The Simplified Frequency Formula

When I was a kid getting into radio and electronics, the only formula I knew to calculate the tuned frequency of an inductor and capacitor wired in combination was the tried and true formula dating back to the dawn of radio:

The result f is the frequency in Hertz of the tuned circuit (logically called cycles per second back in the day), L is the inductance in Henries, and C is the capacitance in Farads. Farads? Henries? Capacitors in the Farads range are huge, like railroad boxcars, they used to say. Coils in the Henries range are heavy (remember those 5U4GB tubed power supply filter chokes?). Conversely, inductance at mediumwave broadcast frequencies is measured in micro henries (millionths of a Henry, or µH) and capacitance is measured in pico farads (millionths of a millionth of a Farad, years ago noted as µµF). Wow, that's a mouthful.

Let's apply this old formula to a commonly-seen tuned circuit, a 240 µH inductor and a 365 pF capacitor wired in parallel. How confusing is this?

f = 1 / (2 * pi * Sqrt(.000240 * .000000000365))   [Sqrt = square root]

Now how many zeros am I supposed to add ahead of the 365 pF capacitor? The inductor?

I remember trying to work this formula before electronic calculators. Lots of zeros. Remember the time before electronic calculators? Okay, well, maybe you don't. Too many zeros to keep track of, in fact, too many zeros to keep track of even if working with a calculator. Oh, use scientific notation? Times ten to the minus what? Please show me (simply), at what frequency a 240 µH inductor and a 365 pf capacitor resonate?

Then one day, years later, I came upon the formula:

L * C * f * f = 25330

In the formula above, f is the frequency in megahertz, C the capacitance in pico farads, L the inductance in micro henries. 25330 will always be the same in every calculation. Inductance times capacitance times frequency times frequency will always equal 25330. Magic. Okay, now we're talking simplicity.

Using this formula, it's also very easy to change the operands around to calculate any one of the values, when the others are known.

Solving for Inductance (in µH):

       L = 25330 / (C * f * f)

Solving for Capacitance (in pF):

       C = 25330 / (L * f * f)

Solving for Frequency (in MHz):

       f = Sqrt(25330 / (L * C))

A few examples:

1. Find the resonant frequency of a 240 µH inductor and a 365 pF capacitor:

       .537 MHz = Sqrt(25330 / (240 * 365))

2. Find the inductance required to resonate at 530 KHz using a 365 pF capacitor:

       247.05 µH = 25330 / (365 * .53 * .53)

3. Find the capacitance required to resonate at 1700 KHz using a 247.05 µH inductor:

       35.47 pF = 25330 / (247.05 * 1.7 * 1.7)

Henry Thoreau said, "Simplify, simplify, simplify!". In this formula we have done that.

Can we make the original formula of old more workable to find frequency? Yes we can. Using our simple whole units of micro henries and pico farads, we use a multiplier at the end to get our result in KHz or MHz.

       f(MHz) = (1 / (2 * pi * Sqrt(240 * 365)) * 1000

       f(KHz) = (1 / (2 * pi * Sqrt(240 * 365)) * 1000000

That's a little easier to manage than the original without all the zeros. Should we rearrange the formula to calculate for inductance or capacitance? Well, we could. But why bother now that we have the simplified formula, L * C * f * f = 25330!

Now, why would we be wanting to calculate the resonant frequency of an inductor and capacitor? Let's say you plan to build a passive loop and want it to tune the mediumwave band. You've decided on a two foot square loop of 11 turns. Your loop calculator widget (or LCR inductance tester for those who can afford one) shows this loop to have a certain inductance. Pair that with your trusty old 365 pF broadcast tuning capacitor and using the frequency formula you can calculate the tuning range of the loop. Or roughly so.

We will have more on the topic of passive loop frequency calculations in a future article, including a custom loop calculator program. Stay tuned.

On the Road Again

"On the road again...." Actually I have been "on the road again" since before I wrote last, and have completed the annual road trip west from New York to Arizona by way of Denver. The last of the Mediumwave Along the Erie Canal series was posted while enroute, from Fraser, Colorado on October 9th, while sitting in six inches of snow in the high Rockies.

Had a good trip. Saw some new mediumwave towers in Kansas and Colorado and photographed them. Am settled now in the southwestern desert an hour north of Old Mexico, just east of the Colorado River, and am back to contemplating our radio hobby and DXing again.

Check back soon for some new posts!

Mediumwave Along the Erie Canal, Part 3

Continuing with our exploration of mediumwave station sites along the Erie Canal.

We are walking the Erie Canal path in the Rochester, New York area, headed west from Lock 33. Within a one mile stretch there are three mediumwave stations lining the canal. In the last part of this series we passed within sight of WHTK-1280.

Hiking along the canal past WHTK-1280 perhaps another half mile brings us to the Clinton Ave. overpass, situated just shy of the four-lane I-390 overpass. Poking above the trees to the north are the four equal height towers of WROC-950, and shown in the photo just below. Three are in direct alignment and the fourth sits off some distance broadside to the northeast. They are each .193 wavelength in height. Take the side path off the canal to the north about 50 yards and you have a good view of the tower site. WROC-950 is one of Rochester's oldest AM radio outlets. Both daytime and nighttime powers are set at 1KW.

The tower configuration at WROC is a bit curious. Three towers are used at night, those three in direct row alignment, broadcasting with a main lobe at 353 degrees, and a very minor lobe at 173 degrees. Gain in the favored direction, north towards the main population center, is 6.2dB.

Daytime coverage is bi-directional, using two towers, tower #3 from the aligned row and tower #4, the tower broadside to the northeast. The main lobes are at 343 degrees and 163 degrees, generating about 4.8 dB gain each.

WROC-AM's call sign is a reference to WROC-TV. While the stations are not and have never been co-owned, WROC-AM has an agreement with WROC-TV to provide local news coverage, and the borrowing of the WROC call signs from WROC-TV is included in this agreement. Coincidentally, the WROC-AM call sign was previously held at WHTK-1280 from 1961 into the 1970s, and was co-owned with WROC-TV while at that location.

WROC-950 began broadcasting in 1947 under the call sign WARC. It was an early affiliate of the ABC radio network, but later changed to a locally programmed, personality-driven popular music station. It was purchased by the B. Forman regional department store chain in 1953 and changed its call letters to WBBF, the last three letters of which stood for "Buy B. Forman". In 1966 it was sold to LIN Broadcasting for what was then a market record of over $2 million, but retained its popular music format and personality lineup until the early 1980s.

As WBBF, 950 AM was a popular Top 40 music station in Rochester, often leading the market in ratings surveys from the 1950s through the early 1970s, and ranking among the city's top stations through the late 1970s even after strong format competition arrived in 1972 from WAXC-1460 and later on the FM band from WPXY. Consistent success was achieved although as a relative latecomer to the AM band in the postwar era, WBBF's coverage area had to be restricted to the east and west to prevent interference with other stations on the same channel. In 1982, as hit music radio listeners were migrating to FM, the station evolved into a talk format. WBBF served a short stint as WEZO from 1998 to 2002, then adopting the WROC call sign. The WBBF calls are now in use in Buffalo, New York.

In September of 2004, WROC signed on with the liberal Air America network, having previously carried conservative talk. In September of 2008, WROC became an ESPN affiliate. The format is closely affiliated with Buffalo sister station WGR-550.

Shown below is WROC-950's nighttime pattern plot. On the plot are shown various co-channel stations around the region and where they fall into WROC's pattern.

This concludes this series. Hope you have enjoyed it.

Mediumwave Along the Erie Canal, Part 2

Continuing with our exploration of mediumwave station sites along the Erie Canal.

We are walking the Erie Canal path in the Rochester, New York area, headed west from Lock 33. Within a one mile stretch there are three mediumwave stations lining the canal. In the last part of this series we passed within sight of WXXI-1370.

Walking just a little further along the canal path, coming into view is the four tower array of WHTK-1280 protruding above the Winton Rd. overpass, and shown in the photo just below. Its equal height .339 wavelength towers stand in a perfectly-aligned row at the northwest corner of Winton Rd. and Henrietta-Townline Rd. behind an industrial park, and just across the canal to the south. Passing under the overpass and walking about 100 yards further we get a great view of WHTK's towers.

WHTK-1280, also known as Sportsradio 1280, obviously airs a sports radio format. It is fully simulcast on WHTK-FM (107.3). WHTK is a Clear Channel Communications affiliate and is owned by Citicasters, Inc., featuring programing from Fox Sports Radio and Westwood One as well as New York Yankees, Rochester Americans and Rochester Red Wings games among other local and national sports. It transmits in AM-HD (IBOC). Both daytime and nighttime powers are set at 5KW.

Daytime coverage is omni-directional, using only one tower. The four tower array in use at night broadcasts with a main lobe at 346 degrees, and a minor lobe at 166 degrees. Gain in the favored direction, north again towards the main population center, is a respectable 7.5dB, pushing an effective 28.3KW towards Rochester.

The station was first known as WVET, signing on in 1947 under ownership of a group of returning World War II veterans calling themselves Veterans' Broadcasting Company. It operated successfully for many years with a personality full service adult popular music format. It changed callsign from WVET to WROC when Veterans bought WROC-TV from Transcontinent Television Corporation in 1961. Simultaneously an FM sister station, WROC-FM, signed on, first playing classical music and later automated jazz and pop standards. Veterans Broadcasting sold all the WROC stations in the mid-1970s. The AM station continued with its full service format until late in the 1970s, when it tried an all-news format first as WROC and then as WPXN (AM). It would later simulcast its FM sister station, by the early 1980s known as WPXY (FM) amd airing the personality contemporary hit music format which it still runs today. Late in the 80s, after changes in ownership, it would migrate to pop standards and then to mostly syndicated "hot talk", a lineup of talk and sports programming meant to appeal to young adult men. At that time it adopted the WHTK callsign (the "HTK" meant to stand for "hot talk") which it still uses today.

Shown below is WHTK-1280's nighttime pattern plot. On the plot are shown various co-channel stations around the region and where they fall into WHTK's pattern.

Stay tuned for Part 3 of this series.