Abstract:
A transmission line for a clock input for a digital device. In the prior art, a clock signal was fed to a digital device on a transmission line. It was found that, when the clock frequency was doubled, the clock pulses received by the device became unacceptable. The invention lengthened the transmission line, rather than shortening it, and thereby removed the unacceptable features of the clock pulses.

Description:
The invention concerns a transmission line for delivery of high-frequency clock signals. 
     BACKGROUND OF THE INVENTION 
     FIG. 1 illustrates a clock  3  which feeds a clock signal to a transmission line  6 , which may take the form of a trace on a printed circuit board (not shown). The trace  6  leads to a clocked device  9 , which uses the clock signal as an input. 
     At low clock frequencies, the apparatus shown encounters no significant problems. For example, when the line  6  is 15.12 inches long, and the clock runs at 33 MHz, clock signal  12  delivered to the clocked device  9  was found to be acceptable, as indicated. 
     However, if the clock frequency is increased, problems can be encountered. For example, if the clock signal is increased to 66 MHz, as in FIG. 2, then the clock signal  16 , measured at the input  17  of device  9 A, was found to be unacceptable, as indicated. FIG. 3 illustrates specific defects found in the clock signal. 
     When the clock signal was supposed to be LOW, as in region  18  in FIG. 3, it failed to remain below the upper limit  20  of the LOW region, as excursions  23  and  26  indicate. When the clock signal was supposed to be HIGH, as in region  29 , it failed to remain above the lower limit  33  of the HIGH region, as excursions  36  and  39  indicate. 
     These defects render the clock signal  16  unacceptable for many purposes. 
     OBJECTS OF THE INVENTION 
     An object of the invention is to provide an improved clock signal in a digital system. 
     SUMMARY OF THE INVENTION 
     In one form of the invention, the transmission line carrying the clock signals is lengthened, in order to cause the reflected impedance of the clocked device, seen by the clock, to be both high and capacitive. For a clocked device having an input capacitance of about 5 to 20 pico-Farads, a line length of about ½ wavelength of the clock pulses provides satisfactory results. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 illustrates a situation found in the prior art, wherein a clock signal at 33 MHz is carried by a 15-inch line to a clocked device. 
     FIG. 2 illustrates that the apparatus of FIG. 1 does not perform acceptably at a clock frequency of 66 MHz. 
     FIG. 3 is a schematic illustration of defects found in the clock signal of FIG.  2 . 
     FIG. 4 illustrates one form of the invention. 
     FIG. 5 defines wavelength. 
     FIG. 6 is a schematic illustration of a clock signal received at point P in FIG.  4 . 
     FIG. 6A illustrates a transmission line and a load. 
     FIG. 7 illustrates reflected impedance for an open-circuited transmission line. 
     FIG. 8 illustrates reflected impedance for a short-circuited transmission line. 
     FIG. 9 illustrates reflected impedance for the system of FIG.  4 . 
     FIG. 10 is an enlargement of region  70  of FIG.  9 . 
     FIG. 11 illustrates reflected impedance for the system of FIG.  1 . 
     FIG. 12 illustrates reflected impedance, for different frequencies, for the system of FIG. 4, for a transmission line of 0.38 meters in length. 
     FIG. 13 illustrates reflected impedance, for different frequencies, for the system of FIG. 4, for a transmission line of 0.9 meters in length. 
     FIGS. 14-16 illustrate reflected impedances for the system of FIG. 4, but for different input capacitances of clocked device  9 C. 
     FIG. 17 illustrates standing waves occurring on a transmission line. 
     FIG. 18 illustrates a specific type of standing wave occurring on the transmission line of FIG.  4 . 
     FIGS. 19A, B, and C illustrate an output stage of a hypothetical clock. 
     FIGS. 20 and 21 illustrate two forms of the invention. 
     FIG. 22 is a copy of the plot of FIG.  9 . 
     FIG. 23 is a plot, illustrating extrapolation of line lengths, for higher frequencies. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Overview 
     FIG. 4 illustrates one form of the invention. A clock  30 , running at 66.66 MHz, feeds a transmission line  33 . The transmission line  33  is 35.98 inches long, which is more than double the length of the line  6  used in FIG.  2 . The clock signals  39  received at point P were found to be acceptable, and are illustrated in FIG.  6 . 
     The improvement in clock signals is believed to be caused by lengthening the transmission line. The length chosen causes the impedance seen by the clock  30  to be high, and also capacitive. For the device of FIG. 4, the length is about one-half the wavelength of the clock signal, which is computed as follows. 
     As indicated in FIG. 5, wavelength, L, is given by the expression 
     
       
           L meters/cycle=( v  meters/sec)/( f  cycles/sec)= v/f   
       
     
     The propagation velocity in the transmission line, v, is roughly forty-five percent the speed of light, which itself is 3.0×10 8  meters/second. Thus, the 33 MHz clock signal has a wavelength of about 2.0 meters: 
     
       
         0.45×3×10 8 /66×10 6 =2.0 meters 
       
     
     2.0 meters equals about 6.2 feet. One-half of this wavelength is about 3.1 feet, which is approximately the length of the 35.98-inch transmission line shown in FIG.  4 . 
     Description in Greater Detail 
     No simple, comprehensive, explanation is readily apparent to explain why the lengthened transmission line  33  in FIG. 4 improves the clock signal detected at point P. However, two plausible hypotheses are the following: 
     1. At 66 MHz, the transmission line  6  in FIG. 2 may present an impedance of zero to the clock  30 . Thus, the clock  30  attempts to drive a short circuit. In so doing, the clock  30  begins to operate outside the linear region for which it was designed, and produces the spikes indicated in FIG.  3 . 
      Increasing the transmission line to a proper length raises the impedance seen by the clock  30 , and may return the clock  30  to its linear design-region. 
     2. At 66 MHz, the transmission line  6  in FIG. 2 may have inductive properties. Capacitors are present elsewhere in the system. Inductor-capacitor combinations are known to “ring” under certain circumstances. The excursions in FIG. 3 may represent ringing behavior. 
      Increasing the transmission line to a proper length can change the inductive properties to capacitive properties, and possibly eliminate the ringing. 
     These hypotheses will be discussed below, but an understanding of “reflected impedance” must first be established. 
     Reflected Impedance 
     FIG. 6A shows a generalized load  45  on a transmission line  48 . The load  45  can represent the clocked device  9 A of FIG.  2 . The load  45  has an impedance, Z, which is a complex number in general, and is the ratio of the voltage, V, to the current, I: 
     
       
         
           Z=V/I. 
         
       
     
     This impedance is not the “reflected impedance” mentioned above, but is the lumped impedance of the load  45  itself. However, when the load  45  is connected to transmission line  48 , and the end  55  of the transmission line  48  is connected to the clock, the clock sees an impedance at end  55  which is a non-linear combination of the load&#39;s lumped impedance, plus the transmission line&#39;s characteristic impedance. 
     This impedance, seen at end  55 , is the reflected impedance. The reflected impedance depends on the length of transmission line  48 , and will change if that length changes. For a lossless line, which is assumed to contain no resistance, the reflected impedance is given by Equation (1):                Z                   (     -   x     )       =       Z   0                Z   L        cos                   (     β                 x     )       +     j                   Z   0        sin                   (     β                 x     )               Z   0        cos                   (     β                 x     )       +     j                   Z   L        sin                   (     β                 x     )                     (   1   )                                
     wherein 
     Z is the reflected impedance, and is complex; 
     x is negative distance from the load, as indicated in FIG. 6; 
     Beta, β, is the propagation constant, in radians per meter; 
     Z 0  is the characteristic impedance of transmission line; 
     Z L  is the impedance of the load; and 
     j is the imaginary operator. 
     This expression is derived in numerous textbooks on electromagnetic wave propagation, including  Electromagnetic Fields and Waves,  by Magdy Iskander (Prentice-Hall, 1992), ISBN 0-13-249442-6, chapter 7. This book is hereby incorporated by reference in its entirety. 
     Equation (1) is somewhat complex, and some of its features will be illustrated through examples. 
     EXAMPLE 1 
     Assume that the load is an open circuit, wherein Z L  equals infinity. Under this assumption, equation (1) reduces to 
       Z (− z )=− jZ   0 cot(β x )  (2) 
     FIG. 7 is a plot of equation (2). (FIG. 7, and other FIGS. herein, were generated by the software package MATHEMATICA, available from Wolfram Research, Inc., Champaign, Ill. The computer code used is given in the FIGS., to assist the reader in duplicating the results, if desired.) 
     Several features of FIG. 7 are significant. One is that the reflected impedance, Z, is purely imaginary. This is a result of the assumption that the line is lossless, and that the load contains no resistance. 
     A second feature is that the impedance, Z, is positive in certain regions. A positive imaginary impedance represents an inductive load. For example, the impedance of a purely inductive load is given by the expression jwL, wherein j is the imaginary operator, w is frequency, and L is the size of the inductance. This impedance is positive, and purely imaginary. 
     A third feature is that the impedance is negative in other regions. A negative imaginary impedance represents a capacitive load. For example, the impedance of a purely capacitive load is given by the expression 1/jwC, wherein j is the imaginary operator, w is frequency, and C is the size of the capacitance. This impedance equals −j/wC, which is negative. 
     A fourth feature is that the transition from inductive to capacitive loading, and capacitive to inductive loading, occurs at asymptote A. 
     A fifth feature is that the magnitude of the reflected impedance depends on the length of the transmission line. For example, the magnitude at the point Bx=1.0 is different from the magnitude at the point Bx=3.0. 
     A sixth feature is that the reflected impedance at the quarter-wavelength point, Π/2, is zero. The point of zero impedance is believed to be particularly significant, and will be elaborated shortly. 
     EXAMPLE 2 
     FIG. 8 is a plot of Equation (1), but under the condition that the load Z L  is a short circuit, wherein Z L =0. The first five features discussed above are also present in FIG.  8 . As to the sixth feature, in FIG. 8, zero impedance occurs at a half-wavelength point, Π, rather than at the quarter-wavelength point, Π/2, as in FIG.  7 . 
     Therefore, FIGS. 7 and 8 illustrate basic features of reflected impedance, including these three, which are particularly significant in the present context: (1) points exist where the reflected impedance is zero, (2) the positions of the zero-impedance points depend on the type of load, and (3) points exist where the reflected impedance is high. 
     EXAMPLE 3 
     Against the previous two Examples as background, this Example will compute the reflected impedance for the situation of FIG.  2 . The input capacitance of the load  9 A is about 16 pico-Farads, and the input resistance is considered to be infinite. FIG. 9 is a plot of Equation (1), for this load. 
     As indicated in FIG. 9, the following parameters were assumed: 
     c: signal propagation velocity, 0.45 of vacuum-speed of light; 
     cap: capacitance of load, 16 pF; 
     w: radian frequency, radians/sec; 
     fr: Hertzian frequency, Hz; 
     B: propagation constant, radians/meter; 
     Zo: characteristic impedance of transmission line, assumed 60 ohms; 
     Zl: load impedance,=1/j*w*cap, for capacitive load; and 
     x: length of transmission line, meters. 
     A significant feature of FIG. 9 lies in region  70 , which is shown enlarged in FIG.  10 . The reflected impedance, Z, reaches zero at about 0.39 meters. This is approximately the length of the transmission line  6  in FIG.  2 . 
     Thus, the reflected impedance of the clocked device  9 A in FIG. 2 could be zero. (The value of zero is treated as a possibility, rather than a certainty, because the plot of FIG. 10 is an approximation. It was computed based on assumed values of propagation velocity, input capacitance, etc. These assumed values may be slightly incorrect. Further, these values can change over time. Therefore, FIG. 10 is taken as an approximation.) 
     If the reflected impedance is zero, the clock  3  would then see a zero-impedance load. As stated above, such a load can perhaps cause the clock to enter a non-linear region of operation, and cause erratic behavior. 
     Alternately, the reflected impedance may not be zero, but the curve of FIG. 10 may actually lie slightly to its left, making the reflected impedance inductive at 0.39 meters. This inductive impedance may induce ringing, as explained above. 
     Therefore, at 66 MHz, the reflected impedance of the 16-pF load is near a zero point, and may be inductive. Both cases are considered undesirable. 
     It is observed that these problems do not arise in the 33-MHz clock of FIG.  1 . FIG. 11 is a plot of the reflected impedance of the clocked device  9 , at 33 MHz. As FIG. 11 indicates, the zero-impedance point lies at a line length of about one meter. But, since the transmission line is about 15 inches long, or about 0.4 meter, the apparatus is operating at point P 2 . At point P 2 , the reflected impedance is not zero, but is finite, and is not inductive, but capacitive. The two hypotheses stated above do not apply to FIG.  1 . 
     FIG. 12 analyzes the situation of FIG. 2 from another perspective. FIG. 12 is a plot of reflected impedance, as a function of frequency, for a constant line length of 15 inches (0.38 meter). (In contrast, FIG. 9 plotted reflected impedance at a constant frequency, for different line lengths.) FIG. 12 indicates that, at about 66 MHz, the reflected impedance may be either zero, or inductive, consistent with the conclusions based on FIGS. 9 and 10. 
     FIG. 13 is a plot of the reflected impedance for the invention of FIG. 4, which uses a line of 0.9 meters. At 66 MHz, the reflected impedance is capacitive, as indicated by point P 3 . The problem of a zero, or inductive, reflected impedance is eliminated. 
     Therefore, in one form of the invention, a length of transmission line is chosen, based on Equation (1), which causes the reflected impedance, seen by the clock  3 , to be capacitive. Further, the length is chosen such that the magnitude of the reflected impedance is equal to, or greater than, the magnitude of the input impedance of the clocked device, at the clock frequency. 
     For example, the 16-pF load of device  9 C in FIG. 4 corresponds to −j150 ohms at 66 MHz: Z=1/j*w*cap=1/j*2*PI*66*10 6 *16*10 −12 =−j150. The length of transmission line is chosen to provide a reflected impedance of this value. Alternately, a line length providing a multiple of this value can be selected. 
     Half-Wavelength Feature 
     It was stated above that the transmission line of FIG. 4 is about one-half wavelength long. However, it appears that this half-wavelength feature is coincidental, and is a result of the particular conditions stipulated for FIG.  4 . Some other conditions will be considered, which indicate that half-wavelength lines do not provide appropriate reflected impedance at 66 MHz. 
     FIG. 9 indicates that the half-wavelength line discussed above presents a reflected impedance of about −j 1400 ohms. FIGS. 14-16 are similar to FIG. 9, but are computed for different input capacitances of the load. 
     In FIG. 14, the input capacitance is very small, at 10 −16  Farad. A length of 0.95 meters, indicated by point P 4 , presents a reflected impedance of about +j 250. This reflected impedance is smaller, and inductive, compared with FIG. 9, for the half-wavelength of line. 
     In FIG. 15, the input capacitance of the load is larger than in FIG. 14, but {fraction (1/16)} that of FIG. 9, at one pico-Farad. Point P 5  indicates the reflected impedance, which is larger than that of FIG. 14, but still smaller than that of FIG. 9, and still inductive. 
     In FIG. 16, the capacitance is about 5 times that of FIG. 9, at 75 pico-Farads. Point P 6  indicates the reflected impedance, which is capacitive, but one or two hundred ohms in size. This impedance is significantly smaller than that of FIG.  9 . 
     Therefore, FIGS. 14-16 indicate that different input capacitances of the load will require lines of different length, for a given frequency. 
     These conclusions are consistent with the plots of FIGS. 7 and 8, for the reflected impedances of open- and short-circuited transmission lines. The plot of FIG. 8 is basically the same as that of FIG. 7, but shifted to the left by 90 degrees. 
     Similarly, it appears that FIG. 14 represents the plot for an extremely small, or zero, input capacitance, and that FIG. 16 represents the plot for a large input capacitance. For an actual input capacitance between these values, the relevant plot will lie between these two extremes. 
     Further Discussion of Reflected Impedance 
     This discussion will explain certain problems which a zero reflected impedance can create for the clock  3  of FIG.  2 . 
     Assume a transmission line which is terminated by a load, as in FIG.  6 A. When a sine wave is projected into the transmission line it reflects at the load. Now, two sine waves exist in the transmission line: one travelling toward the load, and the reflected wave travelling away from the load. These two waves create a standing wave in the transmission line. 
     Equation (3) is a standard equation describing the two waves. V 1  is the magnitude of the voltage of the wave travelling toward the load and gamma, γ, is the propagation constant, which equals α+jβ, wherein α is the attenuation constant, and β is the phase constant. 
       V ( x )= V   1   e   −γ1 ( e   γx +ρ T   e   −γx )  (3) 
     To plot the standing wave graphically, the reflection coefficient, ρ, is re-written as in Equation (4). 
     
       
         ρ T =|ρ T   |e   (jφ     T     )   =e   [−2(p+jq)]   (4) 
       
     
     Equations (5) and (6) are derived from Equation (4). Equation (5) states p in terms of rho, ρ. Equation (6) states q in terms of φ. 
     
       
           p =log θ (1/{square root over (|)}ρ T |)  (5) 
       
     
     
       
           q =−(½)φ T   (6) 
       
     
     Substituting Equations (5) and (6) into Equation (3), and re-arranging, produces Equation (7), wherein k is a constant. 
     
       
         | V ( x )|= k [sin  h   2 (α x+p )+cos 2 (β d+q )] 1/2   (7) 
       
     
     In a lossless line, the attenuation constant, α, is zero, producing Equation (8). 
     
       
         | V ( x )|= k [sin  h   2   p +cos 2 (β d+q )] 1/2   (8) 
       
     
     FIG. 17 is a plot of Equation (8), in the most general case. The “sinh 2 ” term controls the vertical displacement from the horizontal axis, as indicated. The variable “q” is a phase term, and controls the left-right position of the plot. 
     FIG. 17 also shows the standing wave of current, I. 
     FIG. 18 illustrates standing waves wherein the “sinh 2 ” term and “q” are chosen to illustrate problems which a reflected impedance of zero can cause. The reflected impedance is the ratio of the voltage to the current, V/I. 
     At point P 10 , the current is zero, causing the reflected impedance to be infinite. At point P 11 , the voltage is zero, causing the reflected impedance to be zero. At other points, such as point P 12 , the reflected impedance is a finite number, and depends on the ratio of V to I. It is noted that, in FIG. 17, no points of zero reflected impedance exist. 
     In terms of FIG. 18, if the clock  3  of FIG. 2 were connected to a zero-impedance point, it would be connected to a point such as point P 11 . There, the voltage standing wave is always zero: the left- and rightward traveling waves always sum to zero. 
     The output stage of the clock would be clamped to zero voltage, which can cause problems. For example, assume that the output stage of the clock resembles that of FIG.  19 . 
     Ordinarily, this output stage would operate as follows. When the input IN to the stage is HI, as in FIG. 19A, transistor M 2  goes ON and transistor M 1  goes OFF. The OUTPUT is pulled LOW. Conversely, when the input IN to the stage is LOW, as in FIG. 19B, transistor M 2  goes OFF and transistor M 1  goes ON. The OUTPUT is pulled HI. 
     However, if the OUTPUT is clamped to ground, as indicated in FIG. 19C, transistor Ml may have difficulty in pulling the OUTPUT to a HI value. 
     In addition to this problem, the standing wave or the current creates its own problem. As FIG. 18 indicates, the current is maximum at zero-impedance point P 11 . 
     (It should be recognized that FIG. 18 illustrates the envelope of the magnitude of the current, in the phasor sense. That is, the current periodically rises, then peaks, then collapses to zero, then reverses, then peaks at a negative peak, and continues this cycle. FIG. 18 shows the envelope of the maximum positive value.) 
     The output stage of clock  3  must conform to this current. That is, the output stage must alternately supply, and sink, the current I in FIG.  18 . Further, this must be accomplished while the OUTPUT is clamped to ground. This may be impossible. 
     Therefore, FIGS. 17-19 illustrate problems which a clock may encounter in driving a line of zero reflected impedance. 
     One Form of Invention 
     FIG. 20 illustrates a printed circuit board, PCB. It contains a clock  30  and a clocked device  9 C. A trace T carries a clock signal (not shown). The trace T is of a length determined by the present invention. For example, at 66 MHz, the trace is about 36 inches long. In one embodiment, the trace T is at least twice as long as the straight-line-distance, SLD, between the clock  30  and the clocked device  9 C. 
     FIG. 21 illustrates a printed circuit board PCB. Multiple clocked devices  9 C are shown, clocked by a single clock  30 , or a collection of clocks represented by clock  30 . Traces T are all substantially the same length, perhaps at  36  inches. Clock  30  is surrounded by electromagnetic shielding S, which can take the form of a copper, or aluminum, box. This arrangement allows the clock  30 , or collection of clocks, to be remotely located from the clocked devices  9 C. The remote positioning allows the clock  30  to be positioned at a location where space is not at a premium, so that the shielding S does not interfere with other electronic components. 
     Additional Considerations 
     1. It was stated above that the reflected impedance can be equal to that of the input impedance of the clocked device. Alternately, a length of line can be selected which causes the reflected impedance to be higher than that of the clocked device. For example, if multiple devices are to be clocked, their input capacitances effectively add in parallel, and present a large capacitance to the clock. If the traces T in FIG. 21 are chosen of length such that the reflected impedance is four times the impedance of each device, then it is possible for a single clock  30  to drive the four devices, through four traces. 
     In this connection, it is observed that, in the prior-art situation of FIG. 1, the reflected impedance is necessarily less than that of the clocked device. In FIG. 11, in the region to the left of 1.0 meters, the plot monotonically approaches zero. In this region, the maximum value of reflected impedance occurs at a transmission line length of zero. At zero, the reflected impedance equals the lumped impedance of the clocked device. 
     For the reflected impedance to exceed the lumped impedance, and still remain capacitive, a line length of about 2.0 meters, or 6.2 feet, is required! 
     2. The invention accommodates drift which occurs in normal operation. For example, the electronic components of the clock  30 , such as resistors, capacitors, etc., can change with temperature and age, and cause the clock frequency to drift. Also, the characteristic impedance of the transmission line may change, as can the input capacitance of the clocked device. These factors can cause the plot of FIG. 9 to shift in position. 
     FIG. 22 is a copy of FIG.  9 . Assume that point P 20  is chosen as the operating point. It is possible that component drift, and other factors, can cause the operating point to, in effect, move to point P 21 , which is inductive. (Actually, point P 20  does not move, but stays at the same position on the horizontal axis, because the physical length of the transmission line does not change. Instead, the plot moves, because the variables of Equation (1) change, to cause the operating point to jump to point P 21 .) 
     To combat this drift, point P 3  is chosen sufficiently far from the asymptote A to prevent this shift to an inductive impedance. 
     In one embodiment, a distance x 1  is defined in FIG. 22, which is the distance between the operating point and the asymptote A. Distance x 1  is chosen such that (B) (x 1 ) exceeds one of the following: 0.1 radian, 0.2 radian, 0.3 radian, 0.4 radian, or 0.5 radian. “B” is the propagation coefficient, Beta, in radians per meter. 
     From another perspective, if the distance between asymptotes A is D, as indicated, then x 1  is chosen to be at least five percent of D. 
     3. The mathematical analysis given above applied sinusoidal, steady-state analysis to ascertain the reflected impedance of the transmission line. However, a clock signal is a digital signal, and, strictly, the sinusoidal analysis is not applicable. Nevertheless, as FIG. 6 indicates, satisfactory clock signals were obtained. 
     4. In one embodiment, the invention is restricted to use with a continuous clock signal. That is, the invention is not designed for use with occasionally occurring signals, such as an ENABLE signal. One reason is that, upon start-up of the system, the well-behaved waveforms shown in FIG. 6 are not yet established. The clock signals will have some of the defects of FIG.  3 . However, after the first few clock signals, and after certainly no more than ten or twenty signals, the waveform of FIG. 6 becomes established. 
     An occasional signal, such as the ENABLE signal, will behave like the first few signals, and will not deliver a clean pulse. 
     5. It is not necessary that the shortest possible transmission line be selected. For example, point P 21  in FIG. 22 can be chosen as the operating point. 
     6. Calculations indicate that, for input capacitances of loads in the range of about 5 to 20 pF, a half-wavelength transmission line is appropriate, for frequencies of 50 MHz and above. 
     7. The following Tables specify particular line lengths for various frequencies. Table 1 specifies favorable lengths. Table 2 specifies lengths to be avoided. 
     
       
         
               
             
               
               
               
             
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Favored Lengths 
               
             
          
           
               
                   
                 Frequency (MHz) 
                 Favored Lengths (Inches) 
               
               
                   
                   
               
             
          
           
               
                   
                 100 
                 22.6, 27.5 
               
               
                   
                 85 
                 27.3, 33.0 
               
               
                   
                 75 
                 31.6, 37.5 
               
               
                   
                 66 
                 36.2, 43.0 
               
               
                   
                 60 
                 40.7, 47.0 
               
               
                   
                 33 
                 77.4, 88.0 
               
               
                   
                 30 
                 86.5, 97.0 
               
               
                   
                 25 
                 104.9, 117.0  
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
             
               
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 Lengths to Avoid 
               
             
          
           
               
                   
                 Frequency (MHz) 
                 Lengths to Avoid (Inches) 
               
               
                   
                   
               
             
          
           
               
                   
                 100 
                  8.8, 12.6 
               
               
                   
                 85 
                 11.2, 15.3 
               
               
                   
                 75 
                 13.3, 17.6 
               
               
                   
                 66 
                 15.5, 20.1 
               
               
                   
                 60 
                 17.8, 22.6 
               
               
                   
                 33 
                 36.0, 42.5 
               
               
                   
                 30 
                 40.0, 47.5 
               
               
                   
                 25 
                 49.8, 57.5 
               
               
                   
                   
               
             
          
         
       
     
     Lengths for other frequencies can be computed using the equations given above. Alternately, the length can be computed using the data of Table 1. For example, a curve can be plotted, as in FIG. 23, wherein section D is an extrapolation of the data points taken from the Tables. For the frequency used, the required length is found, using the curve. Of course, an actual graphical plot is not necessary; mathematical curve-fitting techniques can be used, based on the data of Table 1. 
     Numerous substitutions and modifications can be undertaken without departing from the true spirit and scope of the invention. What is desires to be secured by Letters Patent is the invention as defined by the following claims.