Can anyone guide me through how to hand draw a correct Bode magnitude plot for
$$G(s) = \dfrac{48000}{s(s+0.1)(s+100)}$$
The MATLAB produced Bode plot can be seen here:
First check the transfer function for poles and zeros. In this case the numerator is a constant, so there are no zeros. The poles are the zeros of the denominator. By inspection we see that they are at s=0, s=-0.1, s=-100. For each pole we get additional -20dB/dec and additional -90 degrees phase shift.
The gain starts to change at the corresponding positive value of the pole. E.g. for s=-100 we will see a change at w=100. The phase changes from about one decade before this frequency to about one decade above this frequency by -90 degrees. Exactly at the corresponding frequency the change because of this pole is -45 degrees.
Before we can draw the diagram we need to find a starting point. This is a little bit tricky because of the 1/s term.
We can rewrite the equation and deal with the parts separately $$ G(s) = \dfrac{48000}{s(s+0.1)(s+100)} = \frac 1s \cdot \dfrac{48000}{(s+0.1)(s+100)} $$ Now we have a 1/s-term that is infinity at zero and 1 at w=1. It will decrease by 20dB per decade. Now we look at the other part $$ \dfrac{48000}{(s+0.1)(s+100)} $$ For s=0 the gain is 48000/(0.1*100) = 4800. At w=0.1 it will start to decrease by 20dB/dec, at w=100 it will start to decrease by 40dB/dec.
Now we know everything about the components and can construct the Bode plot.
The result looks as shown below. The 1/s term is red. The other term is blue and the sum of these two is yellow.
Starting at w=1, we have 0dB for the 1/s term.
The other term has 20*log10(4800) ~ 73dB left to the corner frequency and at w=1 it is 20dB below that value, so we have 53dB.
This is our first point! At this point we have a slope of -40dB/dec. This slope will extend one decade to the left (so we have 93dB there) and then continue with -20dB/dec. To the right it will extend up to w=100 and then continue with -60dB/decade. At w=100 the magnitude is 53dB - 2decades*40dB/dec = -27dB.
Write it in the form \$\dfrac{K}{s(1+s/\omega_1)(1+s/{\omega_2})}\$, then you have the breakpoints and the correct gain term.
As few calculation needed as possible, in other words, just use rules for poles and zeros
There are two more to calculate by setting s = -0.1 (this results in a pole along the sigma axis) and s= -100, resulting in another pole along the sigma axis.
