Bewley's Lattice Diagram - Theory & Advantages

What is Bewley's Lattice Diagram?

Consider a resistive load R_L connected to a generator G having resistance R_g through a transmission line of characteristic impedance Z_c as shown in the future below.

If a voltage or current wave is sent to the load by the generator then, the wave reflects back to the generator after reaching the load R_L. Again due to the presence of resistance, R_g at the generator side, the wave reflects back to the load. Hence, the wave suffers from repeated reflections and to monitor these reflections, Bewley's lattice diagram is drawn, which is also called the zig-zag diagram.

In the lattice diagram, two axes are provided, a horizontal axis representing the distance along with the system and a vertical axis. showing time. The passage of surges is represented by the lines who's slopes provide the time equal to the distance travelled.

The reflected and transmitted waves can be achieved at any point of change impedance by multiplying the magnitudes of incidence waves with their relevant refraction and reflection coefficients.

Lattice diagrams can also be drawn for current, only when the reflection coefficient of current is negative of the reflection coefficient of voltage. For the system shown in the above figure, where a generator unit with internal resistance R_g is switched on a line without attenuation having a characteristic or surge impedance Z_C, with load resistance R_L at its receiving end.

The reflection coefficient at the receiving end is given by,

The reflection coefficient at the sending end is given by,

Let T be the time interval of surge from one end of the line to the other. Now, as soon as the generator unit is switched ON, a step voltage surge of infinite length travels down the line towards the receiving end. This is represented by a line sloping (left to right) as shown in the figure below.

When the surge reaches the load end in time T seconds, a surge of amplitude α_R is generated in the reflection process. This surge is then travelling towards the generator end and reaches the end in time 2T seconds. It is represented by a line sloping (right to left). The reflection at the generator end originates an outward surge of strength α_R α_s. This process continues endlessly and some of its steps are shown in the figure above.

From Bewley's Lattice diagram, it is observed that, at the receiving end, the increment of voltage at each reflection is the sum of the incident and reflected waves. After initiate reflections, the reflection voltage V_R becomes,

If the internal resistance of the source is zero i.e., R_g = 0, then,

Therefore, after infinite reflections, the reflection voltage is equal to the incident voltage (which is unity), which means it reaches a steady-state after infinite reflections.

Advantages of Bewley's Lattice Diagrams :

• Bewley's lattice diagram technique is used to study travelling wave problems.
• It helps in solving the transient problems directly in the time domain.
• Bewley's lattice diagram can be drawn for voltage as well as for current.
• It helps in observing the position and direction of all successive reflections of voltage and current waveforms.
• Bewley's lattice diagram offers the advantage of describing the results for attenuation and wave distortion without any difficulty.
• This technique is efficient for the lossless or distortionless line.
• Good accuracy can be achieved by lumping resistance at one or more points along the line.
• The history of any wave can be determined easily.