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Dielectric 3-1M Stripline transmission line
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R If X >> R Then: ZO = L/C L
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3-2 Equivalent circuit of transmission line
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Transmission line characteristic impedance (Zo) 65 In microwave systems the resistances are typically very low compared with the reactances, so Eq 31 can be reduced to the simplified form: Zo = L C [32]
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Example 3-1 A nearly lossless transmission line (R is very small) has a unit length inductance of 375 nH and a unit length capacitance of 15 pF Find the characteristic impedance Zo Solution: Zo = L C 1H 109 nH
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375 nH 15 pF
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375 10 9 H 15 10 12 F 25 103 = 50
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The characteristic impedance for a specific type of line is a function of the conductor size, the conductor spacing, the conductor geometry (see again Fig 3-1), and the dielectric constant of the insulating material used between the conductors The dielectric constant e is equal to the reciprocal of the velocity (squared) of the wave when a specific medium is used: e= 1 v2 [33]
where e is the dielectric constant (for a perfect vacuum e = 1000) v is the velocity of the wave in the medium (a) Parallel line Zo = 276 2S log d e [34]
66 Transmission lines where Zo is the characteristic impedance, in ohms e is the dielectric constant S is the center-to-center spacing of the conductors d is the diameter of the conductors (b) Coaxial line Zo = where D is the diameter of the outer conductor d is the diameter of the inner conductor (c) Shielded parallel line Zo = where A = s/d B = s/D (d) Stripline Zo = where et is the relative dielectric constant of the printed wiring board (PWB) T is the thickness of the printed wiring board W is the width of the stripline conductor The relative dielectric constant et used above differs from the normal dielectric constant of the material used in the PWB The relative and normal dielectric constants move closer together for larger values of the ratio W/T Example 3-2 A stripline transmission line is built on a 4-mm-thick printed wiring board that has a relative dielectric constant of 55 Calculate the characteristic impedance if the width of the strip is 2 mm Solution: Zo = 377 et T W 377 et T W [37A] (1 B2) 276 log 2A (1 + B2) e [36] 138 D log d e [35]
Transmission line characteristics 67
377 55
4 mm 2 mm
377 (2) = 321 235
In practical situations, we usually don t need to calculate the characteristic impedance of a stripline, but rather design the line to fit a specific system impedance (eg, 50 ) We can make some choices of printed circuit material (hence dielectric constant) and thickness, but even these are usually limited in practice by the availability of standardized boards Thus, stripline width is the variable parameter Equation 32 can be arranged to the form: W= 377 T Zo e [37B]
The impedance of 50 is accepted as standard for RF systems, except in the cable TV industry The reason for this diversity is that power handling ability and low loss operation don t occur at the same characteristic impedance For example, the maximum power handling ability for coaxial cables occurs at 30 , while the lowest loss occurs at 77 ; 50 is therefore a reasonable tradeoff between the two points In the cable TV industry, however, the RF power levels are minuscule, but lines are long The tradeoff for TV is to use 75 as the standard system impedance in order to take advantage of the reduced attenuation factor
Transmission line characteristics
Velocity factor
In the section preceding this section, we discovered that the velocity of the wave (or signal) in the transmission line is less than the free-space velocity (ie, less than the speed of light) Further, we discovered in Eq 33 that velocity is related to the dielectric constant of the insulating material that separates the conductors in the transmission line Velocity factor v is usually specified as a decimal fraction of c, the speed of light (3 108 m/s) For example, if the velocity factor of a transmission line is rated at 066, then the velocity of the wave is 066c, or (066) (3 108 m/s) = 198 108 m/s Velocity factor becomes important when designing things like transmission line transformers, or any other device in which the length of the line is important In most cases, the transmission line length is specified in terms of electrical length, which can be either an angular measurement (eg, 180 or radians), or a relative measure keyed to wavelength (eg, one-half wavelength, which is the same as 180 ) The physical length of the line is longer than the equivalent electrical length For example, let s consider a 1-GHz half-wavelength transmission line A rule of thumb tells us that the length of a wave (in meters) in free space is 030/F, where frequency F is expressed in gigahertz; therefore, a half-wavelength line is 015/F
68 Transmission lines At 1 GHz, the line must be 015 m/1 GHz = 015 m If the velocity factor is 080, then the physical length of the transmission line that will achieve the desired electrical length is [(015 m) (v)]/F = [(015 m) (080)]/1 GHz = 012 m The derivation of the rule of thumb is left as an exercise for the student (Hint: It comes from the relationship between wavelength, frequency, and velocity of propagation for any form of wave) There are certain practical considerations regarding velocity factor that result from the fact that the physical and electrical lengths are not equal For example, in a certain type of phased-array antenna design, radiating elements are spaced a halfwavelength apart, and must be fed 180 (half-wave) out of phase with each other The simplest interconnect is to use a half-wave transmission line between the 0 element and the 180 element According to the standard wisdom, the transmission line will create the 180 phase delay required for the correct operation of the antenna Unfortunately, because of the velocity factor, the physical length for a onehalf electrical wavelength cable is shorter than the free-space half-wave distance between elements In other words, the cable will be too short to reach between the radiating elements by the amount of the velocity factor! Clearly, velocity factor is a topic that must be understood before transmission lines can be used in practical situations Table 3-1 shows the velocity factors for several types of popular transmission line Because these are nominal values, the actual velocity factor for any given line should be measured
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