How to draw a Bode plot on semi-log graph paper
Question: A unity control feedback system has G(s) =80/s(s+2)(s+20). Draw the bode plot. Determine the gain margin and phase margin. Also determine gain cross-over frequency and phase cross-over frequency. Comment on the system stability using this Bode plot.
Solution:
Step 1:
Arrange G(s)H(s)in time constant form as follows:
Step 2:
Identify factors of given H(s)G(s):
1) K=2
2) One pole is at origin. (Because there is ‘s’ in the denominator).
3) Simple pole 1/(1+s/2) with T1 = ½. Hence wc1 = 1/T1 = 2.
4) Similarly simple pole 1/(1+s/20) with T2 = 1/20. wc2 = 1/T2 = 20.
Step 3: Magnitude plot analysis:
1) For K=2, draw a line of 20logK = 20log2 = 6 dB.
2) For one pole at origin. Straight line of slope -20 dB/decade passing through intersection point of w=1 and 0 dB. ( Trick to remember: if there is no pole at origin draw a straight line. For simple pole draw -20dB/decade line, for second order pole i.e. ‘s^2 term in denominator’ draw -40dB/decade line, for third order pole (i.e. S^3)draw -60dB/decade line and so on. And always take intersection point of w=1 and 0 dB. )
3) Shift intersection point of w=1 and 0 dB on 20logK line and draw parallel to -20 dB/decade line drawn. This will continue as a resultant of K and 1/s till first corner frequency occurs. i.e. wc2=2.
4) At wc1 = 2, as there is simple pole it will contribute the rate of -20 dB/decade hence resultant slope after wc1 = 2 becomes -20-20=-40 dB/decade. This is addition of K, 1/s and 1/(1+s/2). This will continue till it intersects next corner frequency line i.e. wc2=20.
5) At wc2 = 20, there is simple pole contributing -20dB/ decade and hence resultant slope after wc2=20 becomes -40-20=-60 dB/decade. This is resultant of overall G(s)H(s). i.e. G(jw)H(jw) the final slope is -60dB/decade, as there is no other factor present.
Step 4: Phase angle plot: Convert G(s)H(s) to G(jw)H(jw)
To draw straight line of -40 dB/decade and -60 dB/decade from wc1 = 2 and wc2 = 20, draw faint lines of slope -20, -40, -60 dB/decade from intersection point of w=1 and 0 dB line and just draw parallel to them from respective points .
Step 5: Bode plot and solution:
You may also like:
- How to draw root locus graph with simple steps.
- Addition of poles and zeros to the forward path transfer function.
- Effects of addition of poles and zeros to closed loop transfer function.
- Mason’s gain formula for determining overall gain of system (with example).
- Block diagram algebra (Block diagram reduction).
If you like this article, please share it with your friends and like or facebook page for future updates. Subscribe to our newsletter to get notifications about our updates via email. If you have any queries, feel free to ask in the comments section below. Have a nice day!