Rational Transfer functions and
the Bode Plot
A1
Autumn 2008
E2.2 Analogue Electronics
Imperial College London – EEE
1
The transfer function
• The transfer function is the Fourier transform of the impulse response
• Filters we can make have a rational transfer function: the transfer function is is a
ratio of two polynomials with real coefficients.
(strictly speaking this is called the “Padé approximation”: it states that any real
function can be approximated by a rational function. The higher the degree of the
polynomials the closer the approximation can be made)
The notation is s=jω. The signals assumed to be sinusoid:
V = V e jωt +φ
0
n
as
P (s) ∑
H (s) =
=
Q (s)
∑b s
n
k =0
m
n
k =0
k
k
=
k
an ( s − z1 )( s − z2 )" ( s − zn )
bm ( s − p1 )( s − p2 )" ( s − pm )
k
• The roots zk of the numerator polynomial are called the “zeroes” of H
• The roots pk of the denominator polynomial are called the “poles” of H
• The pole positions on the complex frequency plane entirely determine the filter
properties.
• Note that since s=jω the denominator is seldom zero unless it has pure imaginary
roots
A1
Autumn 2008
E2.2 Analogue Electronics
Imperial College London – EEE
2
The Bode plot – some maths
Complex numbers can be written in rectangular or polar representation:
z = x + iy = Re jθ
The natural logarithm function works correctly with complex argument:
(
)
ln z = ln Re j (θ + 2 nπ ) = ln R + jθ = ln z + j arg z + j 2π n
It follows that, if we restrict the phase to a circle,
Im ln z = arg z
Since H(s) is rational, we can write:
n
⎛
⎞
−
a
s
z
(
)
k ⎟
⎜ n∏
m
an n
k =1
⎟ = ln + ∑ ln ( s − zk ) − ∑ ln ( s − pk )
ln H ( s ) = ln ⎜
m
bm k =1
⎜b
⎟
k =1
−
s
p
(
)
k ⎟
⎜ m∏
⎝ k =1
⎠
A1
Autumn 2008
E2.2 Analogue Electronics
Imperial College London – EEE
3
The Bode plot – Magnitude plot
Since H(s) is rational, we can write:
an ⎛ n
+ ⎜ ∑ ln s − zk
ln H ( s ) = ln
bm ⎝ k =1
⎞ ⎛ m
⎞
⎟ − ⎜ ∑ ln s − pk ⎟
⎠ ⎝ k =1
⎠
The log of the transfer function is the sum of terms of the form ln s − x
Each term takes two simple limits for high and low frequencies respectively:
lim ln s − x = ln s =lnω , lim ln s − x = ln x
s >> x
s << x
Each of the terms in the numerator contributes a constant at low
frequencies and a line of slope 1 at high frequencies. The corner is at w=zk
Each of the terms in the denominator contributes a constant at low
frequencies and a line of slope -1 at high frequencies. The corner is at w=pk
A1
Autumn 2008
E2.2 Analogue Electronics
Imperial College London – EEE
4
The Bode plot – Phase plot
Phase plot: Make a a linear – log plot of the phase (i.e. the imaginary part
of the logarithm of the transfer function) versus the log of frequency.
m
an n
arg H ( s ) = Im ln H ( s ) = arg + ∑ arg ( s − zk ) − ∑ arg ( s − pk )
bm k =1
k =1
Each term in the numerator contributes (remember s is imaginary!)
lim arg ( s − zn ) = 0
ω →0
lim arg ( s − zn ) = π / 2
ω →∞
lim arg ( s − zn ) = tan
ω → zn
−1
ω
zn
⎞ π 1 ω
1⎛ω
+ ⎜ − 1⎟ + ln
4 2 ⎝ zn ⎠ 4 2 zn
π
ω zn
since ln (1 + ε ) ε . in this case, ε =
•
•
•
•
A1
ω
zn
− 1. also note that log x = ln x / ln10
each term has linear asymptote at 0 radians for small frequencies
each term has linear asymptote at π/2 radians at high frequencies
Each term contributes a linear (-/+) slope near the pole or zero
a factor of 10 in frequency adds/subtracts 45 degrees to phase
Autumn 2008
E2.2 Analogue Electronics
Imperial College London – EEE
5
Bode plot example -poles
Bode Diagram
10
Magnitude plot:
Break point at pole
Slope: -20dB per decade
increase in frequency
Magnitude (dB)
0
-10
-20
-30
-40
Phase plot:
-45 degrees at pole
Slope:
-45 degrees per factor of 5
increase in frequency
Phase (deg)
0
-45
-90
10 -2
10-1
10 0
101
102
Often approximated to
-90 degrees/ 2 decades
Frequency (rad/sec)
A1
Autumn 2008
E2.2 Analogue Electronics
Imperial College London – EEE
6
Bode plots - Zeroes
Bode Diagram
Magnitude (dB)
40
Magnitude plot:
Break point at pole
Slope: 20dB per decade
increase in frequency
30
20
10
0
-10
Phase plot:
45 degrees at pole
Slope:
45 degrees per factor of
5 increase in frequency
Phase (deg)
90
45
0
10 -2
10-1
10 0
101
Frequency (rad/sec)
A1
Autumn 2008
E2.2 Analogue Electronics
102
Often approximated to
90 degrees/ 2 decades
Imperial College London – EEE
7
Bode plot example: 2 poles 1 zero
Bode Diagram
Magnitude (dB)
40
20
0
-20
-40
s  1
Hf 
ss  10
-60
Phase (deg)
0
-45
-90
10 -2
A1
10 -1
Autumn 2008
10 0
10 1
Frequency (rad/sec)
10 2
10 3
E2.2 Analogue Electronics
Imperial College London – EEE
8