Download Bode Plots: Understanding Sinusoidal Steady-State Behavior of Dynamical Systems and more Study notes Electrical and Electronics Engineering in PDF only on Docsity!
Bode Plots
Bode plots are widely used as graphical protrayals of the sinusoidal
steady-state behavior of dynamical systems. Bode plots show the
relationship, in the sinusoidal steady-state , between the input and
output phasors.
Recall that, in s-domain analysis the input and output are related
via the system transfer function. To be specific, consider the input
and output to both be voltages (of course, depending on the
system, they could be temperatures, currents, or forces).
Vo(s) = TF(s) Vi(s)
The sinusoidal steady-state response is given when s → j ω.
Notice when s is replaced by jω, that Vo, TF, and Vi all become
complex functions of ω. That is, they become phasor quantities.
V o(ω)= TF (ω) V i(ω)
Think of these phasor quantities in polar form.
θ θ θ
θ θ θ θ θ
ω
θ θ θ θ ω
o o TF i i
o o o TF o i i i i
o
i
TF o i i
V = TF V
V V
TF = = -
V V
V
TF = (TF is a function of ) V
= - ( is a function of )
There are two Bode plots. 1)The Bode magnitude plot gives the
magnitude (in decibels) of TF as a function of ω. 2)The Bode
phase plot gives θTF as a function of ω.
What good are Bode plots? What are they used for?
Example
Suppose a system has the frequency response shown below.
What is the ouput, if the input is Vi = 10 cos (500t – 15°) volts?
Working with Bode plots helps engineers develop a feel for frequency response. Bode plots are widely used. They will
be used this quarter when discussing control systems.
For these reasons, engineers need to be able to sketch the
frequency response for systems without having to resort to MATLAB.
Bode plots
Begin with the s-domain transfer function. The transfer function
can be written as a ratio of polynomials in s.
z z- n n p p- d d
a s + b s + TF = a s + b s +
L
L
The coefficients are just parameters of the system and so are
real numbers. For polynomials with real coefficients, their
factors are either real or occur in complex conjugate pairs. For
this case, the transfer function can be manipulated in the form
below. This form is called Bode form.
Bode form
2 n z 2 bz1 nz1 nz b (^2) m p 2 bp1 np1 np
s s 2 s ( + 1) ( + s + 1)
TF = K s s 2 s ( + 1) ( + s + 1)
ζ
ω ω ω
ζ
ω ω ω
L L
L L
s is replaced by jω.
2 n (^) z 2 bz1 nz1 nz b (^2) m p 2 bp1 np1 np
j j 2 j ( + 1) ( + j + 1)
= K j s^2 j ( + 1) ( + j + 1)
ω ω ζ ω ω ω ω ω
ω ζ ω ω ω ω ω
TF
L L
L L
Notice that the input-output rela tion, TF , is now complex.
Bode magnitude plot
db^10
db^10 b^10 bz 2 z 2 nz1 nz1 bp
2 p 2 np1 np
= 20 log
j =20 log K 20 n-m log j + 20 log + 1 +
j (^2) j
- 20 log + j + 1 + - 20 log + 1 -
s^2
ω ω ω
ω (^) ζ ω ω ω ω ω
ζ ω ω ω
TF TF
TF L
L L
L
Bode phase plot
b bz 2 z 2 nz1 nz1 bp
2 p 2 np1 np
j = K n-m j + + 1 +
j 2 j
s^2
ω θ ω ω
ω ζ ω ω ω ω ω
ζ ω ω ω
L
L L
L
Both the magnitude and phase plots are sums of individual terms.
Let's consider each of these terms in turn.
Kb
s
2
2 n n
s 2
+ s + 1
2
2 n n
s 2
+ s + 1
Example
200s TF(s) = s + 10
