Overview
This lecture explains how to construct a Bode plot (magnitude and phase) for a given transfer function using step-by-step calculation and graphing on semi-log paper.
Standard Form Conversion
- The transfer function given is ( G(s) = \frac{80}{s(s+2)(s+20)} ) with unity feedback (( H = 1 )).
- Rewrite ( G(s) ) in time constant (standard) form: ( \frac{2}{s(1+\frac{s}{2})(1+\frac{s}{20})} ).
- Identify time constants ( t_1 = \frac{1}{2} ) and ( t_2 = \frac{1}{20} ).
Determining Corner Frequencies
- Corner (break) frequencies found from ( 1 + t_1 s ) and ( 1 + t_2 s ).
- ( \omega_{c1} = 1/t_1 = 2 ) rad/s, ( \omega_{c2} = 1/t_2 = 20 ) rad/s.
- Arrange corner frequencies in ascending order for plotting.
Magnitude Plot (Gain Plot)
- Construct a magnitude and slope table for each element of the transfer function.
- ( 2/s ) term contributes (-20) dB/decade (no corner frequency).
- ( 1/(1+s/2) ) adds (-20) dB/decade after ( \omega_{c1} = 2 ) rad/s; total (-40) dB/decade.
- ( 1/(1+s/20) ) adds another (-20) dB/decade after ( \omega_{c2} = 20 ) rad/s; total (-60) dB/decade.
- Calculate magnitude at sample frequencies using ( 20\log \frac{2}{\omega} ) and the slope changes.
- Example results: at ( \omega=1 ), 6 dB; at ( \omega=2 ), 0 dB; at ( \omega=20 ), (-40) dB; at ( \omega=100 ), (-82) dB.
Phase Plot Analysis
- Substitute ( s = j\omega ) to get ( G(j\omega) ).
- Phase of ( 1/s ) is (-90^\circ ), ( 1/(1+j\omega/a) ) is (-\tan^{-1}(\omega/a) ).
- Calculate phase at selected frequencies ([e.g., 0.1, 1, 2, 10, 100] rad/s).
- Example phase values: at ( \omega=0.1 ), (-93.1^\circ ); at ( \omega=1 ), (-119.4^\circ ); at ( \omega=2 ), (-140.7^\circ ), at ( \omega=100 ), about (-219.3^\circ ).
Bode Plot Construction
- Plot calculated magnitude and phase points on semi-log graph paper.
- Frequency (x-axis) is logarithmic; magnitude (y-axis left) in dB; phase (y-axis right) in degrees.
- Draw lines or smooth curves through calculated points for both plots.
- Mark and use slopes: (-20), (-40), and (-60) dB/decade after each corner frequency.
Gain Margin and Phase Margin
- Draw horizontal line at 0 dB and vertical at (-180^\circ).
- Gain crossover frequency (( \omega_{gc} )): where magnitude crosses 0 dB.
- Phase margin: difference between phase at ( \omega_{gc} ) and (-180^\circ).
- Phase crossover frequency (( \omega_{pc} )): where phase curve crosses (-180^\circ).
- Gain margin: difference between magnitude at ( \omega_{pc} ) and 0 dB.
Key Terms & Definitions
- Bode plot — Graph showing magnitude (dB) and phase (degrees) vs. frequency (log scale).
- Corner frequency (( \omega_c )) — Frequency where slope changes on the Bode plot.
- Gain margin — How much gain can increase before system becomes unstable.
- Phase margin — Additional phase lag at gain crossover frequency before instability.
- Decibel (dB) — Logarithmic unit for magnitude: ( 20 \log_{10}(\text{Amplitude}) )._
Action Items / Next Steps
- Practice drawing Bode plots for other transfer functions.
- Review formulas for magnitude and phase calculations.
- Complete assigned homework and prepare questions for the next class.