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Chapter 1
Graphs of transfer function
1.1 Introduction
SystemInput Output
Figure 1.1: Sketch of a scalar system.
• The transfer function, H , of a invariant scalar linear system is given by:
H(s) =L[Output]L[Input]
L denotes the Laplace transform, s is the Laplace variable with s = σ+i ω.
• A transfer function is a complex fonction H(s) of two real variables σ andω. It is not possible to plot graph of H(s) in a plane.
• For s = i ω, i.e. σ = 0, corresponding to frequencial analysis, a transferfunction is a complex fonction H = H(i ω) (or H(ω)) of a real variable ω.It is possible to plot graph of H = H(ω) in a plane and different types ofgraph can be used.
• Transfer function order is the degree in s (or i ω) of the transfer functiondenominator.
• Poles of transfer function are the roots of the denominator of the transferfunction H(s).
• Zeros of transfer function are the roots of the numerator of the transferfunction H(s).
7
8 Chapter 1. Graphs of transfer function
1.2 Nyquist diagram
1.2.1 Nyquist diagram used by electricians
Orthonormal parametric plot
x = Re H = f(ω) , y = Im H = g(ω) (1.1)
1.2.2 Nyquist diagram used by electrochemists
Orthonormal parametric plot
x = Re H = f(ω) , y = −Im H = g(ω) (1.2)
1.3 Bode diagram
1.3.1 Bode diagram used by electricians
• Modulus diagram: 20 log |H | vs. log ω. |H | is the modulus (or magnitudeor amplitude) of H with |H | =
√(Re H)2 + (Im H)2.
• Phase diagram: φH vs. log ω. φH is the phase of H with φH = arctanIm H
Re H
1.3.2 Bode diagram used by electrochemists
log |H | vs log ω, φH vs. log ω (1.3)
1.4 Black diagram
Parametric plot
x = φH = f(ω) , y = 20 log |H | = g(ω) (1.4)
Not used, to the best of our knowledge, by electrochemists.
1.5 Miscellaneous
• Re H vs. log ω, Im H vs. log ω
• log Im H vs. log Re H [?], log |Im H | vs. log |Re H |
Chapter 2
First-order and generalizedfirst-order transferfunctions
2.1 First-order transfer function [?]
2.1.1 First-order transfer function
H(s) =K
1 + τs, H(ω) =
K
1 + i ω τ
K: static gain, τ : time constant.
2.1.2 Dimensionless first-order transfer function
H∗(S) =H(s)K
=1
1 + S, S = τ s = Σ + i u, Σ = τ σ, u = τ ω
One real pole: Sp = −1 (Fig. 2.1).
H∗(u) =H(ω)
K=
11 + i u
, u = τ ω (2.1)
u: reduced (or dimensionless or nondimensional) angular (or radial) frequency
Re H∗(u) =1
1 + u2, Im H∗(u) = − u
1 + u2, lim
u→0Re H∗(u) = 1
Characteristic frequency: uc = 1 (Fig. 2.1).
9
10 Chapter 2. First-order and generalized first-order transfer functions
�2 0 2log u
0
1
Re
H�,�
ImH�
c
�90 �45 0ΦH� �degrees
�2
�1
0
log�H��
uc�1f
�3 �2 �1 0log�Re H� �
�2
�1
0
log��
ImH��
uc�1f
�2 log uc�0 2log u
�90
�45
0
Φ H��d
egre
es
e
0 0.5 1Re H�
0
0.5
�Im
H�
uc�1a
�2 log uc�0 2log u
�2
�1
0
log�H��
d
�1 0�
0u
Figure 2.1: Cpz, Nyquist (a), log Nyquist (b) Re H∗ vs. log u (c, thick line),−Im H∗ vs. log u (c, thin line), Bode (modulus (c) and phase (d)) and Black dia-grams of the first order transfer function. Arrow always indicates increasing angularfrequencies.
2.2. Generalized first-order transfer functions 11
2.2 Generalized first-order transfer functions
2.2.1 High-pass first-order transfer function
H(s) =K τN s
1 + τD s, H(ω) =
K τN i ω1 + τD i ω
2.2.2 Dimensionless high-pass first-order transferfunction
H∗(S) =H(s)K rτ
=S
1 + S, rτ =
τN
τD, S = τD s = Σ + i u, Σ = τD σ, u = τD w
One real pole: Sp = −1, one zero at the origin: Sz = 0 (Fig. ??).
H∗(u) =H(ω)
rτ=
i u1 + i u
, u = τD w
Re H∗(u) =u2
1 + u2, Im H∗(u) =
u
1 + u2
limu→∞
Re H∗(u) = 1
Characteristic frequency: uc = 1 (Fig. ??).
2.2.3 Generalized first-order transfer function
H(s) =K (1 + τN s)
1 + τD s, H(ω) =
K (1 + τN i ω)1 + τD i ω
2.2.4 Dimensionless generalized first-order transfer func-tion
H∗(S) =H(S)
K=
1 + rτ S
1 + S, rτ =
τN
τD, S = τD s = Σ + i u, Σ = τD σ, u = τD w
One real pole: Sp = −1 = −uc1, one real zero: Sz = −1/rτ = −uc2.
H∗(u) =H(u)K
=1 + i rτ u
1 + i u
Re H∗(u) =1 + rτ u2
1 + u2, Im H∗(u) =
(−1 + rτ ) u
1 + u2
limu→0
Re H∗(u) = 1, limu→∞
Re H∗(u) = rτ
Characteristic frequency: uc1 = 1, uc2 = 1/rτ (φuc1 = φuc2).
rτ < 1 ⇒ Capacitive behaviour (Fig. ??).rτ > 1 ⇒ Inductive behaviour (Fig. ??).
12 Chapter 2. First-order and generalized first-order transfer functions
�2 0 2log u
0
1
Re
H�,�
ImH�
c
90450ΦH� �degrees
�2
�1
0
log�H��
uc�1f
�3 �2 �1 0log�Re H� �
�2
�1
0
log��
ImH��
uc�1f
�2 log uc�0 2log u
90
45
0
Φ H��d
egre
es
e
0 0.5 1Re H�
�0.3
0
�Im
H�
uc�1
a
�2 log uc�0 2log u
�2
�1
0
log�H��
d
�1 0�
0u
Figure 2.2: Cpz, Nyquist (a), log Nyquist (b) Re H∗ vs. log u (c, thick line),−Im H∗ vs. log u (c, thin line), Bode (modulus (c) and phase (d)) and Black dia-grams of the high-pass first-order transfer function. Arrow always indicates increasingangular frequencies.
2.2. Generalized first-order transfer functions 13
�2 0 2log u
0
0.5
1
Re
H�,�
ImH� c
�40 �20 0ΦH� �degrees
�0.5
0
log�H��
f
�1 0log�Re H� �
�1
�0.5
log��
ImH��
b
�2 0 2log u
�40
�20
0
Φ H��d
egre
es
e
0 rΤ 1Re H�
0
0.5
�Im
H�
a
�2 0 2log u
0
�0.5log�H��
d
� uc2 � uc1 0�
0�
Figure 2.3: Cpz, Nyquist (a), log Nyquist (b) Re H∗ vs. log u (c, thick line),−Im H∗ vs. log u (c, thin line), Bode (modulus (c) and phase (d)) and Black dia-grams of the generalized first order transfer function. rτ = 0.2 (rτ < 1), dot: uc1 = 1,circle: uc2 = 1/rτ .
14 Chapter 2. First-order and generalized first-order transfer functions
�2 0 2log u
0
1
2
3
Re
H�,�
ImH�
c
10 200ΦH� �degrees
0.2
0
log�H��
f
0 0.5log �Re H� �
�0.5�0.5
0
log��
ImH��
b
�2 0 2log u
10
20
0
Φ H��d
egre
es
e
0 rΤ1Re H�
�1
0
�Im
H�
a
�2 0 2log u
0.4
0
log�H��
d
� uc2� uc1 0�
0�
Figure 2.4: Cpz, Nyquist (a), log Nyquist (b) Re H∗ vs. log u (c, thick line),−Im H∗ vs. log u (c, thin line), Bode (modulus (c) and phase (d)) and Black dia-grams of the generalized first order transfer function. rτ = 3 (rτ > 1), dot: uc1 = 1,circle: uc2 = 1/rτ .
Chapter 3
Second-order andgeneralized second-ordertransfer functions
3.1 Second-order transfer function
H(s) =K
1 + a1 s + a2 s2
3.1.1 Second-order with real poles
H(s) =K
(1 + τ1 s) (1 + τ2 s), H(ω) =
K
(1 + τ1 i ω) (1 + τ2 i ω)
H∗(S) =H(s)K
=1
(1 + S) (1 + rτ S), S = τ1 s = Σ+i u, Σ = τ1 σ, u = τ1 w, (τ1 > τ2)
Two real poles: Sp1 = 1 = −uc1, Sp1 = 1/rτ = −uc1
H∗(u) =1
(1 + i u) (1 + rτ i u)
Re H∗(u) =1 − u2 rτ
(1 + u2) (1 + u2 rτ2)
, Im H∗(u) = − u (1 + rτ )(1 + u2) (1 + u2 rτ
2)
15
16 Chapter 3. Second-order and generalized second-order transfer functions
�2 0 2log u
0
1
Re
H�,�
ImH� c
�180 �90 0ΦH� �degrees
�3
�2
�1
0
log�H��
f
�3 �2 �1 0log�Re H� �
�2
�1
0
log��
ImH��
f
�2 0 2log u
�180
�90
0
Φ H��d
egre
es
e
0 1Re H�
0
0.5
�Im
H�
a
�2 0 2log u
�2
�1
0
log�H��
d
� uc2 � uc1 0�
0u
Figure 3.1: Cpz, Nyquist (a), log Nyquist (b) Re H∗ vs. log u (c, thick line),−Im H∗ vs. log u (c, thin line), Bode (modulus (c) and phase (d)) and Black dia-grams of the second order transfer function with real poles. rτ = 0.2 (rτ < 1), dot:uc1 = 1, circle: uc2 = 1/rτ .
Chapter 4
Circuits made of twoimpedances
4.1 Circuit (Z1+Z2)
Z1 Z2
Figure 4.1: Circuit (Z1+Z2).
Z = Z1 + Z2, Re Z = Re Z1 + Re Z2, Im Z = Im Z1 + Im Z2
4.2 Circuit (Z1/Z2)
Z1
Z2
Figure 4.2: Circuit (Z1/Z2).
19
20 Chapter 4. Circuits made of two impedances
Z =Z1 Z2
Z1 + Z2(4.1)
Re Z =Im Z2
2 Re Z1 + Re Z2
(Im Z1
2 + Re Z1 (Re Z1 + Re Z2))
(ImZ1 + Im Z2)2 + (Re Z1 + Re Z2)2(4.2)
Im Z =Im Z2
(Im Z1 (Im Z1 + Im Z2) + Re Z1
2)
+ Im Z1 Re Z22
(Im Z1 + Im Z2)2 + (Re Z1 + Re Z2)2(4.3)
Chapter 5
Circuits made of threeimpedances
5.1 Circuit ((Z1/Z2)+Z3)
Z1
Z2
Z3
Figure 5.1: Circuit ((Z1/Z2)+Z3).
Z =Z1 Z2 + Z1 Z3 + Z2 Z3
Z1 + Z2(5.1)
Re Z =Im Z2
2 Re Z1 + Re Z2
(Im Z1
2 + Re Z1 (Re Z1 + Re Z2))
(Im Z1 + Im Z2)2 + (Re Z1 + Re Z2)2+ Re Z3
(5.2)
Im Z =Im Z2
(Im Z1 (Im Z1 + Im Z2) + Re Z1
2)
+ Im Z1 Re Z22
(Im Z1 + Im Z2)2 + (Re Z1 + Re Z2)2+ Im Z3
(5.3)
5.2 Circuit ((Z1+Z2)/Z3)
Z =(Z1 + Z2) Z3
Z1 + Z2 + Z3
21
22 Chapter 5. Circuits made of three impedances
Z1 Z2
Z3
Figure 5.2: Circuit ((Z1+Z2)/Z3).
Re Z =(
(Im Z1 + Im Z2)2 Re Z3+
(Re Z1 + Re Z2)(Im Z3
2 + Re Z3 (Re Z1 + Re Z2 + Re Z3)))
/((Im Z1 + Im Z2 + Im Z3)2 + (Re Z1 + Re Z2 + Re Z3)2
)
Im Z =(
Im Z3
((Im Z1 + Im Z2) (Im Z1 + Im Z2 + Im Z3) + (Re Z1 + Re Z2)2
)+
(Im Z1 + Im Z2) Re Z32)/(
(Im Z1 + Im Z2 + Im Z3)2 + (Re Z1 + Re Z2 + Re Z3)2)
Chapter 6
Circuits made of fourimpedances
6.1 Circuit ((Z1/Z2)+(Z3/Z4))
Z1
Z2
Z3
Z4
Figure 6.1: Circuit ((Z1/Z2)+(Z3/Z4)).
Z =Z1 Z2 Z3 + Z1 Z2 Z4 + Z1 Z3 Z4 + Z2 Z3 Z4
(Z1 + Z2) (Z3 + Z4)
6.2 Circuit ((Z1+(Z2/Z3))/Z4)
Z1
Z2
Z3
Z4
Figure 6.2: Circuit ((Z1+(Z2/Z3))/Z4).
23
24 Chapter 6. Circuits made of four impedances
Z =(Z1 Z2 + Z1 Z3 + Z2 Z3) Z4
Z1 Z2 + Z1 Z3 + Z2 Z3 + Z2 Z4 + Z3 Z4
6.3 Circuit (((Z1+Z2)/Z3)/Z4)
Z1 Z2
Z3
Z4
Figure 6.3: Circuit (((Z1+Z2)/Z3)/Z4).
Z =(Z1 + Z2) Z3 Z4
Z1 Z3 + Z2 Z3 + Z1 Z4 + Z2 Z4 + Z3 Z4