Bode' Plots

        We'll start with second order factors in the denominator, i.e. second order poles.  We're not done with Bode' plots. Remember:
 

• Our Bode' plots so far were all plotted with a log scale on the vertical (gain) axis. Decibels are more often used and you need to learn about them.

• Second order systems have interesting Bode' plots - and it is important to know about them.

        When we introduced Bode' plots, we noted that the vertical scale of a Bode' plot is often in terms of decibels. It's time you got acquainted with decibels if you haven't heard of them before. Here's a start.

Originally, decibels were used to measure power gains.

If a system had an output power, P_o, and an input power, P_i, then the ratio of output power to input power - the power gain - is:

P_o/P_i

The decibel gain is proportional to the logarithm - to the base ten (10) - of the power gain

The gain can be expressed as the logarithm -  to the base ten (10) - of the power gain

Gain = log10(P_o/P_i)

When expressed this way, the units are bels.

A decibel is one tenth of a bel, so the gain expressed in decibels is:

Gain_db = 10 log10(P_o/P_i)

The unit bel is something of a story in itself.

Alexander Graham Bell did a lot of work with the deaf, and he was recognized for his work with an honorary doctorate in 1880 by Gaulladet College in Washington, D.C.(and he also delivered the commencement address) He is more famous for his founding of the National Geographic Society, and other work he did.

Alexander Graham Bell was also honored by having a unit named in his honor - the bel.

Today, the decibel is a commonly used unit to measure sound intensity and it is well known that high decibel levels contribute to deafness - a very ironic closing of the circle.

Today, power is not so much an issue. We're more interested in voltage gain of an amplifier.  There's an interesting transition from power to voltage that will help us understand how gain - expressed in decibels - is viewed today.

In an amplifier, if the amplifier has an input resistance R₁, then the power input to the amplifier is given by:

Power In = V₁²/R₁

Similarly, the output power into a resistor R_o is given by:

Power Out = V_o²/R_o

Now, look at the ratio of output power to input power:

Power Out/Power In = (V_o²/R_o)/(V_i²/R_i)

Now, compute the decibel gain:

               Gain_db = 10 log10(P_o/P_i) = 10 log10(P_o/P_i)
 

                        = 10 log10(V_o²/R_o)/(V_i²/R_i)

                         = 10 log10(V_o²/V_i²) + 10 log10(R_i/R_o)

                         = 20 log10(V_o/V_i) + 10 log10(R_i/R_o)

The final result has a term in it that depends upon the resistors.

Gain_db = 20 log10(V_o/V_i) + 10 log10(R_i/R_o)