266 BELL SYSTEM TECHNICAL JOURNAL 



tors and a lead shield, the values are approximately 6.9 and .36, 

 respectively. 



As with the coaxial circuit, these optimum relations are independent 

 of the diameter and thickness of the shield. Hence they make it 

 possible to find the minimum size of shield necessary for a given value 

 of high-frequency attenuation. The optimum relations are also 

 independent of the frequency, provided the frequency is high enough 

 for the approximate formulas to hold. The inner conductors may be 

 either hollow or solid. 



Condition for Maximum Characieristic Impedance ^^ 



Occasionally it is of interest to know the condition that must be 

 satisfied to obtain maximum high-frequency characteristic impedance 

 for a solid pair with circular shield. At high frequencies the value of 

 \/^[LC approaches a constant value equal to the velocity of light 

 divided by the square root of the ratio of the dielectric constant of the 

 circuit to that of air. Hence the condition for maximum characteristic 

 impedance is also, from equation (11), that for maximum inductance 

 and minimum capacitance. 



Accordingly, the high-frequency characteristic impedance of the 

 shielded solid pair circuit is given by the formula: 



7 -A 



( log. r 2. ^^, 1 - ^^' (1 - 4a^) ) abohms. (35) 



Let it be assumed first that the wires are very small compared with the 

 shield. Then equation (35) may be written 



-7- loge 



I - a- 



H — P loge 2p abohms. (36) 



For a given ratio of inner diameter of shield to outer diameter of 

 conductor, the second term of this expression is constant. By mini- 

 mizing the first term with respect to a, it is found that, so long as the 

 ratio of inner diameter of shield to conductor diameter is large, maxi- 

 mum characteristic impedance is obtained when a has a value of .486. 

 If the conductors are large compared with the shield, equation (36) 

 no longer holds. However, since the capacitance and high-frequency 

 characteristic impedance are inversely proportional to one another, 

 the position of the conductors with respect to the shield must be such as 

 to minimize the capacitance. It is clear that as the conductor diameter 

 approaches the inner radius of the shield, a approaches 0.5 for minimum 

 capacitance. Hence, for any ratio of inner diameter of shield to 



