Tapered Transmission-line

Author: Edmund A. Laport

A transmission line whose characteristic impedance is gradually tapered from one value to another may be used as a coupling transformer between loads and generators of unequal resistive impedances, provided that the change in impedance along the line is sufficiently gradual. The simplest form is a taper that results from converging or diverging straight conductors. The transformation ratio of such an arrangement is somewhat limited, for practical cross-sectional dimensions, as will be evident from the application of the characteristic-imped" ance formulas. If the tapered-line section has an electrical length of two or more wavelengths, the impedance transformation takes place with negligible reflections over a broad band of frequencies, owing to the smooth and gradual change of characteristic impedance with distance along this line. The shortest length for such a section that will give a tolerable minimum of reflection at the lowest of a band of working frequencies depends upon the transformation ratio desired, the difficulty increasing as this ratio becomes larger.

The transformation ratio of a tapered line may be increased by changing the number, spacing, configuration, and size of the wires in the cross-section of the line at different points along the system, thus making possible a greater range of characteristic impedances between the ends of this section. The change of characteristic impedance with distance, if made smoothly and over a sufficient length, will provide a substantially reflectionless impedance match between its two limiting resistance values. The practicality of such an application is determined solely by the structural convenience, once the current and potential requirements have been satisfied.

When it is desired to make a tapered-line transformer with a minimum length, the characteristic impedance must be tapered exponentially between its two limiting values., In such a case, the design of the transforming section must be carefully computed, because at its minimum length, power-factor correcting networks are necessary for perfect matching. As the length of the exponentially tapered transformer section is increased, the more nearly it approaches a resistance-to-resistance direct match without correcting circuits. When sufficiently long, correcting circuits may be omitted, for most practical applications. One can avoid much complicated design computation by using an exponentially tapered line section with a length of at least one-half wavelength at the lowest frequency to be transmitted and connecting it directly between a resistive load and the main feeder.

The following simple procedure may be used for the design of exponentially tapered lines that are one-half wavelength or more in length at the lowest working frequency:

1. On a sheet of semilogarithmic graph paper, mark the logarithmic scale in terms of characteristic impedances and the uniform scale in terms of length of line in wavelengths or in electrical degrees.

2. When this line is to match two resistive impedances R₁ and R₂, with a line of length x degrees (x at the lowest frequency greater than 180 degrees), mark a point on the chart for R₁ at x = 0, and another point for R₂ at x = chosen length of line. Draw a straight line between these points. Read from the logarithmic scale the required characteristic impedance of the tapered line at all intermediate distances along the line.

3. Choose the desired line configurations that provide the range of characteristic impedances required when practical dimensions are used, and apply their formulas at not greater than 20-degree-length intervals from end to end of the tapered section.

Example of Simplified Calculation for a Long Exponentially Tapered Transmission Line. It is desired to match a resistive load of 350 ohms to a feeder of 560 ohms characteristic impedance, over a frequency range of 5 to 12 megacycles. Both ends of the system are balanced to ground. A power of 50,000 watts, amplitude-modulated, is to be transmitted, and ample space is available in the open to use a leisurely tapered exponential transformer section. What should be used?

FIG. 4.42. Exponential taper computed graphically.

Figure 4.42 shows the first step when the problem is stated on semi-logarithmic paper with a length amply chosen to permit the simplified design to be used - in this case, 240 degrees at 5 megacycles. From this figure the values in Table 4.2 are tabulated:

The higher values of characteristic impedance in Table 4.2 are within the normal range for two-wire balanced lines. Figure 4.23 shows that for the lowest impedance of 350 ohms the ratio of wire spacing to wire radius is 18.5. If the wire to be used were of radius 0.102 inch, the center-to-center spacing would be 1.89 inches. This is obviously a very small spacing mechanically and from a potential standpoint for the power to be transmitted would give excessive potential gradients. We may decide to maintain two-wire design from the 550-ohm end to the point where the spacing is 5 1/2 inches, at which point a/ ρ is 54 for a wire of 0.102 inch radius and Z₀ = 478 ohms.

FIG. 4.43. Construction of exponentially tapered matching section from 550 to 350

It is then decided that from this point to the 350-ohm end a four-wire side-connected design will be used with the same size wire. Applying the equation for Z₀ for the type XVI line and maintaining a substantially constant value of a, the values for a and b (tabulated) were obtained., The final electrical design of the tapered line is shown in Fig. 4.43.

It is seen when the arithmetic is developed that the variation of b with distance is very nearly linear with electrical distance from the wide end (low impedance) of the four-wire portion to the place where the two wires in parallel on each side can be soldered together. It is further seen that this particular design is structurally simple to realize with very trivial compromise variations from true exponential continuous taper.