the Thevenin's impedance ZTH [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, ZL → ∞, equation (2.1.86) provides a reflection coefficient ΓL = 1. Thevenin's voltage is obtained from equations (2.1.88a) and (2.1.97):
(2.98)
On replacing Γg from equation (2.1.96), Thevenin's voltage is
(2.1.99)
Thevenin's impedance ZTH 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, ΓL = − 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 Vg or with respect to the input voltage Vs 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
For a lossless transmission line connected to a matched source and a matched load, i.e.γℓ = jβℓ, Zg = Z0, ZTH = Z0, ZL = Z0, Γg = 0 the transfer function is
(2.1.105)
However, if the transfer function is defined by the ratio of the input voltage
2.1.9 Power Relation on Transmission Line
The average power over a time‐period T in any time‐harmonic periodic signal is [J.5, B.10]
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:
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)
Available Power from Generator
Figure (2.9a) shows that the maximum available power from a source is computed by directly connecting the load to it. The average power supplied to the load is
(2.1.113)
The load current is
Therefore, the average power supplied to the load is
(2.1.114)
Under the conjugate matching, XL = −Xg and RL = Rg, the average power supplied to load is maximum:
