439044 1 En Bookbackmatter 773..812

Appendix

A.1. Decimal Preﬁxes of SI Units

International System of units (SI) preﬁxes used to form decimal multiples and submultiples 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. The number following the “E” speciﬁes the number of logarithmic steps per decade. E48, E96 series values are needed for higher accuracy and close tolerance mÀ1 requirements. Derivation is based on M ¼ 10 E , where M is the nominal resistance value at m position, E is a coefﬁcient related to tolerance. 3À1 1 Example The third multiplier in E24 series is M ¼ 10 24 ¼ 1012 ¼ 1:21 ! 1:2

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 8.2 E24 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 9.1 E48 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 9.53 E96 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 ¼À1 eiA ¼ cos A þ i sin A ðEuler's formulaÞ

eiA þ eiA eiA À eiA cos A ¼ ; sin A ¼ 2 2i ln x log x ¼ ; xy ¼ eyÁln x 10 ln 10 ex À eÀx ex þ eÀx sin hx sin hx ¼ ; cos hx ¼ ; tan hx ¼ 2 2 cos hx Appendix 775

Trigonometric Identities sinðÞ¼A þ B sin A cos B þ cos A sin B; sinðÞ¼A À B sin A cos B À cos A sin B cosðÞ¼A þ B cos A cos B À sin A sin B; cosðÞ¼A À B cos A cos B þ sin A sin B

tan A þ tan B tanðÞ¼A þ B 1 À tan A tan B tan A À tan B tanðÞ¼A À B 1 þ tan A tan B 1 sin A cos B ¼ ðÞsinðÞþA þ B sinðÞA À B 2 1 cos A cos B ¼ ðÞcosðÞþA þ B cosðÞA À B 2 1 cos A sin B ¼ ðÞsinðÞÀA þ B sinðÞA À B 2 1 sin A sin B ¼ ðÞcosðÞÀA þ B cosðÞA À B 2 A þ B A À B sin A þ sin B ¼ 2 sin cos 2 2 A þ B A À B cos A þ cos B ¼ 2 cos cos 2 2 A þ B A À B sin A À sin B ¼ 2 cos sin 2 2 A þ B A À B cos A À cos B ¼À2 sin sin 2 2 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 00 010 p pffiffi 30 1 3 p1ffiffi 6 2 2 3 p pffiffi pffiffi 45 2 2 1 4 2 2 p pffiffi pffiffiffi 60 3 1 3 3 2 2 p fi 90 2 1 0 Unde ned 180 p 0 −10 776 Appendix sin2 A þ cos2 A ¼ 1

1 þ cos 2A cos2 A ¼ 2 1 À cos 2A sin2 A ¼ 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 1 À cos A sin ¼Ç 2 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 1 þ cos A cos ¼Ç 2 2 sin 2A ¼ 2 sin A cos A cos 2A ¼ cos2 A À sin2 A ¼ 2 cos2 A À 1 ¼ 1 À sin2 A

Some Power Series Expansions

x2 x3 x4 ex ¼ 1 þ x þ þ þ þ ÁÁÁ 2! 3! 4! x2 x4 x6 cos x ¼ 1 À þ À þ ÁÁÁ 2! 4! 6! x3 x5 x7 sin x ¼ x À þ À þ ÁÁÁ 3! 5! 7! "# x À 1 1 x À 1 3 1 x À 1 5 ln x ¼ 2 þ þ þ ÁÁÁ x þ 1 3 x þ 1 5 x þ 1

Table of Standard Derivatives fxðÞ¼xn; f 0ðÞ¼x nxnÀ1 fxðÞ¼ex ¼ f 0ðÞx fxðÞ¼ax; f 0ðÞ¼x ax ln a ða [ 0Þ fxðÞ¼sin x; f 0ðÞ¼x cosðxÞ fxðÞ¼cos x; f 0ðÞ¼Àx sinðxÞ Appendix 777

1 fxðÞ¼sinÀ1 x; f 0ðÞ¼x pffiffiffiffiffiffiffiffiffiffiffiffiffi ðÞÀ1\x\1 1 À x2 1 fxðÞ¼cosÀ1 x; f 0ðÞ¼Àx pffiffiffiffiffiffiffiffiffiffiffiffiffi ðÞÀ1\x\1 1 À x2 1 fxðÞ¼tanÀ1 x; f 0ðÞ¼x 1 þ x2

L’Hopital’s Rule

If lim fxðÞ¼ lim hxðÞ¼ 0; x!A x!A

f ðxÞ f 0ðxÞ lim ¼ lim x!A hðxÞ x!A h0ðxÞ prime indicating differentiation operation. Also,

If lim fxðÞ¼lim hxðÞ¼1; x!1 x!1

f ðxÞ f 0ðxÞ lim ¼ lim x!1 hðxÞ x!1 h0ðxÞ

Table of Standard Integrals Z xn þ 1 xn dx ¼ þ cn6¼À1 n þ 1 Z 1 dx ¼ ln x þ c x Z ex dx ¼ ex þ c Z ax ax dx ¼ þ c ða [ 0Þ ln a Z sin xdx¼Àcos x þ c 778 Appendix Z cos xdx¼ sin x þ c Z tan xdx¼ lnjj sec x þ c Z x 2x sin2 xdx¼ À sin þ c 2 4 Z x 2x cos2 xdx¼ þ sin þ c 2 4 Z 1 1 x dx ¼ tanÀ1 þ c ða [ 0Þ a2 þ x2 a a Z 1 x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx ¼ÀsinÀ1 þ c ðÀa\x\aÞ a2 À x2 a Z 1 x pffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx ¼ sin hÀ1 þ c ða [ 0Þ a2 þ x2 a

Some Laplace Transform Pairs ftðÞ¼dðÞt ; FsðÞ¼1 eÀas ftðÞ¼utðÞÀ a ; FsðÞ¼ s 1 ftðÞ¼utðÞ; FsðÞ¼ s n! ftðÞ¼tn; FsðÞ¼ sn þ 1 1 ftðÞ¼eat; FsðÞ¼ s À a n! ftðÞ¼tn Á eÀat; FsðÞ¼ ðÞs þ a n þ 1 x ftðÞ¼sin xt; FsðÞ¼ s2 þ x2 s ftðÞ¼cos xt; FsðÞ¼ s2 þ x2 Appendix 779

x2 ftðÞ¼eÀat sin xt; FsðÞ¼ ðÞs þ a 2 þ x2 s þ a ftðÞ¼eÀat cos xt; FsðÞ¼ ðÞs þ a 2 þ x2 2xs ftðÞ¼t sin xt; FsðÞ¼ ðÞs2 þ x2 2

Laplace Transforms, Some Properties

1 ftðÞ FsðÞ¼ R estf ðtÞdt ðdefinitionÞ 0 afðÞþ t bgðÞ t aFðÞþ s bGðÞ s ðLinearityÞ ektf ðtÞ FsðÞðÀ k shift in sÞ f 0ðÞt sFðÞÀ s f ðÞ0 ðfirst derivativeÞ f 00 ðÞt s2FsðÞÀsf ðÞÀ0 f 0 ðÞ0 ðsecond derivativeÞ t 1 R fuðÞdu FsðÞ ðintegralÞ 0 s

HtðÞÀ a ftðÞÀ a easFsðÞ ðshift in tÞ

1 ZT ft T f t eÀstftdt Periodic Function ðÞ¼þ ð Þ ÀsT ðÞ ð Þ 1 À e 0 lim f ðtÞ lim fgsFðÞ s ðinitial valueÞ t!0 s!1 lim ftðÞ limfgsFðÞ s ðfinal valueÞ t!1 s!0

dFsðÞ tfðÞ t À ðFrequency differentiationÞ ds ftðÞ 1Z FðrÞdr ðFrequency integrationÞ t 0 t R f ðt À sÞgðsÞds FsðÞGsðÞ ðConvolutionÞ 0 780 Appendix

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

T ½X¼½x1 x2 x3...xn is the column vector of unknowns, then the system has a unique solution, whose individual values for the unknowns are

detðA Þ x ¼ j j ¼ 1; 2; ...; n j detðAÞ

Aj is the matrix formed by replacing the jth column of [A] by the column vector [Y]. Example ab x1 e detðA1Þ ed À bf detðA2Þ af À ec ¼ ; x1 ¼ ¼ ; x2 ¼ ¼ cd x2 f detðAÞ ad À bc detðAÞ ad À bc

Solving Linear System of Simultaneous Equations of the Form [A].[X]= [Y] in MATLAB Column vector of unknowns, ½Y is a column vector;

%A=(5*5) example 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=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 Appendix 781

Partial Fraction Expansion in MATLAB Consider a transfer function, which is represented by a ratio of two polynomials in s-domain.

bðsÞ b sm þ b smÀ1 þ b smÀ2 þ ÁÁÁ þb Hs 1 2 3 m þ 1 ðÞ¼ ¼ n nÀ1 nÀ2 aðsÞ a1s þ a2s þ a3s þ ÁÁÁ þan þ 1

Such a function can be represented by two vectors, one of them specifying the coefficients of the numerator polynomial, and the other vector specifying the coefficients of the denominator polynomial. For example, assuming that both of these polynomials are fourth-order polynomials, then numerator polynomial coefficients vector is b=[b1b2b3b4b5] and denominator polynomial coefficients vector is a=[a1a2a3a4a5] These vectors specify the coefficients of the polynomials in descending powers of s, and the orders of these polynomials can be different. Partial fraction expansion of this rational function H(s)is

bðsÞ R R R R HsðÞ¼ ¼ 1 þ 2 þ 3 þ 4 þ kðsÞ aðsÞ s À p1 s À p2 s À p3 s À p4

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 R R R R ; p p p p p ; k C C C ¼½ 1 2 3 4 ¼½ 1 2 3 4 ¼½ 2 1 0

R and p are column vectors, while k is a row vector. The command below ﬁnds the partial fraction expansion of the ratio of two polynomials.

[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, and the result is normalized for the leading coefﬁcient in the denominator. 782 Appendix

Example

bðsÞ s4 þ 10s3 þ 40s2 þ 75s þ 50 HsðÞ¼ ¼ aðsÞ s4 þ 10s3 þ 35s2 þ 50s þ 24 b=[110407550] a=[110355024]

its partial fraction expanded form is computed as

[r, p, k] = residue(b, a) r = [-1 -2 2 1] p = [-4 -3 -2 -1] k=1

This means,

À1 À2 2 1 HsðÞ¼ þ þ þ þ 1 s þ 4 s þ 3 s þ 2 s þ 1

The partial fraction expansion for multiple poles: Example

bðsÞ 3 HsðÞ¼ ¼ aðsÞ s3 þ s2

b = [3] a=[1100] its partial fraction expanded form is computed as

[r, p, k] = residue(b, a) r=[3-33] p=[-100] k=[]

This means,

3 3 3 HsðÞ¼ À þ s þ 1 s s2

Note that if a transfer function has multiple poles, then small changes in the data or round-off errors can cause large variations in the resulting poles and residues. Appendix 783

Rules to ﬁnd Thevenin’s Equivalent Circuit Thevenin’s theorem helps to reduce any one-port linear electrical network to a single 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)

Voc RTh ¼ Isc

(1) Find Voc = VTh (2) Connect a 1 A current source ﬂowing from terminal b to terminal a; V V (3) R ¼ oc ¼ oc Th I 1A

The equivalent circuit consists of only RTh (there is neither a current nor a voltage source). Voltage Dividers A voltage Vi is applied to two series connected impedances, Z1; Z2. Let Z2 has a connection to reference (ground) and Z1 has a connection to the ungrounded terminal of the voltage source, Vi. The output voltage Vo is obtained at the junction of Z1; Z2 (Laplace operator s is omitted) (Table A.2).

Fig. A.1 Linear electrical network to a single voltage source and a single impedance 784 Appendix

Table A.2 Voltage dividers Type of voltage divider Voltage transfer function

Resistive Vo R2 ¼ Vi R2 þ R1 Inductive Vo L2 ¼ Vi L2 þ L1 Capacitive Vo C1 ¼ Vi C2 þ C1

Magnitude and Phase of Transfer Functions

NðsÞ a s2 þ a snÀ1 þ ÁÁÁ þa ðÞs À z ðs À z ÞÁÁÁðs À z Þ Hs n nÀ1 o K 1 2 n ðÞ¼ ¼ m mÀ1 ¼ DðsÞ bms þ bmÀ1s þ ÁÁÁ þbo ðÞs À p1 ðs À p2 ÞÁÁÁðs À pmÞ where, an 6¼ 0, bm 6¼ 0, all ai, bi, are real. The magnitude of H(jx) in decibels is deﬁned as

Xn Xm x x x 20 log10jjHjðÞ¼ 20 log10jjK þ 20 log10jjj À zi À 20 log10jjj À pi i¼1 i¼1

The phase in degrees (radians) is deﬁned as

Xn Xn Im½HjðÞx Imðjx À z Þ Imðjx À p Þ u ¼ tanÀ1 ¼ tanÀ1 i À tanÀ1 i Re Hjx Re jx z Re jx p ½ðÞ i¼1 ð À iÞ i¼1 ð À iÞ

Bode Plots Exact manual calculation of magnitude and phase is a laborious process. Approximate sketches of these functions can be easily performed using so called Bode plots, noting that numerator and denominator of a transfer function (in factored 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 as þ b) Following Fig. A.2 displays drawing rules for magnitude and phase graphs for constant term, s,1/s, s + z and 1/(s + p). The number of decades between two frequencies is given as D f2 ; [ f10 ¼ log10 f2 f1 f1

The number of octaves between two frequencies is Appendix 785

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

D f2 ; [ f2 ¼ log2 f2 f1 f1

Duality Dual circuits are the ones which are described by the same characteristic equations with dual quantities interchanged. A dual of a relationship can be written by 786 Appendix

interchanging voltage and current in an expression. The dual expression produced is of 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, Voltage Generator–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, the dual of a star (Y) network of inductors is a delta ðDÞ network of capacitors, which is not the same thing as a star-delta (Y-D) transformation; the transformation results in an equivalent circuit. The dual of a mutual inductance cannot be formed directly, since there is no corresponding capacitive element. In case the circuit conﬁguration is not parallel or series or it contains dependent sources, following steps can be used to construct graphically the dual of a planar circuit: (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 its dual. (d) Assign polarities of sources. A voltage source producing clockwise mesh current 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 vtðÞ¼itðÞ: RiðtÞ¼vtðÞ: G Capacitor–Inductor differential d d iCðtÞ¼C vCðtÞ vLðtÞ¼L iLðtÞ expression dt dt Capacitor–Inductor integral Rt Rt v t V 1 i s ds i t I 1 v s ds expression CðÞ¼ 0 þ C Cð Þ LðÞ¼ 0 þ L Lð Þ 0 0

VCCS–CCVS im ¼ xvn vm ¼ xin

VCVS–CCCS vm ¼ xvn im ¼ xin Resistor conductor R ¼ x X G ¼ x S Capacitor inductor C ¼ x F L ¼ x H Voltage–Current source v ¼ x V i ¼ x A

– Ra Ga Voltage Current division vR ðÞ¼t : v iG ðÞ¼t : i a Ra þ Rb a Ga þ Gb Appendix 787

SPICE Models For Dependent (Controlled) Sources

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

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

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

Current controlled current source CCCS: Fname N1 N2 Vcontrol Value Example F1 0 3 Vm 5 Vm 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 voltage source, 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 voltage source). Operational Amplifier Operational amplifier (op-amp) is a versatile active element that behaves like a voltage-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 operates in the linear range 788 Appendix

Fig.A.4 Opamp model

Ideal op-amp: ÀÁ Vo ¼ A Á Vd ¼ AVp À Vn

in ¼ ip ¼ 0; Vp ¼ Vn

Ro ¼ 0 X; Ri ¼1X

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. The input terminals are not shorted together. Instrumentation Amplifier The instrumentation amplifier is an essential circuit in biomedical electronics. For example, a two terminal sensor produces a signal but neither of its terminals may be connected to the same ground level as with the measuring network. These terminals may be DC biased at relatively large voltages or added to the noise. The differential amplifier acts seletively on measuring the difference between the input terminals. Addition of 2 buffers between the sensor output and the differential amplifier prevents the loading of both sensor and the measuring electronics. The circuit configuration shown here provides gain, as well (Fig. A.5). Vo 2R1 R3 ¼ 1 þ Á ; Vd ¼ V1 À V2 Vd RG R2

Butterworth Polynomials in Factored Form Order Denominator, D(s) 1 s þ 1 pffiffiffi 2 s2 þ 2s þ 1 3 ðs2 þ 1Þðs þ 1Þ 4 ðs2 þ 0:765s þ 1Þðs2 þ :848s þ 1Þ 5 ðs þ 1Þðs2 þ 0:618s þ 1Þðs2 þ 1:618s þ 1Þ pffiffiffi 6 ðs2 þ 0:518s þ 1Þðs2 þ 2s þ 1Þðs2 þ 1:932s þ 1Þ Appendix 789

Fig.A.5 Instrumentation ampliﬁer

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

2 a2sþ a1s þ a0 HðsÞ¼ x s2 þ 0 s þ x2 Q 0

The poles are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 1 p ; p ¼À 0 Æ jx 1 À 1 2 2Q 0 4Q2

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

Table A.4 Second order (biquad) ﬁlter transfer functions Filter type Transfer function Gain Low-pass a0 a0 x 2 0 x2 x2 s þ Q s þ 0 0 2 High-pass a2s a2 x 2 0 x2 s þ Q s þ 0 x Band-pass a1s 0 x Center frequency ¼ a1: 2 0 x2 s þ Q s þ 0 Q 2 x2 : Band-stop s þ 0 dc ðÞ¼ high freq gain a2 a2 x 2 0 x2 s þ Q s þ 0 x 2 0 x2 a All-pass s À Q s þ 0 Flat gain 2 a2 x 2 0 x2 s þ Q s þ 0 790 Appendix

Table A.5 Parameter relations zyh z z y y D h z 11 12 22 À 12 h 12 D D h h z z y y 22 22 21 22 y y 21 11 h 1 À À 21 Dy Dy h22 h22 z22 z12 y y h y À 11 12 1 12 D D h À h z z y y 11 11 z z 21 22 21 11 h D À 21 h Dz Dz h11 h11 y h Dz z12 1 12 h11 h12 y À y h h z22 z22 11 11 21 22 z 1 y D À 21 21 y z z y y 22 22 11 11 D z z z z ; D y y y y ; D h h h h z ¼ 11 22 À 12 21 y ¼ 11 22 À 12 21 h ¼ 11 22 À 12 21

Note that there are other parameter sets to characterize two-port networks other than the three parameter types presented here. However, they are not used in this book. Historical Profiles

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

Alessandro Giuseppe Volta, 1745–1827 Volta was an Italian physicist and chemist. He invented the ﬁrst electrical battery, the Voltaic pile, in 1799. Volta’s invention led to the development of the ﬁeld of electrochemistry. The SI unit of electric potential is named in his honor. In 1778 he managed to isolate methane. Volta studied capacitance, and it was for this work that the unit of electrical potential has been named the volt.

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

Léon Charles Thévenin, 1857–1926 Thévenin was a French electrical engineer. He developed his famous theorem, known as Thévenin’s theorem.

Joseph Henry, 1797–1878 Henry was an American scientist who discovered electromagnetic induction independently of and at about the same time as Michael Faraday. Henry discovered the self-inductance, Unit of inductance is named in his honor.

Michael Faraday, 1791–1867 Faraday invented the electric motor in 1821. He discovered the induction of electric current by magnetism in 1831, and the ability of magnetic fields to change the polarization of light in 1845. The unit of electrical capacitance—the farad (F) —is named in his honor. Faraday discovered the carbon compound benzene, and in 1823, he was the first scientist to liquefy a gas. He was also first to introduce terms such as “electrode,”“cathode” and “ion”. Historical Profiles 793

Georg Simon Ohm, 1789–1854 A German physicist and mathematician. Ohm found that there is a direct proportionality between the potential difference applied across a conductor and the resultant current. This relationship is known as Ohm’s law. The work of Ohm marked the beginning of circuit theory; the unit of resistance is named in his honor.

Gustav Robert Kirchhoff, 1824–1887 Kirchhoff was a physicist. He contributed to the fun- damental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects. Kirchhoff formulated his circuit laws (KCL, KVL) in 1845. He worked at University of Heidelberg in 1854, collaborating in a spectroscopic work with Robert Bunsen, where they discovered cesium and rubidium in 1861.

Ernst Werner von Siemens, 1816–1892 Siemens’s name has been honored as the SI unit of electrical conductance. He invented electrically charged sea mines, worked on perfecting technologies that had already been established, and invented a telegraph that used a needle to point to the right letter, instead of using Morse code. Siemens is the father of the trol- leybus, which was ﬁrst introduced in 1882. 794 Historical Profiles

Bernard D.H. Tellegen, 1900–1990 B. Tellegen was a Dutch electrical engineer. He is the inventor of the pentode tube in 1926 and the gyrator in 1948. Tellegen held 41 US patents and he is also known for a theorem in circuit theory. He received MSEE degree from Delft University in 1923.

Hendrik Wade Bode, 1905–1982 Bode was a Dutch-American engineer. Bode received his Ph.D. from Columbia University in 1935. He is a pioneer of modern control theory and electronic telecommunications. He made important contributions to the design, guidance and control of anti-aircraft systems missiles and anti-ballistic missiles. He also made important contributions for the analysis of stability of linear systems. He is known for the graph that honored on his name, the Bode plot. He held 25 US patents.

Alan Lloyd Hodgkin, 1914–1998 AL Hodgkin was an English physiologist and biophysicist, 1963 Nobel Prize winner in Medicine. He worked in Cambridge University, also held additional administrative 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 theory representing one of the earliest applications of voltage clamping technique. Historical Profiles 795

Andrew Fielding Huxley, 1917–2012 A.F. Huxley was a 1963 Nobel Prize-winning physiologist and biophysicist for his studies on the action potentials with A.L. Hodgkin. He developed interfer- ence microscopy to study muscle ﬁbers. He discovered in 1954 the mechanism of muscle contraction, so called the sliding ﬁlament theory.

William T. Bovie, 1882–1958 W.T. Bowie was an American biophysicist. He invented the electrosurgical generator. He completed a Ph.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, 1860–1927 Einthoven was a Dutch doctor and physiologist. He was the inventor of the first electrocardiogram in 1903. Einthoven received a medical degree from the University of Utrecht. He became a professor at the University of Leiden in 1886. ECG equipment he first used had a string galvanometer and moving roll of photo-sensitive paper, weighting about 270 kg. He received the Nobel Prize in Medicine in 1924 for his contributions in this field. 796 Historical Profiles

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

Hans Berger, 1873–1941 Berger was a German neurologist. Berger received his medical degree from Jena in 1897. He became Rector of Jena University in 1927. He recorded the electrical brain waves, the electroencephalogram (EEG) in 1924. Berger also described the different waves or rhythms which were present in the normal and abnormal brain.

Ian Donald, 1910–1987 Ian 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 at London University in 1937. In 1954, he introduced echo-sounding (the term from sonar) and searched its possible medical applications. Historical Profiles 797

Godfrey Newbold Hounsfield, 1919–2004 He was an English electrical engineer who received the 1979 Nobel Prize with Allan McLeod Cormack for developing the X-ray computed tomography. Hounsfield built a prototype head scanner and tested it on himself. In 1971, the first head scanner was operated in a London hospital. His name is given to Hounsfield scale, a quantitative measure in evaluating CT scans.

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

Raymond V. Damadian, 1936– Damadian was credited as the originator of the whole-body magnetic resonance imaging. He earned his degree in mathematics from the University of Wisconsin–Madison in 1956, and an M.D. degree from the Albert Einstein College of Medicine in New York City in 1960. In 1974, he received the ﬁrst patent in the ﬁeld of MRI. 798 Historical Profiles

William Bennett Kouwenhoven, 1886–1975 Kouwenhoven invented the ﬁrst cardiac deﬁbrillator. He received his BSEE from Brooklyn Polytechnic in New York, and his Ph.D. in electrical engineering from the Karlsruhe Technische Hochschule in Germany in 1913. He joined the faculty of the Johns Hopkins University School of Engineering in 1914. He tested his device on a dog. In 1947, Professor Claude Beck used his device on a 14-year-old boy at Case Western Reserve University.

Paul M. Zoll, 1911–1999 Zoll was a practicing physician and recognized as a Pioneer in Cardiac Pacing. In 1952, Paul M. Zoll described cardiac resuscitation via electrodes on the bare chest with 2-millisecond duration pulses of 100– 150 V across the chest, at 60 stimuli per minute. This became the basis for future clinical pacing develop- ments. In 1956, he published a transcutaneous approach to terminate ventricular ﬁbrillation with a shock voltage up to 750 V, and later described similar termination of ventricular tachycardia.

Karl William Edmark, 1924–1994 A cardiovascular surgeon, inventor. Edmark developed adefibrillator that utilized direct current (DC), which provided lower-energy and more effective shocks. Edmark’s invention, known as the Edmark Pulse Defibrillator, was first used to save the life of a 12-year-old girl in 1961. Historical Profiles 799

James Francis Pantridge, 1916–2004 Professor Pantridge and Dr. John Geddes of the Royal Victoria Hospital in Belfast produced the first portable defibrillator in 1965. A mains (AC) powered defibrillator was powered by an inverter, which converted a 12 V car battery to 230 V. The unit weighed 70 kg. By 1968 he had designed an instrument weighing only 3 kg, incorporating a mini capacitor manufactured for NASA.

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

John G. Webster J.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 of electrical impedance tomography in 1978, and published many books on biomedical engineering. Prof. Webster was a professor emeritus in the College of Engineering 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 Hill Balabanian N, Bickart TA, Seshu S (1969) Electrical network theory. Wiley Basso CP (2016) Linear circuit transfer functions: an introduction to fast ana- lytical techniques. Wiley-IEEE Press Bobrow LS (1987) Elementary linear circuit analysis, Holt Rinehart and Winston Boylestad R (2013) Introductory circuit analysis, 12th edn. Pearson New International Edition Davis AM (1998) Linear circuit analysis. PWS Publishing Company Dorf RC, Svoboda JA (2013) Introduction to electric circuits, 9th edn. International Student Version, Wiley Floyd T (2013) Principles of electric circuits, 9th edn. Pearson New International Edition Hayt 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 circuit analysis, 5th edn. Prentice-Hall, Inc. Nahvi M, Edminister JA (2014) Schaum’s outline of electric circuits, 6th edn. McGraw-Hill Education Nilsson J, Riedel S (2014) Electric circuits with mastering engineering, 10th edn. Pearson Global Edition O’Malley J (1992) Schaum’s outline of basic circuit analysis, 2nd edn. McGraw‐Hill Reddy HC (2002) The circuits and ﬁlters handbook, 2nd edn. CRC Press

Spence R (2008) Introductory circuits. Wiley Thomas RE, Rosa AJ, Toussaint GJ (2015) The analysis and design of linear circuits. Wiley Vlach J (2014) Linear circuit theory: matrices in computer applications. Apple Academic Press Wing O (2009) Classical circuit theory. Springer Circuit Synthesis and Design Anderson BDO, Vongpanitlerd S (1973) Network analysis and synthesis: a modern systems approach. Prentice-Hall Baher H (1984) Synthesis of electrical networks. Wiley, New York Bakshi UA, Bakshi AV (2009) Fundamentals of network analysis and synthesis. Technical Publications Pune Daryanani G (1976) Principles of active network synthesis and design. Wiley, New York Glisson TG (2011) Introduction to circuit analysis and design. Springer Guillemin EA (1977) Synthesis of passive networks: theory and methods appropriate to the realization and approximation problems (Reprint). Huntington, N.Y., R. E. Krieger Pub. Co. Kuo F (1966) Network analysis and synthesis, 2nd edn. Wiley Lamm HY-F (1979) Analog and digital ﬁlters: design and realization. Prentice Hall, Inc. Schaumann R, Valkenburg MEV (2001) Design of analog ﬁlters. Oxford University Press. Temes GG, Lapatra JW (1977) Circuit synthesis and design. McGraw-Hill Valkenburg MEV (1960) Introduction to modern network synthesis. Wiley Weinberg L (1962) Network analysis and synthesis Yarman 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 Media Kandaswamy A, Pittet A (2009) Analog electronics. Prentice Hall India, Learning Pvt. Ltd. Khanchandani S (2007) Power electronics. Tata McGraw-Hill Education Liu Y (2012) Power electronic packaging: design, assembly process, reliability and modeling. Springer Science & Business Media Peyton A, Walsh V (1993) Analog electronics with op-amps: a source book of practical circuits. Cambridge University Press Sedra AS, Smith KC (2004) Microelectronic circuits, 5th edn. Oxford University Press Selected Bibliography 803

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 Interscience Bruce EN (2001) Biomedical signal processing and signal modelling. Wiley Carr JJ, Brown JM (2001) Introduction to biomedical equipment technology, 4th edn. Prentice Hall David Y, Maltzahn WW, Neuman MR, Bronzino JD (2003) Clinical engineering. CRC Press Enderle J, Blanchard S, Bronzino J (2005) Introduction to biomedical engineering, 2nd edn. Elsevier Academic Press Saltzman WM (2015) Biomedical engineering: bridging medicine and technology. Cambridge University Press Semmlow 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 Press Weiss TF (1996) Cellular biophysics, electrical properties, vol. 2. The MIT Press Webster JG (ed) (1998) Medical instrumentation: application and design, 3rd edn. Wiley, New York Webster JG (ed) (2004) Bioinstrumentation. Wiley SPICE, MATLAB and Others Banzhaf W (1989) Computer-aided circuit analysis using SPICE. Prentice Hall Butt R (2009) Introduction to numerical analysis using MATLAB. Jones & Bartlett Learning Hahn BD (2002) Essential MATLAB for scientists and engineers, 2nd edn. Butterworth-Heinemann Rashid MH, Rashid HM (2006) SPICE for power electronics and electric power, 2nd edn. Taylor and Francis Sedra AS, Roberts GW, Smith KC (1992) SPICE for microelectronic circuits. Saunders College Pub. Smythe WR (1989) Static and dynamic electricity, 3rd edn. Taylor and Francis Thorpe TW (1992) Computerized circuit analysis with SPICE: a complete guide to SPICE, with applications. Wiley Yang X-S (2006) An introduction to computational engineering with MATLAB. Cambridge Int. Science Publishing Index

A Biphasic, 335, 336 AC bridge circuit, 406, 407 Bipolar Junction Transistor (BJT), 665 AC circuits, 407 Blood, 34, 35, 240, 429, 459 Acetic acid, 634 Bode plot, 530, 580, 784 AC power Bode’s method, 616, 617 apparent power, 408 Break frequency, 522, 694, 724, 751, 755, 758, average power, 348, 350, 413, 414, 427 760 complex power, 419 Bridged-T filter, 497, 501 effective value, 386 Butterworth filters, 655, 656, 745–747, instantaneous power, 353 752–754 maximum average power transfer, 421 By inspection, 90, 105, 116, 125, 126, 129, power factor, 408, 409, 411, 412 136, 382, 459 power factor correction, 409 power measurement, 335 C rms value, 347–349, 385, 400 Calcium reagents, 240 AC voltage, 365, 370, 372 Cancerous tissues, 633 Additivity property, 142 Canonic, 548, 563 Admittance, 352, 363, 366, 383, 384, 474, 487, Capacitance multiplier, 730, 731 490, 501, 503, 520, 521, 548, 549, 553, Cardiac, 334, 335 586, 599, 601, 607, 611, 615, 619, 629, 630 Cascaded networks, 749, 755 Admittance parameters, 474, 548, 549, 553 Cauchy integral formula, 608, 609 Air-core transformers, 418 Cauchy principal value, 608, 609 Alternating current (AC), 6, 334, 385, 399 Cauer, 553, 555, 557–560, 562, 564, 603, Ammeter, 154, 302 655–656, 658, 660, 662 Ampère, Andre-Marie, 791 Causal, 610–612, 642, 643 Analog computer, 743, 744 Cell counting, 39, 63 Analytic continuation, 607 Ceramic capacitor, 302 Analytic signal, 614 Cervical, 633, 634 Apparent power, 408 Cervix, 633, 634 Average power, 5, 220, 221, 348 Characteristic equation, 60, 305, 306, 318, 330, 338, 421, 459, 535, 737, 743 B Charge, electric, 4, 11, 12, 194, 195 Balanced, 11, 12, 17, 60, 61, 67, 68, 71, 277, Chebyshev filters 407, 712, 714–715 Chebyshev polynomials Battery, electric, 2, 43, 166, 246, 259, 351 Citrated blood plasma, 240 Bessel filter, 652 Clark electrode, 35, 36 Bessel polynomials, 652 Closed-loop gain, 671, 676 Binary weighted ladder, 720 Clot formation, 240 Biomedical instrumentation, 788 Coagulation, 240, 398, 399, 401

Coaxial cylindrical capacitor, 207, 208 Decibel (dB), 595, 784 Coefficient of coupling, 419 Defibrillator, 220, 224, 230, 246, 249, 264, Coil, 278, 282, 285, 296, 298, 299, 301, 302, 270, 311, 312, 316, 333–338 316, 317, 395, 409, 416, 417, 429, 430 Delay circuit, 235 Colposcopy, 634 Delta-to-wye conversion, 16 Complex conjugate, 396, 483 Dependent current source, 274 Complex frequency Dependent voltage source, 274 Complex numbers, 142, 358 Derivatives, 308, 333, 447, 603, 604, 743 Complex power, 419 Determinant, 487, 493, 669, 780 Composite, 26–30, 198, 200, 201, 203, 440, Difference amplifier, 681, 683–685, 717, 733 441, 639 Differential equations, 4, 253, 260, 273, 281, Conductance, 18, 19, 37, 45, 111, 141, 240, 294, 295, 463, 737, 739, 743, 744 241, 261, 280, 366, 384, 500–503, 722 Digital-to-analog converter (DAC), 329, 719 Conductance matrix, 115, 154 Dot convention, 416, 417 Confidence bounds, 257, 259 Double layer, 629, 634 Constraint, 21, 119, 120, 138, 139, 173, 337, Driving-point impedance, 647 339 Duality, 338, 339, 341, 520 Controlled source, 86, 116, 171 Convolution, 468 E Coulomb, 11, 12, 250 Echo-cardiography, 424 Coupling coefficient, 418, 427 EEG, 761, 764 Cramer’s rule, 87–89, 91, 101, 103, 108, 126, Effective medium models, 629-631 128, 130, 133, 159, 160, 379, 425, 459, Effective value, 386 486, 492 Electrode/electrolyte interface, 638 Critically damped case, 306 Electrodes, 11, 13, 33, 37, 47, 48, 63, 249, 317, CT scanner, 329 334, 335, 337, 351, 399, 400, 404, 630, 634 Current, 2, 4–6, 8, 9, 11, 12, 18, 20, 22, 24, 26, Electrodynamics, 791 34, 36, 41, 44, 47, 52, 59, 74, 77, 85, 86, Electrolyte, 35, 37, 63, 224, 632 88, 99–101, 105, 113, 116, 121, 124, 128, Electrolytic, 11, 224, 638 130, 132, 134, 137, 140, 143, 145, 150, Electrolytic capacitor, 639 151, 154, 155, 161, 171, 174, 191, 210, Electromagnetic induction, 792 212, 218, 223, 229, 230, 232, 233, 235, Electromagnetic waves, 751 236, 238, 244, 245, 249, 255, 262, 264, Electrosurgery unit (ESU), 398–402, 404, 405, 265, 267, 277, 282–285, 287, 290, 291, 431 293, 296, 299, 301, 302, 305, 317, 323, Elimination method, 103 328, 334–336, 339, 348, 361, 364, 368, Energy, 3, 4, 6, 8, 9, 12, 213, 214, 216, 220, 369, 371, 372, 382, 389, 399–401, 407, 221, 224, 226, 228–230, 232, 235, 237, 409, 416–418, 421, 432, 465, 468, 633, 249, 266, 267, 291, 301, 309, 311, 312, 670, 671, 678, 680, 682, 694, 707, 712, 314–317, 330, 334–336, 351, 400, 429, 715, 732 430, 456 Current-division principle, 44, 56, 145, 149, Equivalent circuit, 77, 173, 209, 224, 243, 252, 152, 293, 321, 322 253, 402, 406, 428, 454, 455, 492, 516, Curve fitting, 38, 257, 259, 603, 605, 702 559, 626, 632, 640, 643, 651, 664, 732, Cutoff frequency, 612, 726, 746 733, 783, 786 Cytology, 634 Equivalent conductance, 45 Equivalent inductance, 410 D Equivalent resistance, 14, 17, 24, 78–80, 160, Damped natural frequency, 319 161, 170, 208, 242, 418, 531, 729 Damping factor, 307, 330 Equivalent T-circuit, 427, 428 Damping frequency, 317 Euler’s formula, 356 Darlington, 656, 657 Euler’s identities, 630 DC voltage, 23, 46, 173, 185, 249, 258, 262, Excitable cell, 6, 249, 261 669 Extracellular ionic concentration, 250, 251 Index 807

F Hounsfield, 793 Factor inhibitor, 241 Hurwitz, 535, 536, 538 Faraday’s law, 11, 786 H-parameters, 647, 663–666 Faraday, Michael, 11, 250 Fat, 514 I Filters Ideal op-amp, 674, 683, 724, 730, 736, 788 active, 723, 724 Ideal transformers, 417–419, 421, 422, 426, allpass, 724, 766 647, 663 high-pass, 724 Imaginary part, 374, 378, 384, 391, 396, 501, KHN, 764 509, 511, 512, 610, 611, 619, 640–642 low-pass, 724 Immittance parameters, 616 notch (bandstop), 724 Impedance passive, 469, 723, 724 characteristic, 633 Final-value theorem, 273, 446 driving point, 509, 514, 545, 550, 563–565, First-order circuits, 531, 534 567, 568, 574, 576, 578, 582, 583, 585, First-order differential equation, 60 588, 590, 593, 598, 600, 603–605, 613, First-order high-pass filter, 718 634–636, 639, 640 First-order low-pass filter, 752 input, 58, 394, 397, 423, 427, 428, 472, Flat, 42, 316, 423, 479, 751, 764, 766 520, 521, 537, 546, 557–560, 578, 581, Flyback topology, 249 584, 587, 592, 655–657, 661, 664, 706, Foster synthesis, 564, 567, 568, 576, 598, 601, 728, 730, 731 603, 606, 634 load, 389, 396, 397, 414, 652, 724 Four points in-line probe, 643 lossless, 550 Frequency domain, 361–363, 426, 446, 643 matching, 394, 419, 420 Frequency-inverse duals, 549 open circuit impedance, 646 Frequency response, 461, 482, 643, 753 output, 394, 424, 664 Frequency scaling, 747, 752 parameters, 634 Fricke Model, 632 scaling, 524 source, 165, 394, 423, 427, 560, 647, 724, G 783 Gain, 86, 91, 111, 113, 117, 154, 472, spectroscopy, 374, 614, 629, 630, 633, 642 476–479, 525, 532, 580, 596, 664, 676, synthesis from real part, 374 680, 684, 687, 692, 694, 697, 708–710, synthesis from two-port parameters, 647 722, 724–727, 738, 745, 746, 751, 755, Thévenin, 427, 457 756, 758, 760, 762, 765, 768, 788 Impulse function, 253 Gastro-Esophageal, 633 Indefinite integrals, 610 Geiger tube, 262 Independent current source, 161 Gewertz’ method, 616 Inductance, 280, 282, 285, 286, 290, 295, 296, Glycerol, 208 299, 301, 302, 316–318, 335, 386, 387, Ground, 99, 119, 169, 173, 291, 335, 337, 455, 389, 396, 397, 409, 411 490, 733, 744, 783, 786 Inductance, mutual, 316, 416–418, 427, 428, Guillemin, 798 430, 786 Gyrator, 794 Inductance simulator, 728 Inductive, 384, 411, 416, 429, 431, 784 H Inductors, 277, 282–292, 294, 301, 317, 320, Half-power frequencies, 389 321, 323, 325, 326, 329, 334, 348, 361, Healthcare, 337 363, 368, 369, 373, 386, 387, 394, 426, Hematology, 275 537, 559, 562, 724, 728, 786 Henry, Joseph, 417 Infinite network, 187 High-pass filter, 724, 755, 758, 764 Initial-value theorem, 446 Hilbert transform, 610614, 616, 642 Instantaneous power, 4, 311, 314, 353 Hodgin/Huxley, 259, 260 Instrumentation amplifier, 687, 707–710, 716, Homogeneity property, 142 717, 719, 732, 788 Hospital, 10, 399, 409 Integrator, 252, 414, 728, 764, 765, 769 808 Index

Interface, 392, 402, 403, 638, 717 M International Normalized Ratio (INR), 241 Magnetically coupled, 416 International System of Units (SI), 773 Magnetic Resonance Imaging (MRI), 302 Interpolation, 610 Matching, 379, 393, 419, 692 Intracellular ionic concentration, 250, 251 Mathematical formulas, 774 Inverse Laplace transform, 443, 448, 456, 463, MATLAB, 8, 17, 24, 64, 82, 111, 115, 116, 598, 601, 605 130, 214, 257–259, 287, 296, 309–311, Inverting op-amp, 669–671, 675, 676, 687, 315, 349, 350, 359, 379, 381, 382, 403, 689–691, 697, 703, 715, 722, 724, 737, 409, 428, 437–439, 441, 444, 445, 450, 739, 743, 755 476, 493, 498, 502, 504, 505, 511, 512, Ion channel, 259, 261 514, 515, 533, 536, 551–554, 557–559, Isolation transformer, 419 569, 571, 576, 578, 580, 582, 588, 592, 595, 597, 598, 600, 602, 603, 605, 607, K 617, 627, 629, 641, 667, 672, 697, Kirchoff, Gustav Robert, 793 699–701, 703, 733, 735, 750, 760, 780, 781 Kirchoff’s current law (KCL), 20, 86, 146, 159, Matrix, 26, 28–30, 88, 89, 91, 96, 101, 105, 162, 675, 786 107, 114–116, 125, 127, 129, 130, 133, Kirchoff’s voltage law (KVL), 21, 305, 786 135, 136, 154, 162, 165, 184, 200, 425, Kramers/Kronig transform, 608, 609, 616, 642 427, 459, 629, 635, 662, 776 Krawzenski model, 201 Matrix inversion, 651 Maximum average power transfer, 420 L Maximum power transfer, 165, 166, 167, 175, Ladder network, 187 176, 184, 388, 389, 393–397, 402, 403, Ladder network synthesis, 187 420, 424, 652 Lagging power factor, 409 Maxwell/Garnett, 197, 198, 199 Laplace transform, 435, 437, 438, 440, 442, Maxwell/Wagner Model, 632 445, 447, 601, 603 Medical, 302, 336–338, 350, 392, 394, 401, Law of cosines, 411, 412 429, 633 Layer models, 630, 631 Membrane potential, 250, 253, 254, 261, 335 Leaky Integrate-and-Fire (LIF) model, 252, Mesh analysis, 123, 125, 128, 132, 137, 492, 253 496 Left-half plane, 535, 565, 567, 624, 625 Mesh current, 125, 126, 128–131, 135, 136, L’Hopital’s rule, 777 138, 177, 338, 381, 425, 427, 471, 520, 786 Lichtenecher model, 206 Microstructural model, 629, 630 Linear circuit, 154 Midband, 727, 758, 760, 762 Linearity, 27, 64, 77, 140, 142–144, 154, 200, Minimum phase transfer functions, 623 642 Modeling, 802 Linear transformers, 419 Monophasic, 220, 230, 334, 335 Lithotripsy (ESWL), 301 Most Signiﬁcant Bit (MSB), 720 Load, 2, 3, 10, 23, 46, 60, 85, 166, 167, 169, Mutual inductance, 316, 416–418, 427, 428, 173–175, 183, 184, 316, 351, 388, 389, 430, 786 393, 394, 397, 399, 408, 411, 414, 415, 417, 419–423, 426, 428, 430, 495, 496, N 652, 658, 665, 724, 738 Natural frequency, 306, 319, 330, 331 Loop, 32, 123, 174, 279, 302, 411, 425, 724, Natural response, 290, 306, 313, 318, 737 764, 769 Neoplasias, 633 Looyenga model, 201 Nernst equation, 250 Loss, 10, 334, 401, 422, 430, 676 Netlist, 95, 108, 113, 218, 219, 240, 246, 248, Lossless impedance function, 537 326, 753, 762 Lossy, 728, 731 Network function, 614, 616–617, 619 Low-pass ﬁlter, 724, 752 Network stability, 642 Lung ventilator, 71 Network synthesis, 187 L-sections, 187 Neuron model, 252, 253 Index 809

Nodal analysis, 85, 96, 469, 486, 678 Passive filters, 723, 724 Node, 15, 22, 23, 33, 59, 67, 85–87, 89, 90, 92, Passive sign convention, 426 94, 97–112, 114–116, 118–123, 125, 127, Perfectly coupled, 419, 423, 430 135, 137–139, 141, 143, 148, 153, Period, 3, 11, 208, 226, 228, 230 155–158, 160, 162, 164, 169, 170, 178, Periodic function, 8, 212, 240, 255, 257, 285, 181–184, 188, 242, 253, 260, 261, 294, 296, 305, 307, 309, 311, 448, 460 273–275, 300, 301, 333, 338, 342, 343, Permeability, 36, 277, 316, 419, 786 379, 458, 459, 470, 486, 490, 520, 672, Permittivity, 185, 188, 190, 193, 197–201, 203, 677, 678, 710, 715, 729, 731, 734, 736, 206, 207, 209, 630, 786 737, 739, 740, 742, 786 Phasors, 355, 367, 415 Node voltage method, 100, 101, 111, 116 Phosoholipids, 240 Noise, 687, 694, 788 Piecewise linear, 691 Non inverting amplifier, 678, 680, 684, 688, Piecewise linear function, 691 711, 740 Pi network, 649 Norton equivalent circuits, 165 Polar form, 355 Norton’s theorem, 786 Pole, 74, 444, 447–450, 459–462, 466–468, Notch filter, 479, 482, 486, 490, 492, 495–499, 472, 476, 478, 479, 481, 482, 484, 485, 501– 500, 501, 508, 515, 524, 525, 527–530, Nuclear Magnetic Resonance (NMR), 303 532–536, 540, 544, 550, 559, 565–567, Numerical, 23, 24, 39, 50, 64, 91, 99, 110, 165, 593, 595, 615, 616, 622–624, 628, 629, 182, 214, 274, 280, 287, 309, 402, 420, 639, 641, 725, 744, 745, 768, 769, 781, 458, 496, 500, 519, 585, 591, 609, 610, 782, 789 730, 739, 750 Polycrystalline solids, 630 Numerical analysis tools, 214, 309 Polyester capacitor, 189 Numerical integration, 309 Polynomial, 444, 450, 451, 460, 472, 481, 484, Nyquist plot, 374, 578, 580, 596, 638 485, 488, 494, 496, 498, 508, 509, 535, 542, 553, 558, 565, 624, 636, 659, 652, 781 O Polynomial approximation, 257, 475, 476, 601, Objective function, 627 696, 698, 708, 710 Ohm, Georg Simon, 793 Porous carbon-based electrodes, 224 Ohm’s law, 14, 18, 19, 21, 85, 100, 110, 190, Port, 174, 423, 537, 656, 657, 733, 783, 789, 192, 322, 360, 362, 421, 422 790 Open circuit, 150, 151, 165, 166, 169, 170, Positive definite, 535, 537, 539, 547, 567 173, 174, 176, 223, 242, 323, 397, 419, Positive Real Function (PRF), 535, 539, 547, 783, 786 656 Operational amplifier, 693, 694, 738, 766, 787 Potential difference, 13, 33, 40, 57, 63, 193, OP room, 5, 34, 399, 401 194 Optimization, 626, 628, 629 Potentiometer, 70 Oscillator, 768, 770 Power, 3, 5, 7–10, 12, 15, 18, 19, 22, 25, 33, Oscilloscope, 258, 335, 431, 432 43, 54, 56, 60, 167, 173, 175, 176, 183, Oxygen, 34–36 184, 247, 249, 335, 348, 350, 389, 399–401, 403, 404, 408–411, 413, 415, P 419, 422, 429, 431, 451, 487, 491, 493, Parallel capacitors, 189, 196, 198, 199, 209, 652, 691, 700, 724, 730, 732, 733, 735, 394, 405, 406 751, 781 Parallel layer model, 631 Power factor, 409, 411, 412 Parallel resistors, 14, 26, 28, 39, 160, 161, 516 Power factor correction, 409 Parallel resonance, 387 Power measurement, 335 Parallel RLC circuits, 318 Power triangle, 410 Partial fraction expansion, 43, 445, 456, 461, Precancerous, 633 616, 781, 782 Pressure, 34, 36, 59, 71, 75, 76, 198, 294, 337, Passive element, 428, 543, 545 377, 706, 716 810 Index

Primary winding, 419, 422, 430, 432 Resonant frequency, 378, 379, 383, 384, 429, Principle of current division, 44, 56, 85, 145, 430 149, 152, 293, 321, 322 Resonator, 378 Principle of voltage division, 44, 46, 57, 58, 64, Response, 76, 306, 307, 330–332, 335, 400, 150–153, 168, 175, 221, 234, 364, 369, 401, 423, 446, 464, 467, 478, 490, 498, 406, 457, 464, 471, 473, 523, 614, 676, 501, 523, 525–529, 554, 559–562, 574, 707, 731, 733, 738, 769 593, 603, 604, 623, 625–628, 643, 647, Probe, 37, 48, 166, 392, 431, 432, 633, 634 738, 741, 745, 749–752, 764, 766 Proper rational functions, 443, 447–450 Reuss model, 27, 29, 201 Prothrombin time, 240 Rise time, 688, 689 RLC circuits, 305, 318 Q RL circuits, 288 Quadratic, 527, 529, 752 Root Mean Square (RMS) value, 347, 348, Quadrature, 614, 768, 770 349, 400 Quality factor, 378, 386, 388, 390, 393, 764 Roots, 306, 443, 450, 451, 460, 481, 484, 485, 535–537, 540, 624, 738, 744 R Radiation detector, 262 S Rational function, 443, 447–450, 537, 545, Sallen and Key high-pass circuit, 751 550, 553, 564, 582, 639, 781 Sallen and Key low-pass circuit, 748, 749 Rational transfer function, 467, 535 Scaling, 257, 420, 521, 522, 524, 656, 723, 747 RC circuits, 394 Schwartz inequality, 180 Reactance, 366 Secondary winding, 249, 417, 419, 420, Reactive load, 409 422–424, 426, 427, 430–432 Reactive power, 408, 409, 419 Second-order circuits, 305, 469, 546, 764 Realizability, 549, 571 Self-inductance, 416, 792 Realizable, 543, 549, 676 Sensitivity, 63, 74, 76, 205, 207, 634, 698, 700, Reciprocal network, 663 708 Reciprocity, 154, 155 Series, 14, 24, 25, 28, 29, 43, 44, 56, 137, 185, Rectangular form, 28, 39, 189, 351 201, 224, 249, 305–307, 309, 315, 317, Reference node, 786 318, 334, 340–343, 345, 356, 360, 367, Reﬂected, 420, 703 368, 371, 377, 384, 385, 394, 409, 413, Reﬂected impedance, 426 429, 479, 480, 533, 537, 546, 588, 630, Relay, 296, 298, 299 632, 692, 698, 702, 703, 731, 747, 773, Relay circuits, 296, 299 783, 786 Relay delay time, 299 Series capacitors, 334, 589 Residues, 550, 566, 567, 781, 782 Series inductors, 334 Resistance, 6, 25, 28–30, 39–43, 46, 50, 52, 53, Series layer model, 627, 628 55, 57, 60, 62, 63, 65, 69–71, 73–81, 85, Series resonance, 386 97, 132, 154, 726, 731–733, 735, 738, 755, Series RLC circuits, 305, 306, 309–311, 313, 756, 773, 788 318, 331, 332, 383, 385, 386, 388, 393, Resistance bridge, 17, 22, 62, 69, 73, 75 461, 462, 502, 503 Resistance matrix, 154 Sheet resistivity, 48, 50, 644 Resistance measurement, 169 Short circuit, 149, 151, 159, 165, 168, 170, Resistive load, 173, 427 171, 174, 177, 182, 221, 242, 293, 329, Resistivity, 6, 39–42, 47, 48, 50–53, 55, 56, 74, 397, 414, 457, 516, 555, 783, 786 194, 302, 404, 515, 519, 630, 631, 633, Siemens, W.V., 793 643, 644 Signal, 57, 75, 142, 165, 256, 257, 259, 263, Resistors, 14, 23, 24, 26, 28, 29, 44, 56, 75, 76, 284, 287, 303, 329, 347–350, 352–354, 79, 93, 110, 114, 160, 176, 224, 230, 320, 414, 419, 430, 431, 435, 468, 486, 498, 335, 336, 387, 658, 672, 674, 676, 684, 531, 534, 614, 664, 665, 669, 671, 672, 687, 692, 709, 710, 728, 743, 769, 783 676, 679, 683, 686–691, 707, 708, 722, Resonance, 278, 302, 303, 317, 371, 378, 379, 727, 729, 736, 737, 751, 764, 788 383, 384, 386, 387, 395, 398, 429, 522 Signal-to-noise ratio, 683 Index 811

Simultaneous equations, 95, 183, 333, 780 Time-delay, 372 Singly terminated, 656 Tissue, 40, 53, 55, 351, 399–401, 403, 405, Smoothing circuits, 330 430, 514, 633, 634 Sodium ions, 249–251 Titanium oxide, 643 Source-free parallel RLC circuit, 318 Tomography, 329, 797 Source-free RC circuit, 215, 289 Topology, 73, 75, 200, 249, 479, 483, 486, Source-free RL circuit, 290, 295, 302 498, 555, 564, 569, 588, 660, 684, 723, Source-free series RLC circuits, 305 747, 752, 755, 761, 762, 764 Source transformation, 155–164, 168, 170, 274 Toroidal inductor, 431 Spectroscopy, 614, 629, 630, 633 Transducer, 71, 614, 716 SPICE, 15, 16, 22, 23, 30, 90–92, 95, 98, 99, Transfer function, 463, 465–471, 473, 105, 106, 108, 111, 113, 116, 121, 123, 476–479, 482, 484–486, 493, 496–498, 138, 139, 148, 149, 154, 211, 217–219, 501, 523, 524, 526–530, 533–536, 566, 226, 228, 232, 235, 239, 240, 245–247, 567, 585, 592, 617, 622, 623, 641, 655, 253, 254, 262, 265, 267, 268, 271, 273, 703, 746, 747, 749, 752, 761, 769, 781, 275, 283, 285–287, 290, 300, 325–327, 782, 784, 789 329, 330, 332, 365, 370, 372, 422, 423, Transformation ratio, 17, 490, 564, 604, 643, 431, 460, 653, 688–694, 698, 700, 710, 786 725, 745, 753, 756, 760, 762, 767 Transformers Stability, 535, 639, 760, 764, 765 air-core, 418 Step-down transformer, 420 ideal, 417–423, 426, 427 Step response of an RC circuit, 598, 600 isolation, 419 Step response of an RL circuit, 523 linear, 419 Step-up transformer, 422, 431 step-down, 420 Summing ampliﬁer, 722 step-up, 249, 422, 431 Supercapacitor, 224, 233 Transient response, 289 Supermesh, 137, 138 Transistor, 664, 692, 723 Supernode, 118, 120, 122, 123 Transpose, 110, 115 Superposition, 140, 141, 145–151, 153, 158, Transresistance ampliﬁer, 154 159, 188, 415, 416, 681, 765 Triangular wave, 282 Superposition theorem, 149–151, 153, 158 Trigonometric identities, 775 Susceptance, 366, 384, 502, 503, 786 Turns ratio, 417–422, 424, 426, Switching functions, 213 430, 432 Symmetric, 154, 764, 780 Two-phase dispersions, 632 Two-port networks T hybrid parameters, 663 Temperature, 6, 34, 36–38, 59, 60, 64, 65, 69, impedance parameters, 647 74–76, 196, 198, 199, 241, 249, 251, 302, 671, 694, 695, 697, 698, 700, 702, 703, U 709, 716 Ultrasound, 392, 614 Terminals, 13, 22, 60, 166, 167, 169–171, 179, Undamped natural frequency, 330, 331 456, 640, 678, 692, 708, 783, 787, 788 Underdamped case, 306, 310 Thévenin, M. Leon, 165, 167, 168, 171–173, Unit impulse function, 468 176, 177, 180, 181, 389, 396, 426 Unit ramp function, 440 Thévenin’s theorem, 168, 783 Unit step function, 213, 440, 465 Thromboplastin, 240 Unity gain, 717, 725, 755 Thrombosis, 240 Unloaded, 430, 431 Thévenin equivalent circuit, 172, 180, 389, 457 Time constant, 192–195, 207, 210, 220, 221, V 226, 229, 231, 236, 240, 241, 253, 254, Vector, 74, 115, 199, 287, 395, 604, 628, 629, 257, 259, 260, 262, 266, 290, 291, 296, 780, 781 302, 320, 331, 334, 531, 534, 536, 638, 729 Voigt Model, 27, 28, 198, 199, 201, 204 812 Index

Voltage, 4, 8, 9, 13, 15, 18, 19, 21–23, 30, 34, Voltage follower, 265, 272 44, 56, 57, 72, 75, 85, 89, 90, 93, 98, 99, Voltmeter, 13, 154, 385, 411 105, 115, 117, 118, 121, 125, 141, 149, 150, 154, 157, 160, 162, 165, 169, 170, W 172, 173, 183, 185, 195, 211, 212, 214, Warfarin, 240 217, 220, 221, 223, 225, 227, 230, 232, Wattmeter, 409 238–240, 242–246, 253–255, 264, 265, Wheatstone bridge, 22, 39, 60–63, 67–70, 75, 270, 273, 274, 278, 284, 288, 290, 291, 76, 706, 709, 710, 717 301, 305, 307, 329, 334, 336, 345, 368, Winding capacitance, 249 379, 384, 385, 397, 400, 407, 411, 413, Wye to delta transformations, 15, 490 416, 419, 423, 427, 430, 455, 461, 465, 520, 573, 604, 653, 666, 673, 678–680, X 684, 685, 687, 688, 691, 692, 697, 698, X-ray, 329, 797 706, 707, 709, 710, 712, 715–717, 719, 720, 724, 732, 734, 738, 740, 783 Y Voltage divider, 21, 46, 59, 60, 142, 479, 483, Y-parameters, 663 717, 783, 784 Voltage division, 44, 58, 64, 150–153, 175, Z 221, 234, 364, 369, 406, 457, 464, 471, Zero, 715, 744, 787, 788 473, 523, 614, 676, 707, 731, 733, 738, Z-parameters, 648, 649, 653, 663 740, 769