PUL parameters are extracted. Although *L(f)* and *C(f)* do not vary a lot with frequency, it is found that *R(f)* and *G(f)* are highly frequency dependent.

Only when all four of the PUL parameters are used, and their frequency dependence included, can correct CFR calculations for indoor power cables be made. It is shown that conductance is a critical parameter in determining the CFR of a power cable and cannot be neglected.

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