Introduction To Modern Planar Transmission Lines. Anand K. Verma

above equations are written as follows:

(2.1.89)

      In the above equations, the origin is at the load end, i.e. x < 0. The maxima and minima of the voltage and current waves along the line occur due to the phase variation along the line. The voltage maximum occurs at e^{j(ϕ + 2βx)} = + 1. In this case, both the forward and reflected waves are in‐phase. The voltage minimum occurs at e^{j(ϕ + 2βx)} = − 1. In this case, both the forward and reflected waves are out of phase. Finally, the maxima and minima of the voltage on a line are given as follows:

      (2.1.90)

      The reflection coefficient Γ(x) at any location x from the load end is related to the reflection coefficient at the load Γ_L by

      (2.1.91)

      The measurable quantity voltage standing wave ratio (VSWR) is defined as follows:

      (2.1.92)

      For a lossless line, the VSWR is constant along the length of a line. Likewise, the current standing wave ratio is also defined.

The wave reflection also takes place at the sending end when the source impedance Z_g is not matched to the characteristic impedance of a line. The reflection coefficient at x = − ℓ, i.e. at the generator (source) end is defined as Γ(x = − ℓ) = Γ_g. The voltage and current at the generator end are obtained from equation (2.1.88),

(2.1.93)

      The amplitude factor V⁺ is determined by the reflections at both the source and load ends.

      Figure (2.8b) shows that the port voltage and the line current , at the input port – aa, are related to the source voltage and its internal impedance Z_g by

(2.1.94)

On substitution of equation (2.1.93) in (2.1.94), the voltage wave amplitude V⁺ is obtained as follows:

      (2.1.95)

      However, the reflection coefficient at the source end is

(2.1.96)

      Therefore, the amplitude of the voltage wave launched by the source is

(2.1.97)

Equation (2.1.88a and b) give the voltage and current waves on a transmission line with the amplitude factor V⁺. The amplitude factor V⁺ is given by equation (2.1.97).

      2.1.8 Application of Thevenin's Theorem to Transmission Line

Thevenin's theorem is a very popular concept used in the analysis of the low‐frequency lumped element circuits. It is equally applicable to a transmission line network. At the output end of the line, the input source voltage and the line section are replaced by the equivalent Thevenin's voltage, with internal impedance, i.e. the Thevenin's impedance Z_TH [B.12]. Figure (2.8d) shows it. The distance is measured from the load end. Thevenin's voltage is an open‐circuit voltage at the load end. In the case of the open‐circuited load, Z_L → ∞, equation (2.1.86) provides a reflection coefficient Γ_L = 1. Thevenin's voltage is obtained from equations (2.1.88a) and (2.1.97):

      (2.98)

On replacing Γ_g from equation (2.1.96), Thevenin's voltage is

      (2.1.99)

      Thevenin's impedance Z_TH is obtained from equation (2.1.88b) by computing Norton current, i.e. the short‐circuit current at x = 0. Under the short‐circuited load condition at x = 0, Γ_L = − 1, and the Norton current is

      (2.1.100)

      Thevenin's impedance is obtained as follows:

      (2.1.101)

      Transfer Function

      The transmission line section could be treated as a circuit element. Its transfer function is obtained either with respect to the source voltage V_g or with respect to the input voltage V_s at the port‐ aa, as shown in Fig (2.8a). The load current is obtained from Fig (2.8d):

      (2.1.102)

      The voltage across the load is

      (2.1.103)

      The transfer function of a transmission line with respect to the source voltage is

(2.1.104)