his article presents a concept of a wave together with the wave equations and their solutions. The time-varying EM fields and their propagation in both the time and frequency domains is discussed first. Subsequently, the equations for the voltages and currents on the transmission line are obtained. It is shown that these equations and that their solutions represent voltage and current waves propagating along the line.
This means that any function of the form represents a wave traveling in the positive z direction with a velocity
In general, the electric and magnetic fields have three nonzero components, each of them being a function of all three coordinates and time. That is,
Under the uniformity in the plane assumption, if the E field points in the +x direction (usual designation) then the Maxwell’s equations show that the H field is pointing in the + y direction, and
Based on the discussion in Section 1, we recognize the functions f and g, as waves propagating in +z and –z directions, respectively, with a velocity of propagation equal to
Figure 3 shows a forward propagating EM wave in a lossless medium.
The wavelength λ is related to the velocity of propagation and frequency by
Physical object is electrically short if its electrical length z/l ≤ 1/10 or equivalently if its physical length z ≤ l /10.
If the physical object is electrically short, then the lumped-parameter circuit models are an adequate representation of that object. This also means that we can use Kirchhoff’s laws instead of Maxwell’s equations to analyze the circuit models.
To obtain the transmission line equations, let’s consider a single segment of a lossless transmission line shown in Figure 4.
The distributed parameters describing the transmission line are: l – inductance per-unit-length (H/m) and c – capacitance per-unit-length (F/m).
Similar statements are valid for the current waves. The total solution consists of the sum of forward-traveling and backward-traveling waves.
The velocity of the wave propagation along the line is given by
- Bogdan Adamczyk, Foundations of Electromagnetic Compatibility with Practical Applications, Wiley, 2017.