Electricity

and

System theory

for Electrochemists

Erase

January 2, 2003

2

Contents

3

4 Contents

Part I

Transfer function

5

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 .

Part II

Circuit made of impedances

17

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

Bibliography

[1] Fournier, J., Wrona, P. K., Lasia, A., Lacasse, R., Lalancette,

J.-M., Menard, H., and Brossard, L. J. Electrochem. Soc. 139 (1992),2372.

[2] Gille, J.-C., Decaulne, P., and Pelegrin, M. Dynamique de la com-mande lineaire. Dunod, Paris, 1991.

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