Having Fun With Decibels
Having fun with what? Do you think I am some kind of nerd? Well, I have fun with decibels and I am no nerd. I love to go to audiophile cocktail parties and demonstrate how many drinks I can have and still do decibels in midair, no calculator, no chart, and no props at all. I assure you I have not memorized the table many of you use nor have I converted my brain to a logarithmic computer. I still think quite linearly, well at least about decibels. I started the practice some ten years ago out of some simple observations about decibels. You will have to memorize four simple things and understand what the reasoning behind decibels is about. That is probably the most beneficial part of the trick as you will truly understand decibels when you are done with this. You will also be able to pick out the multitude of errors in publications and become a decibel proofreader if you like.
Now let us get a few things straight. I insist that all my students develop the proper language in speaking about electronic matters. I do this because right language promotes right thinking. First, there are no such things as voltage decibels and power decibels, there are simply decibels. That is the elegance of the system. There is indeed a relation for deriving decibels from voltage and another one for power, but once in decibel form, the number stands alone in its simplicity. The formula for decibels is dB=20logV1/V2 which is spoken, “Twenty times the log (base ten) of voltage one over voltage two. Well, I do not talk like that around my house. Besides I may be able to divide two numbers in my head but how am I going to figure out the base 10 log in my head. What is a base 10 log anyway? For those of you that care it is simply the exponent of the number 10 that yields the number you are concerned with in our common base 1 number system. It is like a converter between a base 1 number system and a base 10 system.
So, what are decibels and how are they useful? As engineers we want a way of specifying things in a relative way. Ratios are useful things and easier to deal with than absolute numbers. It is easier to say, "I have half as much," than to say, "Well I first had this much and now I have this much," or to say, "The voltage decreased 10%," instead of, "The voltage went down from 120 volts to 108 volts." If I make a 10 watt amplifier that is 5 watts shy of its rated output, you should be upset with me for cheating you out of half of what was promised. If I make a 250 watt amplifier that is 5 watts shy, I hope to be forgiven for being off by 2%. My experience those of you who have accounting type minds tend to count every bean where those of us with science type minds are happier with the percentage to which things are accurate. If you are an accountant, try my little game on the other side of your brain.
Now if I say my amplifier is flat to within 1 dB, I am saying that it reproduces all frequencies to within 10% of accurate level. Generally, it will do this at any level within the operating range of the amplifier. Since amplifiers do have trouble at full power, we often have another set of specs for that. Now where did I get that 10% figure? That is one of the first things for you to memorize. You can confirm it on a decibel table or on your calculator. The actual figure is 12.2% but I am more interested in making it easy for you. Note that we rarely need to know decibels to more than two places of accuracy. What is more important to me is being able to spot gross errors and understand the system. If you object to my calling 12.2% as 10% then use 12%. I have been quite successful with the rounder numbers.
Here is where some or you may say, "Well that is fine for voltage but what about power?" If you know Ohm's law, you know that power goes as the voltage squared. That is the two martini way of saying P=V squared/R which is a very nerdy thing to say among polite company. So, if you increase something 10% and then 10% again you get 21% more. That is called compound interest in the banking world. So, power is like compounding. Not being a banker and remembering that I cheated you out of 2.2% on the voltage side, I will round this off to 25%. The actual number is 26%. For the math majors, dB = 10 log P1/P2, which makes perfect sense as 10 times the log takes care of the square relationship between voltage and power. Remember that if you have lost 1 dB, you have lost 12% in voltage which will lose you 25% (12% compounded) in power. As far as the amplifier or your ear is concerned it is all the same. Just a simple dB. That is why Alexander Graham Bell worked out the system the way he did instead of percentages.
Now that you know what 1 dB is let us go on to lots of them. You can compound the value of a dB all you want by multiplication but with increasing errors if you approximate as much as I do. Besides, three martinis are my limit for compounding. There are a few exact numbers you should remember to make it a lot easier when the dBs are large. For example, 6 dB is a popular figure and is exactly two times in voltage which means four times in power because 2 x 2 = 4. I told you this is easy! Another one is the ubiquitous 3 dB which is 1.414 that many of you will recognize as the square root of two. So, when something is up 3 dB it is up 1.414 times and when it is down 3 dB it is down to .707 which is 1/1.414. That is what is neat about decibels. Positive dB are ratios greater than 1 and negative are less than 1. To turn positive dB into negative simply invert the ratio. Since we simply square voltage ratios to get power ratios, 3 dB in power is exactly two and the square root of 2, squared, is two, exactly! As you see, I am exact when it matters and rather approximate when the error introduced will be small. I also balance my checkbook to the nearest buck or two rather than the penny.
The last one is the big steps which are exact also. For example, 20 dB is a voltage ratio of 10 and again must be squared for power making it 100 times. Note that a ratio of 10 in power is 10 dB or 1 Bel and that is the basic unit. Do not bother to remember this but note that the voltage ratio that corresponds to this is 3.1622765 which is a silly number (the square root 10) but an interesting number as it appears on every AC voltmeter that is marked with a 10 volt and 3 volt scale system. If you have one run and look at it right now and make a martini on your way back so we can practice dB converting in the air.
So here are the five rules in review, all pertaining to voltage or power ratio:
1 dB = 12% (though I often use 10%)
3 dB = 40 %
6 dB = 100% or two times
10 dB= 3 times
20 dB = 10 times, 40 dB =100 times, 60 dB = 1000 times, etc.
The rest I call, "knowledge of the system". You must know that power goes as voltage squared and that ratios larger than 1 are positive in dB and ratios less than 1 are negative. You must also be prepared to argue with most everyone who will disagree with you from their misunderstanding of this most confusing topic. Just remember, you read the article and you are right. You can carry a dB table in your pocket to prove it.
So, let us do a few conversions. Say you want to convert 46 dB. Notice that it is made up of 40 and 6. I simply convert the 40 dB to 100x and the 6 dB to 2x and note 100 x 2 = 200. So, 40 dB = 200 exactly. You can also see it as 20, 20, and 6 which yields 10 x 10 x 2 = 200. 34 dB is also easy if seen as 6 dB less than 40. Simply take the 40 as 100x and multiply by one half, which makes 34 dB = 50 exactly. What about 12 dB? Just note that 6 + 6 =12 and that converts to 2 x 2 = 4 exactly. You may now be remembering some formerly useless thing you learned in math class that adding logs is the same as multiplying numbers. That is what the dB system is, a convenient way of multiplying numbers by adding. Note that it is the same system a slide rule uses to multiply numbers. I remember the day in physics class that I realized that I was holding a log chart on a stick. Before that I thought a slide rule was some Egyptian artifact.
Now let us take some approximate ones like 1/2 dB, that is just 5% (or 6% is you are fussy). Getting things down to 0.1 dB accuracy is quite fussy as it requires 1% accuracy. That is why most of us use 1% resistors in feedback amplifiers to set the gain to 0.1 dB accuracy. There are two resistors that determine gain in feedback systems so 1% resistors yield 0.2 dB accuracy worst case if the loop gain is high or very constant. What about 2 dB? I call it 20% even though it is 25.9%. You may object but when dealing with big dBs it hardly shows. Take 62 dB for instance. The easiest way to take it apart is into 60 dB and 2 dB. We know that 60 dB is 1000 exactly and the interest on $1,000 at 20% is $200 so it must be $1,200 or 1200x. The exact figure is 1259 and is only off by 4.6% or 0.46 dB. So, we managed to convert a large number to within 1/2 dB in our heads. Remember the purpose of this is to check the believability of published numbers. When someone says an amplifier’s noise is 112 dB below 1 watt you may begin to wonder. Let us look at that.
It takes 2.83 volts to produce 1 watt in an 8 ohm load (V squared/R = power). If we say the signal to noise (S/N) ratio is 112 dB, we mean that the noise is 112 dB below this level. Breaking 112 dB into 100 and 12 we see that the ratio is 100,000 x 4 = 400,000, a large number. The noise is then 2.83 x 1/400,000 which is 7.1 microvolt (mV)s. As an amplifier designer, I want you to know that it is virtually impossible to achieve those levels of output noise at the output terminals of a power amplifier, yet I see numbers routinely reported in this range. It is tricky enough to get them out of a moving coil pre-preamplifier where every microvolt has been accounted for. Typical noise from a power amplifier with the input shorted is 100 uV for a quiet one which is 90 dB. Now how did I get that? That was a tricky one.
Let us practice working the other way. I will note that 2.83 V is about 10 dB above 1 V and 100 uV (0.1 mV) is 80 dB below 1 V. The difference is then 90 dB. As you can see, we can even handle the messy reference level of 2.83 V since I brought it up. As decibels are a relative number, we must give a reference when we use them to express voltages or power directly. It is a bit of double speak to do this, but it has become increasingly popular over the years. I prefer to express voltages and powers directly in volts and watts. The most popular absolute decibels are dBV, which is dB above and below 1 V exactly. This is common on laboratory AC volt meters. In this language 0 dBV = 1 V, -60 dBV = 1 mV, and 60 dBV = 1000 V. The sign is very important especially if your hands are on the terminals.
A popular reference in the recording studio is 0 dBw or 0 dBm = 1 milliwatt (mW). This is tricky as the load resistance must also be stated to convert it into voltage though not to convert its power. In voltage it converts to .775 V which is the voltage required to produce 1 mW in a 600 ohm load. The standard came from the phone company more than 100 years ago and was the reference level for conversations on the line. They built the first VU meters and established this level as "0 VU" (Volume Units) Recording studios use the dBm scale and talk of headroom to +24 dB which is about 12 V. We can also see that is 30 less 6 which is 1000x divided by 4 or 250 mW exactly. Now I hear that +30dB is the standard, a whole watt at the preamp output (if loaded). It is convenient to remember that 10 dB in power is 10x. I always take a moment to set my mind up for either power or voltage when converting. Once converted into dB there is no difference. That is the beauty of the system though many an engineer has insisted that I must tell them if I am stating voltage dBs or power dBs, but those words do not apply. Although recording engineers want this headroom, they generally only want the voltage not the power.
One newest reference is the dBW or dB watt. It is 1000 times bigger than the dBw (milliwatt), so we better be careful about our capital letters. A dBW is 2.83 V into 8 ohms, 2 V into 4 ohms, and 1.414 volts into 2 ohms all of which are 1 watt. Since a watt is a watt is a watt we only need be concerned with the impedance when we must figure the voltage. An ideal transformer will convert our dB watt into any voltage we want which is why it was first adopted for specifying the sensitivity of FM tuners into Femtowatt. Before this, tuners were rated in uV. There was a big difference if you were referring to the 50 ohm or 300 ohm input. Since the antenna power is constant any given input converts to two voltages differing by about 2.5x depending on the impedance. In Femtowatts it is all the same.
Power amplifiers now reference dBW and I see many mistakes. If 1 dBW is 1 watt then 20 dBW is 100 watts and 23 dBW is 200 watts, regardless of impedance. The problem occurs when we test our amplifiers at different impedances. A typical transistor amplifier that delivers 100 watts into 8 ohms will deliver 160 or so into 4 ohms. So, we can say the power with a 4 ohm load goes from 20 dBW to 22 dBW. However, the voltage has dropped from 28.3 V (100 watts into 8 ohms) to 25.3 V (160 watts into 4 ohms). If the technician was simply reading a voltmeter with a dB scale, he would report the output decreased 1 dB and might erroneously report this as 19 dBW. I have seen this routinely in the press and bring it to your attention for your consideration. My logic states that a watt is watt is a watt and dBW refers to power. The interesting thing is you can get the 160 watts (from the 100 watt amplifier) into an 8 ohm speaker by using a 4 ohm to 8 ohm matching transformer. If the transformer is lossless, you will get 22 dBW into your 8 ohm speaker (35.8V).
I hope you now see that the decibel does many things with ease. It can express voltage ratios, current ratios, power ratios, and even quantities of voltage (dBV) or quantities of power (dBw). One hundred years ago a telephone engineer could say, "I have 0 VU here, that is a milliwatt of level," and not say anything about voltage, current, or impedance, the measurement stands alone. If I were designing a system with transformers in audio lines, I would be inclined to do just what the phone company did and create the dB.