Introduction To Modern Planar Transmission Lines. Anand K. Verma
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:
This is the maximum power available from a generator under the matching condition and delivered to a load RL. At this stage, the maximum power delivered to a load is computed in the absence of the transmission line. For a matched terminated lossless line, the maximum available power from the source is delivered to the load. It is examined below.
The voltage and current waves on a line under no reflection case are
(2.1.116)
The average power on the line is
On a lossless line, the average power is independent of the distance x from a source. Physically it makes a sense, as the same amount of power flows at any location on the line. Under the matched load termination, ZL = Z0, the input impedance at the source end is Z0 itself. It is shown in Fig (2.9b). The sending end voltage at the input port – aa of a transmission line is
(2.1.118)
The maximum power is transferred from a generator to the transmission line under the matching condition, Rg = Z0. The maximum available power from the generator to feed the line is given by equation (2.1.115). It is identical to the power determined from equation (2.1.117), as
If the line is not terminated in its characteristic impedance, then a reflection takes place at the load end. The reflected wave travels from the load toward the generator given by
(2.1.119)
The average power in the reflected wave is
(2.1.120)
However, at the load end amplitude of the reflected voltage wave is V− = ΓLV+; where
Figure 2.9 Load connections to a source.
The negative sign (−) shows that the reflected power travels from the load toward the source. Finally, under the mismatched load, the power delivered to the load is obtained from equations