Q3Power System - I

(a) Bewley Lattice Diagram for the Given Open-Circuited Line

A Bewley lattice diagram is a space-time diagram used to track the successive reflections of a traveling voltage (or current) surge as it travels back and forth along a transmission line between two points of impedance discontinuity (here, the sending end and the open-circuited receiving end), with distance plotted horizontally and time plotted vertically (increasing downward), and each diagonal line representing the wave traveling at velocity v = 1/√(LC) in one direction, taking time τ = l/v to traverse the full line length l.

Given data: R = 0.5 Ω/km, G = 10×10⁻⁷ S/km, l = 400 km, and Vs(initial) = 2 p.u. To determine the reflection coefficients, we need the surge/characteristic impedance Zc = √(Z/Y), where for a line dominated by its L and C (R and G here being given for a related loss/attenuation calculation, but the reflection coefficient at the open end depends fundamentally on line termination, not on R, G values themselves).

Reflection coefficient at the receiving (open) end: for an open-circuited termination, the receiving-end impedance Zr = ∞, so the voltage reflection coefficient is:

This means the incident voltage wave is reflected at the open end with the same polarity and full magnitude (ρr = +1 for voltage), while the corresponding current reflection coefficient is -1 (current wave is reflected with reversed polarity, since the terminal current must be zero at an open circuit at all times).

Reflection coefficient at the sending end: this depends on the source/sending-end impedance Zs; if the sending end is represented as an ideal voltage source (Zs = 0, as is a common simplifying assumption when a surge is applied by a source of negligible internal impedance), then:

Lattice diagram construction: starting with the initial incident wave of 2 p.u. launched from the sending end at t=0, it travels down the line, arriving at the open receiving end at time τ = l/v; there it reflects with ρr=+1, so a further +2 p.u. wave (doubling the voltage momentarily at the open end to 4 p.u., since incident+reflected = 2+2 = 4 p.u. at that instant) travels back toward the sending end, arriving at t=2τ; at the sending end it reflects with ρs=-1, producing a -2 p.u. wave heading back toward the receiving end, arriving at t=3τ, where it again reflects with ρr=+1 giving -2 p.u. traveling back, and so on — each round trip alternates the sign of the wave (due to ρs=-1) while the magnitude at the open end doubles the currently arriving wave value at each reflection instant. The lattice diagram is drawn as a zigzag pattern of diagonal lines between the sending-end and receiving-end vertical axes, with the wave magnitude (obtained by multiplying the previous incident wave by the appropriate reflection coefficient) labeled alongside each diagonal segment, and the actual voltage at any point and time obtained by summing all wave components that have already arrived at that point by that time.

(b) Bewley Lattice Diagram for a Bifurcated Line

A bifurcated line is one that splits into two separate line sections of (generally) different surge impedance (Zc2 and Zc3) at some junction point along the original line (Zc1). When the traveling wave, incident on this junction from the main line, reaches the bifurcation point, it partially reflects back along the incoming line and partially refracts (transmits) forward into each of the two branch lines, in general with different transmitted wave magnitudes into each branch since they may have different surge impedances.

At the junction, treating the two branch lines as impedances in parallel (as seen from the incoming line), the equivalent impedance is Zeq = (Zc2·Zc3)/(Zc2+Zc3). Using standard junction formulas (derived from continuity of voltage and current at the junction point), the reflection coefficient of the wave back into the incoming line 1 is:

and the voltage transmission (refraction) coefficient into each of the two branch lines is identical (since the junction voltage is a single common value seen by both branches at that instant):

This transmitted voltage wave of magnitude τ·(incident wave) then propagates independently down each branch line at that branch's own velocity of propagation, while the current that flows into each branch splits in inverse proportion to that branch's own surge impedance (since the same junction voltage divided by a smaller Zc gives a larger current into that branch) — physically, more current flows into the lower-impedance branch, analogous to current division between two parallel resistors.

The lattice diagram for the bifurcated line therefore needs three (or more) vertical reference lines: one for the sending end, one for the junction point, and one for the remote termination of each branch. The incident wave first travels from the sending end to the junction, exactly as in the simple two-terminal case; on arrival at the junction it splits into a reflected component (ρ1, heading back to the sending end) and two transmitted components (τ, one into each branch, heading toward each branch's own remote end). Each transmitted wave then reflects at its own branch's termination according to that termination's own reflection coefficient (open end: +1, short-circuited end: -1, matched/terminated end: 0) and travels back toward the junction; upon return to the junction, this returning wave itself undergoes a further reflection (back into its own branch) and refraction (a portion transmitted into the incoming line 1 and into the other branch), continuing the zigzag lattice pattern independently within each branch section while being coupled to the other sections only through the junction's reflection/refraction coefficients whenever a wave arrives there from any of the three connected line sections. In this way, the total voltage or current at any point and time on any section is obtained, exactly as in the simple case, by summing all the individual wave components (correctly scaled by the cumulative product of reflection/refraction coefficients encountered en route) that have arrived at that point by that time — the bifurcated-line lattice diagram is thus a direct, systematic extension of the two-terminal lattice diagram to a three-way (or higher) branching network of transmission lines.

Practical relevance: bifurcated-line lattice analysis is important in real substations and switching stations, where an incoming overhead transmission line frequently splits into two or more outgoing feeders or cable sections of different surge impedance at a busbar or tee-off point; correctly predicting how an incoming lightning or switching surge divides and reflects at such a junction is essential for correctly rating the insulation (BIL) of the equipment on each branch, and for placing surge arresters at the optimal locations to protect the most sensitive apparatus (transformers, GIS equipment) from excessive transient overvoltage resulting from surges arriving via the overhead system.