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The reflection phenomenon was noted earlier during the coverage of the stepfunction and single-pulse response of a transmission line; the same phenomenon also applies when the transmission line is excited with an ac signal When a transmission
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Transmission line responses 87
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R = 50
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Transmission line (Zo = 50 ) To source
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3-10A Stub matching system
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line is not matched to its load, some of the energy is absorbed by the load and some is reflected back down the line toward the source The interference of incident (or forward ) and reflected (or reverse ) waves creates standing waves on the transmission line If the voltage or current is measured along the line, it will vary, depending on the load, according to Fig 3-11 Figure 3-11A shows the voltage-versus-length curve for a matched line (ie, where ZL = Zo) The line is said to be flat because the voltage (and current) is constant all along the line But now consider Figs 3-11B and 3-11C Figure 3-11B shows the voltage distribution over the length of the line when the load end of the line is shorted (ie, ZL = 0) Of course, at the load end the voltage is zero, which results from zero impedance The same impedance and voltage situation is repeated every half-wavelength down the line from the load end toward the generator Voltage minima are called nodes, and voltage maxima are called antinodes The pattern in Fig 3-11C results when the line is unterminated (open) (ie, ZL = ) Note that the pattern is the same shape as Fig 3-11B (shorted line), but the phase is shifted 90 In both cases, the reflection is 100 percent, but the phase of the reflected wave is opposite Figure 3-11D shows the situation in which ZL is not equal to Zo, but is neither zero nor infinite In this case, the nodes represent some finite voltage, Vmin, rather
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Shorted matching stub (X =
88 Transmission lines
Imax
3-10B Quarter-wavelength stub Length 0 /2
than zero The standing wave ratio (SWR) reveals the relationship between load and line If the current along the line is measured, the pattern will resemble the patterns of Fig 3-11 The SWR is then called ISWR, to indicate the fact that it came from a current measurement Similarly, if the SWR is derived from voltage measurements it is called VSWR Perhaps because voltage is easier to measure, VSWR is the term most commonly used in most radio work VSWR can be specified in any of several equivalent ways: 1 From incident voltage (Vi ) and reflected voltage (Vi ): VSWR = Vi + Vr Vi Vr [346]
2 From transmission line voltage measurements (Fig 3-11D): Vmax VSWR = Vmin
[347]
Transmission line responses 89 3 From load and line characteristic impedances: (ZL > Zo) VSWR = ZL/Zo (ZL < Zo) VSWR = Zo/ZL 4 From incident (Pi ) and reflected (Pr ) power: 1+ VSWR = 1 Pr/ Pi Pr / Pi [350] [348] [349]
5 From reflection coefficient ( ): VSWR = 1+ 1 [351]
It is also possible to determine the reflection coefficient from a knowledge of VSWR: = VSWR 1 VSWR + 1 [352]
The relationship between reflection coefficient and VSWR is shown in Fig 3-11D VSWR is usually expressed as a ratio For example, when ZL is 100 and Zo is 50 , the VSWR is ZL/Zo = 100 /50 = 2, which is usually expressed as VSWR = 2:1 VSWR can also be expressed in decibel form: VSWR = 20 log (VSWR) [353]
Example 3-7 A transmission line is connected to a mismatched load Calculate both the VSWR and VSWR decibel equivalent if the reflection coefficient is 025 Solution: (a) VSWR = = = 1+ 1 1 + 025 1 025 125 = 167:1 075
(b) VSWRdB = 20 log (VSWR) = (20) (log 167) = (20) (022) = 43 dB
90 Transmission lines
ZL = Z o
0 Load end
Toward source
ZL = 0
0 Load end V
Toward source
ZL =
0 Load end
Toward source
3-11 Voltage versus electrical length: (A) Matched impedances, (B) ZL = 0, (C) ZL = infinite
Transmission line responses 91
V ZL Zo
0 Load end
Toward source
3-11 Continued: (D) ZL not equal to Zo
The SWR is regarded as important in systems for several reasons The base of these reasons is the fact that the reflected wave represents energy lost to the load For example, in an antenna system, less power is radiated if some of its input power is reflected back down the transmission line, because the antenna feedpoint impedance does not match the transmission line characteristic impedance The next section covers the problem of mismatch losses
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