# Engineering Graph Paper

In 2019, I began working on designing and printing nomographic charts useful in electrical engineering. I wanted them as a handy complement to computer simulation tools. Attention to typography and graphical precision were the driving motivation for making my own charts. In 2021, I released printable PDFs of charts. They are free to download and use from Pen & Paper Nomographic. This page serves as a reference on the use of reactance paper, and some discussion of the process behind making the charts.

## Reactance paper

In studying passive networks in electronics, it’s important to understand the frequency-dependent nature of components and how they interact when they are connected in series and parallel. One way is to derive a mathematical expression for the impedance or frequency response, but this becomes tedious and complicated, and it’s often no longer illuminating once more than 3 components are involved. Another method is to use a circuit simulator like LTSpice, which can produce exact results very quickly, but it also often obscures how individual components contribute to features in the graph.

A third method used by engineers before the era of digital computers was specialty graph paper called reactance paper (or impedance paper). It permits quick, approximate characterization of passive networks. Reactance paper is a log-log chart of impedance vs. frequency, with diagonal lines for the frequency-dependent impedance of capacitors and inductors.

As an example, to see how a 1 kΩ resistor and 1 nF capacitor interact, draw asymptote lines at the component values. For a series connection, trace along the maximum of the lines. This device works because impedances add in series, and on a logarithmic scale, $\log(x + y) \approx \log x$ when $x \gg y$. The intersection of two lines tells us the corner frequency, and that this combination will act like a 1 nF capacitor well below 160 kHz, and a 1 kΩ resistor well above 160 kHz. The exact impedance chart (shown in orange) doesn’t have a sharp corner and smoothly blends between the asymptotes, but this a good approximation to start with.

While it is the case that “impedances add in series,” it’s only true when they are considered as complex numbers, so a little more care is required when looking at resonant circuits. When the lines for a capacitor and inductor cross on the chart, the impedances are equal and opposite imaginary quantites, and the resulting sum is zero. For an example involving a resonant circuit, consider a series RLC circuit of a 100 pF capacitor, a 100 nH inductor, and a 10 Ω resistor. Drawing the intersection of the inductance and capacitance lines shows the resonant frequency is 50 MHz, and characteristic impedance of this combination is a little over 30 Ω. The impedances cancel each other at this point, which would show as a sharp notch on the chart, except it is bounded below by the series 10 Ω resistor. The ratio of the characteristic impedance and series resistance means the $Q$ of this circuit is about 3. All of this can be read from the chart without doing calculations. It’s also easy to see how things change if we vary the values of the components.

When circuit elements are connected in parallel, the impedances combine as the inverse of reciprocal sums:

$Z = \frac{1}{1/Z_1 + 1/Z_2 + \cdots} = Z_1 || Z_2 || \cdots$

If asymptote lines are drawn at the component values, then the impedance of the parallel combination can be found by tracing along the minimum of lines. This can be seen because the smallest impedance will have the largest inverse, and will dominate the sum of reciprocals.

As a variation of the previous series RC circuit, consider the parallel connection of a 1 kΩ resistor and 1 nF capacitor. Following the minimum of the asymptotes shows that this combination acts resistively below the corner frequency of 160 kHz and capacitively above it. This matches our intuition of the circuit diagram; at low frequencies, the capacitor is nearly an open circuit, and the current must flow through the resistor. At higher frequencies, the impedance of the capacitor decreases and it becomes the easier path for current.

Returning to the previous resonant circuit but with components in parallel, the above formula shows that at the resonant frequency, the impedance of a parallel capacitor and inductor becomes infinite. The ratio of the parallel resistance to the characteristic impedance indicates $Q\approx \frac{1}{3} < \frac{1}{2}$, so the circuit is said to be overdamped and it will not oscillate at all. Instead of a conjugate pole pair at the resonant frequency, the poles separate into a pair of real poles. A result known as the “low-$Q$ approximation” states the poles should be located at about $Q$ and $1/Q$ times the resonant frequency, or approximately 16.7 MHz and 150 MHz. All of this can be read easily from the chart; by the following the minimum of the asymptotes, the overdamped behavior and pole frequencies can easily be seen without doing any algebra.

I found working with reactance paper so helpful for quickly characterizing passive networks that I wanted to have a notepad of them, but I found only mediocre versions online. I made my own using PGF/TikZ, a graphics layer built on $\TeX$ that is well suited for technical illustrations. I borrowed the color scheme used on FAA airspace charts, since they have a useful, but muted contrast. I found that a frequency range of 1 kHz to 1 GHz, and an impedance range of 100 mΩ to 1 MΩ covered the majority of typical analog circuits.

Another well known nomogram used in electrical engineering is the Smith chart. The Smith chart is useful for understanding transmission line dynamics and designing antenna matching networks in RF engineering. Like reactance paper, it allows for solving classes of problems without tedious calculations, but it has also been supplanted by the use of computers that produce exact results quicker. Nonetheless, the Smith chart remains a standard way of displaying reflection measurements on test instruments, and the paper version is occasionally used for quick problems. And admittedly, it just looks cool. In trying to find a printable Smith chart, I was again disappointed in available online versions. Drawing your own is a fun technical and graphical challenge, and PGF/TikZ was again up to the task.

Paper nomogramic charts like the Smith chart and reactance paper reached their peak of popularity just before the advent of digital computers. To evoke the sense of that period, I chose Futura demibold for the textual elements, recognizable from its use in the Apollo program instrumentation panels and its unmistakable mid-century technical look. As a geometric font with simple letter forms, it is legible even at small sizes and high density. I updated the title box from the long out-of-print Analog Instruments charts, and branded them under “Pen & Paper Nomographic” — a nod to the era when engineering was done by hand on notepads and notebooks.

I did a trial run of four pad variants: portrait and landscape versions of reactance paper, and impedance-only and combined impedance-admittance versions of the Smith chart. They were digitally printed on light buff 20 lb. letter paper with 3-hole punching. The choice of digital printing was a compromise since lithographic offset printing produces higher-quality results but it has high setup costs. Unfortunately, the digital press results were a disappointment. The finest lines on the charts were broken, and the lines from the digital press’ “liquid toner” were still waxy enough to smear ink drawn on the paper.

## Sharing the result

Te best way to produce high-quality paper charts is offset printing, but unless I can crowdfund a large enough press run, it’s not economical to go forward with this project. I’ve decided to share the the reactance paper and Smith chart PDFs on a dedicated site, with permission granted for all individual use.

## References

An excellent discussion of graphical analysis techniques using reactance paper, including an extension of the technique to transfer functions, can be found in Fundamentals of Power Electronics by Erickson and Maksimović (under §8.3 Graphical Construction of Impedances and Transfer Functions).