appendix a.1. decimal preﬁxes of si units978-3-319-55101-2/1.pdf · a.1. decimal preﬁxes of si...

Appendix

A.1. Decimal Prefixes of SI Units

International System of units (SI) prefixes used to form decimal multiples andsubmultiples of SI units are given below:

Factor Name Symbol

1015 Peta P

1012 Tera T

109 Giga G

106 Mega M

103 Kilo k

10−3 Milli m

10−6 Micro µ

10−9 Nano n

10−12 Pico p

10−15 Femto f

A.2. Standard Resistance Values (Preferred Values)

“E” series specify the preferred resistance values for various tolerances. Thenumber following the “E” specifies the number of logarithmic steps per decade.E48, E96 series values are needed for higher accuracy and close tolerancerequirements. Derivation is based on M ¼ 10

m�1E , where M is the nominal resistance

value at m position, E is a coefficient related to tolerance.Example The third multiplier in E24 series is M ¼ 10

3�124 ¼ 10

112 ¼ 1:21 ! 1:2

© Springer International Publishing AG 2017A.Ü. Keskin, Electrical Circuits in Biomedical Engineering,DOI 10.1007/978-3-319-55101-2

773

E12 series multipliers (10%)

1.0 1.2 1.5 1.8 2.2 2.7 3.3 3.9 4.7 5.6 6.8 8.2E24 series multipliers (5%)

1.0 1.1 1.2 1.3 1.5 1.6 1.8 2.0 2.2 2.4 2.7 3.0

3.3 3.6 3.9 4.3 4.7 5.1 5.6 6.2 6.8 7.5 8.2 9.1E48 series multipliers (2%)

1.00 1.05 1.10 1.15 1.21 1.27 1.33 1.40

1.47 1.54 1.62 1.69 1.78 1.87 1.96 2.05

2.15 2.26 2.37 2.49 2.61 2.74 2.87 3.01

3.16 3.32 3.48 3.65 3.83 4.02 4.22 4.42

4.64 4.87 5.11 5.36 5.62 5.90 6.19 6.49

6.81 7.15 7.50 7.87 8.25 8.66 9.09 9.53E96 series multipliers (1%)

1.00 1.02 1.05 1.07 1.10 1.13 1.15 1.18

1.21 1.24 1.27 1.30 1.33 1.37 1.40 1.43

1.47 1.50 1.54 1.58 1.62 1.65 1.69 1.74

1.78 1.82 1.87 1.91 1.96 2.00 2.05 2.10

2.15 2.21 2.26 2.32 2.37 2.43 2.49 2.55

2.61 2.67 2.74 2.80 2.87 2.94 3.01 3.09

3.16 3.24 3.32 3.40 3.48 3.57 3.65 3.74

3.83 3.92 4.02 4.12 4.22 4.32 4.42 4.53

4.64 4.75 4.87 4.99 5.11 5.23 5.36 5.49

5.62 5.76 5.90 6.04 6.19 6.34 6.49 6.65

6.81 6.98 7.15 7.32 7.50 7.68 7.87 8.06

8.25 8.45 8.66 8.87 9.09 9.31 9.53 9.76

A.3. Mathematical Formulas and Tables

Exponential Identities

i2 ¼ �1

eiA ¼ cosAþ i sinA ðEuler's formulaÞ

cosA ¼ eiA þ eiA

2; sinA ¼ eiA � eiA

2i

log10 x ¼ln xln 10

; xy ¼ ey�ln x

sin hx ¼ ex � e�x

2; cos hx ¼ ex þ e�x

2; tan hx ¼ sin hx

cos hx

774 Appendix

Trigonometric Identities

sin AþBð Þ ¼ sinA cosBþ cosA sinB; sin A� Bð Þ ¼ sinA cosB� cosA sinBcos AþBð Þ ¼ cosA cosB� sinA sinB; cos A� Bð Þ ¼ cosA cosBþ sinA sinB

tan AþBð Þ ¼ tanAþ tanB1� tanA tanB

tan A� Bð Þ ¼ tanA� tanB1þ tanA tanB

sinA cosB ¼ 12

sin AþBð Þþ sin A� Bð Þð Þ

cosA cosB ¼ 12

cos AþBð Þþ cos A� Bð Þð Þ

cosA sinB ¼ 12

sin AþBð Þ � sin A� Bð Þð Þ

sinA sinB ¼ 12

cos AþBð Þ � cos A� Bð Þð Þ

sinAþ sinB ¼ 2 sinAþB2

cosA� B2

cosAþ cosB ¼ 2 cosAþB2

cosA� B2

sinA� sinB ¼ 2 cosAþB2

sinA� B2

cosA� cosB ¼ �2 sinAþB2

sinA� B2

Following table (Table A.1) lists values of some angles.

Table A.1 Some angles and values

Angle degrees Angle radians sin h cos h tan h

0 0 0 1 0

30 p6

12

ffiffi3

p2

1ffiffi3

p

45 p4

ffiffi2

p2

ffiffi2

p2

1

60 p3

ffiffi3

p2

12

ffiffiffi3

p

90 p2 1 0 Undefined

180 p 0 −1 0

Appendix 775

sin2 Aþ cos2 A ¼ 1

cos2 A ¼ 1þ cos 2A2

sin2 A ¼ 1� cos 2A2

sinA2

� �¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� cosA

2

r

cosA2

� �¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ cosA

2

r

sin 2A ¼ 2 sinA cosA

cos 2A ¼ cos2 A� sin2 A ¼ 2 cos2 A� 1 ¼ 1� sin2 A

Some Power Series Expansions

ex ¼ 1þ xþ x2

2!þ x3

3!þ x4

4!þ � � �

cos x ¼ 1� x2

2!þ x4

4!� x6

6!þ � � �

sin x ¼ x� x3

3!þ x5

5!� x7

7!þ � � �

ln x ¼ 2x� 1xþ 1

þ 13

x� 1xþ 1

� �3

þ 15

x� 1xþ 1

� �5

þ � � �" #

Table of Standard Derivatives

f xð Þ ¼ xn; f 0 xð Þ ¼ nxn�1

f xð Þ ¼ ex ¼ f 0 xð Þf xð Þ ¼ ax; f 0 xð Þ ¼ ax ln a ða[ 0Þf xð Þ ¼ sin x; f 0 xð Þ ¼ cosðxÞf xð Þ ¼ cos x; f 0 xð Þ ¼ � sinðxÞ

776 Appendix

f xð Þ ¼ sin�1 x; f 0 xð Þ ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffi1� x2

p �1\x\1ð Þ

f xð Þ ¼ cos�1 x; f 0 xð Þ ¼ � 1ffiffiffiffiffiffiffiffiffiffiffiffiffi1� x2

p �1\x\1ð Þ

f xð Þ ¼ tan�1 x; f 0 xð Þ ¼ 11þ x2

L’Hopital’s Rule

If limx!A

f xð Þ ¼ limx!A

h xð Þ ¼ 0;

limx!A

f ðxÞhðxÞ ¼ lim

x!A

f 0ðxÞh0ðxÞ

prime indicating differentiation operation. Also,

If limx!1 f xð Þ ¼ lim

x!1 h xð Þ ¼ 1;

limx!1

f ðxÞhðxÞ ¼ lim

x!1f 0ðxÞh0ðxÞ

Table of Standard Integrals

Zxn dx ¼ xnþ 1

nþ 1þ c n 6¼ �1

Z1xdx ¼ ln xþ c

Zex dx ¼ ex þ c

Zax dx ¼ ax

ln aþ c ða[ 0Þ

Zsin x dx ¼ � cos xþ c

Appendix 777

Zcos x dx ¼ sin xþ c

Ztan x dx ¼ ln sec xj j þ c

Zsin2 x dx ¼ x

2� sin

2x4

þ c

Zcos2 x dx ¼ x

2þ sin

2x4

þ c

Z1

a2 þ x2dx ¼ 1

atan�1 x

aþ c ða[ 0Þ

Z1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2 � x2p dx ¼ � sin�1 x

aþ c ð�a\x\aÞ

Z1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2 þ x2p dx ¼ sin h�1 x

aþ c ða[ 0Þ

Some Laplace Transform Pairs

f tð Þ ¼ d tð Þ; F sð Þ ¼ 1

f tð Þ ¼ u t � að Þ; F sð Þ ¼ e�as

s

f tð Þ ¼ u tð Þ; F sð Þ ¼ 1s

f tð Þ ¼ tn; F sð Þ ¼ n!snþ 1

f tð Þ ¼ eat; F sð Þ ¼ 1s� a

f tð Þ ¼ tn � e�at; F sð Þ ¼ n!

sþ að Þnþ 1

f tð Þ ¼ sinxt; F sð Þ ¼ xs2 þx2

f tð Þ ¼ cosxt; F sð Þ ¼ ss2 þx2

778 Appendix

f tð Þ ¼ e�at sinxt; F sð Þ ¼ x2

sþ að Þ2 þx2

f tð Þ ¼ e�at cosxt; F sð Þ ¼ sþ a

sþ að Þ2 þx2

f tð Þ ¼ t sinxt; F sð Þ ¼ 2xs

s2 þx2ð Þ2

Laplace Transforms, Some Properties

f tð Þ F sð Þ ¼ R10estf ðtÞdt ðdefinitionÞ

af tð Þþ bg tð Þ aF sð Þþ bG sð Þ ðLinearityÞ

ektf ðtÞ F s� kð Þ ðshift in sÞf 0 tð Þ sF sð Þ � f 0ð Þ ðfirst derivativeÞ

f00tð Þ s2F sð Þ � sf 0ð Þ � f

00ð Þ ðsecond derivativeÞ

Rt0f uð Þdu 1

sF sð Þ ðintegralÞ

H t � að Þf t � að Þ easF sð Þ ðshift in tÞ

f tþ Tð Þ ¼ f ðtÞ 11� e�sT

ZT

0

e�stf tð Þdt ðPeriodic FunctionÞ

limt!0

f ðtÞ lims!1 sF sð Þf g ðinitial valueÞ

limt!1 f tð Þ lim

s!0sF sð Þf g ðfinal valueÞ

tf tð Þ � dF sð Þds

ðFrequency differentiationÞ

f tð Þt

Z1

0

FðrÞdr ðFrequency integrationÞ

Rt0f ðt � sÞgðsÞds F sð ÞG sð Þ ðConvolutionÞ

Appendix 779

Cramer’s Rule for Solving Equations of the Form [A] . [X] = [Y]If [A] is a symmetric matrix having a nonzero determinant, and the vector

½X� ¼ x1 x2 x3. . .xn½ �T

is the column vector of unknowns, then the system has a unique solution, whoseindividual values for the unknowns are

xj ¼ detðAjÞdetðAÞ j ¼ 1; 2; . . .; n

Aj is the matrix formed by replacing the jth column of [A] by the column vector [Y].

Example

a bc d

� �x1x2

� �¼ e

f

� �; x1 ¼ detðA1Þ

detðAÞ ¼ ed � bfad � bc

; x2 ¼ detðA2ÞdetðAÞ ¼ af � ec

ad � bc

Solving Linear System of Simultaneous Equations of the Form [A] . [X] =[Y] in MATLABColumn vector of unknowns, Y½ � is a column vector;

%A=(5*5) exampleA=[1 2 -1 3 1;0 2 -2 1 2;3 1 -2 1 -1;1 1 0 -1 1;1 0 2 3 -2] Y=[1;-1;0;2;1]; X=A\Y; Y=Y' X=X' ----------------------------------------------------------------------- The print out of the resulting solution for 5x5 linear equations:

(Y and X vectors are transposed for space saving reason) A = 1 2 -1 3 1 0 2 -2 1 2 3 1 -2 1 -1 1 1 0 -1 1 1 0 2 3 -2

Y = 1 -1 0 2 1

X = 2.3333 -2.5000 1.1667 0.6667 2.8333

780 Appendix

Partial Fraction Expansion in MATLABConsider a transfer function, which is represented by a ratio of two polynomials ins-domain.

H sð Þ ¼ bðsÞaðsÞ ¼

b1sm þ b2sm�1 þ b3sm�2 þ � � � þ bmþ 1

a1sn þ a2sn�1 þ a3sn�2 þ � � � þ anþ 1

Such a function can be represented by two vectors, one of them specifying thecoefficients of the numerator polynomial, and the other vector specifying thecoefficients of the denominator polynomial. For example, assuming that both ofthese polynomials are fourth-order polynomials, then numerator polynomial coef-ficients vector is

b = [b1b2b3b4b5]

and denominator polynomial coefficients vector is

a = [a1a2a3a4a5]

These vectors specify the coefficients of the polynomials in descending powersof s, and the orders of these polynomials can be different. Partial fraction expansionof this rational function H(s) is


R1

s� p1þ R2

s� p2þ R3

s� p3þ R4

s� p4þ kðsÞ

R1, R2, R3 and R4 are the residues, and p1, p2, p3 and p4 are the poles. The term k(s) is a polynomial in s. MATLAB representation of these vectors are

R ¼ ½R1 R2 R3 R4�; p ¼ ½p1 p2 p3 p4�; k ¼ ½C2 C1 C0�

R and p are column vectors, while k is a row vector.

The command below finds the partial fraction expansion of the ratio of twopolynomials.

[r, p, k] = residue(b, a)

This command calculates the poles and residues from H(s). On the other hand,the command[b2, a2] = residue(r, p, k)

calculates the coefficients of polynomials if the poles and residues are given, andthe result is normalized for the leading coefficient in the denominator.

Appendix 781

Example


s4 þ 10s3 þ 40s2 þ 75sþ 50s4 þ 10s3 þ 35s2 þ 50sþ 24

b = [1 10 40 75 50]

a = [1 10 35 50 24]

its partial fraction expanded form is computed as


r = [-1 -2 2 1]

p = [-4 -3 -2 -1]

k = 1

This means,

H sð Þ ¼ �1sþ 4

þ �2sþ 3

þ 2sþ 2

þ 1sþ 1

þ 1

The partial fraction expansion for multiple poles:Example


3s3 þ s2

b = [3]

a = [1 1 0 0]

its partial fraction expanded form is computed as


r = [3 -3 3]

p = [-1 0 0]

k = []

This means,

H sð Þ ¼ 3sþ 1

� 3sþ 3

s2

Note that if a transfer function has multiple poles, then small changes in the dataor round-off errors can cause large variations in the resulting poles and residues.

782 Appendix

Rules to find Thevenin’s Equivalent CircuitThevenin’s theorem helps to reduce any one-port linear electrical network to asingle voltage source and a single impedance (Fig. A.1).

(a) When the circuit contains resistors and independent sources:

(1) Find open-circuit voltage Voc = VTh

(2) Find Thevenin’s resistance RTh by deactivating all independent sources(open circuit the current sources and short circuit the voltage sources).

(b) When the circuit contains resistors, dependent sources and independent sources

(1) Find Voc = VTh

(2) Short circuit a–b (output terminals) and determine the current through a–b(Isc = Iab)

RTh ¼ Voc

Isc

(c) When the circuit has resistors and dependent sources (no independent sources)

(1) Find Voc = VTh

(2) Connect a 1 A current source flowing from terminal b to terminal a;

(3) RTh ¼ Voc

I¼ Voc

1A

The equivalent circuit consists of only RTh (there is neither a current nor avoltage source).

Voltage DividersA voltage Vi is applied to two series connected impedances, Z1; Z2. Let Z2 has aconnection to reference (ground) and Z1 has a connection to the ungrounded ter-minal of the voltage source, Vi. The output voltage Vo is obtained at the junction ofZ1; Z2 (Laplace operator s is omitted) (Table A.2).

Fig. A.1 Linear electricalnetwork to a single voltagesource and a single impedance

Appendix 783

Magnitude and Phase of Transfer Functions

H sð Þ ¼ NðsÞDðsÞ ¼

ans2 þ an�1sn�1 þ � � � þ aobmsm þ bm�1sm�1 þ � � � þ bo

¼ Ks� z1ð Þ s� z2ð Þ � � � ðs� znÞs� p1ð Þ s� p2ð Þ � � � ðs� pmÞ

where, an 6¼ 0, bm 6¼ 0, all ai, bi, are real.The magnitude of H(jx) in decibels is defined as

20 log10 H jxð Þj j ¼ 20 log10 Kj j þXni¼1

20 log10 jx� zij j �Xmi¼1

20 log10 jx� pij j

The phase in degrees (radians) is defined as

u ¼ tan�1 Im H jxð Þ½ �Re H jxð Þ½ � ¼

Xni¼1

tan�1 Imðjx� ziÞReðjx� ziÞ �

Xni¼1

tan�1 Imðjx� piÞReðjx� piÞ

Bode PlotsExact manual calculation of magnitude and phase is a laborious process.Approximate sketches of these functions can be easily performed using so calledBode plots, noting that numerator and denominator of a transfer function (in fac-tored form) are made up of the following terms:

(i) Constant term (K)(ii) A root of the origin (s)(iii) A real root (s + p)(iv) Complex conjugete (s2 þ asþ b)

Following Fig. A.2 displays drawing rules for magnitude and phase graphs forconstant term, s, 1/s, s + z and 1/(s + p).

The number of decades between two frequencies is given as

Df10 ¼ log10f2f1

� �; f2 [ f1

The number of octaves between two frequencies is

Table A.2 Voltage dividers

Type of voltage divider Voltage transfer function

Resistive Vo

Vi¼ R2

R2 þR1

Inductive Vo

Vi¼ L2

L2 þ L1Capacitive Vo

Vi¼ C1

C2 þC1

784 Appendix

Df2 ¼ log2f2f1

� �; f2 [ f1

DualityDual circuits are the ones which are described by the same characteristic equationswith dual quantities interchanged. A dual of a relationship can be written by

Fig.A.2 Rules for magnitude and phase graphs for constant term

Appendix 785

interchanging voltage and current in an expression. The dual expression produced isof the same form as the original equation.

Some duals: Open circuit–Short circuit, Switch turns on–switch turns off,Current–Voltage, Parallel connected elements–Serial connected elements, VoltageGenerator–Current Generator, Node voltage–Mesh current, Branch–Branch,Resistance–conductance, Impedance–Admittance, Inductance–Capacitance,Reactance–Susceptance, Kirchhoff’s Voltage Law (KVL)–Kirchhoff’s Current Law(KCL), Thévenin’s Theorem–Norton’s Theorem, Faraday’s Law–Ampére’s Law,Permittivity–permeability, Piezoelectricity–Piezomagnetism, Permanent magnet–Electret.

A dual circuit is not the same thing as an equivalent circuit. For example, thedual of a star (Y) network of inductors is a delta ðDÞ network of capacitors, which isnot the same thing as a star-delta (Y-D) transformation; the transformation results inan equivalent circuit.

The dual of a mutual inductance cannot be formed directly, since there is nocorresponding capacitive element.

In case the circuit configuration is not parallel or series or it contains dependentsources, following steps can be used to construct graphically the dual of a planarcircuit:

(a) At the center of each mesh, place a node for the dual circuit.(b) Reference node of the dual circuit is placed outside of the given circuit.(c) Draw lines between nodes and reference line in such a way that each line

crosses an element of the given circuit. Then, this element is replaced by itsdual.

(d) Assign polarities of sources. A voltage source producing clockwise meshcurrent has its dual current source pointing from ground to non-reference node.

A Table A.3 for dual circuit elements and relationships is shown.

Table A.3 Dual circuit elements and relationships

Ohm’s law v tð Þ ¼ i tð Þ :R iðtÞ ¼ v tð Þ :GCapacitor–Inductor differentialexpression

iCðtÞ ¼ C ddt vCðtÞ vLðtÞ ¼ L d

dt iLðtÞ

Capacitor–Inductor integralexpression vC tð Þ ¼ V0 þ 1

C

Rt0iCðsÞds iL tð Þ ¼ I0 þ 1

L

Rt0vLðsÞds

VCCS–CCVS im ¼ xvn vm ¼ xinVCVS–CCCS vm ¼ xvn im ¼ xinResistor conductor R ¼ xX G ¼ xS

Capacitor inductor C ¼ x F L ¼ xH

Voltage–Current source v ¼ xV i ¼ xA

Voltage–Current division vRa tð Þ ¼ RaRa þRb

: v iGa tð Þ ¼ GaGa þGb

: i

786 Appendix

SPICE Models For Dependent (Controlled) Sources

Voltage controlled voltage source VCVS: Ename N1 N2 +C1 -C2 ValueExample E2 3 4 6 0 12 * load control voltage vc 6 0

Current controlled voltage source CCVS: Hname N1 N2 Vcontrol ValueExample H4 1 3 Vm 12 Vm 4 0 dc 0

Voltage controlled current source VCCS: Gname N1 N2 +C1 -C2 ValueExample G1 3 5 4 6 12* load control voltage vs 4 6

Current controlled current source CCCS: Fname N1 N2 Vcontrol ValueExample F1 0 3 Vm 5Vm 4 0 dc 0

N1 and N2 are the positive and negative terminals of the dependent source,respectively.

+C1 and −C2 are the positive and negative terminals of the controlling voltagesource, respectively.

Vcontrol is the zero value voltage source used to measure the controlling current(the positive current flows into the positive terminal of the controlling voltagesource).

Operational AmplifierOperational amplifier (op-amp) is a versatile active element that behaves like avoltage-controlled voltage source. It is used to perform many mathematical oper-ations, filtering and signal processing.

Key Assumption: The op-amp operates in the linear range (away from satura-tion) (Fig. A.3).

Fig.A.3 The op-amp oper-ates in the linear range

Appendix 787

Ideal op-amp:

Vo ¼ A � Vd ¼ A Vp � Vn� �

in ¼ ip ¼ 0; Vp ¼ Vn

Ro ¼ 0X; Ri ¼ 1 X

A = Open loop voltage gain (sometimes expessed in dB, x dB ¼ 20 log10 x)Ro = Output resistance, Ri = Input resistance (Fig. A.4).The inverting input terminal in this particular configuration is at zero volts which

is referred to as virtual ground. Note that this pin is not actually grounded. Theinput terminals are not shorted together.

Instrumentation AmplifierThe instrumentation amplifier is an essential circuit in biomedical electronics. Forexample, a two terminal sensor produces a signal but neither of its terminals may beconnected to the same ground level as with the measuring network. These terminalsmay be DC biased at relatively large voltages or added to the noise. The differentialamplifier acts seletively on measuring the difference between the input terminals.

Addition of 2 buffers between the sensor output and the differential amplifierprevents the loading of both sensor and the measuring electronics. The circuitconfiguration shown here provides gain, as well (Fig. A.5).

Vo

Vd¼ 1þ 2R1

RG

� �� R3

R2; Vd ¼ V1 � V2

Butterworth Polynomials in Factored FormOrder Denominator, D(s)

1 sþ 1

2 s2 þ ffiffiffi2

psþ 1

3 ðs2 þ 1Þðsþ 1Þ4 ðs2 þ 0:765sþ 1Þðs2 þ :848sþ 1Þ5 ðsþ 1Þðs2 þ 0:618sþ 1Þðs2 þ 1:618sþ 1Þ6 ðs2 þ 0:518sþ 1Þðs2 þ ffiffiffi

2p

sþ 1Þðs2 þ 1:932sþ 1Þ

Fig.A.4 Opamp model

788 Appendix

Second-Order (Biquad) Filter Transfer Functions (Table A.4)A biquadratic filter transfer function is defined as

HðsÞ ¼ a2s2 þ a1sþ a0

s2 þ x0

Q

� �sþx2

0

The poles are

p1; p2 ¼ � x0

2Q� jx0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 1

4Q2

s

Conversion of Two-Port Parameters(Table A.5).

Fig.A.5 Instrumentationamplifier

Table A.4 Second order (biquad) filter transfer functions

Filter type Transfer function Gain

Low-pass a0s2 þ x0

Q sþx20

a0x2

0

High-pass a2s2

s2 þ x0Q sþx2

0

a2

Band-pass a1ss2 þ x0

Q sþx20

Center frequency ¼ a1:x0

Q

Band-stopa2

s2 þx20

s2 þ x0Q sþx2

0

dc ¼ high freq:ð Þ gain a2

All-passa2

s2 � x0Q sþx2

0

s2 þ x0Q sþx2

0

Flat gain a2

Appendix 789

Note that there are other parameter sets to characterize two-port networks otherthan the three parameter types presented here. However, they are not used in thisbook.

Table A.5 Parameter relations

z y h

z z11 z12

z21 z22

y22Dy

� y12Dy

� y21Dy

y11Dy

Dh

h22

h12h22

� h21h22

1h22

y z22Dz

� z12Dz

� z21Dz

z11Dz

y11 y12

y21 y22

1h11

� h12h11

h21h11

Dh

h11h Dz

z22

z12z22

� z21z22

1z22

1y11

� y12y11

y21y11

Dy

y11

h11 h12h21 h22

Dz ¼ z11z22 � z12z21; Dy ¼ y11y22 � y12y21; Dh ¼ h11h22 � h12h21

790 Appendix

Historical Profiles

…in the belief that remembrance adds more human values of respect, appreciation,and progress.

Alessandro Giuseppe Volta, 1745–1827Volta was an Italian physicist and chemist. He inventedthe first electrical battery, the Voltaic pile, in 1799.Volta’s invention led to the development of the field ofelectrochemistry. The SI unit of electric potential isnamed in his honor. In 1778 he managed to isolatemethane. Volta studied capacitance, and it was for thiswork that the unit of electrical potential has beennamed the volt.

Andre-Marie Ampère, 1775–1836Ampère, a French physicist, is the man who created thescience of electrodynamics. The unit of electrical cur-rent—the Ampere (A)—is named in his honor.


791

Léon Charles Thévenin, 1857–1926Thévenin was a French electrical engineer. He devel-oped his famous theorem, known as Thévenin’stheorem.

Joseph Henry, 1797–1878Henry was an American scientist who discoveredelectromagnetic induction independently of and atabout the same time as Michael Faraday. Henry dis-covered the self-inductance, Unit of inductance isnamed in his honor.

Michael Faraday, 1791–1867 Faraday invented theelectric motor in 1821. He discovered the induction ofelectric current by magnetism in 1831, and the abilityof magnetic fields to change the polarization of light in1845. The unit of electrical capacitance—the farad (F)—is named in his honor. Faraday discovered the car-bon compound benzene, and in 1823, he was the firstscientist to liquefy a gas. He was also first to introduceterms such as “electrode,” “cathode” and “ion”.

792 Historical Profiles

Georg Simon Ohm, 1789–1854A German physicist and mathematician. Ohm foundthat there is a direct proportionality between thepotential difference applied across a conductor and theresultant current. This relationship is known as Ohm’slaw. The work of Ohm marked the beginning of circuittheory; the unit of resistance is named in his honor.

Gustav Robert Kirchhoff, 1824–1887Kirchhoff was a physicist. He contributed to the fun-damental understanding of electrical circuits, spec-troscopy, and the emission of black-body radiation byheated objects. Kirchhoff formulated his circuit laws(KCL, KVL) in 1845. He worked at University ofHeidelberg in 1854, collaborating in a spectroscopicwork with Robert Bunsen, where they discoveredcesium and rubidium in 1861.

Ernst Werner von Siemens, 1816–1892Siemens’s name has been honored as the SI unit ofelectrical conductance. He invented electrically chargedsea mines, worked on perfecting technologies that hadalready been established, and invented a telegraph thatused a needle to point to the right letter, instead ofusing Morse code. Siemens is the father of the trol-leybus, which was first introduced in 1882.

Historical Profiles 793

Bernard D.H. Tellegen, 1900–1990B. Tellegen was a Dutch electrical engineer. He is theinventor of the pentode tube in 1926 and the gyrator in1948. Tellegen held 41 US patents and he is alsoknown for a theorem in circuit theory. He receivedMSEE degree from Delft University in 1923.

Hendrik Wade Bode, 1905–1982Bode was a Dutch-American engineer. Bode receivedhis Ph.D. from Columbia University in 1935. He is apioneer of modern control theory and electronictelecommunications. He made important contributionsto the design, guidance and control of anti-aircraftsystems missiles and anti-ballistic missiles. He alsomade important contributions for the analysis of sta-bility of linear systems. He is known for the graph thathonored on his name, the Bode plot. He held 25 USpatents.

Alan Lloyd Hodgkin, 1914–1998AL Hodgkin was an English physiologist and bio-physicist, 1963 Nobel Prize winner in Medicine. Heworked in Cambridge University, also held additionaladministrative posts in the University of Leicester,from 1971 to 1984, and Trinity College, Cambridge,from 1978 to 1985. He worked on experimental mea-surements and developed an action potential theoryrepresenting one of the earliest applications of voltageclamping technique.


Andrew Fielding Huxley, 1917–2012A.F. Huxley was a 1963 Nobel Prize-winning physi-ologist and biophysicist for his studies on the actionpotentials with A.L. Hodgkin. He developed interfer-ence microscopy to study muscle fibers. He discoveredin 1954 the mechanism of muscle contraction, socalled the sliding filament theory.

William T. Bovie, 1882–1958W.T. Bowie was an American biophysicist. Heinvented the electrosurgical generator. He completed aPh.D. in plant physiology from Harvard University.The first use of the electrosurgical device in an oper-ating room was on October 1, 1926, by Harvey W.Cushing at Peter Bent Brigham Hospital in Boston,Massachusetts.

Willem Einthoven, 1860–1927Einthoven was a Dutch doctor and physiologist. Hewas the inventor of the first electrocardiogram in 1903.Einthoven received a medical degree from theUniversity of Utrecht. He became a professor at theUniversity of Leiden in 1886. ECG equipment he firstused had a string galvanometer and moving roll ofphoto-sensitive paper, weighting about 270 kg. Hereceived the Nobel Prize in Medicine in 1924 for hiscontributions in this field.


Nikolai Sergeyevich Korotkov, 1874–1920A Russian inventor of auscultatory technique for bloodpressure measurement which is considered a “goldstandard” for blood pressure measurement. The name“Korotkoff sounds” are given in his honor forpulse-synchronous circulatory acoustic signalsobserved through the stethoscope in auscultation ofblood pressure using a sphygmomanometer.

Hans Berger, 1873–1941Berger was a German neurologist. Berger received hismedical degree from Jena in 1897. He became Rectorof Jena University in 1927. He recorded the electricalbrain waves, the electroencephalogram (EEG) in 1924.Berger also described the different waves or rhythmswhich were present in the normal and abnormal brain.

Ian Donald, 1910–1987Ian Donald was a Scottish and educated in Edinburgh,graduated from the Diocesan College in Cape Town.He then studied medicine and was awarded MB BS atLondon University in 1937. In 1954, he introducedecho-sounding (the term from sonar) and searched itspossible medical applications.


Godfrey Newbold Hounsfield, 1919–2004He was an English electrical engineer who received the1979 Nobel Prize with Allan McLeod Cormack fordeveloping the X-ray computed tomography.Hounsfield built a prototype head scanner and tested iton himself. In 1971, the first head scanner was operatedin a London hospital. His name is given to Hounsfieldscale, a quantitative measure in evaluating CT scans.

Paul C. Lauterbur, 1929–2007Lauterbur was an American chemist. In 2003, theNobel Prize in Physiology or Medicine was awarded toPaul Lauterbur and Sir Peter Mansfield for theirresearch related to MRI. He received a B.S. in chem-istry from the Case Institute of Technology he obtainedhis Ph.D. in 1962 from the University of Pittsburgh.

Raymond V. Damadian, 1936–Damadian was credited as the originator of thewhole-body magnetic resonance imaging. He earnedhis degree in mathematics from the University ofWisconsin–Madison in 1956, and an M.D. degree fromthe Albert Einstein College of Medicine in New YorkCity in 1960. In 1974, he received the first patent in thefield of MRI.


William Bennett Kouwenhoven, 1886–1975Kouwenhoven invented the first cardiac defibrillator.He received his BSEE from Brooklyn Polytechnic inNew York, and his Ph.D. in electrical engineering fromthe Karlsruhe Technische Hochschule in Germany in1913. He joined the faculty of the Johns HopkinsUniversity School of Engineering in 1914. He testedhis device on a dog. In 1947, Professor Claude Beckused his device on a 14-year-old boy at Case WesternReserve University.

Paul M. Zoll, 1911–1999Zoll was a practicing physician and recognized as aPioneer in Cardiac Pacing. In 1952, Paul M. Zolldescribed cardiac resuscitation via electrodes on thebare chest with 2-millisecond duration pulses of 100–150 V across the chest, at 60 stimuli per minute. Thisbecame the basis for future clinical pacing develop-ments. In 1956, he published a transcutaneousapproach to terminate ventricular fibrillation with ashock voltage up to 750 V, and later described similartermination of ventricular tachycardia.

Karl William Edmark, 1924–1994A cardiovascular surgeon, inventor. Edmark developeda defibrillator that utilized direct current (DC), whichprovided lower-energy and more effective shocks.Edmark’s invention, known as the Edmark PulseDefibrillator, was first used to save the life of a12-year-old girl in 1961.


James Francis Pantridge, 1916–2004Professor Pantridge and Dr. John Geddes of the RoyalVictoria Hospital in Belfast produced the first portabledefibrillator in 1965. A mains (AC) powered defibril-lator was powered by an inverter, which converted a12 V car battery to 230 V. The unit weighed 70 kg. By1968 he had designed an instrument weighing only3 kg, incorporating a mini capacitor manufactured forNASA.

Bernard Lown, 1921–A Lithuanian-American, 1985 Nobel Peace Prize lau-reate, Professor of Cardiology Emeritus at the HarvardSchool of Public Health, Boston, developed the directcurrent defibrillator for cardiac resuscitation, andintroduced a new use for lidocaine to control heartbeatdisturbances. In 1961 Lown, Baruch Berkowitz, andcoworkers proved that a specific current waveform(Lown waveform) reversed ventricular fibrillation,without injuring heart.

John G. WebsterJ.G. Webster is a pioneer in biomedical engineering.(1953 BSEE, Cornell University, 1965 MSEE,University of Rochester, 1967 Ph.D., Elec. Eng.University of Rochester). He first proposed the idea ofelectrical impedance tomography in 1978, and pub-lished many books on biomedical engineering. Prof.Webster was a professor emeritus in the College ofEngineering at the University of Wisconsin–Madison(2015).


Selected Bibliography

Following is a list of selected books for further reading.Circuit Analysis

Alexander CK, Sadiku MNO (2013) Fundamentals of electric circuits, 5 edn.McGraw HillBalabanian N, Bickart TA, Seshu S (1969) Electrical network theory. WileyBasso CP (2016) Linear circuit transfer functions: an introduction to fast ana-lytical techniques. Wiley-IEEE PressBobrow LS (1987) Elementary linear circuit analysis, Holt Rinehart andWinstonBoylestad R (2013) Introductory circuit analysis, 12th edn. Pearson NewInternational EditionDavis AM (1998) Linear circuit analysis. PWS Publishing CompanyDorf RC, Svoboda JA (2013) Introduction to electric circuits, 9th edn.International Student Version, WileyFloyd T (2013) Principles of electric circuits, 9th edn. Pearson NewInternational EditionHayt WH, Kemmerly JE (2001) Engineering circuit analysis, 6th edn.McGraw-Hill Book Company Inc.Irwin JD, Nelms RM, Patnaik A (2015) Engineering circuit analysis, 11th edn.International Student Version, Wiley.Johnson DE, Hilburn JL, Johnson JR, Scott PD (1995) Basic electric circuitanalysis, 5th edn. Prentice-Hall, Inc.Nahvi M, Edminister JA (2014) Schaum’s outline of electric circuits, 6th edn.McGraw-Hill EducationNilsson J, Riedel S (2014) Electric circuits with mastering engineering, 10thedn. Pearson Global EditionO’Malley J (1992) Schaum’s outline of basic circuit analysis, 2nd edn.McGraw‐HillReddy HC (2002) The circuits and filters handbook, 2nd edn. CRC Press


801

Spence R (2008) Introductory circuits. WileyThomas RE, Rosa AJ, Toussaint GJ (2015) The analysis and design of linearcircuits. WileyVlach J (2014) Linear circuit theory: matrices in computer applications. AppleAcademic PressWing O (2009) Classical circuit theory. Springer

Circuit Synthesis and Design

Anderson BDO, Vongpanitlerd S (1973) Network analysis and synthesis: amodern systems approach. Prentice-HallBaher H (1984) Synthesis of electrical networks. Wiley, New YorkBakshi UA, Bakshi AV (2009) Fundamentals of network analysis and synthesis.Technical Publications PuneDaryanani G (1976) Principles of active network synthesis and design. Wiley,New YorkGlisson TG (2011) Introduction to circuit analysis and design. SpringerGuillemin EA (1977) Synthesis of passive networks: theory and methodsappropriate to the realization and approximation problems (Reprint).Huntington, N.Y., R. E. Krieger Pub. Co.Kuo F (1966) Network analysis and synthesis, 2nd edn. WileyLamm HY-F (1979) Analog and digital filters: design and realization. PrenticeHall, Inc.Schaumann R, Valkenburg MEV (2001) Design of analog filters. OxfordUniversity Press.Temes GG, Lapatra JW (1977) Circuit synthesis and design. McGraw-HillValkenburg MEV (1960) Introduction to modern network synthesis. WileyWeinberg L (1962) Network analysis and synthesisYarman BS (2010) Gewertz design of ultra-wideband power transfer networks.Wiley, Chichester, UK

Electronics

Heumann K (2012) Basic principles of power electronics. Springer Science &Business MediaKandaswamy A, Pittet A (2009) Analog electronics. Prentice Hall India,Learning Pvt. Ltd.Khanchandani S (2007) Power electronics. Tata McGraw-Hill EducationLiu Y (2012) Power electronic packaging: design, assembly process, reliabilityand modeling. Springer Science & Business MediaPeyton A, Walsh V (1993) Analog electronics with op-amps: a source book ofpractical circuits. Cambridge University PressSedra AS, Smith KC (2004) Microelectronic circuits, 5th edn. OxfordUniversity Press

802 Selected Bibliography

Biomedical Engineering

Aston R (1991) Principles of biomedical instrumentation and measurement.Merrill Publishing Company (Macmillan)Barsoukov E, Macdonald JR (2005) Impedance spectroscopy, theory, experi-ment and applications. Wiley InterscienceBruce EN (2001) Biomedical signal processing and signal modelling. WileyCarr JJ, Brown JM (2001) Introduction to biomedical equipment technology,4th edn. Prentice HallDavid Y, Maltzahn WW, Neuman MR, Bronzino JD (2003) Clinical engi-neering. CRC PressEnderle J, Blanchard S, Bronzino J (2005) Introduction to biomedical engi-neering, 2nd edn. Elsevier Academic PressSaltzman WM (2015) Biomedical engineering: bridging medicine and tech-nology. Cambridge University PressSemmlow JL (2011) Signals and systems for bioengineers. 2nd edn.A MATLAB-Based Introduction (Biomedical Engineering) Academic Press.Street LJ (2011) Introduction to biomedical engineering technology, 2nd edn.CRC PressWeiss TF (1996) Cellular biophysics, electrical properties, vol. 2. The MITPressWebster JG (ed) (1998) Medical instrumentation: application and design, 3rdedn. Wiley, New YorkWebster JG (ed) (2004) Bioinstrumentation. Wiley

SPICE, MATLAB and Others

Banzhaf W (1989) Computer-aided circuit analysis using SPICE. Prentice HallButt R (2009) Introduction to numerical analysis using MATLAB. Jones &Bartlett LearningHahn BD (2002) Essential MATLAB for scientists and engineers, 2nd edn.Butterworth-HeinemannRashid MH, Rashid HM (2006) SPICE for power electronics and electric power,2nd edn. Taylor and FrancisSedra AS, Roberts GW, Smith KC (1992) SPICE for microelectronic circuits.Saunders College Pub.Smythe WR (1989) Static and dynamic electricity, 3rd edn. Taylor and FrancisThorpe TW (1992) Computerized circuit analysis with SPICE: a complete guideto SPICE, with applications. WileyYang X-S (2006) An introduction to computational engineering withMATLAB. Cambridge Int. Science Publishing

Selected Bibliography 803

Index

AAC bridge circuit, 406, 407AC circuits, 407Acetic acid, 634AC power

apparent power, 408average power, 348, 350, 413, 414, 427complex power, 419effective value, 386instantaneous power, 353maximum average power transfer, 421power factor, 408, 409, 411, 412power factor correction, 409power measurement, 335rms value, 347–349, 385, 400

AC voltage, 365, 370, 372Additivity property, 142Admittance, 352, 363, 366, 383, 384, 474, 487,

490, 501, 503, 520, 521, 548, 549, 553,586, 599, 601, 607, 611, 615, 619, 629, 630

Admittance parameters, 474, 548, 549, 553Air-core transformers, 418Alternating current (AC), 6, 334, 385, 399Ammeter, 154, 302Ampère, Andre-Marie, 791Analog computer, 743, 744Analytic continuation, 607Analytic signal, 614Apparent power, 408Average power, 5, 220, 221, 348

BBalanced, 11, 12, 17, 60, 61, 67, 68, 71, 277,

407, 712, 714–715Battery, electric, 2, 43, 166, 246, 259, 351Bessel filter, 652Bessel polynomials, 652Binary weighted ladder, 720Biomedical instrumentation, 788

Biphasic, 335, 336Bipolar Junction Transistor (BJT), 665Blood, 34, 35, 240, 429, 459Bode plot, 530, 580, 784Bode’s method, 616, 617Break frequency, 522, 694, 724, 751, 755, 758,

760Bridged-T filter, 497, 501Butterworth filters, 655, 656, 745–747,

752–754By inspection, 90, 105, 116, 125, 126, 129,

136, 382, 459

CCalcium reagents, 240Cancerous tissues, 633Canonic, 548, 563Capacitance multiplier, 730, 731Cardiac, 334, 335Cascaded networks, 749, 755Cauchy integral formula, 608, 609Cauchy principal value, 608, 609Cauer, 553, 555, 557–560, 562, 564, 603,

655–656, 658, 660, 662Causal, 610–612, 642, 643Cell counting, 39, 63Ceramic capacitor, 302Cervical, 633, 634Cervix, 633, 634Characteristic equation, 60, 305, 306, 318, 330,

338, 421, 459, 535, 737, 743Charge, electric, 4, 11, 12, 194, 195Chebyshev filtersChebyshev polynomialsCitrated blood plasma, 240Clark electrode, 35, 36Closed-loop gain, 671, 676Clot formation, 240Coagulation, 240, 398, 399, 401


805

Coaxial cylindrical capacitor, 207, 208Coefficient of coupling, 419Coil, 278, 282, 285, 296, 298, 299, 301, 302,

316, 317, 395, 409, 416, 417, 429, 430Colposcopy, 634Complex conjugate, 396, 483Complex frequencyComplex numbers, 142, 358Complex power, 419Composite, 26–30, 198, 200, 201, 203, 440,

441, 639Conductance, 18, 19, 37, 45, 111, 141, 240,

241, 261, 280, 366, 384, 500–503, 722Conductance matrix, 115, 154Confidence bounds, 257, 259Constraint, 21, 119, 120, 138, 139, 173, 337,

339Controlled source, 86, 116, 171Convolution, 468Coulomb, 11, 12, 250Coupling coefficient, 418, 427Cramer’s rule, 87–89, 91, 101, 103, 108, 126,

128, 130, 133, 159, 160, 379, 425, 459,486, 492

Critically damped case, 306CT scanner, 329Current, 2, 4–6, 8, 9, 11, 12, 18, 20, 22, 24, 26,

34, 36, 41, 44, 47, 52, 59, 74, 77, 85, 86,88, 99–101, 105, 113, 116, 121, 124, 128,130, 132, 134, 137, 140, 143, 145, 150,151, 154, 155, 161, 171, 174, 191, 210,212, 218, 223, 229, 230, 232, 233, 235,236, 238, 244, 245, 249, 255, 262, 264,265, 267, 277, 282–285, 287, 290, 291,293, 296, 299, 301, 302, 305, 317, 323,328, 334–336, 339, 348, 361, 364, 368,369, 371, 372, 382, 389, 399–401, 407,409, 416–418, 421, 432, 465, 468, 633,670, 671, 678, 680, 682, 694, 707, 712,715, 732

Current-division principle, 44, 56, 145, 149,152, 293, 321, 322

Curve fitting, 38, 257, 259, 603, 605, 702Cutoff frequency, 612, 726, 746Cytology, 634

DDamped natural frequency, 319Damping factor, 307, 330Damping frequency, 317Darlington, 656, 657DC voltage, 23, 46, 173, 185, 249, 258, 262,

669

Decibel (dB), 595, 784Defibrillator, 220, 224, 230, 246, 249, 264,

270, 311, 312, 316, 333–338Delay circuit, 235Delta-to-wye conversion, 16Dependent current source, 274Dependent voltage source, 274Derivatives, 308, 333, 447, 603, 604, 743Determinant, 487, 493, 669, 780Difference amplifier, 681, 683–685, 717, 733Differential equations, 4, 253, 260, 273, 281,

294, 295, 463, 737, 739, 743, 744Digital-to-analog converter (DAC), 329, 719Dot convention, 416, 417Double layer, 629, 634Driving-point impedance, 647Duality, 338, 339, 341, 520

EEcho-cardiography, 424EEG, 761, 764Effective medium models, 629-631Effective value, 386Electrode/electrolyte interface, 638Electrodes, 11, 13, 33, 37, 47, 48, 63, 249, 317,

334, 335, 337, 351, 399, 400, 404, 630, 634Electrodynamics, 791Electrolyte, 35, 37, 63, 224, 632Electrolytic, 11, 224, 638Electrolytic capacitor, 639Electromagnetic induction, 792Electromagnetic waves, 751Electrosurgery unit (ESU), 398–402, 404, 405,

431Elimination method, 103Energy, 3, 4, 6, 8, 9, 12, 213, 214, 216, 220,

221, 224, 226, 228–230, 232, 235, 237,249, 266, 267, 291, 301, 309, 311, 312,314–317, 330, 334–336, 351, 400, 429,430, 456

Equivalent circuit, 77, 173, 209, 224, 243, 252,253, 402, 406, 428, 454, 455, 492, 516,559, 626, 632, 640, 643, 651, 664, 732,733, 783, 786

Equivalent conductance, 45Equivalent inductance, 410Equivalent resistance, 14, 17, 24, 78–80, 160,

161, 170, 208, 242, 418, 531, 729Equivalent T-circuit, 427, 428Euler’s formula, 356Euler’s identities, 630Excitable cell, 6, 249, 261Extracellular ionic concentration, 250, 251

806 Index

FFactor inhibitor, 241Faraday’s law, 11, 786Faraday, Michael, 11, 250Fat, 514Filters

active, 723, 724allpass, 724, 766high-pass, 724KHN, 764low-pass, 724notch (bandstop), 724passive, 469, 723, 724

Final-value theorem, 273, 446First-order circuits, 531, 534First-order differential equation, 60First-order high-pass filter, 718First-order low-pass filter, 752Flat, 42, 316, 423, 479, 751, 764, 766Flyback topology, 249Foster synthesis, 564, 567, 568, 576, 598, 601,

603, 606, 634Four points in-line probe, 643Frequency domain, 361–363, 426, 446, 643Frequency-inverse duals, 549Frequency response, 461, 482, 643, 753Frequency scaling, 747, 752Fricke Model, 632

GGain, 86, 91, 111, 113, 117, 154, 472,

476–479, 525, 532, 580, 596, 664, 676,680, 684, 687, 692, 694, 697, 708–710,722, 724–727, 738, 745, 746, 751, 755,756, 758, 760, 762, 765, 768, 788

Gastro-Esophageal, 633Geiger tube, 262Gewertz’ method, 616Glycerol, 208Ground, 99, 119, 169, 173, 291, 335, 337, 455,

490, 733, 744, 783, 786Guillemin, 798Gyrator, 794

HHalf-power frequencies, 389Healthcare, 337Hematology, 275Henry, Joseph, 417High-pass filter, 724, 755, 758, 764Hilbert transform, 610614, 616, 642Hodgin/Huxley, 259, 260Homogeneity property, 142Hospital, 10, 399, 409

Hounsfield, 793Hurwitz, 535, 536, 538H-parameters, 647, 663–666

IIdeal op-amp, 674, 683, 724, 730, 736, 788Ideal transformers, 417–419, 421, 422, 426,

647, 663Imaginary part, 374, 378, 384, 391, 396, 501,

509, 511, 512, 610, 611, 619, 640–642Immittance parameters, 616Impedance

characteristic, 633driving point, 509, 514, 545, 550, 563–565,

567, 568, 574, 576, 578, 582, 583, 585,588, 590, 593, 598, 600, 603–605, 613,634–636, 639, 640

input, 58, 394, 397, 423, 427, 428, 472,520, 521, 537, 546, 557–560, 578, 581,584, 587, 592, 655–657, 661, 664, 706,728, 730, 731

load, 389, 396, 397, 414, 652, 724lossless, 550matching, 394, 419, 420open circuit impedance, 646output, 394, 424, 664parameters, 634scaling, 524source, 165, 394, 423, 427, 560, 647, 724,

783spectroscopy, 374, 614, 629, 630, 633, 642synthesis from real part, 374synthesis from two-port parameters, 647Thévenin, 427, 457

Impulse function, 253Indefinite integrals, 610Independent current source, 161Inductance, 280, 282, 285, 286, 290, 295, 296,

299, 301, 302, 316–318, 335, 386, 387,389, 396, 397, 409, 411

Inductance, mutual, 316, 416–418, 427, 428,430, 786

Inductance simulator, 728Inductive, 384, 411, 416, 429, 431, 784Inductors, 277, 282–292, 294, 301, 317, 320,

321, 323, 325, 326, 329, 334, 348, 361,363, 368, 369, 373, 386, 387, 394, 426,537, 559, 562, 724, 728, 786

Infinite network, 187Initial-value theorem, 446Instantaneous power, 4, 311, 314, 353Instrumentation amplifier, 687, 707–710, 716,

717, 719, 732, 788Integrator, 252, 414, 728, 764, 765, 769

Index 807

Interface, 392, 402, 403, 638, 717International Normalized Ratio (INR), 241International System of Units (SI), 773Interpolation, 610Intracellular ionic concentration, 250, 251Inverse Laplace transform, 443, 448, 456, 463,

598, 601, 605Inverting op-amp, 669–671, 675, 676, 687,

689–691, 697, 703, 715, 722, 724, 737,739, 743, 755

Ion channel, 259, 261Isolation transformer, 419

KKirchoff, Gustav Robert, 793Kirchoff’s current law (KCL), 20, 86, 146, 159,

162, 675, 786Kirchoff’s voltage law (KVL), 21, 305, 786Kramers/Kronig transform, 608, 609, 616, 642Krawzenski model, 201

LLadder network, 187Ladder network synthesis, 187Lagging power factor, 409Laplace transform, 435, 437, 438, 440, 442,

445, 447, 601, 603Law of cosines, 411, 412Layer models, 630, 631Leaky Integrate-and-Fire (LIF) model, 252,

253Left-half plane, 535, 565, 567, 624, 625L’Hopital’s rule, 777Lichtenecher model, 206Linear circuit, 154Linearity, 27, 64, 77, 140, 142–144, 154, 200,

642Linear transformers, 419Lithotripsy (ESWL), 301Load, 2, 3, 10, 23, 46, 60, 85, 166, 167, 169,

173–175, 183, 184, 316, 351, 388, 389,393, 394, 397, 399, 408, 411, 414, 415,417, 419–423, 426, 428, 430, 495, 496,652, 658, 665, 724, 738

Loop, 32, 123, 174, 279, 302, 411, 425, 724,764, 769

Looyenga model, 201Loss, 10, 334, 401, 422, 430, 676Lossless impedance function, 537Lossy, 728, 731Low-pass filter, 724, 752Lung ventilator, 71L-sections, 187

MMagnetically coupled, 416Magnetic Resonance Imaging (MRI), 302Matching, 379, 393, 419, 692Mathematical formulas, 774MATLAB, 8, 17, 24, 64, 82, 111, 115, 116,

130, 214, 257–259, 287, 296, 309–311,315, 349, 350, 359, 379, 381, 382, 403,409, 428, 437–439, 441, 444, 445, 450,476, 493, 498, 502, 504, 505, 511, 512,514, 515, 533, 536, 551–554, 557–559,569, 571, 576, 578, 580, 582, 588, 592,595, 597, 598, 600, 602, 603, 605, 607,617, 627, 629, 641, 667, 672, 697,699–701, 703, 733, 735, 750, 760, 780, 781

Matrix, 26, 28–30, 88, 89, 91, 96, 101, 105,107, 114–116, 125, 127, 129, 130, 133,135, 136, 154, 162, 165, 184, 200, 425,427, 459, 629, 635, 662, 776

Matrix inversion, 651Maximum average power transfer, 420Maximum power transfer, 165, 166, 167, 175,

176, 184, 388, 389, 393–397, 402, 403,420, 424, 652

Maxwell/Garnett, 197, 198, 199Maxwell/Wagner Model, 632Medical, 302, 336–338, 350, 392, 394, 401,

429, 633Membrane potential, 250, 253, 254, 261, 335Mesh analysis, 123, 125, 128, 132, 137, 492,

496Mesh current, 125, 126, 128–131, 135, 136,

138, 177, 338, 381, 425, 427, 471, 520, 786Microstructural model, 629, 630Midband, 727, 758, 760, 762Minimum phase transfer functions, 623Modeling, 802Monophasic, 220, 230, 334, 335Most Significant Bit (MSB), 720Mutual inductance, 316, 416–418, 427, 428,

430, 786

NNatural frequency, 306, 319, 330, 331Natural response, 290, 306, 313, 318, 737Neoplasias, 633Nernst equation, 250Netlist, 95, 108, 113, 218, 219, 240, 246, 248,

326, 753, 762Network function, 614, 616–617, 619Network stability, 642Network synthesis, 187Neuron model, 252, 253

808 Index

Nodal analysis, 85, 96, 469, 486, 678Node, 15, 22, 23, 33, 59, 67, 85–87, 89, 90, 92,

94, 97–112, 114–116, 118–123, 125, 127,135, 137–139, 141, 143, 148, 153,155–158, 160, 162, 164, 169, 170, 178,181–184, 188, 242, 253, 260, 261,273–275, 300, 301, 333, 338, 342, 343,379, 458, 459, 470, 486, 490, 520, 672,677, 678, 710, 715, 729, 731, 734, 736,737, 739, 740, 742, 786

Node voltage method, 100, 101, 111, 116Noise, 687, 694, 788Non inverting amplifier, 678, 680, 684, 688,

711, 740Norton equivalent circuits, 165Norton’s theorem, 786Notch filter, 479, 482, 486, 490, 492, 495–499,

501–Nuclear Magnetic Resonance (NMR), 303Numerical, 23, 24, 39, 50, 64, 91, 99, 110, 165,

182, 214, 274, 280, 287, 309, 402, 420,458, 496, 500, 519, 585, 591, 609, 610,730, 739, 750

Numerical analysis tools, 214, 309Numerical integration, 309Nyquist plot, 374, 578, 580, 596, 638

OObjective function, 627Ohm, Georg Simon, 793Ohm’s law, 14, 18, 19, 21, 85, 100, 110, 190,

192, 322, 360, 362, 421, 422Open circuit, 150, 151, 165, 166, 169, 170,

173, 174, 176, 223, 242, 323, 397, 419,783, 786

Operational amplifier, 693, 694, 738, 766, 787OP room, 5, 34, 399, 401Optimization, 626, 628, 629Oscillator, 768, 770Oscilloscope, 258, 335, 431, 432Oxygen, 34–36

PParallel capacitors, 189, 196, 198, 199, 209,

394, 405, 406Parallel layer model, 631Parallel resistors, 14, 26, 28, 39, 160, 161, 516Parallel resonance, 387Parallel RLC circuits, 318Partial fraction expansion, 43, 445, 456, 461,

616, 781, 782Passive element, 428, 543, 545

Passive filters, 723, 724Passive sign convention, 426Perfectly coupled, 419, 423, 430Period, 3, 11, 208, 226, 228, 230Periodic function, 8, 212, 240, 255, 257, 285,

294, 296, 305, 307, 309, 311, 448, 460Permeability, 36, 277, 316, 419, 786Permittivity, 185, 188, 190, 193, 197–201, 203,

206, 207, 209, 630, 786Phasors, 355, 367, 415Phosoholipids, 240Piecewise linear, 691Piecewise linear function, 691Pi network, 649Polar form, 355Pole, 74, 444, 447–450, 459–462, 466–468,

472, 476, 478, 479, 481, 482, 484, 485,500, 501, 508, 515, 524, 525, 527–530,532–536, 540, 544, 550, 559, 565–567,593, 595, 615, 616, 622–624, 628, 629,639, 641, 725, 744, 745, 768, 769, 781,782, 789

Polycrystalline solids, 630Polyester capacitor, 189Polynomial, 444, 450, 451, 460, 472, 481, 484,

485, 488, 494, 496, 498, 508, 509, 535,542, 553, 558, 565, 624, 636, 659, 652, 781

Polynomial approximation, 257, 475, 476, 601,696, 698, 708, 710

Porous carbon-based electrodes, 224Port, 174, 423, 537, 656, 657, 733, 783, 789,

790Positive definite, 535, 537, 539, 547, 567Positive Real Function (PRF), 535, 539, 547,

656Potential difference, 13, 33, 40, 57, 63, 193,

194Potentiometer, 70Power, 3, 5, 7–10, 12, 15, 18, 19, 22, 25, 33,

43, 54, 56, 60, 167, 173, 175, 176, 183,184, 247, 249, 335, 348, 350, 389,399–401, 403, 404, 408–411, 413, 415,419, 422, 429, 431, 451, 487, 491, 493,652, 691, 700, 724, 730, 732, 733, 735,751, 781

Power factor, 409, 411, 412Power factor correction, 409Power measurement, 335Power triangle, 410Precancerous, 633Pressure, 34, 36, 59, 71, 75, 76, 198, 294, 337,

377, 706, 716

Index 809

Primary winding, 419, 422, 430, 432Principle of current division, 44, 56, 85, 145,

149, 152, 293, 321, 322Principle of voltage division, 44, 46, 57, 58, 64,

150–153, 168, 175, 221, 234, 364, 369,406, 457, 464, 471, 473, 523, 614, 676,707, 731, 733, 738, 769

Probe, 37, 48, 166, 392, 431, 432, 633, 634Proper rational functions, 443, 447–450Prothrombin time, 240

QQuadratic, 527, 529, 752Quadrature, 614, 768, 770Quality factor, 378, 386, 388, 390, 393, 764

RRadiation detector, 262Rational function, 443, 447–450, 537, 545,

550, 553, 564, 582, 639, 781Rational transfer function, 467, 535RC circuits, 394Reactance, 366Reactive load, 409Reactive power, 408, 409, 419Realizability, 549, 571Realizable, 543, 549, 676Reciprocal network, 663Reciprocity, 154, 155Rectangular form, 28, 39, 189, 351Reference node, 786Reflected, 420, 703Reflected impedance, 426Relay, 296, 298, 299Relay circuits, 296, 299Relay delay time, 299Residues, 550, 566, 567, 781, 782Resistance, 6, 25, 28–30, 39–43, 46, 50, 52, 53,

55, 57, 60, 62, 63, 65, 69–71, 73–81, 85,97, 132, 154, 726, 731–733, 735, 738, 755,756, 773, 788

Resistance bridge, 17, 22, 62, 69, 73, 75Resistance matrix, 154Resistance measurement, 169Resistive load, 173, 427Resistivity, 6, 39–42, 47, 48, 50–53, 55, 56, 74,

194, 302, 404, 515, 519, 630, 631, 633,643, 644

Resistors, 14, 23, 24, 26, 28, 29, 44, 56, 75, 76,79, 93, 110, 114, 160, 176, 224, 230, 320,335, 336, 387, 658, 672, 674, 676, 684,687, 692, 709, 710, 728, 743, 769, 783

Resonance, 278, 302, 303, 317, 371, 378, 379,383, 384, 386, 387, 395, 398, 429, 522

Resonant frequency, 378, 379, 383, 384, 429,430

Resonator, 378Response, 76, 306, 307, 330–332, 335, 400,

401, 423, 446, 464, 467, 478, 490, 498,501, 523, 525–529, 554, 559–562, 574,593, 603, 604, 623, 625–628, 643, 647,738, 741, 745, 749–752, 764, 766

Reuss model, 27, 29, 201Rise time, 688, 689RLC circuits, 305, 318RL circuits, 288Root Mean Square (RMS) value, 347, 348,

349, 400Roots, 306, 443, 450, 451, 460, 481, 484, 485,

535–537, 540, 624, 738, 744

SSallen and Key high-pass circuit, 751Sallen and Key low-pass circuit, 748, 749Scaling, 257, 420, 521, 522, 524, 656, 723, 747Schwartz inequality, 180Secondary winding, 249, 417, 419, 420,

422–424, 426, 427, 430–432Second-order circuits, 305, 469, 546, 764Self-inductance, 416, 792Sensitivity, 63, 74, 76, 205, 207, 634, 698, 700,

708Series, 14, 24, 25, 28, 29, 43, 44, 56, 137, 185,

201, 224, 249, 305–307, 309, 315, 317,318, 334, 340–343, 345, 356, 360, 367,368, 371, 377, 384, 385, 394, 409, 413,429, 479, 480, 533, 537, 546, 588, 630,632, 692, 698, 702, 703, 731, 747, 773,783, 786

Series capacitors, 334, 589Series inductors, 334Series layer model, 627, 628Series resonance, 386Series RLC circuits, 305, 306, 309–311, 313,

318, 331, 332, 383, 385, 386, 388, 393,461, 462, 502, 503

Sheet resistivity, 48, 50, 644Short circuit, 149, 151, 159, 165, 168, 170,

171, 174, 177, 182, 221, 242, 293, 329,397, 414, 457, 516, 555, 783, 786

Siemens, W.V., 793Signal, 57, 75, 142, 165, 256, 257, 259, 263,

284, 287, 303, 329, 347–350, 352–354,414, 419, 430, 431, 435, 468, 486, 498,531, 534, 614, 664, 665, 669, 671, 672,676, 679, 683, 686–691, 707, 708, 722,727, 729, 736, 737, 751, 764, 788

Signal-to-noise ratio, 683

810 Index

Simultaneous equations, 95, 183, 333, 780Singly terminated, 656Smoothing circuits, 330Sodium ions, 249–251Source-free parallel RLC circuit, 318Source-free RC circuit, 215, 289Source-free RL circuit, 290, 295, 302Source-free series RLC circuits, 305Source transformation, 155–164, 168, 170, 274Spectroscopy, 614, 629, 630, 633SPICE, 15, 16, 22, 23, 30, 90–92, 95, 98, 99,

105, 106, 108, 111, 113, 116, 121, 123,138, 139, 148, 149, 154, 211, 217–219,226, 228, 232, 235, 239, 240, 245–247,253, 254, 262, 265, 267, 268, 271, 273,275, 283, 285–287, 290, 300, 325–327,329, 330, 332, 365, 370, 372, 422, 423,431, 460, 653, 688–694, 698, 700, 710,725, 745, 753, 756, 760, 762, 767

Stability, 535, 639, 760, 764, 765Step-down transformer, 420Step response of an RC circuit, 598, 600Step response of an RL circuit, 523Step-up transformer, 422, 431Summing amplifier, 722Supercapacitor, 224, 233Supermesh, 137, 138Supernode, 118, 120, 122, 123Superposition, 140, 141, 145–151, 153, 158,

159, 188, 415, 416, 681, 765Superposition theorem, 149–151, 153, 158Susceptance, 366, 384, 502, 503, 786Switching functions, 213Symmetric, 154, 764, 780

TTemperature, 6, 34, 36–38, 59, 60, 64, 65, 69,

74–76, 196, 198, 199, 241, 249, 251, 302,671, 694, 695, 697, 698, 700, 702, 703,709, 716

Terminals, 13, 22, 60, 166, 167, 169–171, 179,456, 640, 678, 692, 708, 783, 787, 788

Thévenin, M. Leon, 165, 167, 168, 171–173,176, 177, 180, 181, 389, 396, 426

Thévenin’s theorem, 168, 783Thromboplastin, 240Thrombosis, 240Thévenin equivalent circuit, 172, 180, 389, 457Time constant, 192–195, 207, 210, 220, 221,

226, 229, 231, 236, 240, 241, 253, 254,257, 259, 260, 262, 266, 290, 291, 296,302, 320, 331, 334, 531, 534, 536, 638, 729

Time-delay, 372Tissue, 40, 53, 55, 351, 399–401, 403, 405,

430, 514, 633, 634Titanium oxide, 643Tomography, 329, 797Topology, 73, 75, 200, 249, 479, 483, 486,

498, 555, 564, 569, 588, 660, 684, 723,747, 752, 755, 761, 762, 764

Toroidal inductor, 431Transducer, 71, 614, 716Transfer function, 463, 465–471, 473,

476–479, 482, 484–486, 493, 496–498,501, 523, 524, 526–530, 533–536, 566,567, 585, 592, 617, 622, 623, 641, 655,703, 746, 747, 749, 752, 761, 769, 781,782, 784, 789

Transformation ratio, 17, 490, 564, 604, 643,786

Transformersair-core, 418ideal, 417–423, 426, 427isolation, 419linear, 419step-down, 420step-up, 249, 422, 431

Transient response, 289Transistor, 664, 692, 723Transpose, 110, 115Transresistance amplifier, 154Triangular wave, 282Trigonometric identities, 775Turns ratio, 417–422, 424, 426,

430, 432Two-phase dispersions, 632Two-port networks

hybrid parameters, 663impedance parameters, 647

UUltrasound, 392, 614Undamped natural frequency, 330, 331Underdamped case, 306, 310Unit impulse function, 468Unit ramp function, 440Unit step function, 213, 440, 465Unity gain, 717, 725, 755Unloaded, 430, 431

VVector, 74, 115, 199, 287, 395, 604, 628, 629,

780, 781Voigt Model, 27, 28, 198, 199, 201, 204

Index 811

Voltage, 4, 8, 9, 13, 15, 18, 19, 21–23, 30, 34,44, 56, 57, 72, 75, 85, 89, 90, 93, 98, 99,105, 115, 117, 118, 121, 125, 141, 149,150, 154, 157, 160, 162, 165, 169, 170,172, 173, 183, 185, 195, 211, 212, 214,217, 220, 221, 223, 225, 227, 230, 232,238–240, 242–246, 253–255, 264, 265,270, 273, 274, 278, 284, 288, 290, 291,301, 305, 307, 329, 334, 336, 345, 368,379, 384, 385, 397, 400, 407, 411, 413,416, 419, 423, 427, 430, 455, 461, 465,520, 573, 604, 653, 666, 673, 678–680,684, 685, 687, 688, 691, 692, 697, 698,706, 707, 709, 710, 712, 715–717, 719,720, 724, 732, 734, 738, 740, 783

Voltage divider, 21, 46, 59, 60, 142, 479, 483,717, 783, 784

Voltage division, 44, 58, 64, 150–153, 175,221, 234, 364, 369, 406, 457, 464, 471,473, 523, 614, 676, 707, 731, 733, 738,740, 769

Voltage follower, 265, 272Voltmeter, 13, 154, 385, 411

WWarfarin, 240Wattmeter, 409Wheatstone bridge, 22, 39, 60–63, 67–70, 75,

76, 706, 709, 710, 717Winding capacitance, 249Wye to delta transformations, 15, 490

XX-ray, 329, 797

YY-parameters, 663

ZZero, 715, 744, 787, 788Z-parameters, 648, 649, 653, 663

812 Index

appendix a.1. decimal preﬁxes of si units978-3-319-55101-2/1.pdf · a.1. decimal preﬁxes of si...

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