Bode Plots Dr Holbert April 16 2008 Lect

Bode Plots Dr. Holbert April 16, 2008 Lect 21 EEE 202 1

Sinusoidal Frequency Analysis • The transfer function is composed of both magnitude and phase information as a function of frequency where |H(jω)| is the magnitude and (ω) is the phase angle • Plots of the magnitude and phase characteristics are used to fully describe the frequency response Lect 21 EEE 202 2

Bode Plots • A Bode plot is a (semilog) plot of the transfer function magnitude and phase angle as a function of frequency • The gain magnitude is many times expressed in terms of decibels (d. B) d. B = 20 log 10 A where A is the amplitude or gain – a decade is defined as any 10 -to-1 frequency range – an octave is any 2 -to-1 frequency range 20 d. B/decade = 6 d. B/octave Lect 21 EEE 202 3

Bode Plots • Straight-line approximations of the Bode plot may be drawn quickly from knowing the poles and zeros – response approaches a minimum near the zeros – response approaches a maximum near the poles • The overall effect of constant, zero and pole terms Lect 21 EEE 202 4

Bode Plots • Express the transfer function in standard form • There are four different factors: – – Lect 21 Constant gain term, K Poles or zeros at the origin, (j )±N Poles or zeros of the form (1+ j ) Quadratic poles or zeros of the form 1+2 (j )+(j )2 EEE 202 5

Bode Plots • We can combine the constant gain term (K) and the N pole(s) or zero(s) at the origin such that the magnitude crosses 0 d. B at • Define the break frequency to be at ω=1/ with magnitude at ± 3 d. B and phase at ± 45° Lect 21 EEE 202 6

Bode Plot Summary where N is the number of roots of value τ Lect 21 EEE 202 7

Single Pole & Zero Bode Plots Gain ωp Gain 0 d. B +20 d. B – 20 d. B ωz ω Phase 0° +90° – 45° +45° – 90° 0° ω Pole at ωp=1/ Lect 21 One Decade Phase One Decade ω Assume K=1 20 log 10(K) = 0 d. B EEE 202 ω Zero at ωz=1/ 8

Bode Plot Refinements • Further refinement of the magnitude characteristic for first order poles and zeros is possible since Magnitude at half break frequency: |H(½ b)| = ± 1 d. B Magnitude at break frequency: |H( b)| = ± 3 d. B Magnitude at twice break frequency: |H(2 b)| = ± 7 d. B • Second order poles (and zeros) require that the damping ratio ( value) be taken into account; see Fig. 9 -30 in textbook Lect 21 EEE 202 9

Bode Plots to Transfer Function • We can also take the Bode plot and extract the transfer function from it (although in reality there will be error associated with our extracting information from the graph) • First, determine the constant gain factor, K • Next, move from lowest to highest frequency noting the appearance and order of the poles and zeros Lect 21 EEE 202 10

Class Examples • Drill Problems P 9 -3, P 9 -4, P 9 -5, P 9 -6 (handdrawn Bode plots) • Determine the system transfer function, given the Bode magnitude plot below |H(ω)| +20 d. B/decade – 20 d. B/decade 6 d. B – 40 d. B/decade ω (rad/sec) 0. 1 Lect 21 0. 7 2 EEE 202 11 90 11