Anand K. Verma - Introduction To Modern Planar Transmission Lines

(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 special cases of the load termination, i.e. the short‐circuited load and the open‐circuited load, are discussed below. The voltage and current distributions on a transmission line for both the cases are also obtained.

For the short‐circuited load Z L= 0, the line voltage at the load end is zero . However, the voltage on the line is not zero. Equations (2.1.63)and (2.1.64)provide the voltage and current distributions on a short‐circuited line:

(2.1.75)

The input impedance at any distance (ℓ − x) from the source end is

(2.1.76)

At the load end, the voltage is zero. However, the line current is not infinite like the lumped element circuit with a short‐circuited termination at output. A short‐circuited transmission line draws only a finite current from the source. The ℓ < λ/4 short‐circuited line section behaves as an inductive element . The electrical nature of the line section can be controlled by changing its electrical length [B.9–B.15].

The load impedance is Z L→ ∞ for an open‐circuited transmission line and the load current . Again, equations (2.1.63)and (2.1.64)provide the voltage and current distributions on an open‐circuited transmission line. The voltage and current waves and the input impedance at location P from the load end can be computed for the open‐circuited load as follows:

(2.1.77)

(2.1.78)

The ℓ < λ/4 open‐circuited line section behaves as a capacitive element . The electrical nature of the line section can be controlled by changing its electrical length.

The input impedance at any location on a line is Z in(x) = Z 0if it is matched terminated in its characteristic impedance, i.e. Z L= Z 0. Normally, the characteristics impedance of a microwave line is a real quantity. The line terminated in Z 0does not create any reflected wave on a transmission line. However, for the mismatched termination Z L≠ Z 0, there is a reflected wave on a transmission line, traveling from the load end to the source end.

The wave nature of the line voltage and line current becomes more obvious from the exponential form of solutions of the wave equations. The solution of wave equation (2.1.37), for the phasor line voltage and line current, can also be written in the exponential form:

(2.1.79)

The distance x is measured from the source end. The time‐dependent harmonic form of the voltage wave is . Finally, it is written as follows:

(2.1.80)

For an outgoing wave on a lossy line, the wave amplitude decays and its phase lags; whether the distance is measured from source end or load end . It is accounted for by the proper sign of distance x. The amplitude of a wave is exponentially decaying due to the line losses. It is expressed by the attenuation constant α (Np/m).

The expression of the traveling current wave on a line could be written as follows:

(2.81)

If the source at x = 0 is connected to a line of infinite extent, there is no reflection from the load end, as the wave will never reach to the load end to get reflected. Therefore, for the forward traveling voltage and current waves on an infinite line, V −= 0 and the above solutions of the wave equations are written as follows:

(2.1.82)