Equation Solving sans Computer, Calculator or Slide Rule

Richard J. Nelson1

On a boss financed Paradise Island vacation

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Situation: The boss will finance an all expense paid two week vacation on a paradise island which doesn’t allow any electronics: Cell phones, computers, internet, etc. Even a slide rule is considered anti-vacation and

not allowed.

• Problem: You have two days of design work on your

electronics circuit design.

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Take Your Pick

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I Need to Solve Design Equations

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Helicopter Requires Minimum Weight

My Equation Solving Constraints Are:

1. No Power Usage.2. No Weight.3. No Electronics.4. No Hope?

Obviously I have to go Old-school

What Method, What Technique, What Aids do I Use?

Hints

“It” is one word starting with a letter early in the alphabet.

“It” fits Into my Shirt Pocket.

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The Solution is to Use Graphical Aids to Solve the Equations

These are also known as:

• Graphs• Nomographs*• Nomograms

• * This is the term I prefer.

Simple Nomograph Equation Example

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Parallel Resistor Nomograph Relationships

1. R1 & R2 Scales are linear at right angles.2. RT scale is 45 degrees linear but different scale.3. Practical scales should cover one decade.4. RT scale easily scaled. Use R1 = R2.5. Scales can be made to accuracy desired.6. Equation is solved for any variable.

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Nomograph Terminology

Nonlinear Scale Example

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Logarithmic Scale Example

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Multiple Equation Example

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Smith Chart

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Smith Chart Usage

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Nomograph Summary

• Very low cost (paper and ink)• Very easy to use, especially for inexperienced practioners• No power required• Very complex equations are easily solvable• Very fast to use

Advantages

Disadvantages

• Tend to wear out• May be difficult to create• Limited accuracy• Highly specialized to few equations

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References1. Excellent Book – Fig. 12. Computer Program http://

www.projectrho.com/nomogram/cad.html 3 Tutorial on the Smith Chart http://www.antenna-theory.com/tutorial/smith/chart.php

Fig. 1 – Nomograph Book details

Is Old-school Graphical Equation Solving Obsolete?

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NO!

Nomographs are still actively being used in:

(a) The Medical Community(b) The Pilot Community(c) The Engineering Community• 1. Chemical• 2. Mechanical• 3. Electrical, esp Smith Chart

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The End