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October 1999 Issue
Hranac-Notes for the Technologist
By Ron Hranac

The Ongoing dB "d-Bate"

• Hey, we all make mistakes
• Stand by for math
• Spare me further math and suffering
• Still more examples
• Bottom Line

	Ron Hranac

Lets see, where were we?

I believe we had taken a couple of arbitrarily chosen signal powers (0.0000133 watt and 0.000000133 watt) and converted them to voltages using a variation of the formula P = V²/R. In that formula, P is power in watts, V is voltage in volts, and R is resistance, which in this case is the 75-ohm impedance of our coax networks.

Hey, we all make mistakes

As I look at the manuscript for the first installment of this drama, it appears that I inadvertently got my answers backwards (oops!) because 0.0000133 watt is 0.03162 volt (31.62 millivolts) and 0.000000133 watt is 0.003162 volt (3.16 millivolts), not the other way around. If you spent the past few weeks trying to figure out why your calculator gave you the correct answers instead of the ones I gave, consider it a test to see if you were able to stay awake through the whole exercise!

Anyhow, dealing with all those zeroes and decimal places can get confusing and clearly can lead to mistakes (ahem!) when sorting through calculations. This could get even nastier if we had to interpret system signal levels on a signal level meter (SLM) that displayed those levels in volts, millivolts or microvolts.

Fortunately, there is a much easier way to deal with these often cumbersome numbers. In the world of cable TV and our 75-ohm impedance networks, the solution is based on the decibel, but with a reference appended to it: decibel millivolt, or dBmV.

Stand by for math

Invoking good ol ratios, we can compare signal voltages in millivolts to a so-called zero dB reference of 1 millivolt, where 1 millivolt equals 0 dBmV. Mathematically, dBmV = 20log(level in millivolts/1 millivolt). A side note here: Converting from watts to volts via Ohms Law gives us "20log" in the voltage world instead of the "10log" used in the power world.

All that being so, whats 3.16 millivolts expressed as dBmV?

dBmV = 20 x [log(level in millivolts/1 millivolt)]
dBmV = 20 x [log(3.16 mV/1 mV)]
dBmV = 20 x [log(3.16)]
dBmV = 20 x [0.5]
dBmV = 10

Its +10 dBmV. In other words, 3.16 millivolts is 10 dB greater than the 1 millivolt reference. If you plug 31.62 millivolts into the formula, you get +30 dBmV, which simply says 31.62 millivolts is 30 dB greater than the 1 millivolt reference.

dBmV = 20 x [log(31.62 mV/1 mV)]
dBmV = 20 x [log(31.62)]
dBmV = 20 x [1.5]
dBmV = 30

Its all about ratios. One nice thing about working with decibels is that much of the math is reduced to addition and subtraction. Back to that amplifier gain example I used in last months column for a moment: If the input is +10 dBmV (3.16 millivolts, or 0.000000133 watt) and the output is +30 dBmV (31.62 millivolts, or 0.0000133 watt), the gain is 20 dB. Or, (+30 dBmV) - (+10 dBmV) = 20 dB.

Now then, which would you rather deal with: Signal voltages such as 0.003162 volt and 0.03162 volt, or +10 dBmV and +30 dBmV? See how easy all of this becomes using the decibel? By the way, if you feel so inclined, you can convert dBmV to millivolts using the formula:

millivolts = 10(^dBmV/20)

Suppose you want to figure out how many millivolts +6 dBmV is. Plug the number into the formula, and work a little magic with your scientific calculator. Well, not really magic. You have to use the exponent key. On my calculator, its the one marked 10^x. The equations follow:

millivolts = 10(^dBmV/20)
millivolts = 10(^{+6 dBmV/20})
millivolts = 10(^0.3)
millivolts = 1.995

Spare me further math and suffering

Using the formulas discussed so far, you can make yourself a handy dBmV-to-millivolts conversion table. Ive created a small one (see below), including values from -10 dBmV to +10 dBmV.

As you look at the table, it should become obvious that the concept of dBmV is nothing more than ratios of various voltages to a defined reference. For example, 2.5119 millivolts (+8 dBmV) is 8 dB greater than 1 millivolt, 0.3981 millivolt (-8 dBmV) is 8 dB less than 1 millivolt, and so on. While dBmV technically is nothing more than a ratio, it provides an indirect but convenient way to express absolute signal levels using the decibel.

Still more examples

What about other examples where different references have been appended to the decibel? Youve probably seen some of them: dBµV, dBm, and dBW are three that come to mind. These are abbreviations for decibel microvolt, decibel milliwatt, and decibel watt. The 0 dB references are 1 microvolt, 1 milliwatt, and 1 watt respectively. In each case, some signal level is being compared to the 0 dB reference, resulting in a ratio of that signal level to the defined reference. Here, too, the decibel provides an indirect way to express absolute signal levels. The formulas for these three are dBµV = 20log(level in microvolts/1 microvolt), dBm = 10log(level in milliwatts/1 milliwatt), and dBW = 10log(level in watts/1 watt). Note that the two direct expressions of power are 10log-based, while voltage is 20log.

Ron Hranac is vice president of RF engineering for Denver-based High Speed Access Corp. He also is senior technical editor for Communications Technology. He can be reached via e-mail at .

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