Appendix A Useful MATLABrfunctions

The following are useful MATLABrfunctions for circuit analysis and synthesis. The reader is encouraged to consult the reference guide and the “Help” directory in any MATLABrwindow for a detailed description of each function.1 Ax=b: Linear equation Ax=b is solved by invoking x = A\b. See also linsolve. Bode plot: [mag, phs, w]=bode(n, d, w) places the magnitude in vector mag and phase in vector phs at each frequency in vector w of a transfer function whose numerator polynomial is n and denominator polynomial is d. Convolution: y=conv(p,q) obtains the product of two polynomials, represented by their vectors of coefficients p and q. Eigenvalues: [V,D]=eig(A) places the eigenvalues of A in a diagonal matrix D and their corresponding eigenvectors in V. Elliptic functions: [sn cn dn]=ellipj(u,m) yields sn(u,k), cn(u,k), and dn(u,k); and [K, E]=ellipke(m) yields K(k) and E(k), where m = k2. Factors: y=factor(f) obtains the factors of a symbolic polynomial f . Imaginary part: y=imag(z) returns the imaginary part of z. Linear equations: y=linsolve(A,b) obtains the symbolic or numerical solution of linear equation Ax = b, A and b being numeric or symbolic. Matrix exponential: H=expm(At), where A is a matrix of simple rational numbers and t declared symbolic, will obtain the impulse response matrix H(t) in analytic form. Numerator and denominator: [n,d]=numden(f) places the numerator of symbolic rational function f in n and the denominator in d. Ordinary differential equations: dsolve(...) obtains the symbolic solution of an or- dinary differential equation. Polynomial from its roots: p=poly(r) obtain the coefficients of polynomial p in descending order in a row vector whose roots are placed in a column vector r.

1 The Academia area of The MathWorks Web site provides many resources for professors and students. Check there for contributed course materials, recommended products by cur- riculum, tutorials on MATLABrsoftware, and information on MATLABrStudent Version: www.mathworks.com/products/academia/

285 286 A Useful MATLABrfunctions

Polynomial to symbolic: p=poly2sym(q,’s’) converts a vector of coefficients of a polynomial q to its symbolic representation in s. Random numbers: y = m + σ ∗ randn generates a random number taken from a universe of numbers which are normally distributed with mean m and variance σ 2 . Random numbers: y=a + (b-a)*rand generates a random number taken from a universe of numbers which are uniformly distributed over [a,b]. Real part: y=real(z) returns the real part of z. Residues: [r,p,k]=residue(b,a) obtains the partial fraction expansion of rational function b/a where a and b are expressed as vectors of coefficients of polynomi- als a and b. The residues are placed in vector r and the poles in vector p. Vector k contains the coefficients of any excess polynomial. Roots of a polynomial: r=roots(p) obtains a column vector of roots of polyno- mial p expressed as a row vector of coefficients in descending order. Step response: y=step(num,den,t) produces the step response of a transfer func- tion whose numerator polynomial is num and denominator polynomial is den, both expressed as vectors of coefficients, for a vector of time points in t. Simplify: y=simplify(f) obtains a simplified version of a symbolic function f . Substitution: y=subs(f, ’s’, w) replaces each occurrence of s by each member of vector w in the symbolic function f . This function is useful in computing the frequency response of a rational function f if w is a vector of frequency points. Symbolic to polynomial: q=sym2poly(p) converts a symbolic representation of a polynomial p to its representation as a vector q of coefficients. Transfer function: [num,den]=ss2tf(A,B,C,D,k), where A,B,C,D are the matri- ces of a state equation in normal form, with an input component Uk = 1 and all others being zero, will return a vector of transfer functions whose numerator polynomials are in the vector num and whose denominator polynomial, being the same for all, is in den. Transfer function to zeros and poles: [z,p,k]=tf2zp(num,den) places the zeros of a transfer function with numerator num and denominator den in vector z, the poles in vector p and the gain in k. The vector num contains the coefficients of the numerator polynomial and den those of the denominator polynomial. Zeros and poles to transfer function: [num,den]=zp2tf(z,p,k) forms the transfer function with numerator polynomial num and denominator polynomial den from a vector of zeros z, a vector of poles p and a multiplicative constant k. References

1. Abramowitz, M., (ed.), I.A.S.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, Washington, D.C. (1967) 2. Ascher, U.M., Petzold, L.R.: Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia (1998) 3. Bashkow, T.R.: The A matrix, new network description. IRE Trans. on Circuit Theory CT-4, 117–120 (1957) 4. Belevitch, V.: Th´eorie des circuits de t´ele ´communication. Librairie Universitaire Louvain (1957) 5. Belevitch, V.: An alternative derivation of Brune’s cycle. IRE Trans. Circuit Theory CT-6, 389–390 (1959) 6. Belevitch, V.: On the Bott-Duffin synthesis of driving-point impedances. IRE Trans. Circuit Theory CT-6, 389–390 (1959) 7. Blostein, M.L.: Sensitivity analysis of parasitic effects in resistance-terminated LC two-ports. IEEE Trans. on Circuit Theory CT-14, 21–26 (1967) 8. Bode, H.W.: Network Analysis and Amplifier Design. D. Van Nostrand, Princeton (1945) 9. Bott, R., Duffin, R.J.: Impedance synthesis without use of transformers. J. Appl. Phys. 20, 816 (1949) 10. Branin Jr., F.H.: The inverse of the incidence matrix of a tree and the formulation of the algebraic- first-order differential equations of an RLC network. IEEE Trans. on Circuit Theory CT-10, 543–544 (1963) 11. Brown, D.P.: Derivative-explicit differential equations for RLC graphs. J. Franklin Institute 275, 503–514 (1963) 12. Brune, O.: Synthesis of a finite two-terminal network whose driving-point impedance is a prescribed function of frequency. J. Math. Phys. 10, 191–236 (1931) 13. Bryant, P.R.: The explicit form of Bashkow A matrix. IRE Trans. on Circuit Theory CT-9, 303–306 (1962) 14. Carlin, H.J.: A new approach to gain-bandwidth problems. IEEE Trans. on Circuits and Sys- tems CAS-23(4), 170–175 (1977) 15. Carlin, H.J., Civalleri, P.P.: Wideband Circuit Design. CRC Press, Boca Raton (1998) 16. Cauer, W.: Die Verwirklichung von Wechselstromwidersta ¨nden vorgeschriebener Frequenzabha ¨ngigkeit. Arch. Elektrotech. 17(4), 355–388 (1926) 17. Chen, W.K.: Theory and Design of Broadband Matching Networks. Pergamom Press, Oxford (1976) 18. Daniels, R.W.: Approximation Methods for Electronic Filter Design. McGraw-Hill, New York (1974) 19. Darlington, S.: Synthesis of reactance 4-poles which produce prescribed insertion loss char- acteristics. J. Math. Phys. 18, 257–353 (1939)

287 288 References

20. Darlington, S.: Realization of a constant phase-difference. Bell System Technical Journal 29, 94–104 (1950) 21. Director, S.W.: LU factorization in network sensitivity computations. IEEE Trans. on Circuit Theory CT-18, 184–185 (1971) 22. Ellinger, F.: Radio Frequency Integrated Circuits and Technologies. Springer, Berlin; New York (2007) 23. Fano, R.M.: Theoretical limitations on the broad-band matching of arbitrary impedances, Part I. J. Franklin Institute 249(1), 57–83 (1950) 24. Fano, R.M.: Theoretical limitations on the broad-band matching of arbitrary impedances, Part II. J. Franklin Institute 249(2), 139–154 (1950) 25. Filanovsky, I.M.: Sensitivity and selectivity. In: W.K. Chen (ed.) The Circuits and Filters Handbook, chap. 68, pp. 2205–2236. CRC Press, Boca Raton (1995) 26. Foster, R.M.: A reactance theorem. Bell System Journal 3, 259–267 (1924) 27. Fujisawa, T.: Realizability theorem for mid-series or mid-shunt low-pass ladders without mu- tual induction. IRE Trans. PGCT CT-2(4), 320–325 (1955) 28. Guillemin, E.A.: The Mathematics of Circuit Analysis. John Wiley and Sons, New York (1953) 29. Guillemin, E.A.: Synthesis of Passive Networks. John Wiley and Sons, New York (1957) 30. Hazony, D.: An alternate approach to the Bott-Duffin cycle. IRE Trans. Circuit Theory CT-8, 363 (1961) 31. Hazony, D.: Two extensions of the Darlington synthesis procedure. IRE Trans. Circuit Theory CT-8, 284–288 (1961) 32. Hazony, D., Schott, F.W.: A cascade representation of the Bott-Duffin synthesis. IRE Trans. Circuit Theory CT-5, 144 (1958) 33. Huang, Q., Sansen, W.: Design techniques for switched capacitor broadband phase splitting networks. IEEE Transactions on Circuit Theory CT-34, 1096–1102 (1987) 34. Hurwitz, A.: Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit nega- tiven reelen Theilen besitzt. Math. Ann. 46, 273–284 (1895) 35. II, L.W.C.: Digital and Analog Communication Systems. Prentice Hall, Upper Saddle River, NJ (1997) 36. Jeffrey, A.: Handbook of Mathematical Formulas and Integrals. Academic Press (1995) 37. Kim, W.H.: A new method of driving-point function synthesis. Technical report 1, University of Illinois Engineering Experimental Station (1956) 38. Kishi, G., Kida, T.: Energy theory of sensitivity in LCR networks. IEEE Transactions on Circuit Theory CT-14(4), 380–387 (1967) 39. Lathi, B.P.: Modern Digital and Analog Communication Systems, 3rd edn. Oxford University Press, New York (1998) 40. Luck, D.G.C.: Properties of some wide-band phase splitting networks. Proceedings of the IRE 37, 147–151 (1949) 41. Ludwig, R., Bretchko, P.: RF Circuit Design. Prentice Hall, Upper Saddle River, NJ (2000) 42. Miyata, F.: A new system of two-terminal synthesis. J. Inst. Elec. Engrs. (Japan) 35, 211–218 (1952) 43. More,´ J.J.: The Levenberg-Marquardt algorithm: implementation and theory. In: G.A. Watson (ed.) Lecture Notes in Mathematics 630, chap. Numerical Analysis, pp. 105–116. Springer- Verlag, Berlin (1977) 44. Murdoch, J.B., Hazony, D.: Cascade driving-point impedance synthesis by removal of sections containing arbitrary constants. IRE Trans. Circuit Theory CT-9, 56–61 (1962) 45. Nebeker, F.: An interview with Alfred Fettweiss. Oral His- tory 338, IEEE Center for History of Electrical Engineering, http://www.ieee.org/web/aboutus/history center/oral history/oral history.html (1997) 46. Oden, P.H.: On the synthesis of passive networks containing exactly one inductor. Ph.D. thesis, Columbia University (1966) 47. Orchard, H.J.: Synthesis of wideband two-phase networks. Wireless Engineer 27, 72–81 (1950) References 289

48. Orchard, H.J.: Inductorless filters. Electronics Letters 2, 224–225 (1966) 49. Orchard, H.J.: Loss sensitivities in singly and doubly terminated filters. IEEE Transactions on Circuits and Systems CAS-26(5), 293–297 (1979) 50. Pinel, J.F., Blostein, M.L.: Computer techniques for the frequency analysis of linear electrical networks. Proc. IEEE 55, 1810–1826 (1967) 51. Polak, E.: Computational Methods in Optimization. Academic Press, New York (1971) 52. Protonotarios, E.N., Wing, O.: Theory of nonuniform RC lines, Part I: Analytic properties and realizability in the frequency domain. IEEE Transactions on Circuit Theory CT-14(1), 2–12 (1967) 53. Protonotarios, E.N., Wing, O.: Theory of nonuniform RC lines, Part II: Analytic properties in the time domain. IEEE Transactions on Circuit Theory CT-14(1), 13–20 (1967) 54. Rappaport, T.S.: Wireless Communications. Prentice Hall, Upper Saddle River, NJ (1996) 55. Razavi, B.: RF Microelectronics. Prentice Hall, Upper Saddle River, NJ (1998) 56. Rhodes, J.D.: Theory of Electrical Filters. John Wiley and Sons, New York (1976) 57. Richards, P.I.: Universal optimum response curves for arbitrary coupled resonators. Proceed- ings of Institute of Radio Engineers 34, 624–629 (1946) 58. Richards, P.I.: A special class of functions with positive real part in a half-plane. Duke Math. J. 14, 777–788 (1947) 59. Saal, R., Ulbrich, E.: On the design of filters by synthesis. IRE Transactions on Circuit Theory CT-5(4), 284–327 (1958) 60. Saraga, W.: The design of wide-band phase splitting networks. Proceedings of the IRE 38, 754–770 (1950) 61. Schaumann, R., Valkenburg, M.E.V.: Design of Analog Filters. Oxford University Press, New York (2001) 62. Skwirzynski, J.K.: Design Theory and Data for Electric Filters. Van Nostrand, London (1965) 63. Tellegen, B.D.H.: Theorie der electrische Netwerken, chap. Part III. P. Noordhoff N. V., Groningen, Djakarta (1952) 64. Temes, G.C., Orchard, H.J.: First-order sensitivities and worst case analysis of doubly termi- nated reactance two-ports. IEEE Transactions on Circuit Theory CT-20(5), 567–571 (1973) 65. Valkenburg, M.E.V.: Introduction to Modern Network Synthesis. John Wiley and Sons, New York (1960) 66. Valkenburg, M.E.V.: Analog Filter Design. Hold, Rinehart, and Winston, New York (1982) 67. Weaver Jr., D.K.: Design of RC wide-band 90◦ phase-difference network. Proceedings of the IRE 42, 671–676 (1954) 68. Weinberg, L.: Network Analysis and Synthesis. McGraw-Hill, New York (1962) 69. Wing, O.: Ladder network analysis by signal flow graph. IRE Transactions on Circuit Theory CT-2, 289–294 (1956) 70. Youla, D.C.: A new theory of broadband matching. IEEE Trans. on Circuit Theory CT-11(1), 30–50 (1964) 71. Youla, D.C., Castriota, L.J., Carlin, H.J.: Bounded real scattering matrices and the foundations of linear passive network theory. IEEE Trans. Circuit Theory CT-6(1), 102–124 (1959) 72. Zhu, Y.S., Chen, W.K.: Computer-Aided Design of Communication Networks. World Scien- tific, Singapore (2000) 73. Zverev, A.I.: Handbook of Filter Synthesis. John Wiley and Sons, New York (1967) Index

90◦ phase difference circuit, 268 phase difference circuit, 270 RC all-pass circuit, 266 phase response, 262 MATLABrfunction, 6 realizations, 262 Ax = b, with symbolic b, 41 transfer function, 261 [r, p,k] = residue(b,a), 65 all-pass functions, 69 x = A\b solves Ax = b, 27 all-pass lattice, 86 dsolve, symbolic ODE, 52 Ampere,` Andre-Maria,´ 1, 7 eig(A), eigenvalues and eigenvectors, 44 analog computer, 52, 232 expm(M), matrix exponential, 48 simulation of filters, 232 p=poly(r), 173 Ascher, U. M., 26 r=roots(p), 172 auxiliary polynomial, 169 ss2tf(A,B,C,D,k), state space to transfer function, 54 Bode plot, 67 backward Euler method, 27 elliptic function, y = ellip j(u,m), 211 Bashkow, T. R., 35 elliptic integral, y = ellipke(m), 211 Belevitch, V., 2 random numbers, 30 bipolar transistor, 13 Blostein, M. L., 77 Abramowitz, M., 211 Bode plot, 67 active circuit, 23 Bode, Hendrik W., 2, 8, 76 small-signal, 23 Bott, R., 2, 8 additivity, 12 Bott-Duffin synthesis, 122 adjoint circuit, 78 bounded real function, 135 admittance, 62 bounded real matrix, 136 driving point, 63 Branin, F. H., Jr., 43 input, 63 bridged-T, 264 admittance function, 62, 90 constant-resistance, 264 LC, 100 bridged-T two-port, 241 RC, 107 constant resistance, 241 RL, 109 broadband matching, 256, 259 all-pass circuit, 261 Brown, D. P., 43 RC, 266 Brune synthesis, 115, 179 constant-R lattice, 263 Brune, Otto, 2, 8 delay equalizer, 278 Bryant, P. R., 43 group delay, 262 Butterworth lumped delay line, 268 approximation, 194 non-R-lattice, 265 low-pass filter, 194

291 292 Index

C-fundamental loop, 40, 42 convolution, 45 C-subcircuit, 40 coupled inductors, 13, 21, 89 Campbell, George, 2, 8 symmetry, 22 canonical realization, 103 current LC, 103, 104 terminal, 17 RC, 109 current source, 13 RL, 110 capacitor, 12 Daniels, Richard W., 216 capacitor loop, 40, 42 Darlington synthesis, 128, 182 Carlin, Herbert J., 3 Brune section, 179 Cauer filter, 207 C-section, 182, 192 optimality, 226 D-section, 185 sensitivity, 227 Darlington, Sidney, 2, 8, 164 Cauer realization Davidson-Fletcher-Powell method, 251 LC, 103, 104 delay equalizer, 278 RC, 108 all-pass circuit, 279 RL, 110 bridged-T, 281 Cauer, Wilhelm, 2, 8, 104, 207 optimization, 280 characteristic impedance, 161 delay line, 86, 267 Chebysheff lumped approximation, 86, 87 low-pass filter, 171 design parameters, 243 Chebyshev filter, 200, 226 differential-algebraic equations, 26 Chebyshev polynomial, 202 consistent initial conditions, 27 design equations, 205 numerical solution, 26 optimality, 226 Director, S. W., 77 poles and zeros, 204 drain conductance, 16 sensitivity, 227 Duffin, R. J., 2, 8 transmission function, 204 Chen, Wai-Kai, 3, 186 eigenvalues, 44, 45 circuit design repeated, 48 by optimization, 243 eigenvectors, 44 Davison-Fletcher-Powell, 251 elliptic filter, 207 design parameters, 243 derivation, 216 gradient, 245 design equations, 212, 222 Hessian matrix, 246 equi-ripple function, 224 Jacobian matrix, 248 equi-ripple rational function, 208 least squares, 249 optimality, 226 Levenberg-Marquardt, 250 poles and zeros, 211 Newton’s method, 248 synthesis, 213 objective function, 244 elliptic function, 218 one-dimensional search, 247 complex argument, 219 sensitivity function, 253 cosine, cn(u,k), 218 steepest descent, 245 modulus, 211 circuit dynamics, 35 periodic rectangle, 219, 272 cn(u,k), elliptic cosine, 218 periods, 219 compact pole, 155 phase difference circuit, 272 consistent initial conditions, 27 sine or sn(u,k), 211 constant-R lattice, 86, 263 elliptic integral, 211 first order, 264 complementary K’(k), 211 second order, 264 complete, K(k), 211 continued fraction expansion incomplete, 211 LC, 103 energy, 22 RC, 108 RLCcircuit, 23 RL, 110 coupled inductors, 21 Index 293

passive circuit, 23 ideal transformer, 117 ensignant, 94 immittance, 91 exponential excitation, 53 impedance, 62 driving point, 62 Fano, R. M., 3 from its real part, 64 Faraday, Michael, 1, 7 imaginary part, 64 formulation of state equations, 40 input, 62 Foster realization real part, 64 LC, 101 impedance function, 62, 89, 90 RC, 107 from its real part, 83 RL, 110 LC, 99 Foster, R. M., 2, 8, 102 RC, 105 frequency domain analysis, 59 RL, 109 node equations, 60 impulse response, 45, 46 frequency transformation, 233 impulse response matrix, 45 low-pass to band-elimination, 237 independent state variables, 38 low-pass to band-pass, 234 inductance matrix low-pass to high-pass, 233 positive semi-definite, 22 Fujisawa, T., 2, 178, 214 symmetry, 21 fundamental KCL equations, 19 inductor, 12 fundamental KVL equations, 19 inductor cutset, 39, 40, 42 infinite ladder gain and phase LC, 98 analytic relation, 74 RC, 98 Bode’s formula, 76 interconnect, 33, 56 piece-wise linear approximation, 75, 76 RC line, 33, 56 gain of transfer function, 66 gain sensitivity, 77, 254 Jacobian matrix, 248, 249 gain-bandwidth limitations, 145, 147 Jordan form, 49 Gaussian low-pass filter, 254, 258 gradient, 245 K’(k), complementary complete elliptic computation of, 251 integral, 211 ground node, 17 K(k), complete elliptic integral of the first group delay, 70, 71 kind, 211 computation, 79 KCL equations, 19 coupled inductors, 81 fundamental, 19 formula, 79 Kirchhoff’s laws, 11 Guillemin, Ernst A., 8, 165 KCL, 11 KVL, 11 Heaviside, Oliver, 2, 7 Kirchhoff, Gustav, 1, 7 Henry, Joseph, 1, 7 KVL equations, 17, 19 Hermitian matrix, 136 fundamental, 19 Hertz, Heinrich R., 7 Hessian matrix, 245 L-fundamental cut set, 40 computation of, 251 L-subcircuit, 40 positive definite, 247, 251 LC impedance function, 99 high-pass filter, 31 Cauer realization, 103 RC, 32 continued fraction expansion, 103 homogeneity, 12 Foster realization, 101 homogeneous solution, 45 necessary and sufficient conditions, 100 Hurwitz polynomial, 63, 165 non-series-parallel, 105 strictly, 63 partial fraction expansion, 101 Hurwitz, A., 63 reactance function, 102 hybrid equations, 16 least squares method, 249 294 Index

Levenberg-Marquardt, 250, 254 nonlinear element, 12 linear circuit, 14 normal form, 37 definition, 14 normalization factor, 140, 143 linear element, 12 principal, 143 definition, 12 Norton, Edward L., 2, 8 linear phase, 70 numerical solution, 26 Lipschitz’s condition, 37 backward Euler method, 27 loading coils, 2, 87, 161 time step, 26 loss sensitivity, 227 lossless two-port, 136 objective function, 244 scattering matrix, 136 Ohm, Georg, 1, 7 unitary properties, 136 one-dimensional search, 247 low-noise amplifier, 148, 149, 158, 159 open-circuit impedance matrix, 152 low-pass filter, 29 definition, 152 Butterworth, 194 lossless two-port, 154 Cauer, 207 residue condition, 154 Chebyshev, 200 open-circuit impedance parameters, 166 elliptic, 207 input impedance, 166 maximally flat, 194 order, 40 removal of Gaussian noise, 30 state equations, 40 lumped delay line, 267 order reduction, 259 magnitude of transfer function, 66 Pade´ approximation, 267 matching two-port, 149, 159 partial fraction expansion, 65 maximally flat LC, 101 approximation, 194 RC, 107 low-pass filter, 194 RL, 110 maximally flat filter particular solution, 45 design equations, 197 passband, 194 poles and zeros, 196 edge, 194 synthesis, 198 passive circuit, 22 transmission gain function, 197 definition, 23 Maxwell, James Clerk, 1, 7 Petzold, L. R., 26 minimum phase function, 70 phase difference circuit, 268 minimum reactance function, 116 approximation problem, 272 minimum resistance, 116 design parameters, 273 minimum susceptance function, 116 synthesis, 274 Miyata synthesis, 127, 130 phase of transfer function, 66 modified node equations, 16, 23 phase sensitivity, 77 MOS transistor, 13 phase splitting circuit, 270 Pinel, J. F., 77 negative resistance, 23 pn-diode, 13 Newton’s method, 248 positive real function, 90 node equations definition, 91 circuits with transconductances, 61 extended definition, 98 node voltage, 17 irrational, 98 noise minimum real part, 96 Gaussian, 30 necessary and sufficient conditions, 93 through high-pass filter, 31 phase angle, 92 through low-pass filter, 30 pole at infinity, 96 non-constant-R lattice, 265 pole at zero, 96 non-series-parallel LC impedance, 105, 112 poles on jω-axis, 92, 96 nonlinear circuit, 14 properties, 92 definition, 14 positive real matrix, 153 Index 295

open-circuit impedance matrix, 154 positive semi-definite, 136 positive semi-definite, 89 reflection coefficient, 134 power, 22 resistive terminations, 132 private poles, 155, 156 scattering parameters, 134 propagation constant, 161 transmission function, 134, 135 Pupin, Michael, 2, 7, 161 unitary properties, 136 scattering parameters, 133 quadratic form, 22 transmission function, 169 positive semi-definite, 22 second order sensitivity function, 253 sensitivity, 76 raised cosine filter, 258 bounds, 228 random numbers, 30 computation, 77 normally distributed, 30 formula, 78 uniformly distributed, 30 gain, 77 RC impedance function, 105 loss, 227 Cauer realization, 108 passband, 227 continued fraction expansion, 108 phase, 77 Foster realization, 107 transconductance, 88 necessary and sufficient conditions, 106 sensitivity function, 253 partial fraction expansion, 107 computation, 253 RC line, 33, 56 gain, 254 real and imaginary parts second order, 253 analytic relation, 71, 73, 74 short-circuit admittance matrix, 155 reciprocal circuit, 62 short-circuit admittance parameters, 169 reciprocity, 61, 62 input impedance, 169 reflected power gain, 137 Siemens, Ernst Werner von, 1, 7 reflection coefficient, 134 simulation of filters, 232 power gain, 137 single-side-band, 269 residue, 65 small-signal active circuit, 23 residue condition, 154 small-signal equivalent circuit, 15 open-circuit impedance matrix, 154 sn(u,k), elliptic sine, 211 resistor, 12 solution Richards theorem, 122 circuit, 12 RL impedance function, 109 definition, 12 Cauer realization, 110 state, 35 continued fraction expansion, 110 equations, 35 Foster realization, 110 space, 35 necessary and sufficient conditions, 110 trajectory, 35 partial fraction expansion, 110 variables, 35 RLC circuit state equations, 35 definition, 15 analog computer simulation, 52 RLC impedance synthesis, 115 capacitor loop, 38 Bott-Duffin synthesis, 122 capacitor loops, 42 Brune synthesis, 115 characteristic polynomial, 45, 53 Darlington synthesis, 128 eigenvalues, 44, 45 Miyata synthesis, 127 eigenvectors, 44 exponential excitation, 53 scattering matrix, 132, 133 formulation, 40 bounded real, 136 homogeneous solution, 45 definition, 132 impulse response, 45, 46 impedance matrix, 155 distinct eigenvalues, 46 impedance termination, 139 repeated eigenvalues, 51 lossless two-port, 136 impulse response matrix, 45 normalization, 139 inductor cutset, 39, 42 296 Index

Jordan form, 50 synthesis, 163 Lipschitz’s condition, 37 transfer function matrix, 54 normal form, 37, 38 transition band, 194 numerical solution, 52 transmission function, 134, 135, 170 order, 40 power gain, 135 particular solution, 45 scattering parameter, 135, 170 repeated eigenvalues, 48 transmission zeros, 170 solution, 44 transmission line, 160 steady state response, 53 characteristic impedance, 161 symbolic solution, 51 loading coil, 162 unique solution, 36 propagation constant, 161 steady state response, 53 transmission power gain, 135, 164 steepest descent, 245 transmission zeros, 170 Steinmetz, Charles, 2, 8 complex, 185 stop-band, 194 finite frequencies, 174 edge, 194 imaginary, 171 superposition theorem, 14 order of removal, 178 real, 182 Tellegen’s theorem, 20, 23, 28, 90, 153 two-port, 131 multi-terminal elements, 29 lossless, 136 Tellegen, Bernard D. H., 5, 8, 20 scattering matrix, 132 terminal current, 17 two-port functions, 131 terminal voltage, 17 Thevenin,´ Leon,´ 2, 7 unitary matrix, 136 time step, 26 trajectory, 35 transconductance, 16, 61 Van Valkenburg, Mac, 9 bulk, 16 vestigial filter, 243 gate, 16 Volta, Alessandro, 1, 7 transducer power gain, 164 voltage transfer function, 66 node, 17 all-pass, 69, 86 terminal, 17 from its magnitude, 68, 83 voltage source, 13 gain, 66 voltage-controlled current source, 16 group delay, 70, 71 linear phase, 70 Youla, Dante C., 3 magnitude, 66 minimum phase, 70 zero sensitivity, 230, 232 phase, 66 Zhu, Yi-Sheng, 186