Introduction To Modern Planar Transmission Lines. Anand K. Verma

It is expressed through Dirac's delta function as . The lumped line constants R, L, C, G are given for p.u.l.

      The loop and node equations are written below to develop the Kelvin–Heaviside transmission line equations with a current source:

Figure 2.7 Equivalent lumped circuit of a transmission line with a shunt current source.

      The Loop Equation

      The Node Equation

      For Δx → 0, the above equations are reduced to

      (2.1.52)

      The above equations are rewritten below in term of the characteristic impedance (Z₀) and propagation constant (γ) of a transmission line:

      (2.1.53)

      On solving the above equations for the voltage, the following inhomogeneous voltage wave equation, with a current source, is obtained:

(2.1.54)

Away from the location of the current source, i.e. for x ≠ x₀, equation (2.1.54) reduces to the homogeneous equation (2.1.37a). The wave equation for the current wave, with a shunt current source, could also be rewritten.

      

      2.1.7 Solution of Voltage and Current‐Wave Equation

      The voltage and current wave equations in the phasor form are given in equation (2.1.37). The solution of a wave equation is written either in terms of the hyperbolic functions or in terms of the exponential functions. The first form is suitable for a line terminated in an arbitrary load. A section of the line transforms the load impedance into the input impedance at any location on the line. The impedance transformation takes place due to the standing wave formation. The hyperbolic form of the solution also provides the voltage and current distributions along the line. The exponential form of the solution demonstrates the traveling waves on a line, both in the forward and in the backward directions. A combination of the forward‐moving and the backward‐moving waves produces the standing wave on a transmission line.

      The Hyperbolic Form of a Solution

Figure (2.8a) shows a section of the transmission line having a length ℓ. It is fed by a voltage source, with Z_g internal impedance. The general solutions for the line voltage and line current of the wave equation (2.1.37) are

(2.1.55)

At any section on the line, its characteristic impedance Z₀ relates the line voltage and line current . So the constants A₂, B₂ are related to the constants A₁ and B₁. In Fig (2.8a), the point P on the line is located at a distance x from the source end, i.e. at a distance d = (ℓ − x) from the load end. The load is located at d = 0, and the source is located at d = − ℓ. The , and are the input voltage and the input current at the port‐aa, i.e at x = 0. At x = ℓ, i.e. at the port‐bb, and are the load end voltage and current, respectively. The ideal voltage generator has the internal impedance, Z_g = 0, i.e. . The phasor form of the line current, from equations (2.1.32b) and (2.1.55a), is written below:

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 x = 0, the line input voltage is , giving the value of A1:(2.1.59)

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

(2.1.60) Скачать книгу