ELEC 240 Lab
Prelude: Decibels
You've probably heard the term decibel used in conjunction with sound levels:
having a muffler louder than 85 dB will get you a ticket, and listening to
music at 110 dB will damage your hearing. You've probably also noticed the
button marked ATT -20dB on the function generator, so decibels don't just
have to do with sound.
So what exactly is a decibel? It's a logarithmic way of expressing the ratio of two power levels (or sound pressure levels, or voltage levels, or any other kinds of levels). More precisely,
\[ \text{power ratio in dB} = 10\log\left(\frac{P_1}{P_0}\right) \]where \(P_0\) and \(P_1\) are the two powers being compared, and \(\log()\) refers to the common, or base 10 logarithm.
If we have two voltage levels, \(V_0\) and \(V_0\) across the same load resistance, \(R_L\), then
\[ \begin{aligned} 10 \log\left(\frac{P_1}{P_0}\right) &= 10 \log\left(\frac{{V_1}^2/R_L}{{V_0}^2/R_L}\right) \\ &= 20 \log\left(\frac{V_1}{V_0}\right) \end{aligned} \]Why logarithmic? The smallest perceivable sound level corresponds to an acoustic power density of approximately \(10^{-12} \frac{\text{W}}{\text{m}^2}\). But the level at which the sensation of sound begins to give way to the sensation of pain is about \(1 \frac{\text{W}}{\text{m}^2}\). To cope with this large dynamic range without loosing track of the number of zeros after the decimal point, a logarithmic scale is useful. It's important to remember that a decibel measurement expresses a ratio. So it always makes sense to say that a signal \(x\)'s amplitude is so many dB greater (or less) than signal \(y\)'s amplitude. But if we say that a signal is equal to some number of decibels, then there must be a reference level. For sound, that reference level is usually taken as \(10^{-12} \frac{\text{W}}{\text{m}^2}\) which corresponds to a pressure of \(10^{-5} \frac{\text{N}}{\text{m}^2}\). If we call this the reference pressure level, \(p_0\), we get the definition of sound pressure level
\[ \text{SPL} = 20 \log\left(\frac{p}{p_0}\right) \]where we use 20 instead of 10 since power is proportional to the square of the pressure. In a circuit, the choice of a reference level is not quite so obvious. For voltages, the typical choice is 1 V, which gives "decibels relative to 1 Volt" or dBV for short. Other forms you may encounter are dBW (relative to 1 watt) or dBm (relative to 1 mW). It is sometimes stated that the response of a filter falls off at "20dB per decade" or "6dB per octave". This is just another way of saying that the response varies as \(1/f\). In other words, if \(f_2\) is 10 times \(f_1\) (i.e. the frequencies are separated by one decade) then \(\vert H(f_2)\vert\) will be \(1/10\) of \(\vert H(f_2)\vert\). Since \(20 \log(1/10) = -20\) then we have a loss of 20 dB (or a gain of -20 dB) for each decade increase in frequency. Similarly, for two frequencies separated by one octave (a factor of two) we would have \(20 \log(1/2) = -6.02\) or approximately 6 dB per octave. For more information, check out the article on Decibels at UCSC Electronic Music Studios.