While mathematicians, scientists, and engineers have celebrated Pi Day—March 14 (i.e., 3-14)—for many years, this year is extra special. Not only will the date 3-14 correspond to the first three digits of the fundamental constant π —and 3-14-15 to the first five digits—but at 9:26:53 a.m., the time will correspond to the first 10 digits of π : 3.141592653!
Companies like Comcast and others are going all out this year to celebrate the unique occurrence, which serves as a reminder of how all of us can highlight the links between cable technologies and our communities—by becoming involved in a local STEM activity like FIRST Robotics or by giving a lecture at your local high school or middle school about how π is used in cable. Cisco’s Ron Hranac, chair of the SCTE Standards Network Operations Subcommittee, and I came up with a few.
The constant π is used extensively in antenna theory, including the relationship between antenna gain and aperture size, the calculation of free space path loss in wireless communications, and many other equations since they rely on a sphere of expansion of electric waves and the surface area of that sphere is 4 π R2.
In electronics, capacitive reactance XC is given by 1/(2πfC), and inductive reactance XL is given by 2πfL. And no electrical engineer or physicist could do Fourier analysis without radian frequency w = 2πf, especially since in DOCSIS® 3.1, the transmitted symbols are actually formed in the frequency domain.
For the cable upstream, the group delay variation currently limits the order of QAM achievable with wideband signaling at the band edges and is given by:I used the word currently because in DOCSIS® 3.1 OFDM, the subcarriers are much narrower and, thus, far less affected by group delay variation.
Headend and data center techs commonly calculate the I2R loss in a conductor. The resistance R of the wire is the resistivity of the material (either copper or aluminum) times length of the wire divided by the cross-sectional area of the wire, which is πr2, where r is the radius of the wire. Multiplying by the square of the current in the wire gives the power lost due to Joule heating in the wire. That’s why larger diameter wires have lower heating losses and also why the industry is considering 380 V DC in data centers.
An old construction lineman’s trick for guesstimating the length of cable remaining on a reel-end is to multiply the diameter of the center of a cable reel by 3, then multiply that by the number of turns. For better accuracy, use π instead of 3.
Like π itself, the number of ways that the constant can be used in cable engineering seems to have no end.
How do you use it? Let us know!