CHAPTER 1: REVOLUTION!

In the late 1940s, two forces converged to change the field of control systems forever: a team of brilliant young engineers at North American Aviation’s Aerophysics Laboratory and one of their newest leaders, Walter Evans. These engineers were solving some of the most difficult problems of the era—designing precision navigation systems for long-range, unmanned aircraft and submarines that needed to reach the North Pole with pinpoint accuracy.

The challenge was immense. Stability and performance requirements were beyond anything designers had previously faced. Classical analysis methods—while powerful—offered little intuitive guidance, often leaving engineers with an uneasy mix of mathematics, physical intuition, and trial-and-error tinkering.

Then came the Root Locus Method.

Within months of its introduction, engineers at North American Aviation recognized its power. It provided an immediate, visual way to understand how a system’s behavior would change as design parameters varied. No longer were control engineers flying blind—they could now see at a glance, how stability and performance were affected by gain adjustments.

As word spread, Root Locus quickly became the default approach to control system design. Its impact was amplified by the Spirule, a simple but ingenious tool that allowed engineers to construct root locus diagrams with precision and speed—long before computers could automate the task. Together, the method and the Spirule embedded themselves into the culture of control engineering, shaping generations of designers and revolutionizing the way servomechanisms were built.

For many engineers, Root Locus was more than a mathematical tool—it was a breakthrough that transformed the way they worked. One engineer captured its impact with a simple phrase:

"My task was to determine an analog autopilot that would be stable throughout the flight. Using all the classical analysis methods, I was floundering. When Walt first started discussing root locus in his class, I saw a ray of hope."

At a time when control system design was as much an art as a science, Root Locus provided something invaluable—clarity. It bridged the gap between frequency-domain analysis and time-domain specifications, making stability and performance requirements easier to achieve.

Its influence spread rapidly, becoming an essential part of the international culture of automatic control. Seventy-five years later, it remains a foundational design tool, still among the first steps taken when engineers tackle a new control system.

Before the advent of Root Locus, control system design was often a patchwork of methods—part theory, part intuition, part sheer persistence. Engineers relied on frequency-domain techniques‑Nyquist and Bode plots, but these had limitations:

• Nyquist Criterion could confirm whether a system      was stable, but it offered little insight into how to fix an unstable      design.
• Bode plots helped engineers understand gain      and phase margins but adjusting them to meet time-domain specifications      was more art than science.
• Trial and error were common. Engineers manually      tuned resistors, capacitors, and inductors, often testing prototypes to      see what worked.
• Mathematical complexity made high-order systems      difficult to analyze, forcing engineers to rely on simplified      approximations that sometimes failed in practice.

The biggest gap? There was no systematic way to connect pole locations to real-world performance. Engineers understood that poles and zeros dictated stability and transient response, but they lacked a method to manipulate them effectively.

At the heart of the design of any automatic control system is what the natural dynamic behavior of the controlled system will be—how stable it will be, and how quick and well-damped its natural motions will be following any disturbance. (Second in order of importance—but also of course central—is how well the controlled system will follow commands.)

Walter Evans, better known to his colleagues as Walt, was a person of remarkable insight, and a major leader in the understanding of automatic control and how to design excellent systems very well and very quickly. One of his field-leading contributions was the invention of the Root Locus Method for seeing instantly the natural dynamic behavior a linear system will have, seeing it directly in terms of the control parameters at the designer's disposal. The method presents—in seconds—a plot of the system's stability, speed of response, and the damping quality of all of its natural motions. It has been called by one of Evans’s contemporaries as among the greatest contributions of the post-war aerospace industry to engineering. Appendix I-1 is a math-based introduction to the method.

Today's computers can indeed provide Root Locus plots in seconds. From such plots the designer can see at once what a system's natural behavior will be as a function of his choice of parameters and can make fully astute choices very quickly. In 1948 there were no such computers! None at all. (Gordon Moore's Law had not yet been perceived.) Instead, all calculations had to be made with slide rules or Marchant calculators. (Aside: The Marchant Calculating Machine Company was founded in 1911.) A new kind of calculator was needed.

Evans noted that if you need to know whether a given root location is on the locus of roots of a characteristic equation, you simply need to make one calculation: You need to know whether or not the sum of the angles of the components of the factored characteristic equation of the system is180 degrees there.  In retrospect, some observers have called that insight obvious.  If anyone other than Evans saw that, they failed to recognize it as the key that would unlock the door for design engineers.

By 1949, Root Locus had already begun changing the way engineers worked, but official recognition lagged. As the A.I.E.E. (precursor to IEEE) slowly reviewed Evans’ paper, engineers eager to implement the method got their hands on it through North American Aviation’s internal documentation. The practical success of Root Locus forced a subtle but telling shift—the paper’s title changed from “Servomechanism Analysis” to “Control System Synthesis”in recognition of its role in actively shaping designs rather than merely analyzing them. Key portions of AL-787 can be found in Appendix I-2.

At the height of his frustration with academia’s slow embrace, Evans penned a handwritten letter to his first supervisor at General Electric, Orrin Livingston, explaining how he had arrived at the idea. Though its contents proved challenging for many to grasp, Evans kept a copy—a historic artifact, much like Jefferson’s handwritten Declaration of Independence. A copy can be found in Appendix I-3.

"Inventing Root Locus" follows three themes: roots, feedback, and stability.

Part I: Roots – The root locus method was not a sudden flash of insight but the result of deep analytical thinking, nurtured by family influences, exceptional mentors, and a professional environment that demanded innovation. This section explores the foundational ideas and experiences that shaped Walter Evans’s approach to engineering and led to his groundbreaking method.

Part II: Feedback – No invention exists in isolation. Once Evans introduced root locus, it underwent refinement and adaptation through the insights of colleagues, reviewers, and engineers in the field. Was it merely an analysis tool, or could it also guide design? Did it need a dedicated plotting tool like the Spirule? And how should it be introduced to the engineering community? This section examines the critical role of external influences—both resistance and enthusiasm—in shaping the method’s eventual acceptance.

Part III: Stability – By 1954, root locus had not only gained widespread recognition but had also begun transforming the way engineers approached control systems. That same year, Evans’s personal life reached a steady state as well, largely due to the unwavering support of his wife, Arline—his partner in both life and work. The final section of the book quantifies the method’s adoption and explores how the stability it provided to engineers mirrored the stability that defined the Evans household.

These three themes—roots, feedback, and stability—are more than just a structural framework; It’s about the people, the challenges, and the breakthroughs that shaped modern control engineering. 

The story of “Inventing Root Locus” is one of perseverance, insight, and practical ingenuity—qualities that remain essential to engineering today. Seventy-five years after its initial publication, the method still stands. Whether drawn by hand with a Spirule or plotted instantly by modern software, the root locus diagram remains one of the most powerful tools in control design.

This is the story of how it all began.

Two things came together in the late 1940's: The remarkable young people in the auto navigator division of North American Aviation's Aerophysics Laboratory, and one of its new leaders, Walter Evans. This team was solving very difficult engineering problems one after another, to produce, for the first time anywhere, precise navigation systems for very long range, unmanned aircraft, and for submarines that went exactly to the North Pole, among many firsts.

These were very hard systems to achieve. These were very smart young people. They saw at once the power of the Root Locus Method (RLM) and it spread very quickly through their culture at North American Aviation. It became immediately the core approach to every design. And the Method, and its inventor, and the Spirule had very much to do with how quickly it was adopted. It was just very easy to learn and use and place at the center of discussion of every control system design.

From there, the Root Locus Method spread quite swiftly throughout the international culture of automatic control. It is still, 50 years later, typically part of the first design steps, and part of the dialog among designers and users.

The difference is that today the loci of roots can be requested of computer programs everywhere, and the answer shown instantly. The Spirule is no longer needed. But without the Spirule, the RLM would have had to wait another 15 years or so until computer capacity reached the stage where it could quickly produce loci. Again, Evans had drawn on deep, very sharp insight to get a totally generic problem solved in the simplest of ways.

Meanwhile, the speed with which the Evans’ Root Locus Method became a mainstay in the broad automatic control community was still hampered for about another year by a remarkable impedance mismatch between its inventor and the IEEE journal editorial management.  In hindsight, that was a small glitch with no long-lasting historical significance.  Still, it is part of the story.   New ideas routinely confront opposition from those who stand to gain by maintaining the status quo.  Fortunately, Evans understood how much his method would benefit the field and stuck it out.  The rest, as they say, is history.
 

THE ROOT LOCUS METHOD OF WALTER R EVANS

We shall present In this short paper(1) What the Root Locus Method (RLM) is, and why -- since its invention in 1948 -- it has remained a primary player in the design of automatic control systems, and (2) An exciting story about the man, Walter Evans, who invented the RLM, how he did it, (and how hard it was for him to get it into the literature!) and (3) The exciting way in which the Spirule he invented accelerated the RLM into everyday use perhaps 15 years sooner than would have come to pass without it!

At the heart of the design of any automatic control system is what the natural dynamic behavior of the controlled system will be -- how stable it will be, and how quick and well-damped its natural motions will be following any disturbance. (Second in order of importance -- but also of course central -- is how well the controlled system will follow commands.)

Walter Evans, better known to his colleagues as Walt, was a person of remarkable insight, and a major leader in the understanding of automatic control and how to design excellent systems very well and very quickly. One of his field-leading contributions was the invention of the Root Locus Method for seeing instantly the natural dynamic behavior a linear system will have, seeing it directly in terms of the control parameters at the designer's disposal. The method presents -- in seconds -- a plot of the system's stability, speed of response, and the damping quality of all of its natural motions. It has been called by one of Evans’s contemporaries as among the greatest contributions of the post-war aerospace industry to engineering.

Today's computers can indeed provide Root Locus Plots in seconds. From such plots the designer can see at once what a system's natural behavior will be as a function of his choice of parameters and can make fully astute choices very quickly.

EVANS’ INVENTION OF THE SPIRULE

In 1948 there were no such computers! None at all. (Gordon Moore's Law had not yet been perceived.) Instead, all calculations had to be made with slide rules or Marchant calculators.

Evans noted that if you need to know whether a given root location is on the locus of roots of a characteristic equation, you simply need to make one calculation: You need to know whether or not the sum of the angles of the components of the factored characteristic equation of the system is 180 degrees there.  In retrospect, some observers have called that insight obvious.  If anyone other than Evans saw that, they failed to recognize it as the key that would unlock the door for design engineers. Let us review a bit.

The Characteristic Equation (CE) for a linear system

Suppose that we have a set of linear differential equations that completely describe the dynamics of a physical system. To determine the character of the natural dynamic behavior of which the system is capable, we simply assume that the behavior for each variable in the equations will be of the form y = Yest. This assumption always works, simply because when you differentiate each variable you get back the original form multiplied by s:

dy/dt= sYest                                                                                             [1]

With this substitution each differential equation is converted to an algebraic one. Then by combining the resulting algebraic equations to eliminate all the capital letters (Y's) you get one simple algebraic equation in s, of the form:

s5 +A4s+ + A3s3 + A2s2 + A1s + A= 0                             [2]

This is the system's Characteristic Equation (CE). The next step is to factor it:

(s + α) (s + B) (s + σ - j∞) (s + σ +jw) (S + y) =0                       [3]

When the roots a, ẞ, etc. are each put into the original assumed motion for y, they connote quantitatively the character of one of the natural motions the system can have, e.g.:

y=Y1 ext                                                   [4]

Any of the Greek letters might be zero, connoting that a root has the value 0. For the complex pair of roots, the value of @ indicates the frequency of a sinusoidal motion, and the value of o   the motion's rate of damping.

Factoring the Characteristic Equation (CE). This is the crux of the matter: determining each root of the CE.
 

Return to Characteristic Equation [2].

For a control system, this equation can be so written that the part of the equation that contains the control variables (which are at the disposal of the designer) are separated from the rest of the CE. For example, the CE could turn out to be of the form:

s (s+b) (s2 + Bs + C) + K(s+a) = 0                                        [5]

where K and a are the control variables that can be selected. When multiplied out, [5] would give back the original CE, [2].

This equation can be written in the powerful Evans Form:

s (s+b) (s2 + B s + C) (s+a) = -K                                            [6]

Next, the values of s that make this equation correct for each value of K are in fact roots of the system's CE for that K. Plotted in the complex plane (the "s plane:" imaginary part of s vs. real part of s) these values of s form. the LOCUS OF ROOTS for the CE. A simple example is plotted in Figure1.

The way the plot is constructed is by beginning with the roots for the value K=0. These roots are connoted by the points labeled X in Figure 1a. Then in principle -- one is to search for all the other points in the s plane that satisfy Equation [6].

Evans’  Incisive Insight

Following Evans' incisive instinct, the search is simply divided into two parts.

Part 1 is finding the loci of all the locations that simply meet the total phase angle 180 degrees that corresponds to the sign in [6]. That is, each of the vectors shown in Figure 1b represents one of the terms in Equation [6], as the labels connote. For a trial point the angles of the vectors are summed; and if the total is 180 degrees the point will satisfy [6]. The locus of all these points in the s plane is THE LOCUS OF THE ROOTS of Equation [6].

Part 2 is to establish the magnitude of K at important points of the locus—for example, where as K is increased the roots of the systems’ characteristic equation are passing from a stable condition to an unstable one (or vice versa), or where a desired level of damping will be achieved

As noted, today all of the above can be done in seconds on a common powerful laptop computer. But in 1948 there were no digital computers.

So, Evans invented a simple plastic device for doing exactly the above computations in minutes without electricity!  He called it the Spirule.

 

THE SPIRULE 

For plotting the locus of roots quickly and accurately.

Part 1. To make a Spirule, you start with a simple protractor, a circular plastic disk of diameter 4 1/2 inches. Using radial lines, mark directions 0o, 90°, 180°, 270°. (0° is of course synonymous with 360°.) Next, mark a tick mark for each degree around the circumference, and label every 10 degrees with its value.

Now drill a hole in the center of the disk for a small circular clamp with a hole in its center. Now clamp to the protractor and atop it a straight-edge ruler ("arm") that extends out 9 1/4 inches from the center of the hole. Give the clamp just enough friction to permit setting the arm at any angle and having it hold well. Finally, give the bottom edge of the clamp a sharp edge so that when you push down on it, it holds fast to the paper.

You are now ready to carry out Part 1 of the process for plotting the roots of any Characteristic Equation. Always begin by putting the CE in the form of Equation [6] and plotting the roots for K = 0 as X's on the s plane.

Next choose a place where for some value of K a root is likely to be found. This is trial and error. But it can be fast for two reasons: (1) You can precede calculations by Spirule by sketching where loci are likely to go using quick Rules that Evans developed, and (2) each trial takes, literally, only seconds.

Thus, to check whether a given spot in the s plane yields a 180° product of the vectors, you simply (i) draw a straight line horizontally to the left from the spot, (ii) set the straightedge to 0° and place the center of the spirule's hole over that spot, and (iii) add up the angles between the horizontal and each X by swinging the straightedge, holding the disk down while going from horizontal to X, and then letting the disk stick to the arm when swinging back. It's really a quick process to find the angle at the point being measured. You mark the value of the product on the plot. Then try another point until you find a 180° point.

It's really fast!
 

SKETCHING RULES

For the Loci of Roots of a Characteristic Equation

A full presentation of Evans' Sketching Rules is given, for example, in Reference 2 (a text by one of us, RC). In this paper we shall present just the first three rules, and demonstrate them for Equation [6], which will provide a good picture of how to do Part 1.

[Figure 2, along with text to show use of a Spirule in drawing a locus: A much simpler version of Sections 21.2 21.5 from my book.]

Part 2. The magnitude of K at any point. With the Spirule's center sitting on any point on a locus, it is entirely straightforward to obtain the value of K there by measuring the lengths of the vectors (using the scale on the straight edge) and then multiplying the lengths together, as Equation [6] indicates. In 1948 the multiplications could be done by using a slide rule, or by entering the lengths into a Marchant calculator and punching "multiply." Time consuming.

Evans invented a more elegant way. You use the logarithmic spiral on the arm of the Spirule. You have a slide rule; but you don't need to set any numbers into it. Instead, you set in the length of each vector graphically.

To insert the length of a vector, you just put the center hole on its point, and the reference line (R) on its tail, and then -- holding the disk fixed – rotate the Spirule until its curve is on the vector's tail. After you've done this for all the vectors you want to multiply together, then you look at the output arrow and read their product from its scale.

Thus, you now have the locus of roots plotted with the values of K at the points of most interest. The first step in designing a good control system is finished: It will be stable with K selected to provide a good margin of stability despite possible changes in physical parameters. Its natural motions will be acceptably quick, and its oscillations will be acceptably well damped.