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Engineering in the Pre-Computer Era (1956-1980)

How Engineers Designed Tapers
Before Calculators Existed

When Robert Klopfenstein published his optimal taper design in 1956, computers were room-sized and inaccessible. Engineers relied on nomographs (alignment charts),slide rules, and printed tables to perform complex calculations involving Bessel functions and numerical integration.

Historical Timeline

1891

Heaviside's Theory

Oliver Heaviside develops transmission line theory, laying mathematical foundation

1955

Collin's Quarter-Wave Work

Robert E. Collin publishes theory of multi-section quarter-wave transformers in Proc. IRE (narrow-band, ~30% BW)

1956

🎯 Klopfenstein's Breakthrough

Robert W. Klopfenstein publishes "A Transmission Line Taper of Improved Design" in Proceedings of the IRE

Innovation: Mathematically optimal taper using Chebyshev equal-ripple criterion
Key Math: Modified Bessel functions I₁(x), variational calculus
Result: Unlimited bandwidth (high-pass), shortest possible length
Tools Used: Slide rules, log tables, hand-calculated Bessel values
1964

Matthaei's Handbook

"Microwave Filters, Impedance-Matching Networks" includes Klopfenstein nomographs and design tables

1967

First Handheld Calculator

Texas Instruments introduces first portable calculator - still too expensive for most engineers

1970s

Transition to Digital

Programmable calculators (HP-65, TI-59) allow custom programs. Nomographs remain common until ~1980.

📥 Download Printable Nomograph

Get a PDF version perfect for printing and use with a straightedge, just like engineers did in 1956!

Interactive Klopfenstein Nomograph (1956 Style)

How to use this nomograph:

  1. Click on your starting impedance (Z₁) on the left scale
  2. Click on your ending impedance (Z₂) on the right scale
  3. The line connecting them intersects the center scale at parameter A
  4. Use A with the length formula: L = (c×A)/(4π×f_low×√εᵣ)
Klopfenstein Taper Design NomographZ₁ (Ω)1015202530405075100150200A (Design Parameter)1.01.52.02.53.03.54.04.55.05.56.0Z₂ (Ω)1015202530405075100150200Click one point on Z₁, then one point on Z₂ to read A parameter

Experience Both Eras

Try the vintage nomograph above to appreciate the ingenuity of 1950s engineers, then switch to the modern designer to see how technology has transformed the process!