.net barcode reader library Transmission lines in Software

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84 Transmission lines
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In Example 3-5 we discovered that the impedance looking into a lossless (or very low loss) half-wavelength transmission line is the load impedance: Z = ZL [330]
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The fact that line input impedance equals load impedance, is very useful in certain practical situations For example, a resistive impedance is not changed by the line length Therefore, when an impedance is inaccessible for measurement purposes, the impedance can be measured through a transmission line that is an integer multiple of a half wavelength Our next special case involves a quarter-wavelength transmission line, and those that are odd integer multiples of quarter-wavelengths (of course, even integer multiples of a quarter wavelength obey the half-wavelength criteria) 4 Quarter-wavelength lossy lines: Z = (Zo) and 5 Quarter-wavelength lossless or very low loss lines: (Zo)2 Z= Z L [332] ZL + Zo coth (al) Zo + ZL coth (al) [331]
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From Eq 332, you can discover an interesting property of the quarter-wavelength transmission line First, divide each side of the equation by Zo: [Zo]2 Z = Z Z Zo L o = Zo ZL [333]
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[334]
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Transmission line responses 85 The ratio Z/Zo shows an inversion of the load impedance ratio ZL/Zo, or, stated another way, Z 1 = Zo ZL/Zo [335]
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Again, from Eq 332, you can deduce another truth about quarter-wavelength transmission lines: If Z= then ZZL = which means Zo = ZZL [338] Zo [337] Zo ZL [336]
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Equation 338 shows that a quarter-wavelength transmission line can be used as an impedance matching network Called a Q section, the quarter-wavelength transmission line used for impedance matching requires a characteristic impedance Zo if Z is the source impedance and ZL is the load impedance Example 3-6 A 50- source must be matched to a load impedance of 36 Find the characteristic impedance required of a Q section matching network Solution: Z= = = ZZL (50 ) (36 ) 1800 2 = 42
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6 Transmission line as a reactance: Reconsider Eq 328, which related impedance looking in to load impedance and line length: Z = (Zo) ZL + jZo tan (Bl) Zo + jZL tan (Bl) [339]
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Now, for the case of a shorted line (ie, ZL = 0), the solution is Z = (Zo) (0) + jZo tan (B l) Zo + j(0) tan (B l) [340]
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86 Transmission lines jZo tan (Bl) Zo
Z = (Zo)
[341] [342]
Z = j Zo tan (Bl) Recall from Eq 325 that B = WZoC Substituting Eq 343 into Eq 342 produces Z = j Zo tan ( ZoCl) or Z = j Zo tan (2 FZoCl)
[343]
[344]
[345]
Because the solutions to Eqs 344 and 345 are multiplied by the j operator, the impedance is actually a reactance (Z = 0 + jX) It is possible to achieve almost any possible reactance (within certain practical limitations) by adjusting the length of the transmission line and shorting the load end This fact leads us to a practical method for impedance matching Figure 3-10A shows a circuit in which an unmatched load is connected to a transmission line with characteristic impedance Zo The load impedance ZL is of the form Z = R jX, and, in this case, it is equal to 50 j20 A complex impedance load can be matched to its source by interposing the complex conjugate of the impedance For example, in the case where Z = 50 j20, the matching impedance network will require an impedance of 50 + j20 The two impedances combine to produce a result of 50 The situation of Fig 3-10A shows a matching stub with a reactance equal in magnitude, but opposite in sign, with respect to the reactive component of the load impedance In this case, the stub has a reactance of +j20 to cancel a reactance of j20 in the load A quarter-wavelength shorted stub is a special case of the stub concept that finds particular application in microwave circuits Waveguides (Chap 19) are based on the properties of the quarter-wavelength shorted stub Figure 3-10B shows a quarter-wave stub and its current distribution The current is maximum across the short, but wave cancellation forces it to zero at the terminals Because Z = V/I, when I goes to zero, the impedance becomes infinite Thus, a quarter-wavelength stub has an infinite impedance at its resonant frequency, and redundant acts as an insulator This concept may be hard to swallow, but the stub is a metal insulator
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