## PHYS345 Electricity and Electronics

Decibel Measure and Asymptotic Bode Plots

Amplifier gain and filter loss are often specified in decibels (dB), a logarithmic measure of ratios. Most generally dB are specified for power ratios,

power gain in dB= 10 log10 (Pout/Pin)

More generally in this course, we are interested in voltage gain; since P ~ V2,

voltage gain in dB= 20 log10 |Vout/Vin|

dB measure is useful for a number of reasons. First, when dealing with quantities that can vary over many orders of magnitude, the compressive nature of the log function provides a more modest apparent range of numbers. However, keep this compression firmly in mind! For example, the Richter scale for earthquake intensity is logarithmic -- a 7 on the Richter scale actually has an amplitude 10 times more powerful than a 6, corresponding to a factor of about 31-32 times more energy.

Second, in cascaded amplifier/filter systems, the overall gain is the product of each stage's gain. Since log(A.B) = log(A) + log(B), if one wishes to consider the overall gain of several stages, one simple adds the gain of each in dB measure.

Third, as we will see below, the frequency dependence of an amplifier or filter is most often summarized on a Bode plot. For a Bode plot, the log of the |gain| is plotted against the log of the frequency. Thus in dB measure the vertical axis becomes a linear axis.

Let's reconsider the RC low-pass filter. The transfer function is given by:

If we consider the dimensionless frequency w' = wRC, MathCAD plots the magnitude of the low-pass gain as a function of w' as follows:

Let's say that we would like to sketch a Bode plot in an approximate way, without the use of plotting software. Start by considering values of the gain expressed in dB at a few special frequencies.

 w' = 0.01 Vout/Vin = 1.000 = 0 dB 0.1 0.995 = 0 dB 1 0.707 = - 3 dB 10 0.100 = - 20 dB 100 0.010 = - 40 dB

Notice that at low enough frequencies the gain flattens off at unity, 0 dB. At large enough frequencies, the gain is falling off at the rate of - 20 dB per decade of frequency (a factor of 10 increase in frequency). This fall-off is also often referred to as - 6 dB per octave (a factor of 2 increase in frequency.) Convince yourself of the equivalence!

At the so-called cutoff frequency, where w' = 1, the gain is - 3 dB. The - 3 dB point is considered to be the breakpoint. The Bode plot for the RC low-pass filter is often sketched by drawing a horizontal line up to the breakpoint followed by a line falling off at - 20 dB per decade as shown by the blue line in the graph below. The actual gain curve is shown in red for comparison; the largest error in the approximation is at the breakpoint. This type of approximate plot is known as an asymptotic Bode plot.