Discussion of Results

The effects of reflections due to branching elements are observable in the transfer character- istics in the form of notches spaced evenly in the frequency span. The first notch will appear when the phase shift between the direct and reflected waves is exactly one half wavelength, leading to a subtraction. This is especially clear when observing a single branch topology as shown in Figure 4-5(a). For a branch terminated with an open circuit, the first notch will appear at 𝑓0 =𝑣𝑝/4𝑙 [109], [96] and subsequent notches will occur at odd multiples of 𝑓₀ within the fixed bandwidth such that:

𝑓_𝑘= 𝑣𝑝

4𝑙(2𝑘+ 1) 𝑘= 1,2,3,· · ·, (4.27) where𝑣_𝑝 is the propagation speed and 𝑙 is the branch length. The first notch frequency in Figure 4-5(b) is 𝑓0 = 7.5 MHz as expected, and corresponds to a time period of 133 ns.

The subsequent notch satisfying the frequency span is expected at𝑓₁ = 22.5MHz according to (4.27) which is the case as shown in Figure 4-5(b). Beyond 𝑓₁ the notch frequencies lie outside the frequency span considered. Given the upper frequency of the bandwidth,

Tx Rx N1

l=5m

l=10m l=10m

(a) Single-branch

0 5 10 15 20 25 30

−18

−16

−14

−12

−10

−8

−6

−4

−2 0

Frequency (MHz)

H(f) (dB)

measured data PRC model lattice model

(b) Single-branch results

Figure 4-5: Measured frequency response magnitude Experiment 1: (a) Single-branch Topol- ogy and (b) results

30 MHz, notches generated by branches of length less than 1.25 m will exist outside the bandwidth considered. The resultant propagation speed in the cable, due to insulation was found through measurements to be𝑣_𝑝 = 1.5×10⁸ m/s. The first wave that traverses a direct path from𝑇𝑥to𝑅𝑥of 20 m, will thus appear after𝑡= 20/(1.5×10⁸) = 133ns. Considering the test topology of Figure 4-5(a), the second wave will appear after𝑡= 30/(1.5×10⁸) = 200 ns. The time period of the first notch (𝑡0= 133ns) should correspond to twice the difference

l=5m l=5m

l=10m l=10m l=10m

Tx Rx

N1 N2

(a) Two branches

0 5 10 15 20 25 30

−35

−30

−25

−20

−15

−10

−5 0

Frequency (MHz)

H(f) (dB)

measured data PRC model lattice model

(b) Two-branch results

Figure 4-6: Experiment 2: (a) Two-branch Topology and (b) Results

(full wavelength) between these two times2×(200−133) = 134ns. Subsequent time periods corresponding to subsequent notch frequencies can be obtained as follows:

𝑡_𝑘 = 𝑡₀

(2𝑘+ 1) 𝑘= 1,2,3,· · ·. (4.28) In Figure 4-5 through 4-7, the lattice model of three test topologies and comparison with measured data and the parallel resonant circuit (PRC) model of [109] are shown. Evidently, there is a good agreement between the model and measurements. These results validates the

l = 5m l = 5m l = 7m

l = 10m l = 10m l = 10m l = 8m

Tx Rx

N1 N2 N3

(a) Three branches

0 5 10 15 20 25 30

−50

−45

−40

−35

−30

−25

−20

−15

−10

−5 0

Frequency (MHz)

H(f) (dB)

measured data PRC model lattice model

(b) Three-branch results

Figure 4-7: Experiment 3: (a) Three-branch Topology and (b) Results

applicability of the lattice approach in powerline modeling. The structure of the topology of Figure 4-6 is such that the second branch has the same length as the first. This strategic choice of branch lengths has a profound result. According to (4.27), the two branches, having equal lengths, will result in notch occurrences at the same frequencies. However, it is noticed that the depth of the notches has increased. It is therefore valid to deduce that multiple branches of the same length will always result in constant notch distribution in the frequency bandwidth, with the notch depth dependent on the number of such branches. In Figure 4-7, a third branch is added having a different branch length of 7 m. Again, using (4.27) it is expected that the notches will appear at𝑓₀ = 5.36MHz,𝑓₁ = 16MHz, and𝑓₂ = 26.8MHz.

Table 4.2: Determination of First Notch Frequency and Inter-Notch Separation Branch Path (p) 𝜏𝑝 (ns) 𝜏2−𝜏1 (ns) F (MHz) 𝑓0 (MHz)

direct path 𝑙1;5/𝑐= 253

5𝑚 = ∆𝑡= 67 = _Δ𝑡¹ = 15 = ^𝐹₂⁵ = 7.5

second path 𝑙_2;5/𝑐= 320 direct path 𝑙1;7/𝑐= 253

7𝑚 = ∆𝑡= 93 = _Δ𝑡¹ = 10.75 = ^𝐹₂⁷ = 5.38

second path 𝑙2;7/𝑐= 347

From the measurements/simulations, 𝑓₀ ≈5.26 MHz,𝑓₁ ≈15.2 MHz, and𝑓₂ ≈25.6 MHz.

The time taken by the direct wave to traverse 38 m will be given by𝑡= 38/1.5×10⁸= 253.3 ns. The second wave will appear after 𝑡= 48/1.5×10⁸ = 320 ns after travelling along one of the 5m branches. Owing to the two branches having equal lengths and also being the shortest in the network, two signals will appear at the receiver at the same time (𝑡 = 320 ns) provided the network is homogeneous. This explains why branches of the same length results in increased notch depth. If it is desired to determine the time period of the first notch due to the 7 m branch, we can follow the same procedure as before. In fact, the impact of each branch can be studied independently. The notch positions of a given PLC network can be determined in the time domain by evaluating the channel impulse response.

By taking the inverse Fourier transform (IFT) of (4.22) it is possible to analyze the channel in time domain, defined as:

ℎ(𝑡, 𝜏) =

𝐾𝑝

∑︁

𝑝=1

𝛽𝑝𝛿(𝑡−𝜏𝑝)𝑒^𝑗𝜑𝑝, (4.29) where𝛽𝑝 is the amplitude gain, 𝜏𝑝 is the time of arrival (TOA) and 𝜑𝑝 is the phase of the 𝑝th arriving path. The characteristics of the direct path are such that it takes variables 𝛽1, 𝜏1 and 𝜑1. In that case, the receiver distance from the transmitter is 𝑙1 =𝜏1𝜈𝑝. With purely time domain characteristics the position of the first notch𝑓₀as well as the inter-notch spacing𝐹 can be predicted in the frequency domain. For the network of Figure 4-7 having two distinct types of branches, the results are shown in Table 4.2.

The notch locations play an important role in many deterministic approaches such as using resonant circuits or filter combinations. In the transfer function, all the notches due to the three branches are observable. Hence we can view the powerline channel transfer

characteristics as a superposition of several cascaded responses.