**The Decibel and its relationship to S meter displays by Lee Bond, N7KC**

March 4, 2009 Educational Radio Net, PSRG

For the 41st session of the Educational Radio Net, I have chosen to continue a review of basic and important concepts that cannot be avoided when dealing with radio equipment. Earlier sessions dealt with the relationship of energy, power, time, voltage, current, and resistance. The 13 parts of the Impedance series is a good starting point for individual review. This week I will delve into the decibel and its relationship to S meter displays.

Before I start let me say that I was poking around Wikipedia looking for information on ‘bels’ when I ran across the decibel offerings. Turns out that TU or transmission unit was a popular telephone unit of transmission gain or loss in the early days of miles of wire associated with the telephone industry. Eventually the attenuation resulting from about 10 miles of wire loop become known as a ‘Bel’. Since the Bel is a large unit, the decibel, or 1/10 of a bel became a more useful unit of measure. Wide usage of the decibel followed.

We humans like to compare things. For example, how much is a gallon of gasoline in Washington compared to a gallon in Oregon or what is the price of a certain apple at QFC compared to the same apple at Safeway? One might say that Washington gasoline costs 1.5 times as much as Oregon gasoline which is the same as saying that the cost ratio of Washington gas to Oregon gas is 1.5. Notice that there are no units associated with this ratio. Dollars per gallon in the numerator cancels with dollars per gallon in the denominator yielding a sterile number of simply 1.5… that is, a number with no units.

So it is with the decibel. The decibel is a comparison of two like objects in the sense that these objects must have the same units. For example, by using decibel notation you can compare two voltages or two powers. In both cases the units of voltage or power will cancel yielding a sterile ratio. The sterile ratio is processed by the decibel conversion mathematics to produce a number associated with base 10 logarithms. Decibel notation for power comparisons is always 10 log (ratio). Decibel notation for voltage ratios is always 20 log (ratio) since power goes as the square of the voltage.

Uh oh… did I mention logarithms? Well, lets try to make some sense of this mysterious thing called a logarithm or log for short. To be complete I must say that there are two systems of logarithms. One system is based on the number e and is called the Naperian or natural log system and is very important in a certain branch of mathematics. The other is based on 10 and is called the common log system. Our interest is with the common log system so lets get a feel for just what is going on with the use of these logarithms.

The common log is a combination of two things… something called the characteristic in conjunction with something called the mantissa. Once you know the underlying idea behind each of these two parts you can use them very effectively in your radio work. Lets start with the number 100 as an example. We all know that 10x10 equals 100. Another way to say 10x10 is 10 squared so 10 with the little ‘exponent’ 2 equals 100. Guess what… the little exponent number 2 is the log of 100. Now, lets look at the number 1000. We also know that 1000 can be written as 10x10x10 or 10 cubed or 10 with a little exponent 3. Easy enough. The log of 1000 is the exponent 3. We have just determined the characteristic by inspection. Providing the number is positive, the characteristic is one less than the number of digits left of the decimal point. Wait you say… what if I have a number between 100 and 1000… perhaps 300. Clearly the log of 300 must be greater than the log of 100 and less than the log of 1000. Enter the mantissa. The log of 300 is 2.4771 which is simply one way to say 10 to the 2.4771 equals 300. The 4 decimal places to the right of the decimal point is the mantissa so the complete log is the sum of the characteristic and the mantissa. Old timers from the slide rule era will know that numbers on their rules were located according to the logarithm of the number so the log of 3 is located about mid point on the rule. Logs are a good way to represent very large numbers in a compact form. When multiplying scientific notation numbers one just adds the exponents which is consistent with adding lengths on the slide rule. Finally, I am going to divulge the big secret… if the base 10 log of the number N equals Y then this is exactly the same thing as writing 10 raised to the Y power equals N. Exponential and logarithmic notation are completely consistent. Simply different ways of saying the same thing. Before the hand held electronic calculator reigned supreme one would determine the characteristic by inspection and then look in a table of common logs for the mantissa then add the two or use the slide rule to find the log. Today you just punch in a number and the log magically appears on the display.

Back to the decibel. If your transmitter develops 150 watts output power with two watts input power then, clearly, there is some power gain. The ratio is 150/2 or 75. The log of 75 is 1.8751 and 10 times that is 18.751 db. Thus you have two ways of expressing the same phenomenon… one as a ratio of 75:1 or another as +18.751 decibels. Why would you choose one over the other? Probably the best answer is that decibels are additive so, for example, if you have one amplifier following another then the overall power gain is simply the sum of the individual gains expressed as decibels. Decibel notation without any reference power is simply a ratio expressed in logarithmic form.

So much for db’s. What about dbm? Enter the reference power. In the previous example where the transmitter output power was 150 watts, the question becomes what is the ratio relative to 1 milli-watt. Given that 1 milli-watt is defined as 0 dbm we need to find out how 150 watts compares to 1 milli-watt. The answer is found by comparing 150 to 0.001, taking the common log and then multiplying by 10. So, key into your calculator 150 divided by .001, find the common log to be 5.1761, multiply by 10 to end up with 51.761 dbm. We can also convert the 2 watt input power to dbm and we find the answer to be 33.01 dbm. What is the difference between 51.761 dbm and 33.01 dbm? 18.751 db. Notice that we say that the gain of the amplifier is 18.751 db… not 18.751 dbm. Dbm is a measure of absolute power whereas db is a measure of relative power.

Now that we have some feel for decibel notation we can move on to signal strength meter displays and make some sense from what we see. As you know, these meters start with S1 through S9 then progress to S9+10 followed by S9+20 and so on. S8 is defined to be 6 db less than S9, S7 is 6 db less than S8, and so on down to S1. S9+10 is defined as 10 db greater than S9 and so on. The generally unknown fact is that S9 is equivalent to 50 microvolts rms at the receiver input terminals when measuring high frequency signals and 5 microvolts rms when measuring VHF/UHF signals. Both definitions require the receiver input impedance to be 50 ohms. Given that power goes as the square of the voltage… remember Joule’s Law… voltage decibels are defined as 20 log (ratio) and, therefore, voltage amplitude either doubles or halves for every 6 db change. Now we have enough ammunition to compute some numbers. If S9 corresponds to 50 microvolts and S8 is ½ or 6 db lower, then S8 corresponds to 25 microvolts. Going down by 2’s reveals that S7 corresponds to ½ of 25 or 12.5 microvolts and so on. Going the other way above S9 we need to compute what 10 db above 50 microvolts represents. I will not bore you with the computational details so the answer is 158 microvolts. S9+20 db pencils out to be 500 microvolts which is no surprise given that 20 db in voltage terms is a factor of 10 so 50 microvolts becomes 500 microvolts.

Unfortunately early S meter circuitry did not follow these rules perfectly and most meters were notorious for their errors. Many operators would report their S meter readings to be ‘Scotch’ if they were lower than the recipient would like to hear. Modern meters are much better performers when reading input signal strength. Measuring S meter accuracy requires some careful work with fancy test equipment so generally out of the realm of the average amateur operator.

This concludes the set up discussion of the decibel as it relates to S meter readings. Are there any questions or comments with regard to tonight's discussion?

This is N7KC for the Wednesday night Educational Radio Net

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