Introduction To Modern Planar Transmission Lines. Anand K. Verma

Figure 2.8 Transmission line circuit. The distance x is measured from the source end; whereas the distance d is measured from the load.

(2.1.56)

On comparing the coefficients of sinh(γx) and cosh(γx), of equations (2.1.55b) and (2.1.56), two constants A₂ and B₂ are determined:

      (2.1.57)

      The phasor line voltage and line current are written as follows:

(2.1.58)

      The constants A₁ and B₁ are determined by using the boundary conditions at input x = 0 and output x = ℓ.

      At the receiving end, x = ℓ, the load end voltage and current are

(2.1.60)

The constant B₁ is evaluated on substituting and , from equation (2.1.60), in the above equation:

      (2.1.62)

On substituting constants A₁ and B₁ in equation (2.1.58a), the expression for the line voltage at location P, from the source or load end, is

(2.1.63)

      Similarly, the line current at the location P is obtained as follows:

(2.1.64)

      At any location P on the line, the load impedance is transformed as input impedance by the line length d = (ℓ − x):

(2.1.65)

Equations (2.1.65a,b) take care of the losses in a line through the complex propagation constant, γ = α + jβ. However, for a lossless line α = 0, γ = jβ and the hyperbolic functions are replaced by the trigonometric functions shown in equation (2.1.65c). It shows the impedance transformation characteristics of d = λ/4 transmission line section.

Equations (2.1.63) and (2.1.64) could be further written in terms of the generator voltage for the case, Z_g ≠ 0. Figure (2.8b) shows that at the source end x = 0, the load appears as the input impedance Z_in. The sending end voltage is obtained as follows:

      , where .

(2.1.66)

The line voltage, in terms of , and Z_L, is obtained on substituting equation (2.1.66) in equation (2.1.63):

      (2.1.67)

      Likewise, from equations (2.1.64) and (2.1.66), an expression for the line current is obtained:

      (2.1.68)

      The above equations could be reduced to the following equations for a lossless line, i.e. for α = 0, γ = jβ, cosh(jβ) = cos β and sinh(jβ) = j sin β:

      (2.1.69)

      (2.1.70)

      Equation (2.1.65c), for the input impedance, could be obtained from the above two equations. The sending end voltage and current are obtained at the input port – aa, x = 0:

      (2.1.71)

      (2.1.72)

      Likewise, the expressions for the voltage and current at the output port – bb, i.e. at the receiving end for x = ℓ, are obtained:

      (2.1.73)

      (2.1.74)

      Two

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