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Transmission line responses 79
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3-8C ZL<Zo
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3-8D ZL>Zo
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sums with the incident wave along the top of the pulse The reflection coefficient can be determined by examining the relative amplitudes of the two waves The opposite situation, in which ZL is 2Zo, is shown in Fig 3-8D In this case, the reflected wave is in phase with the incident wave, so it adds to the incident wave as shown The cases for a short-circuited load and an open-circuited load are shown in Figs 3-8E and 3-8F, respectively The cases of reactive loads are shown in Figs 3-8G and 3-8H The waveform in Fig 3-8G resulted from a capacitance in series with a 50- (matched) resistance; the waveform in Fig 3-8H resulted from a 50- resistance in series with an inductance
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The ac response of the transmission line
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When a CW RF signal is applied to a transmission line, the excitation is sinusoidal (Fig 3-9), so it becomes useful for us to investigate the steady-state ac response of the line The term steady-state implies a sine wave of constant amplitude, phase, and
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80 Transmission lines
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3-8E ZL = 0
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ZL =
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3-8G
ZL = 50 jXC
Transmission line responses 81
3-8H
ZL = 50 + jXL
3-9 AC-excited transmission line
frequency When ac is applied to the input of the line, it propagates along the line at a given velocity The ac signal amplitude and phase will decay exponentially: VR = Ve yl [319]
82 Transmission lines where VR is the voltage received at the far end of the line V is the applied voltage l is the length of the line y is the propagation constant of the line The propagation constant y is defined in various equivalent ways, each of which serves to illustrate its nature For example, the propagation constant is proportional to the product of impedance and admittance characteristics of the line: y= ZY [320]
or, since Z = R + j L and Y = G + j C, we may write: y (R+ jwl) (G + jwC) [321]
You can also write an expression for the propagation constant in terms of the line attenuation constant a and phase constant B: y = a + jB [322]
If you can assume that susceptance dominates conductance in the admittance term, and reactance dominates resistance in the impedance term (both usually true at microwave frequencies), then we may neglect the R and G terms altogether and write: y = j LC [323]
We may also reduce the phase constant B to B= or B = ZoC rad/m [325] LC [324]
and, of course, the characteristic impedance remains: Zo = LC [326]
Special cases
The impedance looking into a transmission line (Z) is the impedance presented to the source by the combination of load impedance and transmission line characteristic impedance Below are presented equations that define the looking-in impedance seen by a generator (or source) driving a transmission line
Transmission line responses 83 The case where the load impedance, and line characteristic impedance, are matched is defined by ZL = RL + j0 = Zo In other words, the load impedance is resistive and equal to the characteristic impedance of the transmission line In this case, the line and load are matched, and the impedance looking in will be a simple Z = ZL = Zo In other cases, however, we find different situations where ZL is not equal to Zo 1 ZL is not equal to Zo in a random-length lossy line: Z = (Zo) where Z is the impedance looking in, in ohms ZL is the load impedance, in ohms Zo is the line characteristic impedance, in ohms l is the length of the line, in meters y is the propagation constant 2 ZL not equal to Zo in a lossless, or very low loss, random-length line: Z = (Zo) ZL + jZo tan (Bl) Zo + jZL tan (Bl) [328] ZL + Zo tanh (yl) Zo + ZL tanh (yl) [327]
Equations 327 and 328 serve for lines of any random length For lines that are either integer multiples of a half-wavelength, or odd-integer (ie, 1, 3, 5, 7, , etc) multiples of a quarter-wavelength, special solutions for these equations are found and some of these solutions are very useful in practical situations For example, consider 3 Half-wavelength lossy lines: Z = (Zo) ZL + Zo tanh (al) Zo + ZL tanh (al) [329]
Example 3-5 A lossless 50- (Zo) transmission line is exactly one-half wavelength long and is terminated in a load impedance of Z = 30 + j0 Calculate the input impedance looking into the line (Note: in a lossless line a = 0) Solution: Z = (Zo) ZL + Zo tanh (al) Zo + ZL tanh (al)
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