Anand K. Verma - Introduction To Modern Planar Transmission Lines

(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).

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 gfrom equation (2.1.96), Thevenin's voltage is

(2.1.99)

Thevenin's impedance Z THis 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)

The transmission line section could be treated as a circuit element. Its transfer function is obtained either with respect to the source voltage V gor with respect to the input voltage V sat 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)

For a lossless transmission line connected to a matched source and a matched load, i.e.γℓ = jβℓ, Z g= Z 0, Z TH= Z 0, Z L= Z 0, Γ g= 0 the transfer function is

(2.1.105)

However, if the transfer function is defined by the ratio of the input voltage at the port – aa to the output voltage , H(ω) = e −γℓ. It is obtained from equation (2.1.104)for Z g= 0.

The average power over a time‐period T in any time‐harmonic periodic signal is [J.5, B.10]

(2.1.106)

where the time‐harmonic instantaneous voltage and current waveforms are

(2.1.107)

The voltage and current in the phasor form are written as follows:

(2.1.108)

A complex number X = a + jb has its complex conjugate, X *= a − jb. Thus, the real (Re) and imaginary (Im) parts of a complex number are written as follows:

(2.1.109)

On using the above property, the instantaneous voltage and current are written as follows:

(2.1.110)

The average power in phasor form is obtained from equations (2.1.106)and (2.1.110),

(2.1.111)

It can be expressed in the usual AC form,

(2.1.112)